MatBench/layer3/data/step1_materials_without_target.json

[
  {
    "id": "10.1016_j.scriptamat.2015.07.021",
    "DOI": "10.1016/j.scriptamat.2015.07.021",
    "DOI Link": "http://dx.doi.org/10.1016/j.scriptamat.2015.07.021",
    "Relative Dir Path": "mds/10.1016_j.scriptamat.2015.07.021",
    "Article Title": "First principles phonon calculations in materials science",
    "Authors": "Togo, A; Tanaka, I",
    "Source Title": "SCRIPTA MATERIALIA",
    "Abstract": "Phonon plays essential roles in dynamical behaviors and thermal properties, which are central topics in fundamental issues of materials science. The importance of first principles phonon calculations cannot be overly emphasized. Phonopy is an open source code for such calculations launched by the present authors, which has been world-widely used. Here we demonstrate phonon properties with fundamental equations and show examples how the phonon calculations are applied in materials science. (C) 2015 Acta Materialia Inc. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.orgilicenses/by/4.0/).",
    "Times Cited, WoS Core": 8499,
    "Times Cited, All Databases": 8915,
    "Publication Year": 2015,
    "Research Areas": "Science & Technology - Other Topics; Materials Science; Metallurgy & Metallurgical Engineering",
    "UT (Unique WOS ID)": "WOS:000360250700001",
    "Markdown": "Viewpoint Paper  \n\n# First principles phonon calculations in materials science  \n\nAtsushi Togo a,b, Isao Tanaka a,b,c,⇑  \n\na Center for Elements Strategy Initiative for Structure Materials (ESISM), Kyoto University, Sakyo, Kyoto 606-8501, Japan b Department of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan c Nanostructures Research Laboratory, Japan Fine Ceramics Center, Atsuta, Nagoya 456-8587, Japan  \n\n# a r t i c l e i n f o  \n\n# a b s t r a c t  \n\nArticle history:   \nReceived 30 June 2015   \nRevised 17 July 2015   \nAccepted 18 July 2015   \nAvailable online 29 July 2015  \n\nKeywords:   \nFirst principles phonon calculation   \nThermal ellipsoid   \nThermal expansion  \n\nPhonon plays essential roles in dynamical behaviors and thermal properties, which are central topics in fundamental issues of materials science. The importance of first principles phonon calculations cannot be overly emphasized. Phonopy is an open source code for such calculations launched by the present authors, which has been world-widely used. Here we demonstrate phonon properties with fundamental equations and show examples how the phonon calculations are applied in materials science.  \n\n$\\circledcirc$ 2015 Acta Materialia Inc. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).  \n\n# 1. Introduction  \n\nApplication of first principles calculations in condensed matter physics and materials science has greatly expanded when phonon calculations became routine in the last decade. Thanks to the progress of high performance computers and development of accurate and efficient density functional theory (DFT) codes, a large set of first principles calculations are now practical with the accuracy comparable to experiments using ordinary PC clusters. In addition to electronic structure information, a DFT calculation for solids provides energy and stress of the system as well as the force on each atom. Equilibrium crystal structures can be obtained by minimizing residual forces and optimizing stress tensors. When an atom in a crystal is displaced from its equilibrium position, the forces on all atoms in the crystal raise. Analysis of the forces associated with a systematic set of displacements provides a series of phonon frequencies. First principles phonon calculations with a finite displacement method (FDM) [1,2] can be made in this way. An alternative approach for phonon calculations is the density functional perturbation theory (DFPT) [3,4]. The present authors have launched a robust and easy-to-use open-source code for first principles phonon calculations, phonopy [5–20]. This can handle force constants obtained both by FDM and DFPT. The number of users is rapidly growing world-wide, since the information of phonon is very useful for accounting variety of properties and behavior of crystalline materials, such as thermal properties, mechanical properties, phase transition, and superconductivity. In this article, we show examples of applications of the first principles phonon calculations.  \n\nIn Sections 2–4, we take FCC-Al as examples of applications of first principles phonon calculations. For the electronic structure calculations, we employed the plane-wave basis projector augmented wave method [21] in the framework of DFT within the generalized gradient approximation in the Perdew–Burke–Ernzer hof form [22] as implemented in the VASP code [23–25]. A plane-wave energy cutoff of $300\\mathrm{eV}$ and an energy convergence criteria of $10^{-8}\\mathrm{eV}$ were used. A $30\\times30\\times30~k$ -point sampling mesh was used for the unit cell and the equivalent density mesh was used for the supercells together with a $0.2\\mathrm{eV}$ smearing width of the Methfessel–Paxton scheme [26]. For the phonon calculations, supercell and finite displacement approaches were used with $3\\times3\\times3$ supercell of the conventional unit cell (108 atoms) and the atomic displacement distance of $0.01\\mathring{\\mathsf{A}}.$  \n\n# 2. Harmonic approximation  \n\nIn crystals, it is presumed that atoms move around their equilibrium positions ${\\bf r}(l\\kappa)$ with displacements ${\\bf u}(l\\kappa)$ , where $l$ and $\\kappa$ are the labels of unit cells and atoms in each unit cell, respectively. Crystal potential energy $\\Phi$ is presumed to be an analytic function of the displacements of the atoms, and $\\Phi$ is expanded as  \n\n$$\n\\begin{array}{l}{{\\Phi=\\Phi_{0}+\\displaystyle\\sum_{l k}\\displaystyle\\sum_{\\alpha}\\Phi_{\\alpha}(l\\kappa)u_{\\alpha}(l\\kappa)}}\\\\ {{\\mathrm{}+\\displaystyle\\frac{1}{2}\\displaystyle\\sum_{l l^{\\prime}\\kappa\\kappa^{\\prime}}\\displaystyle\\sum_{\\alpha\\beta}\\Phi_{\\alpha\\beta}(l\\kappa,l^{\\prime}\\kappa^{\\prime})u_{\\alpha}(l\\kappa)u_{\\beta}(l^{\\prime}\\kappa^{\\prime})}}\\\\ {{\\mathrm{}+\\displaystyle\\frac{1}{3!}\\displaystyle\\sum_{l l^{\\prime\\prime}\\kappa\\kappa^{\\prime}\\kappa^{\\prime\\prime}}\\displaystyle\\sum_{\\alpha\\beta\\gamma}\\Phi_{\\alpha\\beta\\gamma}(l\\kappa,l^{\\prime}\\kappa^{\\prime},l^{\\prime\\prime}\\kappa^{\\prime\\prime})\\times u_{\\alpha}(l\\kappa)u_{\\beta}(l^{\\prime}\\kappa^{\\prime})u_{\\gamma}(l^{\\prime\\prime}\\kappa^{\\prime\\prime})+\\cdots}}\\end{array}\n$$  \n\nwhere $\\alpha,\\beta,\\cdots$ are the Cartesian indices. The coefficients of the series expansion, $\\Phi_{0},\\ \\Phi_{\\alpha}(l\\kappa),\\ \\Phi_{\\alpha\\beta}(l\\kappa,l^{\\prime}\\kappa^{\\prime})$ , and, $\\Phi_{\\alpha\\beta\\gamma}(l\\kappa,l^{\\prime}\\kappa^{\\prime},l^{\\prime\\prime}\\kappa^{\\prime\\prime})$ , are the zeroth, first, second, and third order force constants, respectively. With small displacements at constant volume, the problem of atomic vibrations is solved with the second-order terms as the harmonic approximation, and the higher order terms are treated by the perturbation theory.  \n\nWith a force $\\begin{array}{r}{F_{\\alpha}(l\\kappa)=-\\frac{\\partial\\Phi}{\\partial u_{\\alpha}(l\\kappa)},}\\end{array}$ an element of second-order force constants $\\Phi_{\\alpha\\beta}(l\\kappa,l^{\\prime}\\kappa^{\\prime})$ is obtained by  \n\n$$\n\\frac{\\partial^{2}{\\Phi}}{\\partial u_{\\alpha}(l\\kappa)\\partial u_{\\beta}(l^{\\prime}\\kappa^{\\prime})}=-\\frac{\\partial F_{\\beta}(l^{\\prime}\\kappa^{\\prime})}{\\partial u_{\\alpha}(l\\kappa)}.\n$$  \n\nCrystal symmetry is utilized to improve the numerical accuracy of the force constants and to reduce the computational cost. The more details on the calculation of force constants are found in Refs. [8,9].  \n\nAs found in text books [27–30], dynamical property of atoms in the harmonic approximation is obtained by solving eigenvalue problem of dynamical matrix $\\mathbf{D(\\pmbq)}$ ,  \n\n$$\n\\mathrm{D}(\\mathbf{q})\\mathbf{e}_{\\mathbf{q}j}=\\omega_{\\mathbf{q}j}^{2}\\mathbf{e}_{\\mathbf{q}j},\\quad\\mathrm{or}\\quad\\sum_{\\beta\\kappa^{\\prime}}D_{\\kappa\\kappa^{\\prime}}^{\\alpha\\beta}(\\mathbf{q})e_{\\mathbf{q}j}^{\\beta\\kappa^{\\prime}}=\\omega_{\\mathbf{q}j}^{2}e_{\\mathbf{q}j}^{\\alpha\\kappa},\n$$  \n\nwith  \n\n$$\nD_{\\kappa\\kappa^{\\prime}}^{\\alpha\\beta}(\\mathbf{q})=\\sum_{l^{\\prime}}\\frac{\\Phi_{z\\beta}(0\\kappa,l^{\\prime}\\kappa^{\\prime})}{\\sqrt{m_{\\kappa}m_{\\kappa^{\\prime}}}}e^{i\\mathbf{q}\\cdot[\\mathbf{r}(l^{\\prime}\\kappa^{\\prime})-\\mathbf{r}(0\\kappa)]},\n$$  \n\nwhere $m_{\\kappa}$ is the mass of the atom $\\kappa$ ; $\\pmb q$ is the wave vector, and $j$ is the band index. $\\omega_{\\mathbf{q}j}$ and ${\\bf e_{q}}_{j}$ give the phonon frequency and polarization vector of the phonon mode labeled by a set $\\{\\mathbf{q},j\\}$ , respectively. Since DðqÞ is an Hermitian matrix, its eigenvalues, $\\omega_{\\mathbf{q}j}^{2}$ , are real. Usually DðqÞ is arranged to be a $3n_{\\mathrm{a}}\\times3n_{\\mathrm{a}}$ matrix [30], where 3 comes from the freedom of the Cartesian indices for crystal and $n_{\\mathrm{a}}$ is the number of atoms in a unit cell. Then ${\\mathbf{e_{q}}}_{j}$ becomes a complex column vector with $3n_{\\mathrm{a}}$ elements, and usually ${\\bf e_{q}}_{j}$ is normalized to be 1, i.e., $\\begin{array}{r}{\\sum_{\\alpha\\kappa}|e_{\\mathbf q_{j}}^{\\alpha\\kappa}|^{2}=1}\\end{array}$ . ${\\bf e_{q}}_{j}$ contains information of collective motion of atoms. This may be understood as a set of atomic displacement vectors,  \n\n$$\n[\\mathbf{u}(l1),\\dots,\\mathbf{u}(l\\kappa)]=\\left[{\\frac{A}{\\sqrt{m_{1}}}}\\mathbf{e_{qj}^{1}}e^{i\\mathbf{q}\\cdot\\mathbf{r}(l1)},\\dots,{\\frac{A}{\\sqrt{m_{n}}}}\\mathbf{e_{qj}^{n_{\\mathrm{{a}}}}}e^{i\\mathbf{q}\\cdot\\mathbf{r}(l\\kappa)}\\right],\n$$  \n\nwhere $A$ is the complex constant undetermined by Eq. (3), and $\\mathbf{e_{q}^{\\kappa}}{}^{\\mathrm{T}}=\\left(e_{\\mathbf{q}j}^{x\\kappa},e_{\\mathbf{q}j}^{y\\kappa},e_{\\mathbf{q}j}^{z\\kappa}\\right)$  \n\nAs a typical example, the phonon band structure and phonon density of states (DOS) of Al are shown in Fig. 1. The phonon DOS is defined as  \n\n$$\ng(\\omega)=\\frac{1}{N}\\sum_{\\mathbf{q}j}\\delta(\\omega-\\omega_{\\mathbf{q}j}),\n$$  \n\nwhere $N$ is the number of unit cells in crystal. Divided by N, $g(\\omega)$ is normalized so that the integral over frequency becomes $3n_{\\mathrm{a}}$ . The phonon band structure can be directly comparable with experimental data by neutron or $\\mathsf{X}.$ -ray inelastic scattering. They often show reasonable agreements [20,31,32]. Frequency data by Raman and infrared (IR) spectroscopy can also be well reproduced [12,33].  \n\n![](images/cd061426c3952aaa5959eeb84741e94d2ec790b892981826c6acf8337ee2be6a.jpg)  \nFig. 1. Phonon band structure and DOS of Al.  \n\nIrreducible representations of phonon modes, which can be used to assign Raman or IR active modes, are calculated from polarization vectors [12,34]. Atom specific phonon DOS projected along a unit direction vector $\\hat{\\bf n}$ is defined as  \n\n$$\ng_{\\kappa}(\\omega,\\hat{\\mathbf{n}})=\\frac{1}{N}{\\sum_{\\mathbf{q}j}\\delta{(\\omega-\\omega_{\\mathbf{q}j})}\\left|\\hat{\\mathbf{n}}\\cdot\\mathbf{e}_{\\mathbf{q}j}^{\\kappa}\\right|}^{2}.\n$$  \n\nThis $g_{\\kappa}(\\omega,\\hat{\\mathbf n})$ can be directly compared with that measured by means of nuclear-resonant inelastic scattering using synchrotron radiation. In Ref. [17], phonon calculations of $\\mathtt{L1}_{0}$ -type FePt projected along the $c$ -axis and basal plane are well comparable to experimental $^{57}\\mathrm{Fe}$ nuclear-resonant inelastic scattering spectra measured at $10\\mathrm{K}$ in the parallel and perpendicular geometries, respectively.  \n\nOnce phonon frequencies over Brillouin zone are known, from the canonical distribution in statistical mechanics for phonons under the harmonic approximation, the energy $E$ of phonon system is given as  \n\n$$\nE=\\sum_{\\mathbf{q}j}\\hbar\\omega_{\\mathbf{q}j}\\left[\\frac{1}{2}+\\frac{1}{\\exp(\\hbar\\omega_{\\mathbf{q}j}/k_{\\mathrm{B}}T)-1}\\right],\n$$  \n\nwhere $T,\\ k_{\\mathrm{B}}$ , and $\\hbar$ are the temperature, the Boltzmann constant, and the reduced Planck constant, respectively. Using the thermodynamic relations, a number of thermal properties, such as constant volume heat capacity $C_{V}$ , Helmholtz free energy $F_{\\ast}$ and entropy S, can be computed as functions of temperature [30]:  \n\n$$\nC_{V}=\\sum_{{\\bf{q}}j}C_{{\\bf{q}}j}=\\sum_{{\\bf{q}}j}k_{\\mathrm{{B}}}\\left(\\frac{\\hbar\\omega_{{\\bf{q}}j}}{k_{\\mathrm{{B}}}T}\\right)^{2}\\frac{\\exp(\\hbar\\omega_{{\\bf{q}}j}/k_{\\mathrm{B}}T)}{\\left[{\\exp(\\hbar\\omega_{{\\bf{q}}j}/k_{\\mathrm{{B}}}T)-1}\\right]^{2}},\n$$  \n\n$$\n\\boldsymbol{F}=\\frac{1}{2}\\sum_{\\mathbf{q}j}\\hbar\\omega_{\\mathbf{q}j}+k_{\\mathrm{B}}T\\sum_{\\mathbf{q}j}\\ln{\\left[1-\\exp(-\\hbar\\omega_{\\mathbf{q}j}/k_{\\mathrm{B}}T)\\right]},\n$$  \n\nand  \n\n$$\n\\begin{array}{c}{{S{=}\\displaystyle\\frac{1}{2T}\\sum_{{\\bf{q}}i}{\\hbar\\omega_{{\\bf{q}}i}\\coth\\left[{\\hbar\\omega_{{\\bf{q}}i}\\o{T}}\\right]}}}\\\\ {{{-}k_{\\mathrm{B}}\\displaystyle\\sum_{{\\bf{q}}i}{\\ln\\left[{2\\sinh({\\hbar\\omega_{{\\bf{q}}i}\\o{T}})}\\right]}.}}\\end{array}\n$$  \n\nThe calculated $F,\\ C_{V}$ , and $S$ for Al are shown in Fig. 2.  \n\n# 3. Mean square atomic displacements  \n\nWith the phase factor convention of the dynamical matrix used in Eq. (4), an atomic displacement operator is written as,  \n\n$$\n\\hat{u}_{\\alpha}({l}\\kappa)=\\sqrt{\\frac{\\hbar}{2N m_{\\kappa}}}{\\sum_{{\\bf q}j}}\\frac{\\hat{a}_{{\\bf q}j}+\\hat{a}_{-{\\bf q}j}^{\\dagger}}{\\sqrt{\\omega_{{\\bf q}j}}}e_{{\\bf q}j}^{\\alpha\\kappa}e^{i{\\bf q}\\cdot{\\bf r}(l\\kappa)},\n$$  \n\n![](images/dec8bb0351bd980087eeb19aefeaf2c4c6798d64c1e999a439e35cb7439f9ed8.jpg)  \nFig. 2. Thermal properties of Al. Entropy, $C_{V}$ , and Helmholtz free energy were calculated with harmonic approximation (Section 2). QHA was employed to obtain $C_{P}$ (Section 4). Physical units are shown with labels of the physical properties, and the value of the vertical axis is shared by them. Dotted curve depicts the experiment of $C_{P}$ [35].  \n\nwhere $\\hat{\\boldsymbol{a}}^{\\dagger}$ and $\\hat{\\boldsymbol{a}}$ are the creation and annihilation operators, respectively. Distributions of atoms around their equilibrium positions are then obtained as the expectation values of Eq. (12). The mean square atomic displacement projected along $\\hat{\\bf n}$ is obtained as  \n\n$$\n\\left\\langle\\left|\\widehat{u}_{\\widehat{\\mathbf{n}}}(\\kappa)\\right|^{2}\\right\\rangle=\\frac{\\hbar}{2\\mathrm{N}\\mathrm{m}_{\\kappa}}\\sum_{\\mathbf{q}j}\\frac{1+2n_{\\mathbf{q}j}}{\\omega_{\\mathbf{q}j}}\\left|\\widehat{\\mathbf{n}}\\cdot\\mathbf{e}_{\\mathbf{q}j}^{\\kappa}\\right|^{2},\n$$  \n\nwhere $n_{\\mathbf q{j}}=\\left[\\exp({\\hbar\\omega_{\\mathbf q{j}}}/{k_{\\mathrm{B}}T})-1\\right]^{-1}$ is the phonon occupation number. Eq. (13) is lattice-point $(l)$ independent since the phase factor disappears. Anisotropic atomic displacement parameters (ADPs) to estimate the atom positions during thermal motion can also be computed and compared with experimental neutron diffraction data. Thermal ellipsoids may be discussed using mean square displacement matrix $\\mathsf{B}(\\kappa)$ defined by  \n\n$$\n\\mathbf{B}(\\kappa)=\\frac{\\hbar}{2N m_{\\kappa}}\\sum_{\\mathbf{q}j}\\frac{1+2n_{\\mathbf{q}j}}{\\omega_{\\mathbf{q}j}}\\mathbf{e}_{\\mathbf{q}j}^{\\kappa}\\otimes\\mathbf{e}_{\\mathbf{q}j}^{\\kappa*}.\n$$  \n\nThe shape and orientation of an ellipsoid is obtained solving eigenvalue problem of this matrix. The method has been applied to show the ORTEP (Oak Ridge Thermal Ellipsoid Plot)-style drawing of ADPs [18]. Ref. [11] shows an example for a ternary carbide $\\mathrm{Ti}_{3}\\mathrm{SiC}_{2}$ having a layered structure known as MAX phases, in which we can see good agreement between calculated and experimental ADPs.  \n\n# 4. Quasi-harmonic approximation  \n\nBy changing volume, phonon properties vary since the crystal potential is an anharmonic function of volume. In this article, the term ‘‘quasi-harmonic approximation (QHA)’’ means this volume dependence of phonon properties, but the harmonic approximation is simply applied at each volume. Fig. 3a shows calculated phonon frequencies of Al at $X$ and $L$ points with respect to ten different unit-cell volumes. Typically phonon frequency decreases by increasing volume, and the slope of each phonon mode is nearly constant in the wide volume range. The normalized slope is called mode-Grüneisen parameter that is defined as  \n\n$$\n\\gamma_{\\mathbf{q}j}(V)=-\\frac{V}{\\omega_{\\mathbf{q}j}(V)}\\frac{\\partial\\omega_{\\mathbf{q}j}(V)}{\\partial V}.\n$$  \n\nOnce dynamical matrix is known, $\\gamma_{\\mathbf{q}j}$ is easily calculated from the volume derivative of the dynamical matrix [29],  \n\n$$\n\\gamma_{\\mathbf q{j}}(V)=-\\frac{V}{2{(\\omega_{\\mathbf q{j}})}^{2}}\\sum_{\\alpha\\beta\\kappa\\kappa^{\\prime}}e_{\\mathbf q{j}}^{\\alpha\\kappa*}\\frac{\\partial D_{\\kappa\\kappa^{\\prime}}^{\\alpha\\beta}(V,\\mathbf q)}{\\partial V}e_{\\mathbf q{j}}^{\\beta\\kappa^{\\prime}}.\n$$  \n\n![](images/6ccd8dee3e6368110724c1f477226865f5a7d1bc24477b106a3e37d5d342c52a.jpg)  \nFig. 3. (a) Phonon frequencies of Al at $X$ and L points with respect to unit cell volume are shown by filled and open circles, respectively. The solid and dotted lines are guides to the eye. (b) $U_{\\mathrm{el}}+F_{\\mathrm{ph}}$ of Al with respect to volume at temperatures from 0 to $800\\mathrm{K}$ with $100\\mathrm{K}$ step are depicted by filled circles and the values are fit by the solid curves. Cross symbols show the energy bottoms of the respective curves and simultaneously the equilibrium volumes. Lines connecting the cross symbols are guides to the eye. (c) Volumetric thermal expansion coefficient of Al. Calculation is shown with solid curve and experiments are depicted by filled circle symbols [36] and dotted curve [37].  \n\nThe quantity can be related to macroscopic Grüneisen parameter $\\gamma$ using mode contributions to the heat capacity $C_{\\mathbf{q}j}$ found in Eq. (9), $\\begin{array}{r}{\\gamma=\\sum_{\\mathbf{q}j}\\gamma_{\\mathbf{q}j}C_{\\mathbf{q}j}/C_{V}}\\end{array}$ [28,30].  \n\nSilicon is known as a famous exception to have large negative mode-Grüneisen parameters. Mode-Grüneisen parameter is a measure of anharmonicity of phonon modes and is related to third-order force constants directly [29]. Therefore crystals that possess large anharmonic terms beyond third-order terms in Eq. (1) can show non-linear change of phonon frequency with respect to volume. This is often observed in crystals that exhibit second- or higher-order structural phase transitions [6].  \n\nThe phonon frequency influences the phonon energy through Eq. (8). The thermal properties are thereby affected. Using thermodynamics definition, thermodynamic variables at constant volume is transformed to those at constant pressure that is often more easily measurable in experiments. Gibbs free energy $G(T,p)$ at given temperature $T$ and pressure $p$ is obtained from Helmholtz free energy $F(T;V)$ through the transformation,  \n\n$$\nG(T,p)=\\operatorname*{min}_{V}[F(T;V)+p V],\n$$  \n\nwhere the right hand side of this equation means finding a minimum value in the square bracket by changing volume $V.$ We may approximate $F(T;V)$ by the sum of electronic internal energy $U_{\\mathrm{el}}(V)$ and phonon Helmholtz free energy $F_{\\mathrm{ph}}(T;V)$ , i.e., $F(T;V)\\simeq U_{\\mathrm{el}}(V)+F_{\\mathrm{ph}}(T;V).\\ U_{\\mathrm{el}}(V)$ is obtained as total energy of electronic structure from the first principles calculation, and the first principles phonon calculation at $T$ and $V$ gives $F_{\\mathrm{ph}}(T;V)$ . The calculated $U_{\\mathrm{el}}(V)+F_{\\mathrm{ph}}(T;V)$ are depicted by the filled circle symbols in Fig. 3b, where the ten volume points chosen are the same as those in Fig. 3a. The nine curves are the fits to equation of states (EOS) at temperatures from 0 to $800\\mathrm{K}$ with $100\\mathrm{K}$ step. Here the Vinet EOS [38] was used to fit the points to the curves though any other functions can be used for the fitting. The minimum values at the temperatures are depicted by the cross symbols in Fig. 3b and are the Gibbs free energies at the temperatures and the respective equilibrium volumes are simultaneously given. Volumetric thermal expansion coefficient, bðTÞ ¼ V1T @V@ðT , is obtained from the calculated equilibrium volumes $V(T)$ at dense temperature points. $\\beta(T)$ for Al is shown in Fig. 3c, where we can see that the calculation shows reasonable agreements with the experiments. In thermodynamics, heat capacity at constant pressure $C_{P}$ is given by  \n\n$$\nC_{P}(T,p)=-T{\\frac{\\partial^{2}G(T,p)}{\\partial T^{2}}}=C_{V}(T,V(T,p))+T{\\frac{\\partial V(T,p)}{\\partial T}}{\\frac{\\partial S(T;V)}{\\partial V}}\\Bigg|_{V=V(T,p)}.\n$$  \n\nIn Eq. (18), the second term of the second equation is understood as the contribution to heat capacity from thermal expansion. $C_{P}$ for Al is shown in Fig. 2. The agreement of the calculation with the experiment is excellent. At high temperatures, the difference between $C_{P}$ and $C_{V}$ is not negligible in Al. Therefore it is essential to consider thermal expansion for heat capacity.  \n\nQHA is known as a reasonable approximation in a wide temperature range below melting point except for temperatures very close to melting point where higher-order terms in Eq. (1) become evident [39].  \n\n# 5. Stability condition and imaginary mode  \n\nAt equilibrium, $\\begin{array}{r}{\\frac{\\partial\\Phi}{\\partial r_{\\alpha}(l\\kappa)}=0}\\end{array}$ , a crystal is dynamically (mechanically) stable if its potential energy always increases against any combinations of atomic displacements. In the harmonic approximation, this is equivalent to the condition that all phonons have real and positive frequencies [29]. However under virtual thermodynamic conditions, imaginary frequency or negative eigenvalue can appear in the solution of Eq. (3). This indicates dynamical instability of the system, which means that the corrective atomic displacements of Eq. (5) reduce the potential energy in the vicinity of the equilibrium atomic positions.  \n\nImaginary mode provides useful information to study displacive phase transition. A typical example is shown in Fig. 4a to c [16]. Imaginary modes can be found only for $\\beta$ -Ti, that has BCC structure, at both $P$ and $N$ points. This indicates that $\\beta$ -Ti is unstable at low temperature. Such imaginary modes cannot be seen for either $\\omega$ -Ti whose crystal structure is shown in Fig. 4d or $\\alpha$ -Ti (HCP). Experimentally $\\beta$ -Ti is known to occur above 1155 K. At such high temperatures, large atomic displacements can stabilize the BCC structure. In such a case, the perturbation approach is invalid. Phonons with large atomic displacements may be treated by self-consistent phonon method [29,40] or by a combination of molecular dynamics and lattice dynamics calculation [41–43], which is not discussed in this article.  \n\nA given structure having imaginary phonon modes can be led to alternative structures through continuous atomic displacements and lattice deformations. The present authors systematically investigated the evolution of crystal structures from the simple cubic (SC) structure [10]. The inset of Fig. 5 shows the phonon band structure of SC-Cu $(P m\\bar{3}m)$ . Imaginary modes can be found at $M(1/2,1/2,0)$ and $X(1/2,0,0)$ points. Then the SC structure is deformed along these directions. In order to accommodate the deformation in the calculation model with the periodic boundary condition, the unit cells are expanded by $2\\times2\\times1$ for the $M$ point and $2\\times1\\times1$ for the $X$ point. The $M$ point deformation breaks the crystal symmetry of SC $(P m\\bar{3}m)$ to $P4/n m m$ . The doubly degenerated instability at the $X$ point leads to Pmma and Cmcm as highest possible symmetries. The deformed crystal structures are relaxed by first principles calculations imposing the corresponding space-group operations. After these procedures, body-centered tetragonal (BCT), simple hexagonal (SH), and FCC are respectively formed. The whole procedure finishes when all crystal structures at the end-points are found to be dynamically stable. Finally a treelike structure of crystal structure relationships was drawn as shown in Fig. 5, where thick lines indicate phase transition pathways (PTPs). The space-group type written near a line is a common subgroup of initial and final structures. The presence of the line indicates that the energy decreases monotonically with the phase transition. In other words, the transition can take place spontaneously without any energy barrier. The line is terminated when the final structure is dynamically stable.  \n\n![](images/fe2c36d9cbdd59049f066f0e814795640c96365d35a97052ec31da0100fb5812.jpg)  \nFig. 4. Phonon band structures of (a) $\\alpha$ -Ti (HCP), (b) $\\beta$ -Ti (BCC), and (c) $\\omega$ -Ti. The figure (d) shows the hexagonal crystal structure of $\\omega$ -Ti.  \n\nIn the line diagram, $\\omega$ is located at the junction of two pathways, i.e., $\\omega\\rightarrow\\mathsf{B C C}\\rightarrow\\mathsf{H C P}$ and $\\omega\\rightarrow{\\sf F C C}$ . The instability of $\\omega$ at the $\\Gamma$ point leads to BCC, which is still dynamically unstable and eventually leads to HCP. Another instability at the $M$ point leads to FCC. The other instability at the $K$ point, which is doubly degenerate, leads to FCC. On the path from $\\omega$ to BCC, the crystal symmetry of $\\omega$ having the space-group type of P6=mmm is once lowered to $P\\bar{3}m{1}$ and then becomes $I m\\bar{3}m$ (BCC) after the geometry optimization. Both x- and BCC-Cu are dynamically unstable, which can be formed only under crystal symmetry constraints. FCC-Cu is, of course, dynamically stable. Several PTPs between BCC and FCC have been proposed in literature. However, they are mostly based only upon investigation of continuous lattice deformation. For example in the classical Bain path, formation of BCT in between BCC and FCC is considered. Formation of SC is taken into account in the ‘‘trigonal Bain path’’. Normal modes of phonon, which should be most adequate to describe the collective atomic displacements, have not been considered. The presence of $\\omega$ as the lowest energy barrier in the BCC–FCC pathway had not been reported before Ref. [10]. The situation is the same for the BCC–HCP transition, known as the Burgers path. The Burgers path was thought to be quite complicated from the viewpoint of the lattice continuity. On the basis of the present study, it can be easily pointed out that the BCC– HCP transition pathway is along the space-group type of Cmcm.  \n\n![](images/f0299d5b318b08b2c4cc1c7bca668af8ca3183e7c759caf1948a1f1f11ebf698.jpg)  \nFig. 5. Line diagram of structural transition pathways in Cu. The inset shows phonon band structure of simple cubic (SC) Cu. Open and filled symbols represent dynamically unstable and stable crystal structures, respectively. Lines connecting these symbols are the phase transition pathways for which space-group types are shown near the lines.  \n\nEvolution diagram was constructed in the same way for $\\mathsf{N a}R\\mathsf{T i O}_{4}$ (R: rare-earth metal) with Ruddlesden–Popper type structure [13]. Inversion symmetry breaking by oxygen octahedral rotations was unambiguously demonstrated. The mechanism is expected to lead to many more families of acentric oxides.  \n\n# 6. Interactions among phonons and lattice thermal conductivity  \n\nUsing the harmonic phonon coordinates, anharmonic terms in Eq. (1) are transformed to a picture of phonon–phonon interactions [8,28]. Lattice thermal conductivity can be accurately calculated by solving linearized Boltzmann transport equation with the phonon– phonon interaction strength obtained using first principles calculation [9,44,45]. Although the computational cost for such calculations is many orders of magnitudes higher than the ordinary DFT calculations of primitive cells, such calculations have already been applied for many simple compounds and computed lattice thermal conductivities show good agreements with experimental data [9,45]. Calculations with special focus on searching thermoelectric materials have also been made [14,20,45].  \n\n# Acknowledgments  \n\nThe research leading to these results was supported by Grant-in-Aid for Scientific Research on Innovative Areas ‘‘Nano Informatics’’ (Grant No. 25106005) and for Young Scientists (B) (Grant No. 26820284) both from JSPS. Support from MEXT through ESISM is also acknowledged.  \n\n# References  \n\n[1] G. Kresse, J. Furthmüller, J. Hafner, Europhys. Lett. 32 (1995) 729.   \n[2] K. Parlinski, Z.Q. Li, Y. Kawazoe, Phys. Rev. Lett. 78 (1997) 4063.   \n[3] P. Giannozzi, S. de Gironcoli, P. Pavone, S. Baroni, Phys. Rev. B 43 (1991) 7231.   \n[4] X. Gonze, C. Lee, Phys. Rev. B 55 (1997) 10355.   \n[5] A. Togo, Phonopy, http://phonopy.sourceforge.net/.   \n[6] A. Togo, F. Oba, I. Tanaka, Phys. Rev. B 78 (2008) 134106.   \n[7] A. Togo, L. Chaput, I. Tanaka, G. Hug, Phys. Rev. B 81 (2010) 174301. [8] L. Chaput, A. Togo, I. Tanaka, G. Hug, Phys. Rev. B 84 (2011) 094302. [9] A. Togo, L. Chaput, I. Tanaka, Phys. Rev. B 91 (2015) 094306.   \n[10] A. Togo, I. Tanaka, Phys. Rev. B 87 (2013) 184104.   \n[11] N.J. Lane, S.C. Vogel, G. Hug, A. Togo, L. Chaput, L. Hultman, M.W. Barsoum, Phys. Rev. B 86 (2012) 214301.   \n[12] A. Togo, F. Oba, I. Tanaka, Phys. Rev. B 77 (2008) 184101.   \n[13] H. Akamatsu, K. Fujita, T. Kuge, A. Sen Gupta, A. Togo, S. Lei, F. Xue, G. Stone, J.M. Rondinelli, L.-Q. Chen, I. Tanaka, V. Gopalan, K. Tanaka, Phys. Rev. Lett. 112 (2014) 187602.   \n[14] J.M. Skelton, S.C. Parker, A. Togo, I. Tanaka, A. Walsh, Phys. Rev. B 89 (2014) 205203.   \n[15] A. Matsumoto, Y. Koyama, A. Togo, M. Choi, I. Tanaka, Phys. Rev. B 83 (2011) 214110.   \n[16] K. Edalati, T. Daio, M. Arita, S. Lee, Z. Horita, A. Togo, I. Tanaka, Acta Mater. 68 (2014) 207.   \n[17] Y. Tamada, R. Masuda, A. Togo, S. Yamamoto, Y. Yoda, I. Tanaka, M. Seto, S. Nasu, T. Ono, Phys. Rev. B 81 (2010) 132302.   \n[18] V.L. Deringer, R.P. Stoffel, A. Togo, B. Eck, M. Meven, R. Dronskowski, CrystEngComm 16 (2014) 10907.   \n[19] Y. Ikeda, . Seko, A. Togo, I. Tanaka, Phys. Rev. B 90 (2014) 134106.   \n[20] Tanaka, G.K.H. Madsen, J. Appl. Phys. 117 (2015) 045102. 2 B 50 (1994) 17953. 2 erdew Burk Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. Kresse, Solids 193 (1995) 222.   \n[24] Kresse Comput. Mater. Sci. 6 (1996) 15. 25 Kresse . Rev. B 59 (1999) 1758.   \n[26] M. Methfess Phys. Rev. B 40 (1989) 3616.   \n[27] . Ziman on Phonons, Oxford University Press, 2001.   \n[28] .P. Srivastava ysics of Phonons, CRC Press, 1990.   \n[29] D.C. Wallace, Thermodynamics of Crystals, Dover Publications, 1998.   \n[30] M.T. Dove, Introduction to Lattice Dynamics, Cambridge University Press, 1993.   \n[31] F. Körmann, A. Dick, B. Grabowski, T. Hickel, J. Neugebauer, Phys. Rev. B 85 (2012) 125104.   \n[32] F. Körmann, B. Grabowski, B. Dutta, T. Hickel, L. Mauger, B. Fultz, J. Neugebauer, Phys. Rev. Lett. 113 (2014) 165503.   \n[33] A. Kuwabara, T. Tohei, T. Yamamoto, I. Tanaka, Phys. Rev. B 71 (2005) 064301.   \n[34] G. Venkataraman, L.A. Feldkamp, V.C. Sahni, Dynamics of Perfect Crystals, MIT press, 1975.   \n[35] M.W. Chase, Jr., NIST-JANAF Thermochemical Tables, Journal of Physical and Chemical Reference Data Monographs, American Inst. of Physics, 1998.   \n[36] A.J.C. Wilson, Proc. Phys. Soc. 53 (1941) 235.   \n[37] F.C. Nix, D. MacNair, Phys. Rev. 60 (1941) 597.   \n[38] P. Vinet, J.H. Rose, J. Ferrante, J.R. Smith, J. Phys.: Condens. Matter 1 (1989) 1941.   \n[39] B. Grabowski, . Ismer, T. Hickel, J. Neugebauer, Phys. Rev. B 79 (2009) 134106.   \n[40] I. Errea, Calandra, F. Mauri, Phys. Rev. B 89 (2014) 064302.   \n[41] C.Z Wang, Chan, K.M. Ho, Phys. Rev. B 42 (1990) 11276.   \n[42] C. Lee, D. Vanderbilt, K. Laasonen, R. Car, M. Parrinello, Phys. Rev. B 47 (1993) 4863.   \n[43] T. Sun, D.-B. Zhang, R.M. Wentzcovitch, Phys. Rev. B 89 (2014) 094109.   \n[44] Chaput, Phys. Rev. Lett. 110 (2013) 265506.   \n[45] A. Seko, A. Togo, H. Hayashi, K. Tsuda, L. Chaput, I. Tanaka, ArXiv e-prints: <arXiv:1506.06439>.  ",
    "Target": null
  },
  {
    "id": "10.1021_acsami.5b07517",
    "DOI": "10.1021/acsami.5b07517",
    "DOI Link": "http://dx.doi.org/10.1021/acsami.5b07517",
    "Relative Dir Path": "mds/10.1021_acsami.5b07517",
    "Article Title": "Origin of Outstanding Stability in the Lithium Solid Electrolyte Materials: Insights from Thermodynamic Analyses Based on First-Principles Calculations",
    "Authors": "Zhu, YZ; He, XF; Mo, YF",
    "Source Title": "ACS APPLIED MATERIALS & INTERFACES",
    "Abstract": "First-principles calculations were performed to investigate the electrochemical stability of lithium solid electrolyte materials in all-solid-state Li-ion batteries. The common solid electrolytes were found to have a limited electrochemical window. Our results suggest that the outstanding stability of the solid electrolyte materials is not thermodynamically intrinsic but is originated from kinetic stabilizations. The sluggish kinetics of the decomposition reactions cause a high overpotential leading to a nominally wide electrochemical window observed in many experiments. The decomposition products, similar to the solid-electrolyte-interphases, mitigate the extreme chemical potential from the electrodes and protect the solid electrolyte from further decompositions. With the aid of the first-principles calculations, we revealed the passivation mechanism of these decomposition interphases and quantified the extensions of the electrochemical window from the interphases. We also found that the artificial coating layers applied at the solid electrolyte and electrode interfaces have a similar effect of passivating the solid electrolyte. Our newly gained understanding provided general principles for developing solid electrolyte materials with enhanced stability and for engineering interfaces in all-solid-state Li-ion batteries.",
    "Times Cited, WoS Core": 1461,
    "Times Cited, All Databases": 1607,
    "Publication Year": 2015,
    "Research Areas": "Science & Technology - Other Topics; Materials Science",
    "UT (Unique WOS ID)": "WOS:000363994700039",
    "Markdown": "# Origin of Outstanding Stability in the Lithium Solid Electrolyte Materials: Insights from Thermodynamic Analyses Based on FirstPrinciples Calculations  \n\nYizhou Zhu,† Xingfeng He,† and Yifei Mo\\*,†,‡  \n\n†Department of Materials Science and Engineering and ‡University of Maryland Energy Research Center, University of Maryland, College Park, Maryland 20742, United States  \n\nSupporting Information  \n\nABSTRACT: First-principles calculations were performed to investigate the electrochemical stability of lithium solid electrolyte materials in all-solid-state Li-ion batteries. The common solid electrolytes were found to have a limited electrochemical window. Our results suggest that the outstanding stability of the solid electrolyte materials is not thermodynamically intrinsic but is originated from kinetic stabilizations. The sluggish kinetics of the decomposition reactions cause a high overpotential leading to a nominally wide electrochemical window observed in many experiments.  \n\n![](images/8a315c8b0e442eee07d103b33b4c5d705b65e78ef5a68f6e9aa5fe5c84bac6ac.jpg)  \n\nThe decomposition products, similar to the solid-electrolyte-interphases, mitigate the extreme chemical potential from the electrodes and protect the solid electrolyte from further decompositions. With the aid of the first-principles calculations, we revealed the passivation mechanism of these decomposition interphases and quantified the extensions of the electrochemical window from the interphases. We also found that the artificial coating layers applied at the solid electrolyte and electrode interfaces have a similar effect of passivating the solid electrolyte. Our newly gained understanding provided general principles for developing solid electrolyte materials with enhanced stability and for engineering interfaces in all-solid-state Li-ion batteries.  \n\nKEYWORDS: lithium ionic conductor, solid electrolyte, electrochemical stability, passivation, solid-electrolyte-interphases, first-principles calculations  \n\n# 1. INTRODUCTION  \n\nThe continued drive for high energy density Li-ion batteries has imposed ever stricter requirements on the electrolyte materials. Current organic liquid electrolytes are flammable, causing notorious safety issues for Li-ion batteries. The limited electrochemical window of the organic liquid electrolytes limits the choice of electrode materials and hence the achievable energy density of the Li-ion batteries. The solid electrolyte materials based on Li-ion conducting ceramics are promising alternatives for the conventional polymer electrolytes to make all-solid-state Li-ion batteries.1,2 Thanks to the recent discovery and development of Li ionic conductor materials such as Li thiophosphates1,3,4 and Li garnet-type materials,5 high Li ionic conductivities of $\\mathrm{1-10~mS/cm}$ comparable to the organic liquid electrolytes have been achieved in the solid electrolyte materials. Moreover, the claimed outstanding stability of ceramic solid electrolyte materials may provide intrinsic safety for the Li-ion batteries and may enable Li metal anode and high-voltage cathodes,1,2 which may significantly increase the energy density for Li-ion batteries.6−8  \n\nThe claimed outstanding stability of the solid electrolyte materials is based on the widely reported electrochemical window of $_{0-5\\mathrm{~V~}}$ from cyclic voltammetry (CV) measurements. $^{1,8-10}$ However, some recent experimental and computational studies questioned the claimed stability of solid electrolyte materials against Li metal and at high voltages. For example, the reduction and oxidation of $\\mathrm{Li}_{10}\\mathrm{GeP}_{2}\\mathrm{S}_{12}$ (LGPS) at low and high potentials, respectively, in contrast to the originally claimed $_{0-5\\mathrm{~V~}}$ electrochemical window, have been demonstrated by first-principles computation11 and the experiments.12 Recent in situ X-ray photoelectron spectroscopy (XPS) experiments have also observed the interfacial decomposition of LiPON,13 lithium lanthanum titanate,14 and NASICON-type15 solid electrolyte materials against Li metal. These reports lead to an outstanding discrepancy, i.e., the wide electrochemical windows of $_{0-5\\mathrm{~V~}}$ reported in many CV experiments are contradictory to the decomposition of the solid electrolyte against Li. Although the experimental evidences for the decompositions have been reported in a range of materials from sulfides to oxides and oxynitrides, it is not clear whether the decomposition of ceramic solid electrolytes is a universal phenomenon and whether some ceramic solid electrolyte can indeed achieve a “true” stability window of $_{0-5\\mathrm{~V~}}$ . It is speculated that the decomposition products form interphases to passivate the solid electrolytes and to inhibit the continuous bulk decompositions.11,14,15 However, little is understood about the fundamental physical and chemical mechanisms governing the decomposition and the passivation of the solid electrolyte materials in the all-solid-state Li-ion batteries. Why only certain materials can be spontaneously passivated but others cannot? In addition, the decomposition products at the interfaces between the solid electrolyte and electrode may cause high interfacial resistances and mechanical failures in the all-solid-state Li-ion batteries.2,16 Therefore, computation methods are needed to identify the potential formation of the interfacial decomposition products and to quantify the electrochemical window of the solid electrolyte with the considerations of the passivation effects.  \n\nTable 1. Electrochemical Window and Phase Equilibria at the Reduction and Oxidation Potentials of the Solid Electrolyte Materials   \n\n\n<html><body><table><tr><td></td><td>reduction potential (V)</td><td>phase equilibria at the reduction potential</td><td> oxidation potential (V)</td><td> phase equilibria at the oxidation potentia</td></tr><tr><td>LiS</td><td>-</td><td>LiS (stable at 0 V)</td><td>2.01</td><td>S</td></tr><tr><td>LGPS</td><td>1.71</td><td>P, Li4GeS4, LiS</td><td>2.14</td><td>LiPS4, GeS2, S</td></tr><tr><td>Li3.25Ge0.25P0.75S4</td><td>1.71</td><td>P, Li4GeS4, LiS</td><td>2.14</td><td>LiPS4, GeS, S</td></tr><tr><td>LiPS4</td><td>1.71</td><td>P, LiS</td><td>2.31</td><td>S, PSs</td></tr><tr><td>Li4GeS4</td><td>1.62</td><td>LiS, Ge</td><td>2.14</td><td>GeS2, S</td></tr><tr><td>LiPS1</td><td>2.28</td><td>LiPS4, P4S9</td><td>2.31</td><td>S, PS5</td></tr><tr><td>LigPS5Cl</td><td>1.71</td><td>P, LiS, LiCl</td><td>2.01</td><td>LiPS4, LiCl, S</td></tr><tr><td>LiPSgI</td><td>1.71</td><td>P, LiS, LiI</td><td>2.31</td><td>LiI, S, PSs</td></tr><tr><td>LiPON</td><td>0.68</td><td>LiP, LiPN, LiO</td><td>2.63</td><td>PNs, LiPO, N</td></tr><tr><td>LLZO</td><td>0.05</td><td>ZrO, LaO3,LiO</td><td>2.91</td><td>LiO, LaO,LiZrO7</td></tr><tr><td>LLTO</td><td>1.75</td><td>LiTisO12, Li7/Ti11/6O4, LaTiO7</td><td>3.71</td><td>O, TiO LaTiO7</td></tr><tr><td>LATP</td><td>2.17</td><td>P, LiTiPO5, AIPO4, LiPO4</td><td>4.21</td><td>O, LiTi(PO4)3, Li4PO7, AIPO4</td></tr><tr><td>LAGP</td><td>2.70</td><td>Ge, GeO,LiPO7,AIPO4</td><td>4.27</td><td>O2, GesO(PO4)6, Li4PO7, AIPO4</td></tr><tr><td>LISICON</td><td>1.44</td><td>Zn, Li4GeO4</td><td>3.39</td><td>LiZnGeO4, LiGeO3,O</td></tr></table></body></html>  \n\nIn this study, we systematically investigated the electrochemical stability of common lithium solid electrolytes using first-principles computation methods. We identified the phase equilibria and decomposition reaction energies of the lithiation and delithiation of the solid electrolyte materials against Li metal and at high voltages. Our computation results determined that most solid electrolyte materials have a limited intrinsic electrochemical window and that the decomposition of most solid electrolyte materials are thermodynamically favorable forming decomposition interphases. The mechanisms were suggested regarding the origins of the high nominal electrochemical window observed in the experimental studies. In addition to the high overpotential due to the sluggish kinetics of the decomposition reactions, the passivation mechanism of the decomposition interphases were illustrated. The extensions of the electrochemical window provided by the interphases were quantified in the first-principles calculations. Similar to the interphases, the coating layer materials artificially applied at the interfaces were demonstrated to stabilize and passivate the solid electrolyte materials. These results establish general guidelines for designing solid electrolyte materials with enhanced stability, which is crucial to enable Li metal anode and high-voltage cathode materials in all-solid-state Li-ion batteries.  \n\n# 2. METHODS  \n\nAll density functional theory (DFT) calculations in this work were performed using the Vienna Ab initio Simulation Package (VASP) within the projector augmented-wave approach, and the Perdew− Burke−Ernzerhof (PBE) generalized gradient approximation (GGA) functional was used. The parameters of DFT calculations, such as the plane-wave energy cutoff and $k$ -points density, were consistent with the parameters used for the Materials Project (MP).17 The energy correction schemes for oxides, transition metals, and gas molecules were included as in the MP.18,19 The energies of most materials in this study were obtained from the MP database,20 and DFT calculations were performed only for the solid electrolyte materials that were not available from the MP database. Details of these solid electrolyte structures were summarized in the Supporting Information. In addition, the calculated reaction energies and voltages neglected the contribution of the PV terms and the entropy terms as in previous studies.11,21  \n\nWe constructed the grand potential phase diagram11,21 to study the electrochemical stability of the solid electrolyte materials. The grand potential phase diagram, which were generated using pymatgen,22 identified the phase equilibria of the material in equilibrium with an opening Li reservoir of Li chemical potential $\\mu_{\\mathrm{Li}}.$ . As in the previous studies,11,23 the applied electrostatic potential $\\phi$ was considered in the Li chemical potential $\\mu_{\\mathrm{Li}}$ as  \n\n$$\n\\mu_{\\mathrm{Li}}(\\phi)=\\mu_{\\mathrm{Li}}^{\\:0}-e\\phi\n$$  \n\nwhere $\\mu_{\\mathrm{Li}}^{0}$ is the chemical potential of Li metal, and the potential $\\phi$ is referenced to Li metal in this study. To quantify the thermodynamic driving force, we calculated the decomposition reaction energy $E_{\\mathrm{D}}$ for the decomposition reactions at applied voltage $\\phi$ as  \n\n$$\n\\begin{array}{r l}&{{E}_{\\mathrm{{D}}}(\\phi)=E(\\mathrm{phase\\equilibria},\\phi)-E(\\mathrm{solid\\electrolyte})}\\\\ &{\\phantom{E_{\\mathrm{{D}}}(\\phi)=E\\left(\\mathrm{phase\\equilibria},\\phi\\right)-}-\\Delta{n}_{\\mathrm{L}{\\mid}}\\mu_{\\mathrm{{Li}}}(\\phi)}\\end{array}\n$$  \n\nwhere E(phase equilibria, $\\phi$ ) is the energy of the phase equilibria at the potential $\\phi,$ $E$ (solid electrolyte) is the energy of the solid electrolyte, and $\\Delta n_{\\mathrm{Li}}$ is the change of the number of Li from the solid electrolyte composition to the phase equilibria composition during the lithiation or delithation reaction.  \n\n# 3. RESULTS  \n\n3.1. Stability of Solid Electrolyte Materials against Li Metal. We first evaluated the electrochemical stability of solid electrolyte materials against Li metal and at low voltages. The phase equilibria, i.e., the phases with the lowest energy, in equilibrium with Li metal were identified by the Li grand potential phase diagrams (Table 1). The solid electrolyte materials are not thermodynamically stable against Li metal (Table 2) and are reduced at low voltages with highly favorable decomposition energy (Figure 1 and Table 2). In contrast, the Li binary compounds, such as LiF, $\\mathrm{Li}_{2}\\mathrm{O},\\mathrm{Li}_{2}\\mathrm{S},\\mathrm{Li}_{3}\\mathrm{P},$ and $\\mathrm{Li}_{3}\\mathrm{N},$ are thermodynamically stable against Li metal (Figure 2a). The lithiation and reduction of $\\mathrm{Li}_{10}\\mathrm{GeP}_{2}\\mathrm{S}_{12}$ (LGPS) starts at $1.71\\mathrm{V},$ and the LGPS in equilibrium with Li metal is eventually lithiated into the phase equilibria consisting of $\\mathrm{Li}_{15}\\mathrm{Ge}_{4},$ $\\mathrm{Li}_{3}\\mathrm{P}_{i}$ , and $\\mathrm{Li}_{2}S$ . The Li reduction of the LGPS into these reaction products has a highly favorable reaction energy of $-1.25\\ \\mathrm{eV}/\\$ atom $(-3014\\mathrm{kJ/mol}$ of LGPS) at $0\\mathrm{v}$ (Figure 1 and Table 2). In agreement with our computation, the reduction of LGPS starting at $1.71\\mathrm{~V~}$ and the formation of Li−Ge alloy after the reduction have been demonstrated in the cyclic voltammetry (CV) and XPS experiments, respectively.12 Other sulfides materials, such as $\\mathrm{Li}_{3.25}\\mathrm{Ge}_{0.25}\\mathrm{P}_{0.75}\\mathrm{S}_{4},$ $\\mathrm{Li}_{3}\\mathrm{PS}_{4},$ $\\mathrm{Li}_{4}\\mathrm{GeS}_{4},$ $\\mathrm{Li}_{6}\\mathrm{PS}_{5}\\mathrm{Cl},$ and $\\mathrm{Li}_{7}\\mathrm{P}_{2}\\mathrm{S}_{8}\\mathrm{I},$ are reduced at a similar voltage of ${\\sim}1.6{-}1.7\\mathrm{~V~}$ . The reduction potential is mostly governed by the reduction of $\\mathrm{\\bfP}$ and Ge in the materials, and the reduction products include $\\mathrm{Li}_{3}\\mathrm{P}$ and $\\mathrm{Li}_{2}S$ at $0\\mathrm{V}$ . For those materials containing Ge, Cl, and I elements, Li−Ge alloy, LiCl, and LiI are formed, respectively, as a part of phase equilibria at $0\\mathrm{V}.$ The $\\mathrm{Li}_{7}\\mathrm{P}_{3}\\mathrm{S}_{11}$ is reduced at a voltage of $2.28\\mathrm{~V~}$ into $\\mathrm{Li_{3}P S_{4}}$ with a small decomposition energy (Figure 1a), and the majority of the reduction starts at 1.71 V due to the lithiation of $\\mathrm{Li}_{3}\\mathrm{PS}_{4}$ (Table 2). The decomposition energy for all these solid electrolyte decreases with the potential to $\\mathrm{{\\sim}}1\\mathrm{{eV}}/$ atom at $0\\mathrm{V}$ (Figure 1a and Table 2), indicating the highly favorable reduction reactions of the sulfide solid electrolytes.  \n\nTable 2. Reduction Reaction of the Solid Electrolyte Materials with Li Metal   \n\n\n<html><body><table><tr><td></td><td>phase equilibria with Li metal</td><td>ED (eV/atom)</td></tr><tr><td>LiS</td><td>LiS (stable)</td><td>0</td></tr><tr><td>LGPS</td><td>LisGe4, LiP, LiS</td><td>-1.25</td></tr><tr><td>Li3.25Ge0.25P0.75S4</td><td>LisGe4, LiP, LiS</td><td>-1.28</td></tr><tr><td>LiPS4</td><td>LiP, LiS</td><td>-1.42</td></tr><tr><td>Li4GeS4</td><td>LisGe4, LiS</td><td>-0.89</td></tr><tr><td>LiPS1</td><td>LiP, LiS</td><td>-1.67</td></tr><tr><td>LigPS5Cl</td><td>LiP, LiS, LiCl</td><td>-0.96</td></tr><tr><td>Li-PS8I</td><td>LiP, LiS, LiI</td><td>-1.26</td></tr><tr><td>LiPON</td><td>LiP, LiN, LiO</td><td>-0.66</td></tr><tr><td>LLZO</td><td>Zr (or ZrO), LaO3, LiO</td><td>-0.021</td></tr><tr><td>LLTO</td><td>TiO, LaO, LiO</td><td>-0.34</td></tr><tr><td>LATP</td><td>TiP, TiAl, LiP, LiO</td><td>-1.56</td></tr><tr><td>LAGP</td><td>LigAl4, LisGe4, LiP, LiO</td><td>-1.99</td></tr><tr><td>LISICON</td><td>LisGe4, LiZn, LiO</td><td>-0.77</td></tr></table></body></html>  \n\nThe reduction of oxide solid electrolyte materials $\\mathrm{Li}_{0.33}\\mathrm{La}_{0.56}\\mathrm{TiO}_{3}$ (LLTO) and $\\mathrm{Li_{1.3}T i_{1.7}A l_{0.3}(P O_{4})_{3}}$ (LATP) starts at a voltage of 1.75 and $2.17\\mathrm{~V},$ respectively. Our predicted reduction potential of LLTO is in good agreement with the value of $1.7\\mathrm{-}1.8\\mathrm{~V~}$ reported in the CV experiments.24,25 The calculations also found the reduction of $\\mathrm{Ti}^{\\bar{4}+}$ in LLTO and LATP into $\\mathrm{Ti}^{3+}$ or lower valences at low voltages (Tables 1 and 2). The reduction of Ti is a widely known problem and is observed at the interfaces of $\\mathrm{LLTO}^{14}$ and $\\mathsf{\\bar{L}A T P}^{15}$ with Li metal by in situ XPS spectroscopy. In addition, t he reduction of Ge-cont aining oxide mat erials $\\mathrm{Li}_{1.5}\\mathrm{Al}_{0.5}\\mathrm{Ge}_{1.5}(\\mathrm{PO}_{4})_{3}$ (LAGP) and $\\mathrm{Li}_{3.5}\\mathrm{Zn}_{0.25}\\mathrm{GeO}_{4}$ (LISICON) starting at 2.7 and $1.4~\\mathrm{V}_{;}$ , respectively, and Li−Ge alloys are formed at low voltages (Figure 1b and Table 2). The reductions of LAGP and LISICON are consistent with the experiment studies.26−28 The good agreements between our computation results and many experiments demonstrated the validity of our computation scheme.  \n\nOur calculations found the Li reduction of the solid electrolyte materials that are thought to be stable against Li. For example, LiPON, which is calculated using $\\mathrm{Li}_{2}\\mathrm{PO}_{2}\\mathrm{N}$ as a representative of the material class (details are provided in the Supporting Information), shows a reduction potential of 0.69 V. The final decomposition products of LiPON in equilibrium with Li metal are $\\mathrm{Li}_{3}\\mathrm{N},$ $\\mathrm{Li}_{2}\\mathrm O_{;}$ and $\\mathrm{Li}_{3}\\mathrm{P}$ (Table 2), which are consistent with the in situ XPS observations.13 Although the calculated decomposition energy of LiPON is as large as $-0.66$ eV/atom at $0\\mathrm{V}$ (Figure 1b and Table 2), LiPON is known to be compatible with Li metal as demonstrated by many experimental studies.10,29 Similarly, $\\mathrm{Li}_{3}\\mathrm{PS}_{4}$ and $\\mathrm{Li}_{7}\\mathrm{P}_{2}\\mathrm{S}_{8}\\mathrm{I},$ which are reported to be compatible with Li metal anode,8,30,31 are reduced against Li metal and at low voltages (Table 1 and Table 2). Therefore, the stability of these solid electrolyte materials against Li metal is not thermodynamically intrinsic.  \n\nThe decomposition products, which form an interphase between the solid electrolyte and electrode, passivate the solid electrolyte and inhibit the continuous decomposition. For example, the decomposition products of LiPON, $\\mathrm{Li}_{3}\\mathrm{PS}_{4},$ and $\\mathrm{Li}_{7}\\mathrm{P}_{2}\\mathrm{S}_{8}\\mathrm{I}$ are Li binary compounds, such as $\\mathrm{Li_{2}O,\\ L i_{2}S,\\ L i_{3}P,}$ $\\mathrm{Li}_{3}\\mathrm{N},$ and LiI, formed at the Li reduction. The interphase consisting of these decomposition products are stable against the high $\\mu_{\\mathrm{Li}}$ of Li metal (Figure 2a), which is beyond the reduction potential (cathodic limit) of the solid electrolyte (Figure 2b). At the equilibrium, the redistribution of $\\mathrm{Li}^{+}$ and other charged carriers (such as electron $\\mathrm{e}^{-\\cdot}$ ) are formed at the interface to account for the potential drop across the electrode−electrolyte interface.34 The electrochemical potential of the highly mobile $\\mathrm{Li}^{+}$ , $\\tilde{\\mu}_{\\mathrm{Li}^{+}},$ which includes the electrostatic potential energy, is constant across the interface. In contrast, the electrochemical potential of the electronic carrier $\\tilde{\\mu}_{\\mathrm{e}^{-}}$ (red line in Figure 2b) decreases significantly in the interphase from the anode to the solid electrolyte, since these interphases have poor electronic mobility and conductivity. Therefore, the Li chemical potential $\\mu_{\\mathrm{Li}}$ (black line in Figure 2b), which equals to the sum of $\\tilde{\\mu}_{\\mathrm{Li}^{*}}$ and $\\tilde{\\mu}_{\\mathrm{e}^{-}},$ decreases in the interphase from the anode to the solid electrolyte. The high value of $\\mu_{\\mathrm{Li}}$ from the anode decreases to be within the electrochemical window of the solid electrolyte after the passivation of the decomposition interphase. As a result, the decomposition of the solid electrolyte has no thermodynamic driving force to continue into the bulk. The solid electrolyte is stabilized by the decomposition interphases, which essentially serve as solidelectrolyte-interphases (SEIs) in all-solid-state Li-ion batteries. In summary, the SEI of the decomposition interphase decreases the high Li chemical potential $\\mu_{\\mathrm{Li}}$ applied on the solid electrolyte and bridges the Li chemical potential gap between Li metal and the solid electrolyte. This passivation mechanism explained the observed Li metal compatibility of LiPON, $\\mathrm{Li}_{3}\\mathrm{PS}_{4},$ and $\\mathrm{Li}_{7}\\mathrm{P}_{2}\\mathrm{S}_{8}\\mathrm{I}.$  \n\n![](images/677ee4b2869c9ff1c7202d9d7be6773ad93811439df65641bd6a4441abde3bc2.jpg)  \nFigure 1. Decomposition energy $E_{\\mathrm{D}}$ of (a) sulfide and (b) oxide solid electrolyte materials as a function of the applied voltage $\\phi$ or Li chemical potential $\\mu_{\\mathrm{Li}}.$  \n\n![](images/41295ceb4464ce4ddcf50864a13ea95890d5c2e29c6f3416950f3ab18e8dac61.jpg)  \nFigure 2. (a) Electrochemical window (solid color bar) of solid electrolyte and other materials. The oxidation potential to fully delithiate the material is marked by the dashed line. (b) Schematic diagram about the change of Li chemical potentials $\\mu_{\\mathrm{Li}}$ (black line), the electrochemical potential $\\tilde{\\mu}_{\\mathrm{Li}^{+}}$ (blue dashed line), and $\\tilde{\\mu}_{\\mathrm{e}}^{-}$ (red dashed line) across the interface between the anode and the solid electrolyte. Since the actual profile of $\\tilde{\\mu}_{\\mathrm{e}}^{\\mathrm{~-~}}$ determined by the charge carrier distribution may be complicated,32,33 the profiles of chemical and electrochemical potential shown here are schematic and may not be linear. The vertical scale is for the electrostatic potential or the voltage referenced to Li metal and is reversed for the chemical potential or electrochemical potential (eq 1).  \n\nThe passivation mechanism relies on the electronic insulating properties of the decomposition interphase layers to stabilize the solid electrolyte and is not active if the interphase layer is electronically conductive. For example, the reduction of LGPS, LAGP, and LISICON with Li metal forms electronically conductive Li−Ge alloys, and the lithiation of LLTO and LATP forms titanates with Ti of $^{3+}$ or lower valences. The decomposition interphases for these solid electrolytes at Li reductions are mixed electronic and ionic conductors. The electronic conductivity in the interphase cannot account for the drop of $\\tilde{\\mu}_{\\mathrm{e}^{-}}$ across the interface regardless of the specific electron transport mechanism being metallic, band, or polaronic conduction. These mixed conductor interphases cannot account for the $\\mu_{\\mathrm{Li}}$ drop as the change of both $\\tilde{\\mu}_{\\mathrm{Li}^{+}}$ and $\\tilde{\\mu}_{\\mathrm{e}^{-}}$ would be small across the interphase. As a result, the solid electrolyte is still exposed to the high $\\mu_{\\mathrm{Li}}$ of the anode, and the reduction reaction continues into the bulk. In addition, the mixed electronic and ionic conductor interphase facilitate the kinetic transport of Li ion and electrons for the decomposition reactions.14 The absence of the passivation mechanism explains the lithation and reduction of LGPS, LLTO, LATP, LAGP, and LISICON observed in the CV experiments.  \n\nIt is worth noting that garnet LLZO shows the lowest reduction potential of as low as $0.05\\mathrm{V}$ against Li and the least favorable decomposition reaction energy of only $0.021\\ \\mathrm{~eV}/\\$ atom $(49\\mathrm{~kJ/mol}$ of LLZO) at $0\\mathrm{v}$ among all solid electrolyte materials examined (Figure 1 and Table 2). Given such small reaction energy, the Li reduction of garnet is likely to be kinetically inhibited, and the reduction products of $\\mathrm{Li}_{2}\\mathrm O$ , $\\mathrm{Zr}_{3}\\mathrm{O},$ and $\\mathrm{La}_{2}\\mathrm{O}_{3}$ (Table 1) may provide passivation to the material. These explain the widely reported $_{0-5\\mathrm{~V~}}$ window of garnet from the CV measurements in the literature.5,35 The reduction of garnet at $0.05\\mathrm{~V~}$ forms $\\mathrm{Zr}_{3}\\mathrm{O}$ (Table 1), following another plateau at $0.004\\mathrm{V}$ to form $\\mathrm{Zr}$ (Table 2 and Table S2) based on the DFT GGA energies. Since these small values of energy and voltage is below typical accuracy of DFT and the approximations of the scheme (see section 2), it is inconclusive whether the garnet LLZO is reduced to $\\mathrm{Zr}_{3}\\mathrm{O}$ or $Z\\mathbf{r}$ at $0\\mathrm{v}$ or against Li metal. Nevertheless, the formation of $\\mathrm{Zr}$ would be thermodynamically favorable at a potential significantly lower than $0\\mathrm{V},$ which corresponds to applying high current density at the Li−LLZO interface. Recent report of instability of garnet against Li at elevated temperatures of $300~^{\\circ}\\mathrm{C}$ may be an indication of the limited stability of garnet against Li metal,36 as the diffusion and phase nucleation are facilitated at high temperatures.  \n\n3.2. Stability of Solid Electrolyte Materials at High Voltages. The oxidation reactions of the solid electrolyte materials were investigated using the same method in section 3.1. The LGPS material is delithiated and oxidized starting at $2.14\\mathrm{V}$ (Table 1 and Figure 1), and the final oxidation products of $\\mathrm{P}_{2}S_{5},$ ${\\mathrm{GeS}}_{2},$ and S are formed at the equilibrium oxidation potential of 2.31 V (Table 3). The oxidation potential of the  \n\nTable 3. Oxidation Reaction of the Solid Electrolyte Materials at 5 V   \n\n\n<html><body><table><tr><td colspan=\"2\">phase equilibria at 5 V Ep (eV/atom)</td></tr><tr><td>LiS</td><td>S -1.99</td></tr><tr><td>LGPS GeSz, PSs, S</td><td>-1.12</td></tr><tr><td>Li3.25Ge0.25P0.75S4 PSs, S, GeS2</td><td>-1.08</td></tr><tr><td>LiPS4 S, PSs</td><td>-1.01</td></tr><tr><td>Li4GeS4 GeSz, S</td><td>-1.27</td></tr><tr><td>LiPS1 S, PS5</td><td>-0.92</td></tr><tr><td>LigPS5Cl PSs, S, PCl</td><td>-1.33</td></tr><tr><td>Li-PS8I PSs, S, I</td><td>-1.04</td></tr><tr><td>LiPON PNO, POs, N</td><td>-0.69</td></tr><tr><td>LLZO OLaO,LaZrO</td><td>-0.53</td></tr><tr><td>LLTO O,TiO, LaTiO7</td><td>-0.15</td></tr><tr><td>LATP O,TiPO7,Ti5P4O20,AIPO4</td><td>-0.065</td></tr><tr><td>LAGP</td><td>Ge5O(PO4)6, GePO7, AIPO4, O -0.056</td></tr><tr><td>LISICON ZnGeO4, GeO, O</td><td>-0.57</td></tr></table></body></html>  \n\nLGPS is confirmed by the CV experiment.12 Similar to $\\mathrm{Li}_{2}S,$ all sulfide solid electrolytes such as $\\mathrm{Li}_{3.25}\\mathrm{Ge}_{0.25}\\mathrm{P}_{0.75}\\mathrm{S}_{4},$ $\\mathrm{Li}_{3}\\mathrm{PS}_{4},$ $\\mathrm{Li}_{4}\\mathrm{GeS}_{4},\\mathrm{Li}_{7}\\mathrm{P}_{3}\\mathrm{S}_{11},$ and $\\mathrm{Li}_{7}\\mathrm{P}_{2}\\mathrm{S}_{8}\\mathrm{I}$ are oxidized at $2{-}2.5\\mathrm{V}$ to form S (Table 1 and Table 3). The other elements, such as $\\mathrm{\\bfP}$ and Ge, are usually oxidized into $\\mathrm{P}_{2}\\mathrm{S}_{5}$ and ${\\mathrm{GeS}}_{2},$ respectively. The oxidation reactions of sulfide solid electrolytes are highly favorable at $\\ensuremath{5\\mathrm{~V~}}$ as described by the decomposition energy (Figure 1a and Table 3).  \n\nOxide solid electrolyte materials have higher oxidation potential than sulfides. The oxidation of LLZO, LISICON, and LLTO starts at 2.91, 3.39, and $3.71~\\mathrm{V}_{;}$ , respectively. The NASICON materials, LATP and LAGP, show the best resistance to oxidation with the highest oxidation potential of 4.21 and $4.28~\\mathrm{~V},$ respectively (Table 1), and the lowest decomposition energy of only $\\mathrm{\\sim-}0.06~\\mathrm{eV/}$ atom at $_{5\\mathrm{~V~}}$ (Figure 1b and Table 3). The delithiation reactions continue at higher voltages, and $\\mathrm{O}_{2}$ gas is released during the oxidation at high voltages for all oxide solid electrolytes (Table 3). The oxidation of these solid electrolyte materials is not surprising, given that $\\mathrm{Li}_{2}\\mathrm O$ is oxidized at $2.9\\mathrm{V}$ and that the $\\mathrm{O}_{2}$ gas is released by the further oxidation of ${\\mathrm{Li}}_{2}{\\mathrm O}_{2}$ . LiPON starts oxidation at $2.63\\mathrm{~V~}$ with the $\\mathbf{N}_{2}$ gas release. Our computation results are consistent with the experiments by $\\mathrm{Yu}$ et al.,10 in which the onset of LiPON oxidation at ${\\sim}2.6\\mathrm{~V~}$ in the $I{-}V$ measurements and the microsized gas bubbles in the LiPON material were observed after applying a high voltage of $6\\mathrm{V}$ .  \n\nA significant overpotential to the calculated thermodynamic equilibrium potential is expected for the oxidation reaction processes, which are likely to have slow kinetics. The kinetic limitations of the oxidation reactions may come from multiple aspects. Most decomposition products at high voltages (Table 3) are electronically insulating, and the diffusion of non-Li elements is usually slow in solids. Furthermore, the nucleation and release of $\\mathrm{O}_{2}$ and $\\mathbf{N}_{2}$ gas molecules are likely to have sluggish kinetics. For example, a significant overpotential of $^{>1}$ V is often observed in the oxygen evolution reactions in metalair batteries.37 Therefore, the overpotential of the decomposition reactions may provide a higher nominal oxidation potential of ${>}5\\mathrm{~V~}$ and a wider nominal electrochemical window observed in the CV experiments.1,8−10  \n\n![](images/00bbd9eb45f3a8e42f0067205e6b8af475ef2980d3b6d7cf63699146ced97e27.jpg)  \nFigure 3. (a) Electrochemical stability window (solid color bars) of commonly used coating layer materials. The oxidation potential to fully delithiate the material is marked by the dashed line. The line at $3.9\\mathrm{~V~}$ represents the equilibrium voltage of the $\\mathrm{LiCoO}_{2}$ cathode material. (b) Schematic diagram about the change of Li chemical potentials $\\mu_{\\mathrm{Li}}$ (black line) and the electrochemical potential $\\tilde{\\mu}_{\\mathrm{Li}^{+}}$ (blue dashed line) and $\\tilde{\\mu}_{\\mathrm{e}}^{-}$ (red dashed line) across the interface between the solid electrolyte and the cathode material.  \n\n3.3. Extend the Stability of Solid Electrolytes by Applying Coating Layers. Currently, the interfacial resistance has become a critical problem for the performance of allsolid-state Li-ion batteries. The engineering of the interface, such as the application of interfacial coating layers, is used to improve interfacial protection and to reduce interface resistance. In this section, we investigated the electrochemical stability of the coating layer materials, such as $\\operatorname{Li}_{4}\\operatorname{Ti}_{5}\\operatorname{O}_{12},$ 38,39 $\\mathrm{LiTaO_{3}^{'40}\\ L i N b O_{3}^{41,\\ddagger_{2}}\\ L i_{2}S i O_{3}^{'43}}$ and $\\mathrm{Li_{3}P O_{4}},^{44}$ wh5ich were demonstrated to suppress the mutual diffusion of non-Li elements and to reduce the interfacial resistance at the solid electrolyte−cathode interfaces in all-solid-state Li-ion batteries. $^{2,\\mathrm{i}6,40,45}$ Our calculations show that these coating layer materials have an electrochemical window from the reduction potential of $0.7\\mathrm{-}1.7\\mathrm{~V~}$ to the oxidation potential of $3.7\\mathrm{-}4.2\\mathrm{~V~}$ (Figure 3a). Therefore, the coating layer materials are stable between 2 and $4\\mathrm{V}_{i}$ , the usual voltage range during the cycling of Li-ion batteries. In addition, the coating layer materials have poor electronic conductivity and can serve as artificial SEIs to passivate the solid electrolyte through the same mechanisms illustrated in section 3.1 (Figure 3b). Given that the sulfide solid electrolyte materials are oxidized at as low as $2\\mathrm{V}$ and are not thermodynamically stable at the voltage of $4\\mathrm{V}_{i}$ , the coating layers serve as critical passivations through the same mechanism illustrated in section 3.1. The coating layers mitigate the low Li chemical potential $\\mu_{\\mathrm{Li}}$ from the cathode material applied on the solid electrolyte materials. As a result, the oxidation and delithiation of the solid electrolyte at the cathode interface is stopped, and the oxidation potential (anodic limit) of the solid electrolyte is extended by the artificial coating layer. Therefore, the coating layer effectively extended the anodic limit of the sulfide solid electrolyte from ${\\sim}2{-}2.3\\mathrm{~~V~}$ to ${\\sim}4\\mathrm{~V~}$ . The overpotential to oxidize the coating layers may further extend the nominal stability window. Similar strategy of applying artificial coating layers has been employed at the anode side for the protection and stabilization of Li metal anode. For example, Polyplus46 has applied coating layers between Li metal and LATP electrolyte to protect the LATP materials against Li metal. The passivation mechanism of the coating layer at the anode side is the same as the decomposition interphase demonstrated in section 3.1.  \n\n# 4. DISCUSSION  \n\nOur thermodynamic analyses based on first-principles calculations indicate that most solid electrolyte materials have a limited electrochemical window. In contrast to the widely held perception about the outstanding stability of the solid electrolyte materials, the solid electrolyte materials are reduced and oxidized at low and high potentials, respectively, and are not thermodynamically stable against Li metal. The sulfide solid electrolytes based on thio-phosphates are reduced at ${\\sim}1.6{-}1.7\\$ $\\mathrm{\\DeltaV}$ and oxidized at ${\\sim}2{\\-}{-}2.3\\ \\mathrm{V}.$ . The stability window of oxide solid electrolytes varies greatly from one material to another. Although some oxides have high reduction potential as sulfides, most oxide solid electrolytes have a significantly higher oxidation potential and are not oxidized until $>3\\mathrm{~V~}$ . In particular, the NASICON materials, LATP and LAGP, are thermodynamically stable up to ${\\sim}4.2\\mathrm{V}.$ . Among all these oxides investigated, the Li garnet materials, such as LLZO, have the best resistance to Li reduction. Overall, the oxide solid electrolyte materials have significantly wider electrochemical window than sulfides. The reduction and oxidation potentials as well as the decomposition products of solid electrolytes predicted from our calculations are in good agreement with prior experimental studies, confirming that our computation method based on the Li grand potential phase diagram is a valid scheme in evaluating the electrochemical stability of materials.  \n\nOur calculation results demonstrated that the good stability of the solid electrolyte materials is originated from the kinetic stabilizations. First, the wide, nominal electrochemical window observed in many CV experiments can be partially attributed to the significant overpotential of the sluggish kinetics during the decomposition reactions (Figure 4). The decomposition reactions though kinetically sluggish are still thermodynamically favorable at the applied overpotential and may happen over an extended period of time, leading to the deterioration of the batteries. This kinetic stabilization from the sluggish kinetics of the reactions is different from the passivation mechanisms illustrated in section 3.1. The passivation mechanism of the interphases is the origin of the outstanding stability in the solid electrolyte. The decomposition interphases with good stability and poor electronic transport are effectively the SEIs in the allsolid-state Li-ion batteries to passivate the solid electrolytes (Figure 4). The interphases, which are stable against solid electrolytes and electrodes, mitigate the Li chemical potential discrepancy between the electrolyte and electrode at the interfaces. As a result, the anodic/cathode limits and the electrochemical window of the solid electrolyte are significantly extended by the extra electrochemical window provided by the interphases (Figure 4). The effective electrochemical window of the solid electrolyte materials is its own intrinsic electrochemical window plus the electrochemical window of the interphases (Figure 4).  \n\n![](images/b619fdac14a7665f4d320d5a4e0eebf8f61c9e93addd0671397382d3242a3d10.jpg)  \nFigure 4. Schematic diagram about the electrochemical window (color bars) and the Li chemical potential profile (black line) in the all-solidstate Li-ion battery. The profile of chemical potential is schematic in this plot and may not be linear. The high $\\mu_{\\mathrm{Li}}$ in the anode (silver) and low $\\mu_{\\mathrm{Li}}$ in the cathode (blue) are beyond the stability window of the solid electrolyte (green). The observed nominal electrochemical window is extended by the overpotential (dashed line) and by the interphases (orange and yellow), which account for the gap of $\\mu_{\\mathrm{Li}}$ between solid electrolyte and electrodes across the interfaces.  \n\nIn this study, the electrochemical window of the solid electrolyte and the extensions by the interphases were calculated using the first-principles methods. Our computation scheme evaluated the electrochemical window based on the equilibrium of the neutral Li, which is a necessary condition for the equilibrium at the interface. As suggested by Goodenough,47 the electrochemical window of the electrolyte can also be estimated by the difference between the lowest unoccupied molecular orbital (LUMO) and highest occupied molecular orbital (HOMO) states of the electrolyte based on the equilibrium of electrons across the interfaces. These equilibrium conditions of carriers other than neutral Li also need to be satisfied at the interfaces. The equilibria of the charged carriers such as electrons or $\\mathrm{Li^{+}}$ are subject to the formation of polarizations and interfacial space charge layers, which are dependent on the defect chemistry and the structures of the interface.32,33 In some cases, a significant amount of electrons or holes may accumulate in the interphases due to the charge redistribution, defect chemistry, or special interfacial structures and may activate the electronic conductions in the interphase deactivating the passivation effects. Nevertheless, our results based on the equilibrium of neutral Li are in good agreement with many experimental studies, suggesting the validity of our scheme.  \n\nThe interphase stabilization mechanism provides guidance for the development of solid electrolyte materials. The formation of the decomposition interphases plays an essential role in the stability of the solid electrolyte and should be considered in the design of solid electrolyte materials. Our calculations have shown that the reduction of the solid electrolytes is generally governed by the reduction of the cations, and the interphases formed by the reduction of these cations often control the interfacial stability. For example, LGPS, LAGP, LATP, and LLTO solid electrolyte materials form electronically conductive interphases at low voltages, such as Li−Ge alloys or Li titanates, which cannot provide the passivation for the solid electrolyte materials. Therefore, our results suggest that certain cations or dopants, such as Ti and Ge, in the solid electrolyte materials, negatively affect the stability against Li metal. Other cations, such as Si, Sn, Al, and $Z\\mathrm{n},$ , may have a similar effect. However, doping with anions does not have such limitations for the stability of the solid electrolyte at low voltages. The Li reduction products of common anions, such as O, S, F, Cl, and I, are usually Li binary materials, such as lithium chalcogenides and lithium halides, which are thermodynamically stable against Li and are good electronic insulators. The passivation provided by these materials is the origin of Li metal compatibility for LiPON, $\\mathrm{Li}_{3}\\mathrm{PS}_{4},$ and $\\mathrm{Li}_{7}\\mathrm{P}_{2}\\mathrm{S}_{8}\\mathrm{I}$ solid electrolyte materials.8,10,13,30 Doping lithium halides is a highly effective method in the design of solid electrolyte to simultaneously achieve improved ionic conductivity and Li metal stability.30,48,49  \n\nIn addition, the properties of the decomposition interphases significantly affect the performance of all-solid-state Li-ion batteries. The decomposition interphases with electronic conductivity may enable the continuous decomposition of the solid electrolyte during the cycling of the batteries. For example, a recent experimental study12 has identified that the reduction and oxidation products of the LGPS can be reversibly cycled. Therefore, the interphases formed due to the decomposition of the solid electrolyte may effectively become a part of active electrode materials of the battery. Such decomposition of the solid electrolyte materials during the cycling of the battery may cause degradations of the interfaces, leading to high interfacial resistance, low coulombic efficiency, and poor reversibility, which are major limiting factors in the performance of all-solid-state Li-ion batteries. While the good electronic insulation of the decomposition products are preferred to achieve good stability and low thickness of the interphases, the high Li ionic conductivity is important for achieving low interfacial resistance. For example, $\\mathrm{Li}_{3}\\mathrm{N}$ and $\\mathrm{Li}_{3}\\mathrm{P}$ formed at the LiPON−Li interface are phases with high Li ionic conductivity,50,51 which may explain the good interfacial conductance for LiPON−Li interface.  \n\nHowever, the properties of the decomposition interphases may not always be as desired, since these critical interfacial properties are determined by the spontaneous decomposition of the solid electrolytes and electrode materials.16 The undesired electronic conductivity of the decomposition products may cause continuous decompositions of the solid electrolyte materials, since the electronic insulation of the decomposition interphases is essential in stabilizing the solid electrolyte. The engineering of the interface, such as the application of artificial coating layer, is a demonstrated method for the interfacial protection and to reduce interfacial resistance if the spontaneously formed SEI layers have unsatisfactory properties (e.g., high electronic conductivity and low ${\\mathrm{Li}}^{+}$ conductivity). Our calculation results showed that the coating layer materials passivate the solid electrolyte against the oxidation at high voltages. The outstanding stability of the coating layer against both solid electrolyte and electrode also impedes the mutual diffusion of non-Li elements, such as Co and $s,$ at the interface, which is a known problem for the degradation of interfaces between the sulfide electrolyte and $\\mathrm{LiCoO}_{2}$ .16 Furthermore, the coating layer artificially applied through thin film deposition is as thin as a few nanometers,38,42 while the interphase layer formed by the spontaneous decomposition can be as thick as $100\\ \\mathrm{nm}.^{\\cdot16,42,45}$ The thinner coating layer of less than $10\\ \\mathrm{~nm}$ yields significantly lower interfacial resistance.16,42 In addition, as the applied coating layer bridges the differences of Li chemical potential between the solid electrolyte and the cathode material, the formation of space-charge layers is mitigated40 to reduce the interfacial resistance. Therefore, applying artificial coating layer provides multiple advantages compared to the interphases formed by the spontaneous decompositions. The development of materials processing techniques to engineer the interphases is critical for improving the performance of all-solid-state Li-ion batteries.  \n\n# 5. CONCLUSIONS  \n\nOur first-principles calculation results indicate that most solid electrolyte materials have limited electrochemical window in contrast to the widely held perception about the outstanding stability of the solid electrolyte materials. Most solid electrolyte materials are not thermodynamically stable against Li metal and are reduced and oxidized at low and high potentials, respectively. Sulfide-based solid electrolytes have significantly narrower electrochemical window than the oxide-based solid electrolytes. Our calculation results show that the good stability of the solid electrolyte materials is not thermodynamically intrinsic but is rather originated from the kinetic stabilization. This kinetic stabilization is achieved due to the sluggish kinetics of the decomposition reactions and the decomposition interphases with poor electronic transport similar to the SEIs. We illustrated the stabilization mechanisms of the decomposition interphases, which passivate the solid electrolytes by mitigating extreme Li chemical potential from the electrodes. Our results suggest that the decomposition interphases of the solid electrolyte and the engineering of the interface are critical for the performance of all-solid-state Li-ion batteries. The interphases with good electronic insulation and high Li ionic conductivity are preferred to achieve an interface with good stability and low resistance. The application of artificial coating layers is a promising method for the stabilizing interfaces and for reducing interfacial resistance. Our study demonstrated the computation scheme to evaluate the electrochemical stability and the decomposition interphases of solid electrolyte materials and provided the fundamental understanding to guide the future design of solid electrolytes and interphases in all-solidstate Li-ion batteries.  \n\n# ASSOCIATED CONTENT  \n\n# $\\otimes$ Supporting Information  \n\nThe Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.5b07517.  \n\nDescriptions about the solid electrolyte materials investigated in the computation and the calculated phase equilibria for the lithiation and delithiation of solid electrolyte and coating materials (PDF)  \n\n# AUTHOR INFORMATION  \n\nCorresponding Author $^{*}\\mathrm{E}$ -mail: yfmo@umd.edu.  \n\n# Notes  \n\nThe authors declare no competing financial interest.  \n\n# ACKNOWLEDGMENTS  \n\nWe thank Prof. Chunsheng Wang and Fudong Han for helpful discussions. This work was supported by U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, under Award No. DE-EE0006860. This research used computational facilities from the University of Maryland supercomputing resources and from the Extreme Science and Engineering Discovery Environment (XSEDE) supported by National Science Foundation Award No. TG-DMR130142.  \n\n# REFERENCES  \n\n(1) Kamaya, N.; Homma, K.; Yamakawa, Y.; Hirayama, M.; Kanno, R.; Yonemura, M.; Kamiyama, T.; Kato, Y.; Hama, S.; Kawamoto, K.; Mitsui, A. A Lithium Superionic Conductor. Nat. Mater. 2011, 10, 682−686. (2) Takada, K. Progress and Prospective of Solid-State Lithium Batteries. Acta Mater. 2013, 61, 759−770. (3) Seino, Y.; Ota, T.; Takada, K.; Hayashi, A.; Tatsumisago, M. A Sulphide Lithium Super Ion Conductor Is Superior to Liquid Ion Conductors for Use in Rechargeable Batteries. Energy Environ. Sci. 2014, 7, 627−631. 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Python Materials Genomics (Pymatgen): A Robust, Open-Source Python Library for Materials Analysis. Comput. Mater. Sci. 2013, 68, 314−319.   \n(23) Ong, S. P.; Mo, Y.; Richards, W. D.; Miara, L.; Lee, H. S.; Ceder, G. Phase Stability, Electrochemical Stability and Ionic Conductivity of the $\\mathrm{Li}_{10\\pm1}\\mathrm{MP}_{2}\\mathrm{X}_{12}$ ( $\\mathbf{\\tilde{M}}=\\mathbf{Ge},$ , Si, Sn, Al or ${\\bf P}_{i}$ and $X=\\mathrm{{O}}$ , S or Se) Family of Superionic Conductors. Energy Environ. Sci. 2013, 6, 148− 156.   \n(24) Chen, C. H.; Amine, K. Ionic Conductivity, Lithium Insertion and Extraction of Lanthanum Lithium Titanate. Solid State Ionics 2001, 144, 51−57.   \n(25) Stramare, S.; Thangadurai, V.; Weppner, W. Lithium Lanthanum Titanates: A Review. Chem. Mater. 2003, 15, 3974−3990. (26) Feng, J. K.; Lu, L.; Lai, M. O. Lithium Storage Capability of Lithium Ion Conductor Li1.5Al0.5Ge1.5(PO4)3. J. Alloys Compd. 2010, 501, 255−258.   \n(27) Alpen, U. v.; Bell, M. F.; Wichelhaus, W.; Cheung, K. Y.; Dudley, G. J. Ionic Conductivity of Li14Zn(GeO4)4 (LISICON). Electrochim. Acta 1978, 23, 1395−1397.   \n(28) Knauth, P. Inorganic Solid Li Ion Conductors: An Overview. Solid State Ionics 2009, 180, 911−916. (29) West, W. C.; Whitacre, J. F.; Lim, J. R. Chemical Stability Enhancement of Lithium Conducting Solid Electrolyte Plates Using Sputtered LiPON Thin Films. J. Power Sources 2004, 126, 134−138. (30) Rangasamy, E.; Liu, Z.; Gobet, M.; Pilar, K.; Sahu, G.; Zhou, W.; Wu, H.; Greenbaum, S.; Liang, C. An Iodide-Based Li7P2S8I Superionic Conductor. J. Am. Chem. Soc. 2015, 137, 1384−1387. (31) Lepley, N. D.; Holzwarth, N. A. W.; Du, Y. A. Structures, $\\mathrm{Li}+$ Mobilities, and Interfacial Properties of Solid Electrolytes Li3PS4 and Li3PO4 from First Principles. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 104103.   \n(32) Leung, K.; Leenheer, A. How Voltage Drops Are Manifested by Lithium Ion Configurations at Interfaces and in Thin Films on Battery Electrodes. J. Phys. Chem. C 2015, 119, 10234−10246.   \n(33) Haruyama, J.; Sodeyama, K.; Han, L.; Takada, K.; Tateyama, Y. Space−Charge Layer Effect at Interface between Oxide Cathode and Sulfide Electrolyte in All-Solid-State Lithium-Ion Battery. Chem. Mater. 2014, 26, 4248−4255.   \n(34) Weppner, W., Fundamental Aspects of Electrochemical, Chemical and Electrostatic Potentials in Lithium Batteries. In Materials for Lithium-Ion Batteries; Julien, C., Stoynov, Z., Eds.; Springer Netherlands: Dordrecht, The Netherlands, 2000; Chapter 20, pp 401− 412.   \n(35) Kotobuki, M.; Munakata, H.; Kanamura, K.; Sato, Y.; Yoshida, T. Compatibility of Li7La3Zr2O12 Solid Electrolyte to All-Solid-State Battery Using Li Metal Anode. J. Electrochem. Soc. 2010, 157, A1076− A1079.   \n(36) Wolfenstine, J.; Allen, J. L.; Read, J.; Sakamoto, J. Chemical Stability of Cubic Li7La3Zr2O12 with Molten Lithium at Elevated Temperature. J. Mater. Sci. 2013, 48, 5846−5851.   \n(37) McCloskey, B. D.; Scheffler, R.; Speidel, A.; Girishkumar, G.; Luntz, A. C. On the Mechanism of Nonaqueous Li−O2 Electrochemistry on C and Its Kinetic Overpotentials: Some Implications for Li−Air Batteries. J. Phys. Chem. C 2012, 116, 23897−23905.   \n(38) Ohta, N.; Takada, K.; Zhang, L.; Ma, R.; Osada, M.; Sasaki, T. Enhancement of the High-Rate Capability of Solid-State Lithium Batteries by Nanoscale Interfacial Modification. Adv. Mater. 2006, 18, 2226−2229.   \n(39) Kitaura, H.; Hayashi, A.; Tadanaga, K.; Tatsumisago, M. Improvement of Electrochemical Performance of All-Solid-State Lithium Secondary Batteries by Surface Modification of LiMn2O4 Positive Electrode. Solid State Ionics 2011, 192, 304−307.   \n(40) Takada, K.; Ohta, N.; Zhang, L.; Fukuda, K.; Sakaguchi, I.; Ma, R.; Osada, M.; Sasaki, T. Interfacial Modification for High-Power Solid-State Lithium Batteries. Solid State Ionics 2008, 179, 1333−1337. (41) Ohta, N.; Takada, K.; Sakaguchi, I.; Zhang, L.; Ma, R.; Fukuda, K.; Osada, M.; Sasaki, T. LiNbO3-Coated LiCoO2 as Cathode Material for All Solid-State Lithium Secondary Batteries. Electrochem. Commun. 2007, 9, 1486−1490.   \n(42) Kato, T.; Hamanaka, T.; Yamamoto, ${\\mathrm{K}}.{\\mathrm{}}$ Hirayama, T.; Sagane, F.; Motoyama, M.; Iriyama, Y. In-Situ Li7La3Zr2O12/LiCoO2 Interface Modification for Advanced All-Solid-State Battery. J. Power Sources 2014, 260, 292−298.   \n(43) Sakuda, A.; Kitaura, H.; Hayashi, A.; Tadanaga, K.; Tatsumisago, M. Improvement of High-Rate Performance of All-Solid-State Lithium Secondary Batteries Using LiCoO2 Coated with Li2O−SiO2 Glasses. Electrochem. Solid-State Lett. 2008, 11, A1−A3.   \n(44) Jin, Y.; Li, N.; Chen, C. H.; Wei, S. $\\mathsf{Q}.$ Electrochemical Characterizations of Commercial LiCoO2 Powders with Surface Modified by Li3PO4 Nanoparticles. Electrochem. Solid-State Lett. 2006, 9, A273−A276.   \n(45) Kim, K. H.; Iriyama, Y.; Yamamoto, K.; Kumazaki, S.; Asaka, T.; Tanabe, K.; Fisher, C. A. J.; Hirayama, T.; Murugan, R.; Ogumi, Z. Characterization of the Interface between LiCoO2 and Li7La3Zr2O12 in an All-Solid-State Rechargeable Lithium Battery. J. Power Sources 2011, 196, 764−767.   \n(46) Visco, S.; Nimon, V.; Petrov, A.; Pridatko, K.; Goncharenko, N.; Nimon, E.; De Jonghe, L.; Volfkovich, Y.; Bograchev, D. Aqueous and Nonaqueous Lithium-Air Batteries Enabled by Water-Stable Lithium Metal Electrodes. J. Solid State Electrochem. 2014, 18, 1443−1456. (47) Goodenough, J. B.; Kim, Y. Challenges for Rechargeable Li Batteries. Chem. Mater. 2010, 22, 587−603.   \n(48) Rangasamy, E.; Li, J.; Sahu, G.; Dudney, N.; Liang, C. Pushing the Theoretical Limit of Li-CFx Batteries: A Tale of Bifunctional Electrolyte. J. Am. Chem. Soc. 2014, 136, 6874−6877.   \n(49) Deiseroth, H.-J.; Kong, S.-T.; Eckert, H.; Vannahme, J.; Reiner, C.; Zaiß, T.; Schlosser, M. Li6PS5X: A Class of Crystalline Li-Rich Solids with an Unusually High $\\mathrm{Li^{+}}$ Mobility. Angew. Chem., Int. Ed. 2008, 47, 755−758.   \n(50) Alpen, U. v.; Rabenau, A.; Talat, G. H. Ionic Conductivity in Li3N Single Crystals. Appl. Phys. Lett. 1977, 30, 621−623.   \n(51) Nazri, G. Preparation, Structure and Ionic Conductivity of Lithium Phosphide. Solid State Ionics 1989, 34, 97−102.  ",
    "Target": null
  },
  {
    "id": "10.1063_1.4908244",
    "DOI": "10.1063/1.4908244",
    "DOI Link": "http://dx.doi.org/10.1063/1.4908244",
    "Relative Dir Path": "mds/10.1063_1.4908244",
    "Article Title": "Characterization of Lorenz number with Seebeck coefficient measurement",
    "Authors": "Kim, HS; Gibbs, ZM; Tang, YL; Wang, H; Snyder, GJ",
    "Source Title": "APL MATERIALS",
    "Abstract": "In analyzing zT improvements due to lattice thermal conductivity (kappa(L)) reduction, electrical conductivity (sigma) and total thermal conductivity (kappa(Total)) are often used to estimate the electronic component of the thermal conductivity (kappa(E)) and in turn kappa(L) from kappa(L) = similar to kappa(Total) - L sigma T. TheWiedemann-Franz law, kappa(E) = L sigma T, where L is Lorenz number, is widely used to estimate kappa(E) from sigma measurements. It is a common practice to treat L as a universal factor with 2.44 x 10(-8) W Omega K-2 (degenerate limit). However, significant deviations from the degenerate limit (approximately 40% or more for Kane bands) are known to occur for non-degenerate semiconductors where L converges to 1.5 x 10(-8) W Omega K-2 for acoustic phonon scattering. The decrease in L is correlated with an increase in thermopower (absolute value of Seebeck coefficient (S)). Thus, a first order correction to the degenerate limit of L can be based on the measured thermopower, vertical bar S vertical bar, independent of temperature or doping. We propose the equation: L = 1.5 + exp [ -vertical bar S vertical bar/116] (where L is in 10(-8) W Omega K-2 and S in mu V/K) as a satisfactory approximation for L. This equation is accurate within 5% for single parabolic band/acoustic phonon scattering assumption and within 20% for PbSe, PbS, PbTe, Si0.8Ge0.2 where more complexity is introduced, such as non-parabolic Kane bands, multiple bands, and/or alternate scattering mechanisms. The use of this equation for L rather than a constant value (when detailed band structure and scattering mechanism is not known) will significantly improve the estimation of lattice thermal conductivity. (C) 2015 Author(s).",
    "Times Cited, WoS Core": 1564,
    "Times Cited, All Databases": 1632,
    "Publication Year": 2015,
    "Research Areas": "Science & Technology - Other Topics; Materials Science; Physics",
    "UT (Unique WOS ID)": "WOS:000353828400013",
    "Markdown": "# Characterization of Lorenz number with Seebeck coefficient measurement  \n\nHyun-Sik Kim, Zachary M. Gibbs, Yinglu Tang, Heng Wang, and G. Jeffrey Snyder  \n\nCitation: APL Materials 3, 041506 (2015); doi: 10.1063/1.4908244   \nView online: http://dx.doi.org/10.1063/1.4908244   \nView Table of Contents: http://scitation.aip.org/content/aip/journal/aplmater/3/4?ver=pdfcov   \nPublished by the AIP Publishing   \nArticles you may be interested in   \nTowards a predictive route for selection of doping elements for the thermoelectric compound PbTe from first  \nprinciples   \nJ. Appl. Phys. 117, 175102 (2015); 10.1063/1.4919425   \nInterplay of chemical expansion, Yb valence, and low temperature thermoelectricity in the YbCu2Si2−xGex   \nsolid solution   \nJ. Appl. Phys. 117, 135101 (2015); 10.1063/1.4916786  \n\nComputational modeling and analysis of thermoelectric properties of nanoporous silicon J. Appl. 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Jeffrey Snyder1,a   \n1Department of Materials Science, California Institute of Technology, Pasadena,   \nCalifornia 91125, USA   \n2Materials Research Center, Samsung Advanced Institute of Technology, Samsung   \nElectronics, Suwon 443-803, South Korea   \n3Division of Chemistry and Chemical Engineering, California Institute of Technology,   \nPasadena, California 91125, USA  \n\n(Received 31 December 2014; accepted 1 February 2015; published online 18 February 2015)  \n\nIn analyzing $z T$ improvements due to lattice thermal conductivity $(\\kappa_{L})$ reduction, electrical conductivity $(\\sigma)$ and total thermal conductivity $(\\kappa_{T o t a l})$ are often used to estimate the electronic component of the thermal conductivity $(\\kappa_{E})$ and in turn $\\kappa_{L}$ from $\\kappa_{L}=\\sim\\kappa_{T o t a l}-L\\sigma T$ . The Wiedemann-Franz law, $\\kappa_{E}=L\\sigma T$ , where $L$ is Lorenz number, is widely used to estimate $\\kappa_{E}$ from $\\sigma$ measurements. It is a common practice to treat $L$ as a universal factor with $2.44\\times10^{-8}\\mathrm{W}\\Omega\\mathrm{K}^{-2}$ (degenerate limit). However, significant deviations from the degenerate limit (approximately $40\\%$ or more for Kane bands) are known to occur for non-degenerate semiconductors where $L$ converges to $1.5\\times10^{-8}\\mathrm{W}\\Omega\\mathrm{K}^{-2}$ for acoustic phonon scattering. The decrease in $L$ is correlated with an increase in thermopower (absolute value of Seebeck coefficient (S)). Thus, a first order correction to the degenerate limit of $L$ can be based on the measured thermopower, $|S|$ , independent of temperature or doping. We propose the equation: $\\begin{array}{r}{L=1.5+\\exp\\left[-\\frac{\\left|S\\right|}{116}\\right]}\\end{array}$ (where $L$ is in $10^{-8}\\mathrm{W}\\Omega\\mathrm{K}^{-2}$ and $s$ in $\\upmu\\mathrm{V/K})$ ) as a satisfactory approximation for $\\bar{L}$ . This equation is accurate within $5\\%$ for single parabolic band/acoustic phonon scattering assumption and within $20\\%$ for PbSe, PbS, PbTe, $\\mathrm{Si}_{0.8}\\mathrm{Ge}_{0.2}$ where more complexity is introduced, such as non-parabolic Kane bands, multiple bands, and/or alternate scattering mechanisms. The use of this equation for $L$ rather than a constant value (when detailed band structure and scattering mechanism is not known) will significantly improve the estimation of lattice thermal conductivity. $\\circleddash$ 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4908244]  \n\nA semiconductor with large Seebeck coefficient, high electrical conductivity, and low thermal conductivity is a good candidate for a thermoelectric material. The thermoelectric material’s maximum efficiency is determined by its figure of merit $\\begin{array}{r}{z T=\\frac{S^{2}\\sigma T}{\\kappa_{E}+\\kappa_{L}}}\\end{array}$ , where $T,S,\\sigma,\\kappa_{E}$ , and $\\kappa_{L}$ are the temperature, Seebeck coefficient, electrical conductivity, and the electronic and lattice contributions to the thermal conductivity, respectively. Because the charge carriers (electrons in $n$ -type or holes in $p$ -type semiconductors) transport both heat and charge, $\\kappa_{E}$ is commonly estimated using the measured $\\sigma$ using the Wiedemann-Franz law: $\\kappa_{E}=L\\sigma T$ , where $L$ is the Lorenz number. Once $\\kappa_{E}$ is known, $\\kappa_{L}$ is computed by subtracting the $\\kappa_{E}$ from the total thermal conductivity, $K_{T o t a l}=\\kappa_{E}+\\kappa_{L}$ . For this method, the bipolar thermal conductivity $\\left(\\kappa_{B}\\right)$ will also be included which can be written $\\kappa_{B}+\\kappa_{L}=\\kappa_{T o t a l}-L\\sigma T$ .  \n\nSince a high $z T$ requires low $\\kappa_{T o t a l}$ but high $\\sigma$ simultaneously, one of the more popular routes towards improving $z T$ has been to reduce $\\kappa_{L}$ .1 However, depending on the value of $L$ , which maps from $\\sigma$ to $\\kappa_{E}$ , the resulting $\\kappa_{L}$ can often be misleading. For instance, in the case of lanthanum telluride, incautious determination of $L$ can even cause $\\kappa_{L}$ to be negative, which is not physical.2 Therefore, careful evaluation of $L$ is critical in characterizing enhancements in $z T$ due to $\\kappa_{L}$ reduction.  \n\nFor most metals, where charge carriers behave like free-electrons, $L$ converges to $\\begin{array}{r}{\\frac{\\pi^{2}}{3}\\left(\\frac{k_{B}}{e}\\right)^{2}}\\end{array}$ $=2.44\\times10^{-8}\\ \\mathrm{W}\\Omega\\mathrm{K}^{-2}$ (degenerate limit). Although some heavily doped semiconductor thermoelectric materials have an $L$ very close to the degenerate limit, properly optimized materials often have charge carrier concentrations between the lightly doped (non-degenerate) and heavily doped (degenerate) regions3 $(\\xi_{o p t i m u m}$ is near the band edge where $\\xi$ is the electronic chemical potential) which can result in errors of up to ${\\sim}40\\%$ .4  \n\nDirect measurement of $L^{5}$ requires high mobility—typically beyond that attainable at the temperatures of interest $(>300\\mathrm{K})$ . Thus, $L$ is typically estimated either as a constant $(2.44\\times10^{-8}\\mathrm{W}\\Omega\\mathrm{K}^{-2})$ or by applying a transport model—such as the single parabolic band (SPB) model obtained by solving the Boltzmann transport equations—to experimental data.  \n\nFor example, Larsen et al. proposed an approximate analytical solution of $L$ based on the SPB model as a function of carrier concentration $(n)$ and $(m^{*}T)^{-3/2}$ (where $m^{*}$ is the effective mass) along with various sets of parameters for distinct carrier scattering mechanisms.6 However, when the Hall carrier concentration, $n_{H}$ , of a material is not available, the use of the approximate solution by Larsen is not possible. It can be shown that for the SPB model with acoustic phonon scattering (SPB-APS), both $L$ and $s$ are parametric functions of only the reduced chemical potential $(\\eta=\\xi/k_{B}T$ , where $k_{B}$ is Boltzmann constant); thus, no explicit knowledge of temperature $(T)$ , carrier concentration $(n)$ , or effective mass $(m^{*})$ is required to relate them.7 We have utilized this correlation between $L$ and measured $s$ to estimate $\\kappa_{L}$ for a few known thermoelectric materials including: PbTe,8–10 Zintl materials,11–13 co-doped $\\mathrm{FeV_{0.6}N b_{0.4}S b}$ Half Heusler,14 $\\mathrm{La}_{3-\\mathrm{x}}\\mathrm{Te}_{4}$ ,2 resulting in much more satisfactory values for $\\kappa_{L}$ than the degenerate limit result $(L=2.44\\times10^{-8}\\mathrm{W}\\Omega\\mathrm{K}^{-2})$ would have.  \n\nWhile the SPB model works well to estimate $L$ , a transcendental set of equations is needed to solve for $L$ in terms of $S$ —requiring a numerical solution. Considering that the typical measurement uncertainty for $\\kappa_{T o t a l}$ is $10\\%$ and that SPB-APS is only an approximation, a much simpler equation would supply sufficient accuracy. Here, we propose the equation  \n\n$$\nL=1.5+\\exp\\left[-{\\frac{\\left|S\\right|}{116}}\\right]\n$$  \n\n(where $L$ is in $10^{-8}\\mathrm{W}\\Omega\\mathrm{K}^{-2}$ and $s$ in $\\mu\\mathrm{V/K},$ ) as a satisfactory approximation for $L$ .  \n\nEquation (1) allows for a facile estimation of $L$ from an experimental $S$ only without requiring a numerical solution. We characterize the effectiveness of this estimate for $L$ using some experimental data from relevant thermoelectric materials (PbSe,15 PbS,16 PbTe,17,18 Zintl material $\\left(\\mathrm{Sr}_{3}\\mathrm{GaSb}_{3}\\right)$ ,11 Half Heusler (ZrNiSn),19 and $\\mathrm{Si}_{0.8}\\mathrm{Ge}_{0.2}{}^{20\\cdot},$ .  \n\nFor a single parabolic band, $L$ and $S$ are both functions of reduced chemical potential $(\\eta)$ and carrier scattering factor $(\\uplambda)$ only  \n\n$$\nL=\\left(\\frac{k_{B}}{e}\\right)^{2}\\frac{\\left(1+\\lambda\\right)\\left(3+\\lambda\\right)F_{\\lambda}\\left(\\eta\\right)F_{\\lambda+2}\\left(\\eta\\right)-\\left(2+\\lambda\\right)^{2}F_{\\lambda+1}\\left(\\eta\\right)^{2}}{\\left(1+\\lambda\\right)^{2}F_{\\lambda}\\left(\\eta\\right)^{2}},}\\\\ {S=\\frac{k_{B}}{e}\\left(\\frac{\\left(2+\\lambda\\right)F_{\\lambda+1}\\left(\\eta\\right)}{\\left(1+\\lambda\\right)F_{\\lambda}\\left(\\eta\\right)}-\\eta\\right).}\\end{array}\n$$  \n\nWhere $F_{j}\\left(\\eta\\right)$ represents the Fermi integral,  \n\n$$\nF_{j}\\left(\\eta\\right)=\\int_{0}^{\\infty}{\\frac{\\epsilon^{j}d\\epsilon}{1+\\mathrm{Exp}\\left[\\epsilon-\\eta\\right]}}.\n$$  \n\nBy assuming that the carrier relaxation time is limited by acoustic phonon scattering (one of the most relevant scattering mechanisms for thermoelectric materials above room temperature17,21), Eqs. (2) and (3) can be solved numerically for $L$ and the corresponding $S$ as shown in Fig. 1 along with the proposed approximation (Eq. (1)).  \n\n![](images/4533120a46c986c4d1ff6701bcc582b7feeb716ca6eb5cfe941b7e7a6ffc98ce.jpg)  \nFIG. 1. Thermopower dependent Lorenz number calculated by the SPB model with APS and Eq. (1). For comparison, the degenerate limit of $2.44\\times\\mathrm{10^{-8}W}\\Omega\\mathrm{K}^{-2}$ is also presented in a red dashed line.  \n\nAccording to the Fig. 1, the degenerate limit of $L$ $(2.44\\times10^{-8}\\ \\mathrm{W}\\Omega\\mathrm{K}^{-2})$ ) is valid with errors less than $10\\%$ for materials whose thermopower is smaller than $50~\\mu\\mathrm{V/K}$ (highly degenerate). In contrast, if the thermopower is large, the discrepancy with the degenerate limit can be up to $40\\%$ .  \n\nTo decide an appropriate value of $L$ with a known $s$ easily, rather than graphically extracting it from Fig. 1, Eq. (1) can be used to quickly estimate $L$ , given a measured thermopower. Equation (1) is accurate within $5\\%$ for single parabolic band where acoustic phonon scattering is dominant scattering mechanism when $|S|>{\\sim}10~\\mu\\mathrm{V/K}$ . For $\\vert S\\vert<10~\\mu\\mathrm{V/K}$ , while the SPB model converges to the degenerate limit, Eq. (1) increases exponentially, thus reducing the accuracy of the Eq. (1). Although estimation of $L$ with an accuracy within $0.5\\%$ for SPB-APS is possible, this requires an approximate equation more complex than Eq. (1).22  \n\nExceptions are known where $L$ has been found to be outside the uncertainty described above for SPB-APS which are presented in Fig. 2 and Table I.22 These exceptions typically involve either non-parabolic band structures (PbTe, PbSe, and PbS) or alternative scattering mechanisms (other than acoustic phonons). Narrow-gap semiconductors (lead chalcogenides, for example) are often better described by the non-parabolic Kane model which yields a different $\\eta$ dependence of $L$ and $s$ which depends on the non-parabolicity parameter: $\\begin{array}{r}{\\alpha=\\frac{k_{B}T}{E_{g}}}\\end{array}$ ( ${\\mathrm{~\\it~E}}_{g}$ is the gap between conduction and valence band).23,24 For well-studied lead chalcogenides (PbTe, PbSe, and $\\mathrm{Pb}\\mathrm{S}$ ), a reasonable range of $\\alpha$ is from 0.08 $(300~\\mathrm{K})$ to 0.16 (850 K).25 Figure 2 shows that $L$ is at most ${\\sim}26\\%$ lower than that of the SPB-APS and Eq. (1) results over the entire range of temperatures. In other words, $\\kappa_{L}$ estimates will maintain the order: $\\kappa_{L,d e g}<\\kappa_{L,S P B-A P S}<\\kappa_{L,S K B-A P S}$ with the largest errors being for the degenerate limit when applied in the non-degenerate case.22  \n\n![](images/5ce7ed9ef3d0c5be47c3bb0285c7fee0e6cf6abd6d86a994b81cfcba0254e983.jpg)  \nFIG. 2. Thermopower dependent Lorenz number obtained from materials whose band structure and scattering assumptions are different from those assumed in SPB-APS along with Eq. (1) calculation. For comparison, the degenerate limit of $2.44\\times10^{-8}\\mathrm{W}\\Omega\\mathrm{K}^{-2}$ is also presented in a red dashed line.  \n\nTABLE I. Estimated maximum error to Eq. (1) for $L$ with different band structure and scattering assumptions.   \n\n\n<html><body><table><tr><td>Banda</td><td>Scatteringb</td><td>Examples</td><td>Maximum error (%)</td></tr><tr><td>P</td><td>AP</td><td>Sr3Gao.93Zno.07Sb311</td><td>4.4</td></tr><tr><td>2P</td><td>AP+ⅡI</td><td>Sio.8Geo.220</td><td>7.5</td></tr><tr><td>K</td><td>AP</td><td>PbtTe0.98810.0121</td><td>19.7</td></tr><tr><td>K</td><td>AP+PO</td><td>Pb1.002Se0.998Bro.00215</td><td>19.5</td></tr><tr><td></td><td></td><td>PbSD.9978Cl0.002 ZNiSno.99sb6.0 19</td><td>19.4</td></tr><tr><td>K</td><td>AP+PO+AL</td><td></td><td>25.6</td></tr><tr><td>2K+P</td><td>AP</td><td>PbTe0.85Se0.1517</td><td>14.9</td></tr></table></body></html>\n\naBand is the type and number of bands involved in evaluating $L$ . For instance, $\\mathrm{^{*}2K+P^{*}}$ means two non-parabolic Kane bands ${\\mathrm{(K)}}$ and a parabolic band (P). bScattering is the type of scattering mechanism assumed in estimating $L$ . AP, II, PO, and AL are acoustic phonon, ionized impurities, polar, and alloy scattering, respectively. For example, $\\ensuremath{{}^{\\mathrm{s}}\\mathrm{AP}}+\\ensuremath{\\mathrm{PO}}^{\\ensuremath{\\prime}}$ means that both acoustic phonon and polar scatterings are assumed in calculating $L$ .  \n\nAlternative scattering mechanisms can also yield deviations from the SPB-APS. For example, when ionized impurity scattering dominates $(\\uplambda=2)$ , the $L$ actually increases with increasing $S$ ; however, this example is not particularly prevalent in materials which have high dielectric constants (including the lead chalcogenides)26 or at high temperatures. However, when the ionized impurity scattering and acoustic phonon scattering are both considered, the deviation from the SPB-APS is not significant $\\mathrm{\\mathrm{Si}}_{0.8}\\mathrm{Ge}_{0.2}$ in Table I)–although limited data is available. For $\\mathrm{ZrNiSn_{0.99}S b_{0.01}}$ (Table I), acoustic phonon scattering and two other scattering mechanisms (polar and alloy scatterings) are taken into account; these result in a larger deviation as the Seebeck becomes larger. At low temperatures $(<100\\mathrm{K})$ , as $s$ approaches zero, it is expected that $L$ converges to the degenerate limit regardless of carrier scattering mechanism7 and parabolicity of bands involved in transport.22 However, a pronounced inelastic electron-electron scattering due to high mobility of carriers decreases $L$ from the degenerate limit, even for strongly degenerate materials. In case of $n$ -type PbTe, $L$ at $100~\\mathrm{K}$ is approximately $40\\%$ lower than its value at $300\\mathrm{K}$ .24  \n\nMultiple band behavior (present in $p$ -type $\\mathrm{PbTe}_{0.85}\\mathrm{Se}_{0.15}$ and $n$ -type $\\mathrm{Si}_{0.8}\\mathrm{Ge}_{0.2}$ , Fig. 2) can also lead to deviations in the thermopower-dependence of the Lorenz number. In the case of PbTe, hole population of both the light and heavy bands yields a more complicated relationship between $L$ and S; it is not simply a parametric function of $\\eta$ and depends on the specific effective mass and mobility contributions from each band.  \n\nOne last, prevalent source of error occurs because the Wiedemann-Franz law does not take the bipolar thermal conductivity into consideration. $\\kappa_{L}$ calculated from the difference between κTotal and $\\kappa_{E}$ does include varying portion of bipolar conduction with respect to temperature and band structure of materials (which can become important for lightly doped materials with narrow gaps at high temperatures27).  \n\nAn equation for $L$ entirely in terms of the experimentally determined $S$ is proposed and found to be accurate (within $20\\%$ ) for most common band structures/scattering mechanisms found for thermoelectric materials. Use of this equation would make estimates of lattice thermal conductivity much more accurate without requiring additional measurement. Therefore, $z T$ improvement due to lattice thermal conductivity reduction can be calculated with much improved accuracy and access.  \n\nThe authors would like to acknowledge funding from The Materials Project: supported by Department of Energy’s Basic Energy Sciences program under Grant No. EDCBEE, DOE Contract No. DE-AC02-05CH11231 and as part of the Solid-State Solar-Thermal Energy Conversion Center  \n\n# (S3TEC), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, and Basic Energy Sciences under Award No. DE-SC0001299.  \n\n1 G. J. Snyder and E. S. Toberer, Nat. Mater. 7, 105 (2008).   \n2 A. F. May, J. P. Fleurial, and G. J. Snyder, Phys. Rev. B 78, 125205 (2008).   \n3 Y. Pei, H. Wang, and G. J. Snyder, Adv. Mater. 24, 6124 (2012).   \n4 E. S. Toberer, L. L. Baranowski, and C. Dames, Annu. Rev. Mater. Res. 42, 179 (2012).   \n5 K. Lukas, W. Liu, G. Joshi, M. Zebarjadi, M. Dresselhaus, Z. Ren, G. Chen, and C. Opeil, Phys. Rev. B 85, 205410 (2012).   \n6 E. Flage-Larsen and $\\varnothing$ . Prytz, Appl. Phys. Lett. 99, 202108 (2011).   \n7 A. F. May and G. J. Snyder, in Thermoelectrics and its Energy Harvesting, edited by D. M. Rowe (CRC Press, London,   \n2012), Vol. 1 Chap. 11.   \n8 Y. Pei, A. LaLonde, S. Iwanaga, and G. J. Snyder, Energy Environ. Sci. 4, 2085 (2011).   \n9 Y. Pei, J. Lensch-Falk, E. S. Toberer, D. L. Medlin, and G. J. Snyder, Adv. Funct. Mater. 21, 241 (2011).   \n10 Y. Pei, N. A. Heinz, A. Lalonde, and G. J. Snyder, Energy Environ. Sci. 4, 3640 (2011).   \n11 A. Zevalkink, W. G. Zeier, G. Pomrehn, E. Schechtel, W. Tremel, and G. J. Snyder, Energy Environ. Sci. 5, 9121 (2012).   \n12 A. F. May, J.-P. Fleurial, and G. J. Snyder, Chem. Mater. 22, 2995 (2010).   \n13 A. Zevalkink, E. S. Toberer, W. G. Zeier, E. Flage-Larsen, and G. J. Snyder, Energy Environ. Sci. 4, 510 (2011).   \n14 C. Fu, Y. Liu, H. Xie, X. Liu, X. Zhao, G. J. Snyder, J. Xie, and T. Zhu, J. Appl. Phys. 114, 134905 (2013).   \n15 H. Wang, Y. Pei, A. D. Lalonde, and G. J. Snyder, Proc. Natl. Acad. Sci. U.S.A. 109, 9705 (2012).   \n16 H. Wang, E. Schechtel, Y. Pei, and G. J. Snyder, Adv. Energy Mater. 3, 488 (2013).   \n17 Y. Pei, X. Shi, A. LaLonde, H. Wang, L. Chen, and G. J. Snyder, Nature 473, 66 (2011).   \n18 A. D. LaLonde, Y. Pei, and G. J. Snyder, Energy Environ. Sci. 4, 2090 (2011).   \n19 H. Xie, H. Wang, C. Fu, Y. Liu, G. J. Snyder, X. Zhao, and T. Zhu, Sci. Rep. 4, 6888 (2014).   \n20 C. B. Vining, J. Appl. Phys. 69, 331 (1991).   \n21 C. Wood, Rep. Prog. Phys. 51, 459 (1988).   \n22 See supplementary material at http://dx.doi.org/10.1063/1.4908244 for an estimation of L with an accuracy within   \n$0.5\\%$ for SPB-APS; more details about Fig. 2; more details regarding the L for the non-parabolic band model; and   \nnon-parabolicity parameter dependent L as S approaches zero.   \n$^{23}\\mathrm{C}$ . M. Bhandari and D. M. Rowe, J. Phys. D: Appl. Phys. 18, 873 (1985).   \n24 Y. I. Ravich, B. A. Efimova, and I. A. Smirnov, Semiconducting Lead Chalcogenides (Plenum Press, New York, 1970), Vol.   \n299, p. 181.   \n$^{25}\\mathrm{H}$ . Wang, Ph.D. thesis, California Institute of Technology, 2014.   \n$^{26}\\mathrm{P}.$ Zhu, Y. Imai, Y. Isoda, Y. Shinohara, X. Jia, and G. Zou, Mater. Trans. 46, 2690 (2005).   \n27 Z. M. Gibbs, H.-S. Kim, H. Wang, and G. J. Snyder, Appl. Phys. Lett. 106, 022112 (2015).  ",
    "Target": null
  },
  {
    "id": "10.3762_bjnullo.6.181",
    "DOI": "10.3762/bjnullo.6.181",
    "DOI Link": "http://dx.doi.org/10.3762/bjnullo.6.181",
    "Relative Dir Path": "mds/10.3762_bjnullo.6.181",
    "Article Title": "nullotechnology in the real world: Redeveloping the nullomaterial consumer products inventory",
    "Authors": "Vance, ME; Kuiken, T; Vejerano, EP; McGinnis, SP; Hochella, MF; Rejeski, D; Hull, MS",
    "Source Title": "BEILSTEIN JOURNAL OF nullOTECHNOLOGY",
    "Abstract": "To document the marketing and distribution of nullo-enabled products into the commercial marketplace, the Woodrow Wilson International Center for Scholars and the Project on Emerging nullotechnologies created the nullotechnology Consumer Products Inventory (CPI) in 2005. The objective of this present work is to redevelop the CPI by leading a research effort to increase the usefulness and reliability of this inventory. We created eight new descriptors for consumer products, including information pertaining to the nullomaterials contained in each product. The project was motivated by the recognition that a diverse group of stakeholders from academia, industry, and state/ federal government had become highly dependent on the inventory as an important resource and bellweather of the pervasiveness of nullotechnology in society. We interviewed 68 nullotechnology experts to assess key information needs. Their answers guided inventory modifications by providing a clear conceptual framework best suited for user expectations. The revised inventory was released in October 2013. It currently lists 1814 consumer products from 622 companies in 32 countries. The Health and Fitness category contains the most products (762, or 42% of the total). Silver is the most frequently used nullomaterial (435 products, or 24%); however, 49% of the products (889) included in the CPI do not provide the composition of the nullomaterial used in them. About 29% of the CPI (528 products) contain nullomaterials suspended in a variety of liquid media and dermal contact is the most likely exposure scenario from their use. The majority (1288 products, or 71%) of the products do not present enough supporting information to corroborate the claim that nullomaterials are used. The modified CPI has enabled crowdsourcing capabilities, which allow users to suggest edits to any entry and permits researchers to upload new findings ranging from human and environmental exposure data to complete life cycle assessments. There are inherent limitations to this type of database, but these modifications to the inventory addressed the majority of criticisms raised in published literature and in surveys of nullotechnology stakeholders and experts. The development of standardized methods and metrics for nullomaterial characterization and labelling in consumer products can lead to greater understanding between the key stakeholders in nullotechnology, especially consumers, researchers, regulators, and industry.",
    "Times Cited, WoS Core": 1345,
    "Times Cited, All Databases": 1533,
    "Publication Year": 2015,
    "Research Areas": "Science & Technology - Other Topics; Materials Science; Physics",
    "UT (Unique WOS ID)": "WOS:000359834400001",
    "Markdown": "# Nanotechnology in the real world: Redeveloping the nanomaterial consumer products inventory  \n\nMarina E. Vance\\*1, Todd Kuiken2, Eric P. Vejerano3, Sean P. McGinnis4, Michael F. Hochella Jr.5, David Rejeski2 and Matthew S. Hull1  \n\n# Full Research Paper  \n\nAddress:  \n\n1Institute for Critical Technology and Applied Science, Virginia Tech, 410 Kelly Hall (0194), 235 Stanger St., Blacksburg, VA 24061, United States, 2Woodrow Wilson International Center for Scholars, One Woodrow Wilson Plaza - 1300 Pennsylvania Ave., NW, Washington, DC 20004, United States, 3Department of Civil & Environmental Engineering, Virginia Tech, 418 Durham Hall (0246), Blacksburg, VA 24061, United States, 4Department of Materials Science and Engineering, Virginia Tech, Holden Hall (0237), Blacksburg, VA 24061, United States and 5Department of Geosciences, Virginia Tech, 4044 Derring Hall (0420), Blacksburg, VA 24061, United States  \n\nEmail: Marina E. Vance\\* - marinaeq@vt.edu  \n\nBeilstein J. Nanotechnol. 2015, 6, 1769–1780. doi:10.3762/bjnano.6.181  \n\nReceived: 28 March 2015   \nAccepted: 07 August 2015   \nPublished: 21 August 2015  \n\nThis article is part of the Thematic Series \"Nanoinformatics for environmental health and biomedicine\".  \n\nGuest Editor: R. Liu  \n\n$\\circledcirc$ 2015 Vance et al; licensee Beilstein-Institut.   \nLicense and terms: see end of document.  \n\n\\* Corresponding author  \n\n# Abstract  \n\nTo document the marketing and distribution of nano-enabled products into the commercial marketplace, the Woodrow Wilson International Center for Scholars and the Project on Emerging Nanotechnologies created the Nanotechnology Consumer Products Inventory (CPI) in 2005. The objective of this present work is to redevelop the CPI by leading a research effort to increase the usefulness and reliability of this inventory. We created eight new descriptors for consumer products, including information pertaining to the nanomaterials contained in each product. The project was motivated by the recognition that a diverse group of stakeholders from academia, industry, and state/federal government had become highly dependent on the inventory as an important resource and bellweather of the pervasiveness of nanotechnology in society. We interviewed 68 nanotechnology experts to assess key information needs. Their answers guided inventory modifications by providing a clear conceptual framework best suited for user expectations. The revised inventory was released in October 2013. It currently lists 1814 consumer products from 622 companies in 32 countries. The Health and Fitness category contains the most products (762, or $42\\%$ of the total). Silver is the most frequently used nanomaterial (435 products, or $24\\%$ ); however, $49\\%$ of the products (889) included in the CPI do not provide the composition of the nanomaterial used in them. About $29\\%$ of the CPI (528 products) contain nanomaterials suspended in a variety of liquid media and dermal contact is the most likely exposure scenario from their use. The majority (1288 products, or $71\\%$ ) of the products do not present enough supporting information to corroborate the claim that nanomaterials are used. The modified CPI has enabled crowdsourcing capabilities, which allow users to suggest edits to any entry and permits researchers to upload new findings ranging from human and environmental exposure data to complete life cycle assessments. There are inherent limitations to this type of database, but these modifications to the inventory addressed the majority of criticisms raised in published literature and in surveys of nanotechnology stakeholders and experts. The development of standardized methods and metrics for nanomaterial characterization and labelling in consumer products can lead to greater understanding between the key stakeholders in nanotechnology, especially consumers, researchers, regulators, and industry.  \n\n# Introduction  \n\nAdvancements in the fields of nanoscience and nanotechnology have resulted in myriad possibilities for consumer product applications, many of which have already migrated from laboratory benches into store shelves and e-commerce websites. Nanomaterials have been increasingly incorporated into consumer products, although research is still ongoing on their potential effects to the environment and human health. This research will continue long into the future.  \n\nTo document the penetration of nanotechnology in the consumer marketplace, the Woodrow Wilson International Center for Scholars and the Project on Emerging Nanotechnology created the Nanotechnology Consumer Product Inventory (CPI) in 2005, listing 54 products [1]. This first-of-its-kind inventory has become one of the most frequently cited resources showcasing the widespread applications of nanotechnology in consumer products. In 2010, the CPI listed 1012 products from 409 companies in 24 countries. Even though it did not go through substantial updates in the period between 2010 and 2013, it continued being heavily cited in government reports [2] and the scientific literature – the website http://www.nanotechproject.org has been cited over 2,580 times in articles according to Google Scholar – and became a popular indicator of the prevalence of nanotechnology in everyday life and the need to further study its potential social, economical, and environmental impacts [3-6]. The CPI has also been criticized due to its lack of science-based data to support manufacturer claims. Other longstanding suggestions for improvement included: more frequent updates, indications when products were no longer available for purchase by consumers, and the inclusion of more product categories to improve the searchability of the CPI database [7].  \n\nSince the creation of the CPI, other nanotechnology-related inventories have been developed around the world. In 2006, a German company launched a freely accessible internet database of nanotechnology products [8]. The website associated with this database was not accessible at the time of this writing and its last available record is from May 2014, when 586 products were listed. In 2007, Japan’s National Institute of Advanced Industrial Science and Technology created an inventory of “nanotechnology-claimed consumer products” available in Japan [2]. This inventory is freely accessible online and it acknowledges the CPI in its website. At the time of this writing, the inventory listed 541 product lines and 1241 products; its last update occurred in 2010 [9]. In 2009, two European consumer organizations, the European Consumers Organization (BEUC) and the European Consumer Voice in Standardization (ANEC), joined efforts to develop an inventory of “consumer products with nano-claims” available to consumers in Europe [10]. A new inventory was generated annually from 2009 to 2012, but the 2011 and 2012 versions focused exclusively on products containing silver nanoparticles (nanosilver); the latest version in 2012 listed 141 nanosilver products. This inventory does not provide a searchable online database, but it can be downloaded for free as an Excel spreadsheet. In 2012, the Danish Consumer Council and Ecological Council and the Technical University of Denmark’s Department of Environmental Engineering launched “The Nanodatabase”, an inventory of products available for purchase that are claimed to contain nanomaterials and are available in the European consumer market [11]. This inventory has been continually updated and it currently lists 1423 products.  \n\nThese worldwide efforts to understand the transition of nanotechnology from the laboratory bench to the commercial marketplace substantiate the need for applying the concept of nanoinformatics to a nanotechnology-enabled consumer products database, which is to determine the most relevant and useful information needed by a variety of stakeholders and to develop tools for its most effective use [12]. Databases such as the CPI offer information useful and relevant to a variety of stakeholders who are interested in a) understanding which consumer products incorporate nanotechnology and b) developing strategies, tools, and policies that may be needed to ensure safe and responsible use of those products.  \n\nNanomaterials are regulated without specific provisions in the U.S. as hazardous chemical substances and pesticides, under the EPA’s Toxic Substances Control Act (TSCA) [13] and the Federal Insecticide, Fungicide, and Rodenticide Act (FIFRA) [14]. When used as food additives, drugs, or cosmetics, nanomaterials are regulated under the Federal Food, Drug, and Cosmetic Act (FFDCA).  \n\nIn the European Union, nanomaterials are regulated under the Concerning the Registration, Evaluation, Authorization and Restriction of Chemicals (REACH) and the Classification, Labeling, and Packaging (CLP) regulations when those are classified by the Commission as hazardous chemical substances [15]. The Biocidal Products Regulation (BPR) has special provisions for biocidal materials that consist of nanoparticles, aggregates, or agglomerates in which at least $50\\%$ of primary particles have at least one dimension between 1 and $100~\\mathrm{{nm}}$ , with no provisions for “novel properties” stemming from their small size [16]. Cosmetics that contain nanomaterials are also regulated by the European Commission, and although the use of nanoscale titanium dioxide is permitted, zinc oxide is not [17]. The German Federal Environment Agency performed an Impact Assessment of a European Register of Products Containing Nanomaterials and determined that when compared to the implementation of a variety of national registries, an unified European registry would bring many advantages, including a lower cost for industries and, ultimately, a registry would benefit consumers, companies, and governments [18].  \n\nThe objective of this work was to modify the CPI to improve its functionality, reliability, and utility to the diverse group of stakeholders who have come to depend on it as a critical resource for current information on nano-enabled consumer products. Specific objectives were (1) to update the CPI data to gain an insight into the penetration of nanotechnology in the consumer products market over the past decade; (2) to determine and implement improvements to the CPI based on the scientific literature and a survey of nanotechnology experts and CPI users; and to (3) develop a sustainable model to facilitate future CPI maintenance using crowdsourcing tools.  \n\nBelow, we present a brief history of this inventory over a decade of existence. We also describe the specific changes made in the inventory during this project (referred here as CPI 2.0). Finally, we present an overview of the current data present in the CPI after the completion of this project.  \n\n# Results and Discussion CPI growth over time  \n\nTable 1 lists the growth of the CPI since 2005. In 2011, before this current project, the CPI described 1314 products. Since then, 489 products that are no longer available or marketed as containing nanotechnology have been archived and 500 products have been added. The new total of 1814 products as of March 2015 represents a thirty-fold increase over the 54 products originally listed in 2005 – which is not a complete representation of the growth of this market, as our methodology has also evolved over time. Based on our review, the CPI is the largest online inventory of nanotechnology consumer products available. Products come from 622 companies located in 32 countries (Supporting Information File 1, Table S1).  \n\nThe products listed on the $\\mathrm{CPI}2.0$ satisfy three criteria: (1) they can be readily purchased by consumers; (2) they are claimed to contain nanomaterials by the manufacturer or another source; and (3) their claim to contain nanomaterials appears reasonable to CPI curatorial staff.  \n\nAlthough the steady growth of the inventory indicates that the popularity of products claimed to incorporate nanotechnology is continually increasing, not all products have persisted in the consumer market. In the past seven years, $34\\%$ of the entries in the inventory have been archived because the product is not currently available in the market or their claim to contain nanotechnology can no longer be verified. One example of a claim that can no longer be verified is a product that is still available for purchase on a manufacturer’s website but no longer references, explicitly, the incorporation of nanotechnology into that product. Even after archiving, a product can return to the main inventory listing if a third party makes the claim that the product indeed contains nanomaterials or if the manufacturer restates their nanomaterial claim.  \n\n<html><body><table><tr><td colspan=\"6\">Table 1: Number of products in the CPl over time.</td></tr><tr><td>Year</td><td></td><td>Total products Products added</td><td>Products archived</td><td>Data collection notes</td></tr><tr><td>2005</td><td>54</td><td>54</td><td>0</td><td>Beginning of CPl as a static pdf document.</td></tr><tr><td>2006</td><td>356</td><td>302</td><td>0</td><td>Launch of the online CPl.</td></tr><tr><td>2007</td><td>580</td><td>278</td><td>0</td><td>Nanoscale silver emerged as most cited nanomaterial.</td></tr><tr><td>2008</td><td>803</td><td>223</td><td>0</td><td>Health and fitness products represented 6o% of the inventory.</td></tr><tr><td>2009</td><td>1015</td><td>212</td><td>107</td><td>Added archiving function to the CPl.</td></tr><tr><td>2010</td><td>1015</td><td>0</td><td>0</td><td>No data collected.</td></tr><tr><td>2011</td><td>1015</td><td>0</td><td>0</td><td>No data collected.</td></tr><tr><td>2012</td><td>1438</td><td>426</td><td>0</td><td>Beginning of CPl 2.0 project, focus on adding new products.</td></tr><tr><td>2013</td><td>1628</td><td>190</td><td>288</td><td>Launch of crowdsourcing component. Extensive effort put into adding and archiving products.</td></tr><tr><td>2014</td><td>1814a</td><td>238a</td><td>223a</td><td>Extensive effort put into adding and archiving products.</td></tr></table></body></html>\n\naThe CPI now has crowdsourcing capabilities, so these numbers are a snapshot in time and will not represent the CPI at the time of reading.  \n\nIn the CPI, entries are grouped under eight generally accepted consumer goods categories that are loosely based on publicly available consumer product classification systems (Figure 1) [19]. The Health and Fitness category includes the largest listing of products in the CPI, comprising $42\\%$ of listed products (excluding archived products). Within the Health and Fitness category, Personal Care products (e.g., toothbrushes, lotions, and hairstyling tools and products) comprise the largest subcategory $39\\%$ of products). Starting in 2012, a large continual effort has been put into periodically checking products for their current availability and current claim to contain nanotechnology. This effort resulted in archiving 316 products in the Health and Fitness category – mainly in the Personal Care and Clothing subcategories – with 86 and 78 products archived between 2012 and 2014, respectively.  \n\n![](images/9541ed43044947d0faa27bb1802e77fb2bd35377d63a71779a9c65a9a94886b1.jpg)  \nFigure 1: Number of available products over time (since 2007) in each major category and in the Health and Fitness subcategories.  \n\n# New nanomaterial descriptors  \n\nEight new product descriptors were introduced to facilitate the use of this database by a variety of stakeholders (namely industry and the scientific and regulatory communities):  \n\n1. main nanomaterial composition or type,   \n2. nanomaterial shape and size,   \n3. nanomaterial coating or stabilizing agent,   \n4. nanomaterial location within the product,   \n5. nanomaterial function in the product,   \n6. potential exposure pathways,   \n7. “how much we know”,   \n8. “researchers say”.  \n\nThe experimental section of this paper describes all new product descriptors. The results of the five new quantitative descriptors are presented and discussed below. Since the “nanomaterial shape and size”, “coating and stabilizing agent”, and the “researchers say” categories are text-entry data fields, thus qualitative information at this point, we have not included their analysis in this paper.  \n\n# Nanomaterial composition  \n\nOf the 1814 products listed in the CPI, $47\\%$ (846 products) advertise the composition of at least one nanomaterial component and 62 of those products list more than one nanomaterial component (e.g., a product comprised of both silver and titanium dioxide nanomaterials). There are 39 different types of nanomaterial components listed in the inventory (listed in Supporting Information File 1, Table S2), which have been grouped into five major categories in Figure 2 and Figure 4, to improve their legibility: metal, carbonaceous, silicon, not advertised, and other. Nominally, metals and metal oxides comprise the largest nanomaterial composition group advertised in the inventory, listed in $37\\%$ of products.  \n\nTitanium dioxide $\\mathrm{(TiO}_{2}\\mathrm{)}$ ), silicon dioxide, and zinc oxide are the most produced nanomaterials worldwide (on a mass basis) and the global annual production of silver nanoparticles represents only $2\\%$ of that of $\\mathrm{TiO}_{2}$ [20,21]. However, silver nanoparticles are the most popular advertised nanomaterial in the CPI, present in 438 products $(24\\%)$ . The CPI reports the numbers of different consumer products and product lines available in the market, so there is no implication on mass, volume, or concentration of nanomaterials incorporated into products or the production volume of each product.  \n\nOf carbonaceous nanomaterials (89 products), the majority of products listed contains carbon nanoparticles (sometimes described as carbon black, 39 products) and single- or multiwalled carbon nanotubes (CNT, 38 products). Unfortunately,  \n\n891 $(49\\%)$ of the products included in the CPI do not present the composition or a detailed description of the nanomaterial used (Figure 2).  \n\n![](images/1be3cd7a31c486d8f784b50f166c8e887b9c5f3f7a6400aef74b02e9ed86ded6.jpg)  \nFigure 2: (a) Claimed composition of nanomaterials listed in the CPI, grouped into five major categories: not advertised, metal (including metals and metal oxides), carbonaceous nanomaterials (carbon black, carbon nanotubes, fullerenes, graphene), silicon-based nanomaterials (silicon and silica), and other (organics, polymers, ceramics, etc.). (b) Claimed elemental composition of nanomaterials listed in the metals category: silver, titanium, zinc, gold, and other metals (magnesium, aluminum oxide, copper, platinum, iron and iron oxides, etc.). (c) Claimed carbonaceous nanomaterials (CNT $\\mathbf{\\tau}=\\mathbf{\\tau}$ carbon nanotubes).  \n\nThe percentages of nanomaterial compositions in the CPI 2.0 are somewhat in agreement with those of the Danish Nanodatabase. The Nanodatabase also lists a high fraction of products with unknown nanomaterial composition (944 products or $66\\%$ ) and, among known compositions, silver is also the most frequently advertised nanomaterial component, with 207 products or $14.5\\%$ [11]. Silver nanoparticles are popular consumer product additives due to their well-documented antimicrobial properties [22].  \n\nFigure 3 shows how the availability of these major nanomaterial composition groups changed over time. Since the start of the $\\mathrm{CPI}2.0$ project (2012), products with unknown (not advertised) nanomaterial compositions have decreased by $12\\%$ , which is partially due to these products being archived and of their composition being identified and added to the inventory. Products advertising to contain metal and metal oxide nanomaterials, silicon-based nanomaterials (mostly $\\mathrm{SiO}_{2}$ nanoparticles), and a variety of other nanomaterial components (organics, ceramics, polymers, clays, nanocellulose, liposomes, nano micelles, carnauba wax, etc.) have been growing in popularity. During the same period, carbonaceous nanomaterials have remained stable at around 50 products available in the market.  \n\nOf the 846 products listed in the CPI for which we were able to determine a nanomaterial composition, 61 products $(7\\%)$ advertise to contain more than one main nanomaterial component. Figure 4 presents 11 nanomaterial components that were most frequently listed with others in the same product.  \n\n![](images/4d28b21e8528a9a07dd44cc5a99c5fa89ec31886c476b691ccb795e25c828158.jpg)  \nFigure 3: Major nanomaterial composition groups over time. Carbon $\\mathbf{\\tau}=\\mathbf{\\tau}$ carbonaceous nanomaterials (carbon black, carbon nanotubes, fullerenes, graphene). Other $\\mathbf{\\tau}=\\mathbf{\\tau}$ organics, ceramics, polymers, clays, nanocellulose, liposomes, nano micelles, carnauba wax, etc. Note the difference in scale between the top and bottom panels in this plot.  \n\nSilver and titanium dioxide are the nanomaterial components most likely to be combined with other nanomaterials in consumer products, with 35 and 30 product combinations, respectively. Silver and titanium dioxide were paired with each other in 10 products (cosmetics and electronics); titanium dioxide and zinc oxide were paired in 10 products (sunscreens, cosmetics, and paints). The European Commission’s Cosmetics Regulation has permitted the use of nanoscale titanium dioxide in sunscreens, but not zinc oxide [17].  \n\nCalcium and magnesium were listed together in dietary supplements. Nano-ceramics and silver are used in combination in water filtration products, cosmetics, and a humidifier. These results demonstrate the use of nanohybrids [23] in consumer products and indicate that the use of nanotechnology-based consumer products in the home may, in some cases, lead to multiple exposures from a combination of nanomaterial compositions. These results suggest the need to examine nanomaterial toxicity effects that could be synergistic, additive, or even antagonistic.  \n\n![](images/7cc82540f892fa7d7b30d4bc5aeee4368e4a1168e52871b9b4c55f374403f31d.jpg)  \nFigure 4: Major nanomaterial composition pairs in consumer products Carbonaceous nanomaterials (carbon black, carbon nanotubes, fullerene, and graphene) were combined into the same category (carbon). Grey boxes in the diagonal represent the total times each nanomaterial composition has been listed with other compositions in the same product.  \n\nirons, textiles). Figure 5 shows the location of nanomaterials for which a composition has been identified [24].  \n\nThe majority $(64\\%)$ of carbonaceous nanomaterials are embedded in solid products, whereas products of all other compositions are more commonly suspended in liquid. Of the few bulk nanomaterials that are available for purchase by consumers, the largest group $(42\\%)$ consists of metal and metal oxide nanomaterials. Metals and metal oxides were also the largest composition for surface-bound particles and those suspended in liquid products. The majority $(67\\%)$ of products with nanostructured surfaces consist of nanomaterials of undetermined composition. An example of such product is a liquid or spray products that forms a nanofilm upon application over a surface. Of nanostructured bulk materials, the majority $(57\\%)$ are siliconbased nanomaterials (e.g., computer processor parts). It is interesting to note that we expect nano-electronics to exist now in massive numbers of consumer products, such as mobile devices, where field effect transistors, the heart of chip technology, have components (sources, gates, collectors, channels) that are now in the nanoscale [25] and would fit into the nanostructured bulk category. However, because most of these products do not advertise their use of nanomaterials, we believe that they are grossly underrepresented in the CPI.  \n\n# Nanomaterial location  \n\nAbout $29\\%$ of consumer products in the CPI (528 products) contain nanomaterials suspended in a variety of fluids (e.g., water, skin lotion, oil, car lubricant). The second largest group in this category – with 307 products – comprises solid products with surface-bound nanoparticles (e.g., hair curling and flat  \n\n# Nanomaterial function  \n\nOf the 1814 inventory entries, 1244 were grouped according to the expected benefits of adding such nanomaterials to the product (Figure 6). A significant portion of products in the CPI $31\\%$ of products analyzed) utilize nanomaterials – mostly silver nanoparticles, but also titanium dioxide and others – to confer antimicrobial protection. Nanomaterials such as titanium dioxide and silicon dioxide are used to provide protective coatings $(15\\%)$ and for environmental treatment (to protect products against environmental damage or to treat air and water in the home, $15\\%$ ). Cosmetic products $(12\\%)$ are advertised to contain a variety of nanomaterials such as silver nanoparticles, titanium dioxide, nano-organics, gold, and others. A wide variety of nanomaterial compositions (silver, nano-organics, calcium, gold, silicon dioxide, magnesium, ceramics, etc.) were also advertised to be used for health applications, such as dietary supplements $(11\\%)$ .  \n\n![](images/92059e0ef8d6a64f916a4a3fb7bfcc83edce106f5d8cc91e8eb81188a2eadb64.jpg)  \nFigure 5: Locations of nanomaterials in consumer products for which a nanomaterial composition has been identified.  \n\n![](images/a5e9b638649292e369e596f1091db5d23ce9c8d7eda401961183504ddb4cf3dd.jpg)  \nFigure 6: Expected benefits of incorporating nanomaterial additives into consumer products.  \n\n# Potential exposure pathways  \n\nSince critical information such as nanomaterial size and concentration are not known for most products listed on the CPI, the actual health risks of these products remain largely unknown. Nevertheless, the CPI may be useful for inferring potential exposure pathways from the expected normal use of listed products. To investigate this utility, we analyzed a subset of 770 products from the CPI to determine their most likely route(s) of exposure (Figure 7).  \n\n![](images/ecd4ed4245d9ebe464cc1baf9e27bd81bc25f1879727441776b75bce2daa8a47.jpg)  \nFigure 7: Potential exposure pathways from the expected normal use of consumer products, grouped by major nanomaterial composition categories.  \n\nWe identified the skin as the primary route of exposure for nanomaterials from the use of consumer products $58\\%$ of products evaluated). This is because many entries in the CPI consist of (1) solid products that contain nanomaterials on their surfaces and are meant to be touched or (2) liquid products containing nanomaterial suspensions which are meant to be applied on the skin or hair. Of the products evaluated, $25\\%$ present nanomaterials that can possibly be inhaled during normal use (e.g., sprays and hair driers) and $16\\%$ contain nanomaterials that may be ingested (e.g., supplements and throat sprays). Hansen et al. developed a framework for exposure assessment in consumer products. In this framework, products that contain nanomaterials suspended in liquid and products that may emit airborne nanoparticles during use are expected to cause exposure [26].  \n\nSince metals and metal oxides are the most common nanomaterial composition in the CPI, they are also the most likely materials to which consumers will be exposed during the normal use of product via dermal, ingestion, and inhalation routes. Products containing nanomaterials of unknown composition are most likely to lead to exposure via the dermal route.  \n\nBerube et al. [7] offered a critique of the original CPI in 2010, which focused primarily on the lack of data pertinent to the dosages of nanomaterials to which consumers might be exposed through CPI-listed products. This is a valid criticism given that information used to populate the CPI is based primarily on marketing claims made by manufacturers. However, the most recent modifications of the CPI offer a potential remedy for data gaps through the contributions of third-party research teams. These modifications are especially timely as there is a growing number of published studies assessing consumer exposure to nanomaterials released during the use of nanotechnologyenhanced consumer products [27], such as cosmetic powders [28], sprays [29,30], general household products [31], and products for children [32,33]. One challenge is that there are no standardized methods for assessing consumer risks from using nanotechnology-enabled consumer products or a set of agreedupon metrics for characterizing nanomaterials to determine environmentally relevant concentrations [34]. The development of such standards is seen as a top strategy for safe and sustainable nanotechnology development in the next decade [35]. The Consumer Product Safety Commission recently requested \\$7 million to establish the Center for Consumer Product Applications and Safety Implications of Nanotechnology to help develop methods to identify nanomaterials in consumer products and to understand human exposure to those materials [36].  \n\n# How much we know  \n\nThrough the “How much we know” descriptor, inventory en tr ies ar e r ated acco r d in g to th e r eliab ility o f th e manufacturer’s claim that products contain nanomaterials. We evaluated 1259 products present in the inventory for the “How much we know” descriptor and the majority $(71\\%)$ of products are not accompanied by information sufficient to support claims that nanomaterials are indeed used in the products, such as a manufacturer datasheet containing technical information about nanomaterial components (e.g., median size, size distribution, morphology, concentration). Only nine products have been classified in Category 1, “Extensively verified claim” due to the availability of scientific papers or patents describing the nanomaterials used in these products (Figure 8). The experimental section, below, presents a full description of these categories.  \n\n![](images/5d5a5387f95d0bf7ed5e27e0a1ede7f725ab721c284b81130a15cd1cc1d9cee9.jpg)  \nFigure 8: Distribution of products into the “How much we know” categories.  \n\nHansen [37] performed interviews with 26 nanotechnology stakeholders who agreed on an incremental approach to nanomaterial regulation in consumer products, including classification and labeling. The European Commission’s Classification, Labeling, and Packaging (CLP) regulation covers nanomaterials that are classified by the Commission as hazardous chemical substances [15]. Becker [38] reported that there are diverging opinions in the nanotechnology industry with regards to labeling, ranging from ‘‘If it’s a nano-scale material, people should know, hands down” to not supporting labeling because “it wouldn’t accurately inform consumers of anything and would be bad for business because it would scare consumers.”  \n\n# Nanotechnology expert survey  \n\nAppropriate nanomaterial labeling containing sufficient technical information (i.e., at a minimum, nanomaterial composition, concentration, and median size) would better inform consumers and highly benefit researchers interested in understanding consumers’ exposure and nanomaterial fate and transport in the environment.  \n\n# Crowdsourcing  \n\nSince October 29, 2013, when the modified inventory (CPI 2.0) was released, 557 new user accounts have been requested. Of these, only approximately 10 users who were not directly or indirectly involved in the research team performing the CPI upgrade and maintenance suggested updates or edits to CPI entries. These edits have all been suggested by users from industry and academia.  \n\nFuture work is needed to better educate users on their role as curators of $\\mathrm{CPI}~2.0$ and the importance of the data they contribute. Providing the supporting technical data required to verify the nature and quantity of nanomaterial components in CPI-listed products is a massive undertaking, and no single laboratory can accomplish it on its own or within a short amount of time. A long-term solution is to promote the importance of crowd-sourcing data collection and implementing standard data collection and reporting best practices that can help reliably populate the CPI with much needed supporting data. The new crowd-sourcing capability can also be used to provide high school-, undergraduate- and graduate-level educators with meaningful assignments that can help teach students about the prevalence of nanotechnology in everyday products and will contribute to the continued growth of this resource.  \n\nThe survey was submitted to 147 people who have published research papers or reports in the applications of nanotechnology in consumer products and its potential impacts, participated in recent conferences in the field, or were notably involved in the field of nanotechnology and the consumer products industry. The survey had a $46\\%$ response rate (68 respondents), which is in the expected range for this type of survey [39]. The majority of respondents $(59\\%)$ ) had six to ten years of experience working with nanotechnology and $38\\%$ of respondents had more than ten years of experience. Half $(51\\%)$ of respondents work in academic institutions and $25\\%$ work in governmental agencies. Most respondents $(88\\%)$ ) have previously used the CPI in their work, and all respondents believe they will or may use it again in the future.  \n\nResults convey a general belief or hope that the CPI will become more useful after the modifications reported in this publication. When asked the following open-ended questions: “How did you use the CPI in your work?” and “To what end do you think you might use the CPI in the future?”, answers could be easily grouped into three main categories: (1) for raising awareness, teaching, or for urging the need for regulation, (2) to justify the need for research in research proposals or papers, and (3) to use the inventory data for research (Figure 9).  \n\nHalf the respondents $(51\\%)$ have used the CPI in the past to gather data for research (e.g., searching for consumer products of a certain nanomaterial composition to understand their potential applications or consumer exposure) while $74\\%$ believe they will use the CPI for that purpose in the future. The majority $(79\\%)$ of survey respondents believed the modified CPI would present more products than its previous version, which indicates their belief in the growing prevalence of nanotechnology in consumer products.  \n\n![](images/7d09be830354f75f37af35c0406e29c6fdcd4a64166bb9d9a25bef07e2006671.jpg)  \nFigure 9: Nanotechnology survey answers on how respondents have used the CPI in the past and how they might use it in the future.  \n\nSurvey respondents suggested a number of new categories of information for the CPI 2.0, including nanomaterial type or composition, location of nanomaterial within the product, nanomaterial size, relevant scientific publications that describe the products in the inventory, a summary of known toxicity of the advertised nanomaterial, supply chain information, volume produced, and life cycle assessment information.  \n\nMost of these suggestions were included in the CPI 2.0 as the new categories described in this work. Others, such as known nanomaterials toxicity were not pursued since toxicity can vary greatly depending on particle size, coating, and exposure route (e.g., inhalation versus ingestion).  \n\nPiccinno et al. and Keller et al. provide global estimates for production and major applications of nanomaterials [20,21]. We recommend that future work associated with this inventory or others include information on the production volumes for each product, since this information is presently unavailable.  \n\nAdditional results from this survey are available in Supporting Information File 1.  \n\nability of the data associated with each entry. Finally, the CPI 2.0 has enabled crowdsourcing capabilities, which allow registered users to upload new findings such as basic product composition information, human and environmental exposure data, and complete life cycle assessments. There are inherent limitations to this type of database, but recent improvements address the majority of issues raised in published literature and in a survey of nanotechnology experts.  \n\nImprovements to the CPI were motivated, in part, by the recognition that it represents and will continue to represent an important information resource for a broad range of stakeholders, especially consumers and the academic and regulatory communities. The CPI is a useful interactive database for educating consumers and legislators on the real-world applications of nanotechnology. Michaelson stated that the CPI transformed “the face of nanotechnology away from innovations in the realm of science fiction to the iconic images of everyday consumer products” [2]. The academic community can continue to make use of this inventory to help prioritize, for example, which types of products or nanomaterial components to evaluate in human exposure or toxicity studies, life cycle assessments, and nanomaterial release studies.  \n\nThe CPI is useful for policy makers interested in regulating nanotechnology in consumer products by understanding their increasing numbers in the market, the main nanomaterial components that are chosen by manufacturers, and the likelihood for exposure. Beaudrie et al. [40] urge that there should be regulatory reforms to improve oversight of nanomaterials throughout their life cycle.  \n\nFinally, the current lack of global standardized methods and metrics for nanomaterial characterization and labeling in consumer products is an issue that, if addressed, can lead to greater understanding between the key stakeholders in nanotechnology, especially researchers, regulators, and industry. Further, as we recognize the growing importance of tools like the CPI for the needs of diverse stakeholder groups, steps should be taken to help ensure that those tools are fully developed and refined to meet those needs.  \n\n# Conclusion  \n\nThe modified version of the Wilson Center’s nanotechnology consumer products inventory (CPI 2.0) was released in October 2013. We improved the searchability and utility of the inventory by including new descriptors for both the consumer products and the nanomaterial components of those products (e.g., size, concentration, and potential exposure routes). The updated CPI 2.0 now links listed products to published scientific information, where available, and includes a metric to assess the reli  \n\n# Experimental Nanotechnology expert survey  \n\nTo determine potentially useful improvements for the CPI, we developed a web-based survey to gather the informed opinions of nanotechnology experts – mostly in US-based academic institutions, governmental agencies, and research centers. Their answers guided the CPI modifications and provided an idea of the expectations related to the inventory. The survey questions are presented in the Supporting Information File 1.  \n\n# New descriptors  \n\nTo improve the utility and searchability of this database, seven product descriptors were created. Entries in the inventory were revised to go beyond a categorization of the consumer products and instead, to include more information on the nanomaterials themselves. We searched for this information mainly on the internet – on manufacturer’s websites, retailer’s websites, news sites and blogs, patents – and, when available, product labels.  \n\n# Nanomaterial composition  \n\nThe main composition of the nanomaterials used. This information, when available, was added to the database in the form of a check-box list, in which more than one nanomaterial composition can be selected for each consumer product.  \n\n# Nanomaterial shape and size  \n\nBecause there are many different ways in which manufacturers can measure and describe the shape and size of nanomaterials in consumer products (i.e., units of nanometers or micrometers, thickness of nanofilms, diameter or length of fibers or tubes, diameter or radius of nanoparticles, maximum, median, average, or minimum size), this descriptor was added as a text entry field in the database, which allows for any form of data entry but makes data analysis cumbersome.  \n\n• Suspended in liquid: Nanomaterials suspended in a liquid product (e.g., disinfecting sprays, liquid supplements) • Suspended in solid: Nanomaterials suspended in a solid matrix, usually plastic or metal (e.g., composites of carbon nanotubes in a plastic matrix to confer strength).  \n\n# Nanomaterial function  \n\nWe created a metric to describe the reason why nanotechnology was added to each consumer product or the function it performs within each product. We investigated a subset of 1244 products in the CPI for each product’s intended use, the manufacturer claims, and, most importantly, the type or composition of nanomaterials used to infer potential nanomaterial functions (e.g., antimicrobial protection, hardness and strength, pigment).  \n\n# Potential exposure pathways  \n\nUsing methodology similar to that applied for the “nanomaterial functions” category, we investigated the CPI entries for possible exposure scenarios resulting from the expected normal use of each consumer product. Entries were only populated when a potential exposure risk was identified.  \n\nWe created another text entry field in the CPI to include any available information on the coatings or stabilizing agent used along the nanomaterials in each product.  \n\n# Nanomaterial location  \n\nTo assist CPI users in understanding the potential for nanomaterial release and exposure scenarios from the use of these consumer products, we created a qualitative descriptor for the location of nanomaterials within each product. We adapted the categorization framework for nanomaterials from Hansen et al. [24] to determine the following nanomaterial locations within products:  \n\n# Coatings  \n\n• Bulk: Nanomaterials sold in powder form or in liquid suspensions   \n• Nanostructured bulk: Products or parts that contain nanostructured features in bulk (e.g., nanoscale computer processors)   \n• Nanostructured surface: Products or parts that contain nanostructured features on their surface (e.g., nanofilmcoated products)   \n• Surface-bound particles: Nanoparticles added to the surface of a solid product or part (e.g., a computer keyboard coated with silver nanoparticles for antimicrobial protection)  \n\n# How much we know  \n\nIn an effort to verify the data associated with each product listed on the CPI, we created a metric called “How much we know”. Products were divided into five categories based on the information available to substantiate manufacturer claims that a particular product contains nanomaterial components (Table 2). Category 4, “Unsupported claim”, is the default category for products added to the CPI based soley on a manufacturer’s marketing claims. A product can rise in ranking according to the amount of information that is available to corroborate the manufacturer’s claim that the product contains nanomaterials. If the manufacturer provides supporting information (e.g., a datasheet containing electron micrographs showing the nanomaterials or a particle size distribution), the product is placed in Category 3, “Manufacturer-supported claim”. If a third-party further supports the information provided by the manufacturer, such as through a publication or technical report, then the product can be placed into Category 2, “Verified claim”. If a product is backed by multiple science-based sources (e.g., a peerreviewed scientific paper or patent documentation), it is then placed in Category 1, “Extensively verified claim”. Category 5, “Not advertised by the manufacturer”, is a special class for products that have been shown to contain nanomaterials but the manufacturer does not advertise this fact anywhere in product labeling or other informational materials. Category 5 has been added in recognition of the fact that not all nano-enabled products are marketed by manufacturers as such.  \n\nTable 2: “How much we know” categorization, based on the information available to substantiate manufacturer claims that a particular product   \n\n\n<html><body><table><tr><td colspan=\"5\">contains nanomaterial components.</td></tr><tr><td>Category</td><td>Manu faturer hlaimgyo Mupfrtiginr provides</td><td></td><td>Third-party information is available</td><td>Compelling information from multiple sources is available</td></tr><tr><td>1.Extensively verified claim</td><td>yes</td><td>yes</td><td>yes</td><td>yes</td></tr><tr><td>2.Verified claim</td><td>yes</td><td>yes</td><td>yes</td><td></td></tr><tr><td>3. Manufacturer-supported claim</td><td>yes</td><td>yes</td><td></td><td></td></tr><tr><td>4. Unsupported claim</td><td>yes</td><td></td><td></td><td></td></tr><tr><td>5.Not advertised by manufacturer</td><td></td><td></td><td>yes</td><td></td></tr></table></body></html>  \n\n# Researchers say  \n\nIn order to add available scientific information to the inventory, we created a text-entry database field named “Researchers say”, which makes it possible to include an extract from a research paper (such as the abstract), author citation, and a link to the paper.  \n\n# Crowdsourcing  \n\nWe added a new crowdsourcing capability to the CPI website so that consumers, manufacturers, and the greater scientific community can contribute new information on nanomaterial composition of CPI products to the inventory. New contributors must request an account by completing a form with their contact information, and they must provide a reason why they would like to gain access to this crowdsourcing tool. Accounts are manually reviewed. Access is granted to all requesters who complete the form and have a legitimate purpose for contributing information. Once an account is created, users may sign in and suggest edits to any product (including the archiving of products no longer available or no longer advertising to contain nanomaterials) or suggest new products to the inventory. As a quality control measure, suggestions and new product forms contributed by registered users must be approved by a CPI curator before updates or revisions are posted to the inventory.  \n\n# Supporting Information  \n\n# Supporting Information File 1  \n\nA compilation of company and product numbers listed by country of origin. A list of all nanomaterial components included in the inventory. Nanotechnology expert survey questions. Additional nanotechnology expert survey results. [http://www.beilstein-journals.org/bjnano/content/ supplementary/2190-4286-6-181-S1.pdf]  \n\n# Acknowledgements  \n\nFunding for this work was provided by the Institute for Critical Technology and Applied Science (ICTAS) at Virginia Tech and the Virginia Tech Center for Sustainable Nanotechnology (VTSuN). We acknowledge the important help of J. Rousso, E. Bruning, S. Guldin, J. Wang, D. Yang, X. Zhou, L. Marr, the VTSuN graduate students in updating inventory entries, and the Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech. We also acknowledge the Center for the Environmental Implications of Nanotechnology, funded under NSF Cooperative Agreement EF-0830093, for helping to inform our understanding of the broad world of manufactured nanomaterials. Co-author M. Hull acknowledges helpful discussions with A. Maynard of the Arizona State University Risk Innovation Lab that provided important motivation for this work.  \n\n# References  \n\n1. 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Res. 2014, 16, 2724. doi:10.1007/s11051-014-2724-4   \n28. Nazarenko, Y.; Zhen, H. J.; Han, T.; Lioy, P. J.; Mainelis, G. Environ. Health Perspect. 2012, 120, 885–892. doi:10.1289/ehp.1104350   \n29. Nazarenko, Y.; Han, T. W.; Lioy, P. J.; Mainelis, G. J. Exposure Sci. Environ. Epidemiol. 2011, 21, 515–528. doi:10.1038/jes.2011.10   \n30. Quadros, M. E.; Marr, L. C. Environ. Sci. Technol. 2011, 45, 10713–10719. doi:10.1021/es202770m   \n31. Benn, T.; Cavanagh, B.; Hristovski, K.; Posner, J. D.; Westerhoff, P. J. Environ. Qual. 2010, 39, 1875–1882. doi:10.2134/jeq2009.0363   \n32. Quadros, M. E.; Pierson, R.; Tulve, N. S.; Willis, R.; Rogers, K.; Thomas, T. A.; Marr, L. C. Environ. Sci. Technol. 2013, 47, 8894–8901. doi:10.1021/es4015844   \n33. Tulve, N. S.; Stefaniak, A. B.; Vance, M. E.; Rogers, K.; Mwilu, S.; LeBouf, R. F.; Schwegler-Berry, D.; Willis, R.; Thomas, T. A.; Marr, L. C. Int. J. Hyg. Environ. Health 2015, 218, 345–357. doi:10.1016/j.ijheh.2015.02.002   \n34. Holden, P. A.; Klaessig, F.; Turco, R. F.; Priester, J. H.; Rico, C. M.; Avila-Arias, H.; Mortimer, M.; Pacpaco, K.; Gardea-Torresdey, J. L. Environ. Sci. Technol. 2014, 48, 10541–10551. doi:10.1021/es502440s   \n35. Savolainen, K.; Backman, U.; Brouwer, D.; Fadeel, B.; Fernandes, T.; Kuhlbusch, T.; Landsiedel, R.; Lynch, I.; Pylkkänen, L. Nanosafety in Europe 2015-2025: Towards Safe and Sustainable Nanomaterials and Nanotechnology Innovations. Finnish Institute of Occupational Health: Helsinki, Finnland, 2013; http://www.ttl.fi/en/publications/Electronic_publications/Nanosafety_in_ europe_2015-2025/Documents/nanosafety_2015-2025.pdf.   \n36. Consumer Product Safety Commission Fiscal Year 2016 Performance Budget Request. http://www.cpsc.gov/Global/About-CPSC/Budget-and-Performance/FY 2016BudgettoCongress.pdf (accessed March 2, 2015).   \n37. Hansen, S. F. J. Nanopart. Res. 2010, 12, 1959–1970. doi:10.1007/s11051-010-0006-3   \n38. Becker, S. J. Nanopart. Res. 2013, 15, 1426. doi:10.1007/s11051-013-1426-7   \n39. Baruch, Y.; Holtom, B. C. Hum. Relat. 2008, 61, 1139–1160. doi:10.1177/0018726708094863   \n40. Beaudrie, C. E. H.; Kandlikar, M.; Satterfield, T. Environ. Sci. Technol. 2013, 47, 5524–5534. doi:10.1021/es303591x  \n\n# License and Terms  \n\nThis is an Open Access article under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.  \n\nThe license is subject to the Beilstein Journal of Nanotechnology terms and conditions: (http://www.beilstein-journals.org/bjnano)  \n\nThe definitive version of this article is the electronic one   \nwhich can be found at:   \ndoi:10.3762/bjnano.6.181  ",
    "Target": null
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  {
    "id": "10.1002_jcc.24300",
    "DOI": "10.1002/jcc.24300",
    "DOI Link": "http://dx.doi.org/10.1002/jcc.24300",
    "Relative Dir Path": "mds/10.1002_jcc.24300",
    "Article Title": "LOBSTER: A Tool to Extract Chemical Bonding from Plane-Wave Based DFT",
    "Authors": "Maintz, S; Deringer, VL; Tchougréeff, AL; Dronskowski, R",
    "Source Title": "JOURNAL OF COMPUTATIONAL CHEMISTRY",
    "Abstract": "The computer program LOBSTER (Local Orbital Basis Suite Towards Electronic-Structure Reconstruction) enables chemical-bonding analysis based on periodic plane-wave (PAW) density-functional theory (DFT) output and is applicable to a wide range of first-principles simulations in solid-state and materials chemistry. LOBSTER incorporates analytic projection routines described previously in this very journal [J. Comput. Chem. 2013, 34, 2557] and offers improved functionality. It calculates, among others, atom-projected densities of states (pDOS), projected crystal orbital Hamilton population (pCOHP) curves, and the recently introduced bond-weighted distribution function (BWDF). The software is offered free-of-charge for noncommercial research. (C) 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.",
    "Times Cited, WoS Core": 2363,
    "Times Cited, All Databases": 2467,
    "Publication Year": 2016,
    "Research Areas": "Chemistry",
    "UT (Unique WOS ID)": "WOS:000374023100008",
    "Markdown": "# LOBSTER: A Tool to Extract Chemical Bonding from Plane-Wave Based DFT  \n\nStefan Maintz,[a] Volker L. Deringer,[a] Andrei L. Tchougr\u0002eeff,[a,b,c] and Richard Dronskowski\\*[a,d]  \n\nThe computer program LOBSTER (Local Orbital Basis Suite Towards Electronic-Structure Reconstruction) enables chemicalbonding analysis based on periodic plane-wave (PAW) densityfunctional theory (DFT) output and is applicable to a wide range of first-principles simulations in solid-state and materials chemistry. LOBSTER incorporates analytic projection routines described previously in this very journal [J. Comput. Chem. 2013, 34, 2557] and offers improved functionality. It calculates, among others, atom-projected densities of states (pDOS), projected crystal orbital Hamilton population (pCOHP) curves, and the recently introduced bond-weighted distribution function (BWDF). The software is offered free-of-charge for noncommercial research. $\\circledcirc$ 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.  \n\nDOI: 10.1002/jcc.24300  \n\n# Introduction  \n\nMethods for electron partitioning in molecules have been around in quantum chemistry since Mulliken’s ingenious approach for assigning electrons to atoms and bonds.[1] These models and concepts are likewise helpful for periodic systems, so an analogous scheme was introduced within non-variational extended H€uckel (eH) theory[2] and dubbed Crystal Orbital Overlap Population (COOP); pioneered in the 1980s, it proved powerful ever since.[3] With the advent of variational density-functional theory (DFT), the Crystal Orbital Hamilton Population (COHP) scheme was suggested, which partitions energies rather than electrons but otherwise resembles COOP in that it allows to extract chemical interactions between atoms from band-structure calculations.[4] COHPs have been implemented and widely used within TB-LMTO-ASA theory,[5] which is DFT-type but shares with eH the use of localized basis sets for periodic solids.  \n\nAs of today, many condensed-matter quantum-mechanical codes employ plane waves (PW), which naturally (and effectively) fulfill Bloch’s theorem but are delocalized by their very nature, making bondanalytical approaches such as COOP and COHP unavailable in PW frameworks. Nonetheless, there are ways to transfer PW to localized functions using projection schemes as pioneered by S\u0002anchez-Portal et al.[6] For large atomic numbers, however, PW become impractical for the near-core regions, so the success of the pseudopotential (PP) approach[7] in computational materials science is easily understandable. Nowadays, Blo€chl’s projector-augmented wave (PAW) method is the most powerful of the PP descendants.[8] To project PAW functions[8] onto localized orbitals (say, of the Slater type), we have recently developed an analytical formalism[9] to apply bond-analytic tools even though the system was brought to selfconsistency in a PW basis; other bond-analytical approaches[10] exist as well. Our technique makes COOP or COHP analyses feasible beyond densely packed systems[9] (such as intermetallics[11]) and can, other than before, be seamlessly applied to scenarios such as molecular crystals[12] or even amorphous matter.[13]  \n\nTo facilitate chemical-bonding analyses and other methods for a multitude of systems and applications, we are offering the LOBSTER (Local Orbital Basis Suite Towards Electronic-Structure Reconstruction) software free-of-charge for non-commercial purposes at http://www. cohp.de. In this article, we describe recently developed methodology and functionality added to LOBSTER. A number of illustrative applications are presented, and directions for further reading are given.  \n\n# Methods  \n\nBefore describing recent developments which have found their way into LOBSTER, we refer the reader to our initial publication[9] for a more comprehensive account of the underlying formalisms, and to the manual shipped with the code for any practical questions. A (simplified) scheme of what LOBSTER does is presented in Figure 1.  \n\n# Local basis sets  \n\nThe initial step in projection is finding a suitable choice of local auxiliary basis functions. For reasons of simple chemical  \n\nThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.  \n\n![](images/039483c0e51bc67c636f283d29f3357d238c2435f9849244d717d532cccdf209.jpg)  \nFigure 1. Overview of LOBSTER’s functional principle: a quantum-chemical system, characterized by its one-electron (Bloch) wavefunctions $\\varPsi_{j}$ and the according eigenvalues $\\varepsilon_{j}$ (band energies), has been brought to self-consistency using some plane-wave DFT program. A local auxiliary basis is then selected to determine the overlap matrix S and the transfer matrix T between the delocalized and localized representations. From those, the projected coefficient and Hamiltonian matrices $\\pmb{c}^{(\\mathsf{p})}$ and ${\\pmb H}^{(\\mathsf{p})}$ , respectively, are accessible, which allow for various bond-analytic tools. The LOBSTER logo is copyrighted by the Chair of Solid-State and Quantum Chemistry at RWTH Aachen University.  \n\ninterpretation, LOBSTER employs minimal basis sets that nonetheless carry the correct nodal behavior in the core region, which is necessary to fit PAW wavefunctions. LOBSTER first came with contracted primitive Slater-type orbitals (STOs) fitted to atomic functions,[14] a reasonable choice for postprocessing bonding information. There are also systems, however, where the bonding situation requires additional basis functions which deviate from those of the corresponding free atoms. In elemental beryllium, for example, its 2p levels are unoccupied in the free atom. For bulk Be, however, the 2p levels do get involved into bonding and define the metallic character, so the Be basis set must also include 2p functions.  \n\nFor demonstration, let’s look at the high-temperature phase,[15] body-centered cubic beryllium, $\\beta$ -Be. Its electronic structure was calculated with ABINIT employing the JTH atomic datasets[16] and the GGA-PBE parametrization for exchange and correlation.[17] On the LOBSTER side, the original basis set[14a] and its basis functions (1s and 2s) somehow manage to reconstruct the PAW electronic structure but with an unacceptable absolute charge spilling of roughly $19\\%$ (see below for definitions). For analysis, the differences between the original and projected wavefunctions were calculated and an isosurface at $65\\%$ of the maximum resulting density was plotted for several bands at $T_{\\cdot}$ The fourth band showed enormous deviations (Fig. 2, left) because the basis lacks an orbital of $\\mathsf{p}$ -symmetry, as reasoned before. Adding a 2p function strongly reduces the absolute charge spilling to $1.73\\%$ . For comparison, the $65\\%$ densitydifference maxima decrease by two orders of magnitude (Fig. 2, right). While the functions were added by fitting VASP data, they turn out to be general enough to easily fit other PAW wavefunctions, for example, those calculated by ABINIT, too.  \n\nHence, free-atom calculations in large supercells have been performed for all elements up to $Z=96$ (curium) using GGA  \n\nPBE[17] as implemented in VASP.[18] In nearly all cases up to $Z=$ 80 (mercury) did the new basis functions match the previously given ones well, and they allowed us to numerically fit and add missing (polarization) functions. Obviously, these new functions had to be orthogonalized with regard to the existing functions of the same l azimuthal quantum number to enlarge the basis sets already available in LOBSTER. In the next step, visual evaluation of the calculated PAW atomic orbitals yielded wavefunctions of the desired symmetry and shape; hence they were corroborated as a valid basis choice.  \n\nWhile the basis functions are aligned with the Cartesian axes by default, LOBSTER 2.0.0 supports user-defined rotations of the basis functions as has been described recently;[19] this new feature can be especially useful when isolated, orbitalwise interactions must be studied.  \n\n![](images/4e27de778ad151b1b504537c4f6cbc764ea4dad63d365669faabb408776ff4fa.jpg)  \nFigure 2. Isosurfaces (in $\\mathring{\\mathsf{A}}^{-3})$ at $65\\%$ of the differences between the ABINIT-based PAW densities and the LOBSTER-projected densities for the fourth band of $\\beta$ -Be at $T$ . On the left side, the basis contains only 1s and 2s functions as given by Bunge et al., whereas on the right this basis was enlarged by 2p functions. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]  \n\n# Improved measures for projection quality  \n\nBefore analyzing the projected wavefunctions, one must ensure that the auxiliary basis suffices. To do so, S\u0002anchezPortal et al. introduced the so-called “spilling” and “chargespilling” criteria,[6b] which measure the percentage of electronic density lost during projection; this approach was also used in former LOBSTER versions.[9]  \n\nNonetheless, in a localized basis, unwanted effects such as the basis-set superposition error[20] or counterintuitive orbital mixing[21] can lead to projected wavefunctions with a norm artificially larger than unity; note that the original spilling criterion correctly assumes that the norm of a projected wavefunction is bound to unity. If this condition is broken, averaging the spilling over multiple bands $j$ and $\\vec{k}$ points may lead to error cancellation such that the projection quality looks better than it actually is. To counteract, LOBSTER 2.0.0 comes with an improved definition which we dub “absolute spilling” $s$ and “absolute charge spilling”, ${\\cal{S}}_{\\cal{Q}},$ defined analogously to its predecessor,[6b] but averaging absolute values:  \n\n$$\nS_{\\boldsymbol{Q}}\\mathrm{=}\\frac{1}{N_{j}}\\sum_{\\vec{k}}^{N_{\\vec{k}}}{w_{\\vec{k}}\\sum_{j}^{N_{j}}{\\mathrm{abs}(1-O_{j j}(\\vec{k}))}},\n$$  \n\nwith  \n\n$$\n\\bullet(\\vec{k})=\\bullet^{(\\mathfrak{p})}{}^{\\dagger}(\\vec{k})\\bullet(\\vec{k})\\bullet^{(\\mathfrak{p})}(\\vec{k}),\n$$  \n\nwhere $\\mathbf{C}^{(\\mathsf{p})}(\\vec{k})$ and $\\pmb{\\mathsf{S}}(\\vec{k})$ are the coefficient and overlap matrices, respectively, and $w_{\\vec{k}}$ denotes the normalized $\\vec{k}$ point weights. The absolute charge spilling, $s,$ is collected only over occupied bands, that is, those with nonzero occupation numbers, $f_{j}(\\vec{k})\\neq0$ .  \n\nAnother way to assess the deviation is given by the rootmean-square error (RMS) of projected wavefunctions. If the PW part of the PAW functions is given on a reciprocal grid at values of ${\\vec{k}}+{\\vec{G}},$ one may define the RMS of the projection (RMSp) by comparing the projected LCAO function $|\\vec{k}+\\vec{G}|X_{j}\\rangle$ to its PAW reference $|\\vec{k}+\\vec{G}|\\Psi_{j}\\rangle$ :  \n\n$$\n{\\sf R M S p}=\\sqrt{\\frac{1}{N}\\sum_{\\vec{k},\\vec{G},j}{\\sf a b s}(|\\vec{k}+\\vec{G}|\\varPsi_{j}\\rangle-|\\vec{k}+\\vec{G}|X_{j}\\rangle)^{2}}.\n$$  \n\nThe sum runs over all $N$ vectors $\\vec{k}{+}\\vec{G}$ at each band $j$ in the plane-wave basis. Using the reciprocal representation of the wavefunctions enables us to rewrite the difference in the former equation:  \n\nthe projection by LOBSTER anyway [such as the LCAO coefficients $C_{\\mu,j}^{(\\mathsf{p})}(\\vec{k})$ and the Fourier-transforms of the local basis functions $\\tilde{\\chi}_{\\mu}(\\vec{k})],$ or they can be obtained by a Fourier–Bessel transform  \n\n$$\n\\bar{\\phi}_{i}(\\vec{\\kappa}){=}4\\pi\\mathsf{i}^{I_{i}}\\int\\bar{\\phi}_{i}(r)j_{I_{i}}(\\mathsf{a b s}(\\vec{\\kappa})r)r\\mathsf{d}r Y_{m_{i}}^{I_{i}}(\\hat{\\kappa}),\n$$  \n\nwhere $\\bar{\\phi}_{i}(\\boldsymbol{r})$ is a shorthand notation for the difference between the all-electron and pseudo-space partial-waves in the PAW method and $\\scriptstyle{\\vec{\\kappa}}={\\vec{k}}+{\\vec{G}}$ . $j_{I}(x)$ designates the spherical Bessel function, and $Y_{m}^{I}(\\vartheta,\\varphi)$ is a spherical harmonic.  \n\nIn contrast to the originally defined spilling, the RMSp method is bound to zero independent of assumptions, which makes it a well-suited optimization criterion. If desired, one may normalize RMSp to the range of reference data, viz. $\\mathsf{m a x}(|\\vec{k}+\\vec{G}|\\varPsi_{j}\\rangle){\\-}\\mathsf{m i n}(|\\vec{k}+\\vec{G}|\\varPsi_{j}\\rangle)$ , for example, when comparing results from primitive unit cells to those from supercell models, a rather practical real-world scenario.  \n\n$$\n\\begin{array}{c}{{\\displaystyle{|\\vec{k}+\\vec{G}|\\:\\psi_{j}\\rangle-|\\vec{k}+\\vec{G}|X_{j}\\rangle=C_{\\vec{G},j}^{\\mathsf{P W}}(\\vec{k})+\\sum_{i}\\bar{\\phi}_{i}(\\vec{k}+\\vec{G})\\langle\\tilde{p}_{i}|\\tilde{\\psi}_{j}\\rangle}}}\\\\ {{{-\\displaystyle\\sum_{\\mu}C_{\\mu,j}^{(\\mathsf{p})}(\\vec{k})\\tilde{\\chi}_{\\mu}(\\vec{k}+\\vec{G}).}}}\\end{array}\n$$  \n\nAll of these expressions are either known directly from the PAW calculation (like the plane-wave coefficients $C_{\\vec{G},j}^{\\mathsf{P W}}(\\vec{k})$ and so-called wavefunction characters $\\langle\\tilde{p}_{i}|\\tilde{\\Psi}_{j}\\rangle$ , evaluated during  \n\n# Orthonormalization  \n\nAs stated before, the targeted analytic methods are bound to minimal basis sets on purpose and hence prevent using sophisticated multi- $\\cdot\\zeta$ basis sets easily. Consequently, we apply L€owdin’s symmetric orthonormalization (LSO) to the projected wavefunctions.[9] Even if significant (but comparable) amounts of charge are lost around every atom, this technique was found to ensure properly projected densities of states (pDOS), a crucial ingredient for other bond-analytical tools in the sequel.  \n\nWithin LOBSTER 2.0.0, LSO is also applied to the basis functions themselves. We note that traditional COHP analysis by means of TB-LMTO-ASA theory[5] works with an intrinsically orthogonal basis set which overlaps due to the atomicspheres-approximation: likewise, the basis functions in LOBSTER do overlap. To improve correspondence between traditional COHP and its projected analogue, the projected Hamiltonian matrix ${\\pmb H}^{(\\mathrm{p})}(\\vec{k})$ is now reconstructed after likewise applying LSO to the basis functions, yielding  \n\n$$\n{\\pmb{H}}^{(\\mathrm{p})}(\\vec{k}){\\pmb{C}}^{(\\mathrm{p})\\prime}(\\vec{k}){=}{\\pmb{C}}^{(\\mathrm{p})\\prime}(\\vec{k}){\\pmb{\\varepsilon}}(\\vec{k}),\n$$  \n\nwhere ${\\pmb C}^{(\\mathsf{p})\\prime}(\\vec{k})$ designates the coefficient matrix within the orthogonalized set of basis functions. In contrast, all other bond-analytic tools such as projected COOP still use $\\pmb{C}^{(\\mathfrak{p})}(\\vec{k})$ since the overlap populations would be rendered meaningless in an orthogonal basis set.  \n\n# Visualization  \n\nExamining the causes of an imperfect projection is a nontrivial task but may be significantly simplified by visual inspection. To do so, the internal development version of LOBSTER writes the values of both projected and PAW wavefunctions on a user-defined, linearly equidistant grid which can either be an arbitrarily oriented line or a threedimensional grid within a cuboid, both bounded by the unit cell. This enables density-difference plots as shown in Figure 2, but can also be used to examine the signed wavefunctions directly.  \n\n![](images/77ee341dac72e7a8e170608613a0ff1b68c71d5ce645b060234f8fbd687712ea.jpg)  \nFigure 3. LOBSTER analysis of a diffusion pathway through crystalline GeTe, as originally mapped out using nudged-elastic-band (NEB) theory[35a] and previously analyzed by pCOOP analysis in the thesis of one of us.[38] Top: structural drawing of the supercell setup, with only selected atoms shown for clarity.[38] A germanium atom jumps from one octahedron into an adjacent one; a second one further away serves as reference. Bottom: pCOHP analysis for the sum of the three short Ge–Te bonds shown, respectively. Energy is shifted so that the Fermi level eF equals zero.  \n\n# Technical aspects  \n\nLOBSTER is written in modern, object-oriented $\\mathsf{C}++$ and uses the famous Boost library[22] for various algorithms and concepts for object or memory management, as well as their organization. Even though many of them were incorporated into the $\\mathsf{C}++11$ standard, Boost is still a valuable asset for mathematical special functions and interaction with the operating system. Furthermore, LOBSTER has evolved into a multiplatform tool, supporting Linux, Windows and OS X. Wherever possible, it employs matrix or vector algebra to employ the data structures and algorithms, for example, for matrix decompositions, provided by the highly efficient Eigen library.[23] Results of computationally expensive but repeatedly evaluated functions are cached internally. LOBSTER is parallelized using  \n\nOpenMP, still making efficient use of its internal caches through multiple-readers/single-writer lock patterns. To optimize CPU usage, LOBSTER uses memory mapped I/O to read large chunks of input data from the file system when available and beneficial.  \n\nAt present, LOBSTER 2.0.0 processes and analyzes PAW results from two third-party codes, VASP[18] and ABINIT,[24] but further interfaces are possible. Due to lack of direct tabulation in the case of ABINIT, the projector functions $\\tilde{p}_{i}(r)$ must be transformed to reciprocal space once using the Fourier–Bessel transform, as stated above.  \n\n# Program Features  \n\nOnce the coefficient matrix has been reconstructed (Fig. 1), LOBSTER can readily calculate pDOS and pCOOP by $\\vec{k}$ space integration.[9] Additionally, it writes their energy-integrated counterpart IpDOS (which yields the total number of electrons of the respective atoms, Mulliken’s gross population) and IpCOOP (Mulliken’s overlap population). Based on these IpCOOP values, bonding analysis can be applied even to amorphous structures by means of the recently proposed bond-weighted distribution function (BWDF).[13] The coefficient matrices have also been used directly to analyze orbital mixing (or hybridization in the physicists’ language).[25] Reconstructing the Hamiltonian matrix in a second step facilitates pCOHP analysis.[9,26] Energy integration up to the Fermi level yields IpCOHP, which might likewise serve as a bondweighting indicator for BWDF.[13] For further detail on each specific method, we redirect to the original literature.  \n\n# Applications  \n\nSince its initial publication, LOBSTER has found diverse applications. Being interfaced to state-of-the-art DFT codes such as VASP, it allows to process output from modern methods such as hybrid-DFT results.[27] LOBSTER has begun to play its part in surface chemistry: exploring oxide catalysts,[28] square-planar carbon at transition-metal surfaces,[29] or local structural fragments at quartz-type $\\mathsf{G e O}_{2}$ surfaces.[30] Less-than-densely packed three-dimensional (3D) networks have been of interest as well: complex clathrate structures,[31] hydrogen bonding in molecular crystals,[12] or the stability ranking of metal azide polymorphs.[32]  \n\nWe round out this article by presenting two representative applications from fields of current research interest.  \n\n# Atomic motion in phase-change materials  \n\nPhase-change materials (PCMs) can be switched between crystalline and amorphous phases, and thus be used to encode “ones” and “zeroes” in digital data storage.[33] Atomistic simulations, generally a cornerstone of PCM research,[34] have recently been concerned with transition pathways and atomic motion in crystalline PCMs.[35]  \n\nThe prototypical PCM germanium telluride (GeTe) has been studied using COHPs more than a decade ago using TB-LMTO-ASA theory.[36] With the new projection techniques at hand, we may now explore not only its crystalline and amorphous phases,[13] but also the formation and diffusion of vacancies on the crystalline (distorted rocksalt-type) lattice, as visualized in Figure 3 (top). The well-known presence of antibonding interactions in PCMs[37] is also seen in LOBSTER output (Fig. 3, bottom). While these regions $(-\\mathsf{p c o H P}<0)$ are there both for the moving atom (red) and away from the transition state (gray), they critically depend on the local environment: the antibonding peaks are much more dominant in the transition-state geometry (Fig. 3, bottom left). This reflects the moving Ge atom which has to “squeeze” through an octahedral face to leave its groundstate position, which costs over $100~\\mathrm{kJ/mol}$ according to DFT simulations.[35a] Further away, on the contrary, the bonding situation seems relatively unperturbed (Fig. 3, bottom right) and compares well to what is found in the ground-state structure. The bonding analysis also allows us to rationalize previously made observations regarding Fermi level tuning: lowering it makes the transition state $(\\ddagger)$ less unfavorably bonded, and thus reduces the activation energy for the Ge hop significantly.[35a,38]  \n\n![](images/ed6fbf9e41f9aecdc992904fb5910682bd4cecc52ce8f37d1486f09ecc6ad7f6.jpg)  \nFigure 4. Bonding analysis of $\\alpha$ -iron using (left) COHP as implemented in TB-LMTO-ASA in a non-magnetic (NM) setup and (middle) using ${\\mathsf{p c o H P}}$ based on PAW results by VASP processed with LOBSTER. By allowing for spin polarization (SP, right), the resulting exchange splitting affects the chemical bonding between the Fe atoms which becomes stronger. COHP and ${\\mathsf{p c o H P s}}$ are given as the sum over all symmetry-equivalent bonds in the unit cell. Energy is shifted so that the Fermi level eF equals zero. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]  \n\n![](images/56d0e59b2063366d9657e5259588f2039bc08f3c37987f51a579b62f89374277.jpg)  \nFigure 5. Spin polarization and chemical bonding in a two-dimensional Ru sheet supported on an Ag(001) surface. The left-hand side shows the supercell setup, and a large “vacuum” area is clearly visible, as commonly used in PW based DFT simulations of surfaces and nanomaterials. The right-hand side shows the LOBSTER-computed ${\\mathsf{p c o H P}}$ curve for a single nearest-neighbor Ru–Ru contact in the spin-polarized case; compare with Figure 4 (right). Energy is shifted so that the Fermi level $\\varepsilon_{\\mathsf{F}}$ equals zero. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]  \n\n# The orbital origins of ferromagnetism, revisited: from $\\pmb{\\alpha}$ -iron to Ru monolayers  \n\nFerromagnetism of $\\alpha$ -Fe can be chemically understood as a consequence of orbital interactions which has previously been rationalized at the hand of COHP analysis,[39] and this concept later served as a predictive guideline for the design of more complex magnetic materials.[11,40] Figure 4 (left) presents a COHP analysis based on a traditional TB-LMTO-ASA calculation at the spin-restricted (non-magnetic, NM) GGA level showing occupied antibonding levels at the Fermi level which destabilize the system. The middle and right of Figure 4 offer analogous analyses, but employ the methods and frameworks given in this work. In the non-magnetic case, the VASP/LOBSTER combination recovers what has been known before. By switching on spin-polarization, the majority (a) electrons lower in energy (and its associated spin orbitals spatially contract) while the minority $(\\beta)$ electrons increase energetically (and its orbitals expand);[39] as a result, antibonding states are diminished and the chemical bonding strengthens.  \n\nWhile the above is merely validation, the new method (in contrast to TB-LMTO-ASA theory) can easily handle “open” systems such as two-dimensionally extended surface structures. Magnetism in such systems has been explored earlier, and one of them is shown in Figure 5 (left): a monolayer of ruthenium atoms supported on a slab of Ag. As originally predicted by Bl€ugel,[41] ruthenium becomes ferromagnetic in this configuration. Figure 5 (right) now delves into the chemical-bonding nature again, and it suggests an explanation that is principally analogous to the $\\alpha$ -iron case: namely significant exchange splitting and strengthening of the Ru–Ru bonds while becoming ferromagnetic, now so easily rationalized using LOBSTER.  \n\n# Conclusions  \n\nWe have presented new developments in the LOBSTER software for chemical-bonding analysis. LOBSTER processes delocalized PAW wavefunctions calculated with VASP or ABINIT and performs projection into an auxiliary LCAO basis, which makes bond-analytic tools such as pDOS, pCOOP, and pCOHP accessible for state-of-the-art plane-wave based PAW simulations.  \n\nTo reliably assess the quality of the projection, we here introduced two modified criteria, dubbed absolute (charge) spilling and root-mean-square of the projection (RMSp). Additionally, visual evaluation of either the PAW or the projected wavefunctions has been demonstrated. A new and improved basis set available in LOBSTER has also been described. Finally, to improve correspondence to traditional COHP analysis based on LMTO theory, we now also apply L€owdin’s symmetric orthogonalization to the basis functions.  \n\n# Acknowledgments  \n\nWe thank Marc Esser for reviewing the basis rotation code. We are also thankful to Janine George and all other group members who carefully tested the internal code versions, as well as to many users worldwide for their ongoing and valuable feedback. The work of V.L.D. and R.D. on LOBSTER analyses of phasechange materials has been supported by Deutsche Forschungsgemeinschaft (DFG) within SFB 917 (Nanoswitches). V.L.D. gratefully acknowledges a scholarship from the German National Academic Foundation (Studienstiftung des deutschen Volkes).  \n\nKeywords: chemical bonding $\\cdot\\cdot$ plane waves $\\cdot^{\\ast}$ DFT $\\cdot^{\\cdot}$ projection $\\cdot\\cdot$ COHP  \n\nHow to cite this article: S. Maintz, V. L. 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    "Target": null
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  {
    "id": "10.1080_08927022.2015.1010082",
    "DOI": "10.1080/08927022.2015.1010082",
    "DOI Link": "http://dx.doi.org/10.1080/08927022.2015.1010082",
    "Relative Dir Path": "mds/10.1080_08927022.2015.1010082",
    "Article Title": "RASPA: molecular simulation software for adsorption and diffusion in flexible nulloporous materials",
    "Authors": "Dubbeldam, D; Calero, S; Ellis, DE; Snurr, RQ",
    "Source Title": "MOLECULAR SIMULATION",
    "Abstract": "A new software package, RASPA, for simulating adsorption and diffusion of molecules in flexible nulloporous materials is presented. The code implements the latest state-of-the-art algorithms for molecular dynamics and Monte Carlo (MC) in various ensembles including symplectic/measure-preserving integrators, Ewald summation, configurational-bias MC, continuous fractional component MC, reactive MC and Baker's minimisation. We show example applications of RASPA in computing coexistence properties, adsorption isotherms for single and multiple components, self- and collective diffusivities, reaction systems and visualisation. The software is released under the GNU General Public License.",
    "Times Cited, WoS Core": 1435,
    "Times Cited, All Databases": 1532,
    "Publication Year": 2016,
    "Research Areas": "Chemistry; Physics",
    "UT (Unique WOS ID)": "WOS:000362190200001",
    "Markdown": "This article was downloaded by: [University of Nebraska, Lincoln]   \nOn: 14 April 2015, At: 03:10   \nPublisher: Taylor & Francis   \nInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,   \n37-41 Mortimer Street, London W1T 3JH, UK  \n\n# Molecular Simulation  \n\nPublication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gmos20  \n\n# RASPA: molecular simulation software for adsorption and diffusion in flexible nanoporous materials  \n\nDavid Dubbeldama, Sofía Calerob, Donald E. Ellisc & Randall Q. Snurrd   \na Van 't Hoff Institute of Molecular Sciences, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands   \nb Department of Physical, Chemical and Natural Systems, University Pablo de Olavide, Sevilla 41013, Spain   \nc Department of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA   \nd Department of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA   \nPublished online: 26 Feb 2015.  \n\nTo cite this article: David Dubbeldam, Sofía Calero, Donald E. Ellis & Randall Q. Snurr (2015): RASPA: molecular simulation software for adsorption and diffusion in flexible nanoporous materials, Molecular Simulation, DOI: 10.1080/08927022.2015.1010082  \n\nTo link to this article:  http://dx.doi.org/10.1080/08927022.2015.1010082  \n\nPLEASE SCROLL DOWN FOR ARTICLE  \n\nTaylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Versions of published Taylor & Francis and Routledge Open articles and Taylor & Francis and Routledge Open Select articles posted to institutional or subject repositories or any other third-party website are without warranty from Taylor & Francis of any kind, either expressed or implied, including, but not limited to, warranties of merchantability, fitness for a particular purpose, or non-infringement. Any opinions and views expressed in this article are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor & Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.  \n\nThis article may be used for research, teaching, and private study purposes. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions  \n\nIt is essential that you check the license status of any given Open and Open Select article to confirm conditions of access and use.  \n\n# RASPA: molecular simulation software for adsorption and diffusion in flexible nanoporous materials  \n\nDavid Dubbeldama\\*, Sofı´a Calerob1, Donald E. Ellisc2 and Randall Q. Snurrd3 aVan ’t Hoff Institute of Molecular Sciences, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands; bDepartment of Physical, Chemical and Natural Systems, University Pablo de Olavide, Sevilla 41013, Spain; cDepartment of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA; dDepartment of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA  \n\n(Received 7 October 2014; final version received 16 January 2015)  \n\nA new software package, RASPA, for simulating adsorption and diffusion of molecules in flexible nanoporous materials is presented. The code implements the latest state-of-the-art algorithms for molecular dynamics and Monte Carlo (MC) in various ensembles including symplectic/measure-preserving integrators, Ewald summation, configurational-bias MC, continuous fractional component MC, reactive MC and Baker’s minimisation. We show example applications of RASPA in computing coexistence properties, adsorption isotherms for single and multiple components, self- and collective diffusivities, reaction systems and visualisation. The software is released under the GNU General Public License.  \n\nKeywords: molecular simulation; Monte Carlo; molecular dynamics; adsorption; diffusion; software  \n\n# 1. Introduction  \n\nMolecular sieves are selective, high-capacity adsorbents because of their high intracrystalline surface areas and strong interactions with adsorbates. Molecules of different size generally have different diffusion properties in a given molecular sieve, and molecules can be separated on the basis of their size and structure relative to the size and geometry of the apertures of the sieve. Much progress has been made in understanding the subtle interaction between molecules and the confinement, and much of this understanding comes from computer simulations that are able to analyse the chemistry and physics at the atomistic level.  \n\nThe two main computational approaches to tackle these systems are (i) quantum mechanical calculations and (ii) force field-based simulations. The first approach is required for studying properties such as bond breakage and formation and is available in many excellent commercial and non-commercial packages. The second approach is useful for studying larger systems and for calculating a wide variety of thermodynamic and dynamic properties. Force field-based approaches include Monte Carlo (MC) simulations, molecular dynamics (MD) simulations and energy minimisations. We introduce here a code, RASPA, that focuses on MC, MD and minimisation of systems described by classical force fields.  \n\nThe RASPA code was written as a collaboration among Northwestern University (USA), the University of Amsterdam (The Netherlands) and the University Pablo de Olavide (Spain), with recent contributions also from the University of Delft (The Netherlands). The code evolved initially from the post-doc project (2006–2009) of David Dubbeldam at Northwestern University, where the Snurr group had another MUltipurpose SImulation Code (MUSIC), which was written in object-oriented Fortran 90.[1,2] MUSIC provides functionality for performing MD and MC simulations in a number of different ensembles, minimisations and free energy calculations for bulk and adsorbed phases using a variety of force fields, but not for treating flexible adsorbent frameworks and hence RASPA was developed. Version 1.0 of this code has been used internally by the authors and a growing list of collaborators. In this paper, we present version 2.0 available for public use. Its main areas of utility are thermodynamic properties of liquids and gases, and adsorption/diffusion behaviour of adsorbates in crystalline nanoporous materials.  \n\nExamples of nanoporous materials are clays, carbon nanotubes, zeolites and metal-organic frameworks (MOFs). MOFs are a relatively new class of materials composed of metal nodes connected by organic linkers. MOFs possess almost unlimited structural variety because of the many combinations of building blocks that can be imagined. The building blocks self-assemble during synthesis into crystalline materials that, after evacuation of the structure, may find applications in adsorption separations, gas storage and catalysis.[3–5] MOFs have crystal structures that exhibit unusual flexibility. An extreme example is the ‘breathing MOF’ MIL-53 that expands or shrinks to admit guest molecules such as $\\mathrm{CO}_{2}$ and water.[6] For simulation of zeolites, it is common practice to keep the positions of the framework atoms fixed, but this assumption is not valid for many large-pore MOFs. New algorithms and a new code were developed to handle these systems.  \n\nRASPA is a serial code. A single point of an isotherm can be obtained within hours for a simple system and in days for more complicated systems. MC codes are ideally suited for task-farm parallelism. Here, simulations are independent and are run as batches of serial simulations that differ in temperature, pressure, etc. For example, (assuming no hysteresis), each point of an isotherm can be run independently. Memory requirements of MC codes are modest.  \n\nPrograms can be written in various ways, but often it is true that the fastest codes are probably the hardest to read, while programs strictly based on readability lack efficiency. RASPA (being a ‘research’ code) chooses the middle-ground and is based on the following ideas:  \n\nCorrectness and accuracy. For all techniques and algorithms available in RASPA, we have implemented the ‘best’ ones (in our opinion) available in the literature. For example, RASPA uses configurational-bias Monte-Carlo (CBMC) and continuous fractional component Monte Carlo (CFCMC); it uses the Ewald summation for electrostatics; MD is based on symplectic and measure-preserving integrators. . Functional design. Examining the source code, one can notice that there are not a large number of files. The program is split up according to its function: ‘grid.c’ contains the code to make and use an energy/ force grid of a framework, ‘ewald.c’ handles all the electrostatic calculations, ‘mc_moves.c’ contains all the moves to be used in MC, ‘potentials.c’ contains all the van der Waals potentials, etc. Input made easy. The requirements for the input files are kept as minimal as possible. Only for more advanced options are extra commands in the input file needed. Also the format of the input is straightforward. Fugacity coefficients and excess adsorption are automatically computed. Integrated simulation environment. The code is built up of many functions and routines which can be easily combined. MD can be used in MC and vice versa. Extension and modification of the code is relatively straightforward.  \n\nThis article provides an overview of the application areas of the code. For a detailed description of the algorithms themselves and the inner working of MC and MD codes, we refer to Refs.[7,8]  \n\n# 2. Units, input and conventions  \n\nA small set of internal units needs to be chosen. A convenient set, which is chosen in DLPOLY [9], RASPA, and many other codes, is the following:  \n\n(1) The unit of length $l_{0}$ is chosen as the A˚ ngstrom, i.e. $l_{0}=10^{-10}\\mathrm{m}$   \n(2) The unit of time $t_{0}$ is chosen as the picosecond, i.e. $t_{0}=10^{-12}$ s   \n(3) The unit of mass $m_{0}$ is chosen as the atomic mass unit, i.e. $m_{0}=1.660540210^{-27}{\\mathrm{kg}}$   \n(4) The unit of charge $q_{0}$ is chosen as the unit of proton charge, i.e. $m_{0}=1.6021773310^{-19}\\mathrm{C}$ .  \n\nAll other units follow from this choice. For example, one Pascal $[\\mathrm{Pa}=\\mathrm{mass}/(\\mathrm{length}\\times\\mathrm{time}^{2})]$ is $1.66054\\times10^{7}$ in internal units. A pressure input of $10\\mathrm{Pa}$ in the input file is converted to ‘internal units’ by dividing by $1.66054\\times10^{7}$ . Similarly, at output the pressure in internal units is converted to Pa by multiplying by $1.66054\\times10^{7}$ .  \n\nRASPA used three generic ‘types’ or ‘groups’ for the particles: (1) ‘framework atoms’, (2) ‘adsorbates’ and (3) ‘cations’. (The classification was done with respect to porous materials, for pure fluids the meaning of ‘adsorbates’ reduces to ‘molecules’.) The advantage is that the different components of the total energy are available and the interactions can be examined (also the energies in the Ewald Fourier part are split [10]). Cations are considered as part of the framework (they are included in the total mass of the framework). Another example is when using thermostats, e.g. in LTA5A, a different thermostat can operate on the framework atoms, the adsorbates and the cations (these all vibrate/move at different length- and time scales). There are no restrictions on the number of molecules or the number of components. This allows for example an adsorption simulation of a mixture of $\\mathrm{CO}_{2}$ and $\\Nu_{2}$ in LTA5A with Na and Ca ions. In this example, there are four components: two adsorbate components, $\\mathrm{CO}_{2}$ and $\\Nu_{2}$ , and two cation components, Na and Ca. For each component the MC move types and relative attempt frequency can be specified. In this case, $\\mathrm{CO}_{2}$ and ${\\bf N}_{2}$ can undergo particle insertion and deletion moves, while Na and Ca only use translation (the cations can be simulated as mobile). A typical input for such a simulation at 298 K, 1 bar (1:3 mixture of $\\mathrm{CO}_{2}/\\mathrm{N}_{2}$ ) looks like:  \n\nSimulationType MonteCarlo NumberOfCycles 250000 NumberOfInitialisationCycles 100000 PrintEvery 1000  \n\nForcefield GenericZeolites ModifyOxgensConnectedToAluminium yes  \n\nFramework 0   \nFrameworkName LTA5A   \nRemoveAtomNumberCodeFromLabel yes   \nUnitCells 1 1 1   \nExternalTemperature 298.0   \nExternalPressure 10000.0  \n\nComponent 0 MoleculeName sodium MoleculeDefinition Cations TranslationProbability 1.0 RandomTranslationProbability 1.0 ExtraFrameworkMolecule yes CreateNumberOfMolecules 32  \n\nComponent 1 MoleculeName calcium MoleculeDefinition Cations TranslationProbability 1.0 RandomTranslationProbability 1.0 ExtraFrameworkMolecule yes CreateNumberOfMolecules 32  \n\nComponent 2 MoleculeName CO2 MoleculeDefinition TraPPE MolFraction 0.25 BlockPockets yes BlockPocketsFilename LTA TranslationProbability 1.0 RotationProbability 1.0 ReinsertionProbability 1.0 SwapProbability 2.0 ExtraFrameworkMolecule no CreateNumberOfMolecules 0  \n\nComponent 3 MoleculeName N2 MoleculeDefinition TraPPE MolFraction 0.75 BlockPockets yes BlockPocketsFilename LTA TranslationProbability 1.0 RotationProbability 1.0 ReinsertionProbability 1.0 SwapProbability 2.0 ExtraFrameworkMolecule no CreateNumberOfMolecules 0  \n\nNumbering of frameworks, components, etc., is based on the C-convention, i.e. starting from zero. The ‘SimulationType’ line sets the simulation type (e.g. MC, MD, transition state theory (TST), minimisation). RASPA ‘cycles’ for MD is the number of integration steps. For MC a cycle is $\\operatorname*{max}(20,\\ N)$ -move-attempts with $N$ being the number of molecules. In one cycle each molecule experiences on average one MC move attempt (either accepted or rejected). More molecules require more MC moves and the use of a ‘cycle’ allows for a specification of the simulation length that is relatively insensitive to the number of molecules. Once a line like ‘Framework $0^{\\circ}$ has been read, the lines below it refer to that framework until another line like ‘Framework $\\mathbf{\\Omega}_{1}\\mathbf{\\cdot}\\mathbf{\\Omega}$ is encountered (it is possible to define multiple frameworks per system). Similarly, the MC-move probabilities are set for a specific component (set with the ‘Component [number] MoleculeName [string]’ line). For each component you can specify mole fraction, fugacity coefficient, MC moves, whether the component is an adsorbate or a cation, the initial number of molecules, etc. The MC-move probabilities are appropriately normalised by the code and therefore only have to be given relative to each other. In this example, twice the number of ‘swap’-moves (i.e. insertion/deletion) will be used compared to translation, rotation and reinsertion. If no fugacity coefficients are given in the input, then the Peng–Robinson equation of state will be used to convert pressure to fugacity. Therefore, if a fugacity coefficient of unity is specified, then adsorption will be computed as a function of fugacity instead of pressure. Any specified initial number of molecules will be created at start up using the CBMC algorithm. This avoids the need to create the initial positions by hand.  \n\nIn addition to this input file, force-field files need to be created. If one uses a generic force field, then a simple ‘Forcefield GenericZeolites’ is sufficient. If one creates force field files in the current directory, then these files are read instead of the generic files. This is convenient for force field fitting where one needs to change the parameters frequently. The first two force field files are ‘force_field_ mixing_rules.def’ and ‘force_field.def’. The first is read to construct an initial force field based on the parameters for each atom-type and using mixing rules. The second file allows you to overwrite an interaction pair directly. Both ways of specifying a force field occur in the zeolite and MOF literature. A third file making up the force field is ‘pseudo_atoms.def’ which defines atom properties such as the name, atomic weight, charge and so on.  \n\nNext, for each component a definition must be provided. Many molecules have been already been created for general use and for these one can simply specify, e.g. ‘MoleculeDefinition TraPPE’. The atom names from this file are also those used in the files for defining the molecules. These list all the bond, bend and torsion interactions. For defining a flexible framework, the bond, bend and torsions are specified by type and an algorithm searches automatically for all occurrences of these. Details and examples are given in the manual accompanying the source code.  \n\n# 3. Vapour–liquid coexistence  \n\nRASPA was initially developed to simulate porous materials. However, it can also be used to model vapour– liquid equilibrium (VLE) as illustrated in this section.  \n\n# 3.1 Coexistence properties  \n\nThe enthalpy of vapourisation $\\Delta H_{\\mathrm{vap}}$ (or heat of vapourisation) is the enthalpy change required to transform a given quantity of a substance from a liquid into a gas at a given pressure. The enthalpy of vapourisation is given without approximation by  \n\n$$\n\\begin{array}{r}{\\Delta H_{\\mathrm{vap}}=U(\\mathrm{gas})-U(\\mathrm{liquid})+p[V(\\mathrm{gas})}\\\\ {-V(\\mathrm{liquid})],\\qquad}\\end{array}\n$$  \n\nwhere $U$ is the internal energy per molecule, $p$ is the pressure and $V$ is the volume per molecule. $\\Delta H_{\\mathrm{vap}}$ can be conveniently computed in the canonical Gibbs ensemble. In the canonical Gibbs ensemble, the two fluid phases (i.e. vapour and liquid) are explicitly simulated in two separate simulation boxes. Martin and Biddy noted that, for properties such as enthalpy of vapourisation that involve the pressure, it is preferable to calculate the pressure from the vapour phase.[15] The observed error bars on liquid box pressures are quite large in a molecular simulation and an equilibrated Gibbs ensemble simulation has the same pressure in both boxes. The Gibbs ensemble method, which is implemented in RASPA, is ideally suited to compute properties such as the vapour pressure, gas and liquid densities, compressibility, heat of vapourisation, second virial coefficients, boiling and critical points.  \n\n# 3.2 NVT Gibbs ensemble for vapour–liquid equilibrium  \n\nThe Gibbs ensemble MC simulation technique allows direct simulation of phase equilibria in fluids.[16,17] NVT (also called ‘canonical’) Gibbs ensemble simulations are performed in two separate microscopic regions, each with periodic boundary conditions. The temperature $T$ is held constant, as a well as the total volume $V$ and the total number of particles $N.$ The equilibrium conditions are (i) equal temperature, (ii) equal pressure and (iii) equal chemical potential for each species in the two boxes. The equal temperature in both boxes is imposed via the MC scheme for configurational equilibration (MC moves like translation and rotation). Condition (ii) is enforced by a volume move, and condition (iii) by a particle transfer move. The volume move makes one box larger and the other box smaller, which leads to pressure equilibration. The transfer of particles between the boxes leads to equal chemical potential.  \n\nVan der Waals parameters are very difficult to obtain from experiment or from ab initio calculations. However, the VLE curves are very sensitive to, e.g. the strength parameter 1 and size parameter $\\sigma$ of the Lennard-Jones potentials. Martin and Siepmann developed the transferable potentials for phase equilibria (TraPPE) force field for a large variety of molecules. It includes (and historically started with) united-atom linear and branched alkanes. [12,13] Figure 1 shows simulation data calibrated to experimentally available VLE data. The simulation results of RASPA agree very well with the simulation data of Martin and Siepmann. By fitting sequentially to methane $\\mathrm{(CH_{4})}$ , ethane $\\left(\\mathrm{CH}_{3}\\right)$ , propane $\\left(\\mathrm{CH}_{2}\\right)$ , isobutane (CH) and neopentane (C), the five atom types in the model can be uniquely fitted. Other data for alkanes can then be used to validate the force field. These types of simulations are also useful for re-optimising force fields for a different cut-off or change from tail-correction to a shifted potential type. [18,19] The chief advantage of VLE-fitted force fields is that over a large range of pressures and temperatures, the density of the fluid is accurately reproduced.  \n\nFor pure component systems, the Gibbs phase rule requires that only one intensive variable (usually the temperature) can be independently specified when two phases coexist. The density and pressure are obtained from the simulation. By contrast, for multi-component systems, pressure can be specified in advance, with the total system being considered at constant NpT. The only change necessary is that the volume changes in the two regions are now independent.  \n\n![](images/e6812db8e2e925a30ee41c1caa8dfee6f729264abaf59cb1598d1e1f15504242.jpg)  \nFigure 1. (Colour online) Vapour–liquid coexistence curves for methane, ethane, propane, isobutane and neopentane computed in the Gibbs ensemble. Line, experimental data taken from the NIST database [11]; closed symbols, previous simulation data of Martin and Siepmann [12,13] using the Towhee code [14], open symbols, this work using RASPA. The order of the data from top-to-bottom is the same as the order in the legend.  \n\n# 4. Adsorption  \n\n# 4.1 Adsorption in the NpT Gibbs ensemble  \n\nThe fundamental concept in adsorption science is the adsorption isotherm. It is the equilibrium relationship between the quantity of the molecules adsorbed and the pressure or concentration in the bulk fluid phase at constant temperature.[21] The Gibbs ensemble method can be used to compute adsorption isotherms in nanoporous materials.[16,22] One of the boxes contains the framework, while the other box contains the fluid phase (either gas or liquid) that is in equilibrium with the adsorbed phase. For adsorption of a system of $n$ components, the Gibbs phase rule requires that $n+1$ intensive variables be set, if you consider the adsorbent as an additional component. These $n+1$ variables are conveniently taken as the temperature, the pressure of the fluid phase, and $n{-}1$ mole fractions in the fluid phase. The system is then simulated using the NpT Gibbs ensemble. The fluid-phase box is maintained at constant pressure (and temperature) by applying volume moves. For adsorption in a flexible framework, the adsorbed-phase box is also maintained at constant pressure, but volume moves in a multi component Gibbs ensemble are performed independently for each box. For the simulation of adsorption in a rigid framework, the volume moves on the adsorbed-phase box are switched off; there is no requirement for mechanical equilibrium.[16] The equilibrium constraints are equal temperature in both systems and equal chemical potentials in the bulk and in the interior of the framework (similar to the VLE, the chemical potential equilibrium is enforced by particle swap moves between the boxes).  \n\nFigure 2(a) shows adsorption isotherms of xylenes in MIL-47 using the NpT-Gibbs ensemble. Although we use a slightly different model (all-atom OPLS/DREIDING/UFF model,[23]) the results generally agree with the previous simulation data of Castillo et al. [24]. Note that experiments measure ‘excess adsorption’ while simulations calculate ‘absolute adsorption’. Excess adsorption is the number of molecules in the nanopores in excess of the amount that would be present in the pore volume at the equilibrium density of the bulk gas.[25,26] RASPA calculates absolute adsorption and, if the pore-volume fraction is given (can be computed separately), it also calculates the excess adsorption for convenience. With this type of modelling, the experimental results are well reproduced. The snapshots from the simulation (Figure 3) explain the ortho-selectivity: because ortho-xylene is commensurate with the size of the channel, it forms two layers of molecules that stack very efficiently.[24,23] The advantage of the Gibbs adsorption method is that the reservoir is explicitly simulated and hence the conversion from pressure to fugacity is consistently computed with the same force field. This avoids having to find and use an accurate equation of state for the adsorbates. Downsides include having to explicitly simulate the fluid phase (which can be expensive, especially in the liquid phase), and also the computed fugacity coefficient depends on the quality of the chosen force field and representation of the adsorbates.  \n\n![](images/dede9fa609c93d73c0f8436159be7687d9ad883b5e251810425ed397251553b2.jpg)  \nFigure 2. (Colour online) Pure component isotherms of $o\\mathrm{-}$ -, $m\\cdot$ and $p$ -xylene in MIL-47 at $423\\mathrm{K}$ : closed symbols represent experimental data [20], open symbols represent simulation data.  \n\n![](images/6fcca9a525c710f453067b9f7f6eb5fd80c1265bd164dc1efd51eeec9ff8b5a6.jpg)  \nFigure 3. (Colour online) Snapshot of $o$ -xylene in MIL-47 at $433\\mathrm{K}$ , (left) view along the channel, (right) side view with the channel $45^{\\circ}$ rotated around the channel axis (the line in the centre is an edge of the unit cell). The 1D channels of MIL-47 are about $8.5\\mathring\\mathrm{A}$ in diameter, which optimally stacks molecules that are commensurate with this dimension (i.e. $o$ -xylene). Figure courtesy of Ariana Torres Knoop.  \n\n# 4.2 Adsorption in the grand-canonical ensemble  \n\nIn the limit of low pressure, fugacity and pressure are equal (i.e. the fugacity coefficient is unity). There is, therefore, no need to explicitly simulate the fluid phase. But also if an accurate equation of state is available, or if the fugacity coefficient is known experimentally, or if one is simply interested in adsorption as a function of fugacity, then the reservoir computation is not necessary. In the grand-canonical (GC) ensemble $\\mu V T$ ensemble), the chemical potential is imposed at fixed temperature in a fixed volume (determined in this case by the crystallographic definition of the host framework). Insertion and deletion moves are used in the $\\mu V T$ ensemble to equilibrate the system at the fixed value of the chemical potential which is directly related to the fugacity $f$ by $\\beta\\mu=\\beta\\mu_{i d}^{0}+l n(\\beta f)$ (where $\\mu_{i d}^{0}$ is the reference chemical potential).  \n\nFigure 2(b) shows the results for the xylene-MIL-47 system using the GC ensemble. The results of the Gibbs ensemble and grand canonical Monte Carlo (GCMC) simulations agree very well. In the GCMC simulations, the pressure was converted to fugacity using the Peng– Robinson equation of state, which uses the critical temperature, critical pressure and the ‘acentric factor’ that has been tabulated for many compounds. As output it gives whether under these conditions the fluid is a gas or liquid (or metastable), as well as properties such as the fugacity coefficient, the compressibility and the density of the bulk fluid phase. The density is needed to convert absolute adsorption to excess adsorption and vice versa.  \n\n![](images/b1e1159194712795e8243b75d8e07c5b78e782a399ad4ef22ad22098058ca780.jpg)  \nFigure 4. (Colour online) Pure component adsorption isotherms of benzene in MFI at 603 and $703\\mathrm{K}$ . Closed symbols, previous simulation result of Hansen [27] using BIGMAC [28]; open symbols, this work using RASPA.  \n\nIn theory and simulation, it is common to plot loading as a function of fugacity because these plots are unaffected by the gas–liquid transition. For validation of the GCMC capabilities of RASPA, we show in Figure 4 adsorption results for benzene in MFI-type zeolite compared to the previous simulation results of Hansen [27] using the BIGMAC code.[28] Both simulations use fugacity and absolute loadings, and the agreement is excellent.  \n\nSimulations of mixture adsorption require the specification of chemical potentials for each component. For gas phase adsorption, an empirical equation of state can be employed. The same treatment is often not readily generalisable to liquid mixtures because of the lack of accurate activity models.[29] A convenient simulation setup for such systems is to use NpT Gibbs-ensemble simulations using three boxes: (i) the adsorbed phase with the host framework, (ii) the solution phase and (iii) a vapour phase transfer medium. The molecules are not swapped directly between the adsorbed phase and the liquid phase but instead rely on the vapour phase as an intermediate transfer medium.[29]  \n\nMeasuring mixed-gas adsorption experimentally is difficult. The ideal adsorption solution theory (IAST) of Myers and Prausnitz [30] is often used to estimate the mixture loading from the pure component isotherms. The validity of IAST can be checked using simulations. Figure 5 shows single component isotherms, and results for an equimolar four-component mixture of para-, meta-, ortho-xylene and ethylbenzene in MAF- $^{\\mathbf{\\nabla}_{\\mathbf{X}}8}$ at 433 K.[23] The IAST prediction is validated with explicit mixture simulations and for this system the IAST is applicable. Another reason to validate IAST is because it is convenient to use IAST as the input for breakthrough simulations.[31]  \n\n# 4.3 Adsorption in the ${\\pmb{\\mu}}_{1}\\mathbf{N}_{2}\\mathbf{PT}$ ensemble (flexible frameworks)  \n\nThe $\\mu_{1}N_{2}P T$ ensemble [35,36] is the natural ensemble to compute adsorption for flexible frameworks. The system is considered as two components, where the chemical potential of component 1 (the guest species) is kept constant (and has variable particle number), while component 2 (the framework) has constant particle number. As in GCMC, only the adsorbed phase is simulated, but now the volume moves are included to hold the pressure constant. For a single component system, it is not possible to vary three intensive variables independently because of the Gibbs–Duhem relation (from which Gibbs’ phase rule follows) which relates them. However, for two (or more) species systems, it is possible to derive, rigorously, a statistical ensemble in which $T$ , $P$ and $\\mu_{\\mathrm{ads}}$ and $N_{\\mathrm{host}}$ are held fixed. For this ensemble, $\\mu_{\\mathrm{ads}}$ is the chemical potential of the adsorbate and $N_{\\mathrm{host}}$ is the fixed number of atoms of the framework (host). This is a hybrid statistical ensemble which has some properties similar to the single species $(N p T)$ and $(\\mu V T)$ ensembles. In $(\\mu_{1}N_{2}p T)$ MC simulation, one carries out (at least) three distinct types of trial procedures [35,36] (i) the conventional configuration change moves, (ii) the change of volume and/or size of the system and (iii) a creation or deletion move.  \n\n![](images/638d75dec71c2c9ea1bd6ff7a5ce926206924e5492662c491de3be982a1079b6.jpg)  \nFigure 5. (Colour online) Xylene separation at $433\\mathrm{K}$ using MAF- $\\mathbf{\\sigma}\\cdot\\mathbf{x}8$ ,[23] (a) single components fitted with dual-site Langmuir– Freundlich isotherms, (b) equimolar mixture simulations and IAST prediction based on single component isotherms.  \n\nFigure 6 shows simulated adsorption results of $\\mathrm{CO}_{2}$ in a flexible IRMOF-1 compared to simulations using a rigid structure (the energy-minimised structure with the same force field), and also compared to experimental data. The results for the rigid and flexible model are very similar for this system and in excellent agreement with experimental data. The computation of adsorption in the flexible structure was feasible because the IRMOF-1 structure stays relatively close to its equilibrium structure. The framework motions are efficiently sampled using the MC/ MD-hybrid move.[37,38]  \n\n![](images/3fc6f0083f2499cebc6d6fb6fae22289a80572ccb696221bd5d5d541db1c0a31.jpg)  \nFigure 6. (Colour online) $\\mathrm{CO}_{2}$ adsorption in IRMOF-1 showing step-like isotherms [32]. Lines, experimental data [33];, filled coloured points, rigid framework model [34];, open symbols, flexible framework model.[34]  \n\n# 4.4 Efficient algorithms for open ensembles  \n\nA system where the number of molecules varies is called an open system. All open-ensemble methods suffer from a major drawback: the insertion and deletion probabilities become vanishingly low at high densities. This problem is particularly severe for long chain molecules. For adsorption simulations, the fraction of successful insertions into the pores becomes too low. To increase the number of successfully inserted molecules, the CBMC technique was developed in the early 1990s.[7,39–41] Instead of generating ideal gas configurations and trying to insert the molecule as a whole, the CBMC method inserts chains part by part, biasing the growth process towards energetically favourable configurations, and therefore significantly reduces overlap with the framework and other particles.  \n\nAn alternative scheme to remedy the insertion problem is the recently developed CFCMC method of Shi and Maginn [42–44]. The system is expanded with an additional particle whose interactions with the other atoms in the system are scaled by a parameter $\\lambda$ , where $0\\leq\\lambda\\leq1$ . Note that only the inter-molecular energy is scaled (not the intra-molecular energy). Many variations on the algorithm are possible. For example $\\lambda$ can be changed per molecule or per atom. Both methods slowly ‘inflate’ and ‘deflate’ the molecule like a balloon but differently.  \n\nCFCMC uses conventional NVT MC for thermalisation (such as translation, rotation and/or MC–MD hybrid moves), but in addition attempts to change $\\lambda$ of the fractional molecule using $\\lambda(\\mathrm{new})=\\lambda(\\mathrm{old})+\\Delta\\lambda.\\ \\Delta\\lambda$ is chosen uniformly between $-\\Delta\\lambda^{\\mathrm{max}}$ and $+\\Delta\\lambda^{\\mathrm{max}}$ and scaled to achieve around $50\\%$ acceptance. However, many systems show a behaviour where $\\lambda$ -changes are hard. An additional bias $\\eta$ on $\\lambda$ can be used. This bias will be removed by the acceptance rules. A careful calibration of $\\eta$ can make $\\lambda$ histograms flat and hence can avoid that the system gets stuck in a certain range of $\\lambda$ . There are three possible outcomes of a change of $\\lambda({\\mathrm{old}})$ to $\\lambda(\\mathrm{new})$ : (i) $\\lambda$ remains between 0 and 1; there is no change in the number of particles, nor in the positions, nor in the intra-molecular energies. Only $\\lambda$ and the inter-molecular energy have changed. (ii) $\\lambda$ becomes larger than 1; when $\\lambda$ exceeds unity, $\\lambda=1+\\varepsilon$ , the current fractional molecule is made fully present $\\lambda=1\\dot{}$ ), and a new fractional molecule is randomly inserted with $\\lambda=\\varepsilon$ . Shi and Maginn used a methodology where a rigid conformation is chosen from a ‘reservoir’ of ideal gas molecules generated before the simulation. (iii) $\\lambda$ becomes smaller than 0: when $\\lambda$ falls below 0, $\\lambda=-\\varepsilon$ , the current fractional molecule is removed from the system $\\lambda=0$ Þ, and a new fractional molecule is chosen with $\\lambda=1-\\varepsilon$ .  \n\nRASPA implements both CBMC and CFCMC, but also a combination (named CB/CFCMC) of the two developed by Torres-Knoop et al. [45]. The basic CFCMC algorithm is used with $\\lambda$ -biasing, but the insertion and deletion moves are performed using configurational biasing. Figure 2(c) shows that smoother curves are obtained by using this method. The method leads to very reliable results. Other implemented methods in RASPA to improve the efficiency of MC simulations are parallel-tempering and mole fraction replica exchange.[46,47]  \n\n# 5. Screening  \n\nContinued research and investments in high-performance computing have produced computing platforms that are now fast enough to permit predictive simulations and largescale screening studies. Simulation (virtual) screening is significantly cheaper than experimental screening and can be used to increase the successful hit rate. A hierarchical or step-wise approach is often used.  \n\n. Initial screening (millions of structures). Screening on the basis of properties that can be computed very quickly. Example properties are pore volume, surface area, pore-size distribution, Henry coefficients and heats of adsorption at infinite dilution. . High throughput screening (hundreds of thousands of structures). In the pressure range of practical interest, the heat of adsorption and loadings are simulated (usually using relatively short runs). This allows a comparison of structures and an elucidation of structure–property relationships.[48–52] Detailed analysis (tens or hundreds of structures). Detailed analysis of the most promising structures could include simulations of single-component and mixture isotherms, diffusivities and efficiency estimates of the performance in a fixed-bed adsorber using breakthrough simulations.[23,53]  \n\nFigure 7 shows two examples of screening. Go´mezGualdr´on et al. [48] investigated physical limits for methane storage and delivery in nanoporous materials, with a focus on whether it is possible to reach a methane deliverable capacity of $315\\mathrm{cm}^{3}(\\mathrm{STP})/\\mathrm{cm}^{3}$ in line with the adsorption target established by the ARPA-E agency. Using GCMC simulations, methane adsorption and delivery properties were studied in a population of 122,835 hypothetical pcu MOFs and 39 idealised carbon-based porous materials. From the simulation results, an analytical equation was obtained that delimits the necessary material properties to reach specific methane deliverable capacity targets. This high-throughput analysis elucidates how both geometric and chemical properties, such as void fraction, volumetric surface area and heat of adsorption, impact methane deliverable capacity.  \n\n![](images/3e31fbc059fa87eda7972c5431737bd7dab43478d558bf5051ec31a597995dea.jpg)  \nFigure 7. (Colour online) Screening results for (a) physical limits for methane storage and delivery in about hundred thousand nanoporous materials,[48] (b) fixed bed performance of para-selective MOFs.[23] Figure (a) courtesy of Diego A. Go´mez-Gualdro´n.  \n\nThe second example is a detailed analysis by TorresKnoop et al. on separation of benzene, toluene, meta-, ortho-, para-xylene and ethylbenzene (BTEX process).[23] Many ortho-xylene selective structures have been found, but finding para-selective structures is much harder. Small pore structures are able to separate para-xylene using ‘sieving’ (the smaller molecules fit into the structure, but the larger molecules are excluded), but these are unable to separate para-xylene from ethylbenzene (same smallest dimension) and are usually diffusion limited. TorresKnoop et al. studied about 30 structures in full detail and elucidated why some structures are ortho-selective and others are para-selective. Using snapshots, the reason for a selectivity was explained. Snapshots in Figure 3 show that strong ortho-selectively can be obtained by a two-layer molecular stacking in the MIL-47 structure. The orthoxylene fits in perfectly, while meta-xylene fits less well. Para-xylene and ethylbenzene are too long and are forced to align obliquely. In their work, a para-selective structure was sought. Using the same mechanism, it is then required that the channel dimension are perfectly commensurate with the para-xylene dimensions. The screening found a strongly para-selective structure (MAF- $\\mathbf{\\nabla}\\cdot\\mathbf{x}8_{}$ ). The single component isotherms were computed, the IAST prediction was validated with mixture simulations and the IAST solution was the input for simulating breakthrough curves. These breakthrough simulations have the ‘cycle-time’ as output, i.e. the time needed before needing to start the expensive desorption process. As can be seen from Figure 7, the MAF$^{\\mathbf{\\delta}_{\\mathbf{X}8}}$ would be better than the currently used technology (BaX). The study also revealed that other para-xyleneselective structures, such as MIL-125 and JUC-77, would be diffusion limited (decreasing their performance).  \n\nRASPA provides perl scripts to submit jobs for screening purposes. One specifies the list of adsorbates, structures, temperatures, pressure range and number of pressure points, whether fugacity or pressure is used, whether the points are equally spaced in normal or logscale, etc. The script then generates all the necessary input files and has as output the job scripts needed to submit it to a cluster (with a single command).  \n\n# 6. Reactive Monte Carlo  \n\nThe RxMC method allows computation of equilibrium properties for chemically reacting or associating fluids. [54,55] The method samples forward and backward reactions using MC moves directly (without going through the transition states). No chemical potentials or chemical potential differences need to be specified for the reaction steps, just the stoichiometry of the reactions. Essentially, the method enhances the GCMC with a ‘forward’ and ‘backward’ reaction step, which ensures that the chemical reaction equilibria between the reactants and the products are maintained.  \n\nAs an example, the industrially important propene metathesis is described by three equilibrium reactions [57] $2\\mathrm{C}_{3}\\mathrm{H}_{6}\\leftrightarrow\\mathrm{C}_{2}\\mathrm{H}_{4}+$ trans- $\\mathrm{{C_{4}H_{8}}}$ $\\mathrm{2C_{3}H_{6}\\leftrightarrow C_{2}H_{4}+c i s–C_{4}H_{8}}$ cis- ${\\mathrm{C}}_{3}{\\mathrm{H}}_{6}\\leftrightarrow$ trans- $\\mathrm{C_{4}H_{8}}$  \n\nOnly two reactions are independent and need to be included. In addition to the MC moves associated with simulating a chosen ensemble, also ‘reaction’ moves are performed:  \n\n(1) randomly choose a reaction,   \n(2) randomly choose whether to do a forward or backward reaction (this determines the ‘reactant’ and ‘product’ molecule types),   \n(3) randomly select the reactant molecules and remove them from the system,   \n(4) insert the product molecules at random positions,   \n(5) accept or reject the reaction step with the appropriate acceptance probability.  \n\nInserting molecules at high densities is difficult and even more so when one needs to insert several molecules at the same time. To overcome these difficulties, Hansen et al. [56] and Jakobtorweihen et al. [58] combined the $\\mathbf{RxMC}$ method with CBMC. Recently, Rosch and Maginn combined the CFCMC method with the RxMC method. [44] For the propene metathesis reactions simulated with reactions 2 and 3, six fractional reaction molecules are added to the system. Each reaction has an associated reaction $\\lambda$ (between 0 and 1) and MC moves are performed trying to change $\\lambda_{o}$ to $\\lambda_{n}$ . When $\\lambda_{n}>1$ a forward reaction is performed and, if accepted, $\\lambda$ is set to $\\lambda_{n}-1$ . If $\\lambda_{n}<0$ a backward reaction is performed and, if accepted, $\\lambda$ is set to $\\lambda_{n}+1$ . The insertions and deletions are biased in $\\lambda$ which allows the method to efficiently overcome insertion and deletion difficulties.  \n\nFigure 8 shows the results from RASPA using the CFCMC–RxMC method compared to the CBMC–RxMC simulation results of Hansen et al. [56]. The RASPA results are in excellent agreement. Previously, Rosch and Maginn [44] validated their implementation with the results of Hansen et al. and also found excellent agreement.  \n\n# 7. Diffusion  \n\n# 7.1 Molecular dynamics  \n\nIn many applications of nanoporous materials, the rate of molecular transport inside the pores plays a key role in the overall process. The size of the pores is usually of the same order as the size of the adsorbates. Diffusion properties of guest molecules in nanoporous materials can therefore be quite sensitive to small differences between different host materials. Molecular-level modelling has become a useful tool for gaining a better understanding of diffusion in nanoporous materials. Many different diffusion coefficients can be defined for guest molecules in nanoporous materials, but it is useful to put them into two general classes: transport diffusivities and self-diffusivities. The transport (or Fickian) diffusivity describes the transport of mass and the decay of density fluctuations in the system, while self-diffusion describes the diffusive motion of a single particle. By omitting the thermodynamic contribution in the transport diffusion, the so-called ‘corrected diffusivity’ (also called ‘collective diffusivity’) is obtained. The self-, corrected and transport diffusivities are equal only in the limit of zero loading.  \n\n![](images/75a5b22767d8d3c421932648db74ee1443eba58449c7c3a75ece90e49967b7b6.jpg)  \nFigure 8. (Colour online) Selectivity of the propene metathesis reaction system in the temperature range between 300 and $600\\mathrm{K}$ . Closed symbols, previous simulation result of Hansen et al. [56]; open symbols, this work using RASPA.  \n\nIn an equilibrium molecular dynamics simulation, the self-diffusion coefficient $D_{\\alpha}^{S}$ of component $\\alpha$ is computed by taking the slope of the mean-squared displacement (MSD) at long times  \n\n$$\nD_{\\alpha}^{S}=\\frac{1}{2\\mathrm{d}N_{\\alpha}}\\operatorname*{lim}_{t\\rightarrow\\infty}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\Bigg\\langle\\sum_{i=1}^{N_{\\alpha}}(r_{i}^{\\alpha}(t)-r_{i}^{\\alpha}(0))^{2}\\Bigg\\rangle,\n$$  \n\nwhere $N_{\\alpha}$ is the number of molecules of component $\\alpha,d$ is the spatial dimension of the system, $t$ is the time and $r_{i}^{\\alpha}$ is the centre-of-mass of molecule $i$ of component $\\alpha$ . The order- $\\cdot n$ algorithm for measuring MSDs conveniently and efficiently captures correlations over short, medium and long times.[59] In crystalline materials the MSDs become linear beyond $\\lambda^{2}$ , where $\\lambda$ is the repeating distance (usually the unit cell distance). Since the MSD accuracy rapidly decreases over increasing times, a good practice is to fit the diffusivities from a few data points after the MSD has become linear.  \n\nFor a single adsorbed component, the transport diffusion coefficient $D^{T}$ is given by  \n\n$$\nD^{T}=\\frac{\\Gamma}{2\\mathrm{d}N}\\operatorname*{lim}_{t\\rightarrow\\infty}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\Bigg\\langle\\Bigg(\\sum_{i=1}^{N}(r_{i}(t)-r_{i}(0))\\Bigg)^{2}\\Bigg\\rangle.\n$$  \n\nNote that here first the distances are summed and then squared. Hence collective diffusion can be considered a ‘centre-of-mass’ diffusion. The thermodynamic factor $\\Gamma$ is  \n\n$$\n\\Gamma=\\left(\\frac{\\partial\\ln f}{\\partial\\ln c}\\right)_{T}=\\frac{\\left<N\\right>}{\\left<N^{2}\\right>-\\left<N\\right>^{2}},\n$$  \n\nwhere $c$ denotes the concentration (adsorbate loading in the framework), and can be obtained from the adsorption isotherm or from the fluctuation formula.[60] The omission of the thermodynamic factor in Equation (3) leads to the ‘corrected diffusivity’ (also called ‘collective diffusivity’) $D^{C}$ . The concept of collective diffusivity can be extended to multi-component systems using  \n\n$$\n\\begin{array}{c l}{{\\displaystyle\\Delta_{\\alpha\\beta}=\\frac{1}{2\\mathrm{d}N_{\\alpha}}\\operatorname*{lim}_{t\\rightarrow\\infty}\\frac{\\mathrm{d}}{\\mathrm{d}t}\\Bigg\\langle\\left(\\sum_{i=1}^{N_{\\alpha}}(\\mathbf{r}_{i}^{\\alpha}(t)-\\mathbf{r}_{i}^{\\alpha}(0))\\right)}}\\\\ {{\\displaystyle\\times\\left(\\sum_{i=1}^{N_{\\beta}}(\\mathbf{r}_{i}^{\\beta}(t)-\\mathbf{r}_{i}^{\\beta}(0))\\right)\\Bigg\\rangle,}}\\end{array}\n$$  \n\nwhere the $\\Delta$ elements for components $\\alpha$ and $\\beta$ are known as the Onsager $\\Delta$ elements. Maxwell –Stefan diffusivities are related to the elements of $\\left[B\\right]=\\left[\\Delta\\right]^{-1}$ and can be obtained by matrix inversion.[61,62] Equations relating $\\Delta_{\\alpha\\beta}$ to the Fickian diffusion coefficients can also be derived.[63] From a phenomenological point of view, there are three different approaches to setting up the flux–driving force relationship for diffusion in nanoporous materials under non-equilibrium conditions. The Fickian-, Maxwell–Stefan- and Onsager formulations are strictly equivalent, and all three viewpoints are needed for different purposes.  \n\nIf the framework is kept fixed, the potential energy surface induced by the framework can be pre-computed. [64,65] Instead of looping over all framework atoms in order to compute the host-adsorbate energy at each time step, one can construct a 3D grid before the simulation and then obtain the energy by interpolation during the simulation. The more points in the grid the higher the accuracy. RASPA implements the triclinic grid interpolation scheme in three dimensions of Lekien and Marsden [45,66]. The algorithm is based on a specific $64\\times64$ matrix that provides the relationship between the derivatives at the corners of the elements and the coefficients of the tricubic interpolant for this element. The cubic interpolant and its first three derivatives are continuous and consistent. The same grids can, therefore, be used for both MC and MD with no additional energy drift besides the drift due to the integration scheme, i.e. the energy gradients are the exact derivatives of the energy at each point in the element.  \n\nRASPA uses symplectic or measure-preserving and reversible integrators. Symplectic integrators tend to preserve qualitative properties of phase space trajectories: trajectories do not cross, and although energy is not exactly conserved, energy fluctuations are bounded. Since symplectic integrators preserve the topological structure of trajectories in phase space, they are more reliable for longterm integration than non-symplectic integrators. The implemented NVE integrator is the symplectic and time reversible integrator for molecules with an arbitrary level of rigidity, developed by Miller et al. [67] based on a novel quaternion scheme. Thermo- and barostats can be combined with the Miller scheme to control the temperature and pressure.[68,70,71] Figure 9 shows that this type of scheme has excellent energy conservation over many nanoseconds. The $\\mathrm{CO}_{2}$ is modelled as rigid and integrated using the quaternion integration scheme of Miller et al. [67]. Separate thermostats are used to thermostat the translation and the rotation of the molecules. The framework–molecule interactions are computed using a grid interpolation scheme.[45,66] The grid spacing was $\\mathrm{0.1\\mathring{A}}$ . A separate grid is used for each van der Waals interaction of the O and the C of the adsorbate molecule, and another grid is used for the real part of the Ewald summation. In the Fourier part of the Ewald summation, the contribution of the rigid framework atoms is pre-computed at the start of the simulation.[10]  \n\nFigure 10 shows the self- and collective diffusivities of small gases $(\\mathrm{H}_{2}$ , $\\mathbf{N}_{2}$ , Ar, $\\mathrm{CH}_{4}$ and $\\mathrm{CO}_{2}\\mathrm{\\cdot}$ ) in IRMOF-1 at $298\\mathrm{K}$ as a function of loading using the force field of Skoulidas and Sholl [69]. The results of RASPA agree quantitatively with previous simulation results of Skoulidas and Sholl. Self-diffusivities can be computed very accurately because it is a single particle property (which can be averaged over all particles). The collective diffusivity is much more difficult to compute because it is a system property. The self-diffusivities are more strongly influenced by correlation effects (kinetic and vacancy correlations) than the collective diffusivities. Correlations between the particles increase with loading.  \n\n# 7.2 Dynamically corrected transition state theory  \n\nFor some systems, the molecules move too slowly and the diffusion coefficients cannot be calculated reliably using MD. An alternative approach is to use transition state theory (TST). In the TST approximation, one computes a rate constant for hopping between states A and B by computing the equilibrium particle flux through the dividing surface. The dividing surface should partition the system into two well-defined states along a reaction coordinate, which describes the progress of the diffusion event from state $A$ to state $B$ . In many nanoporous materials, the reaction coordinate follows directly from the geometry of the confinement. For example, in Figure 11 the reaction coordinate for methane in LTL (Linde Type L)-type zeolite is shown: the projection of the position on the channel-axis. The location of the dividing surface is denoted by $q^{*}$ . In ‘dynamically corrected’ TST, one computes the hopping rate over the barrier in two steps [72,73]:  \n\n(1) the relative probability $P(q^{*})=\\mathrm{e}^{-\\beta F(q^{*})}/$ $\\int_{\\mathrm{cageA}}{\\mathrm{e}^{-\\beta F(q)}}\\mathrm{d}q$ is computed to find a particle at the dividing surface $q^{*}$ relative to finding it in state $A$ ,   \n(2) the average velocity at the top of the barrier is computed as $\\sqrt{k_{B}T/2\\pi m}$ (assuming that the particle  \n\n![](images/65eba08de5fe3e9e54a62739bc17b57b52870df3fa2aa8afa4bef0e1e27c9f3f.jpg)  \nFigure 9. (Colour online) Molecular dynamics of $\\mathrm{CO}_{2}$ at 20 molecules per IRMOF-1 unit cell $(2\\times2\\times2$ system) at room temperature using the quaternion integration scheme of Miller et al. [67]: (a) the individual contributions to the conserved quantity, (b) the instantaneous and average energy drift. The temperature is maintained by using a Nose´ –Hoover chain.[68] The molecule–framework interactions are computed using a grid-interpolation scheme [45,66].  \n\n![](images/b2c5a36cbedcfb6fcf260e1abacf99ccc2440580aa0c3cc4111f0f0101535d43.jpg)  \nFigure 10. (Colour online) Simulated diffusivities of small gases in IRMOF-1 at $298\\mathrm{K}$ . Closed symbols, previous simulation result of Skoulidas and Sholl [69]; open symbols, this work using RASPA.  \n\nvelocities follow a Maxwell–Boltzmann distribution), and the probability $\\kappa$ (dynamical correction) that the system ends up in state $B$ is obtained by running short MD trajectories from the dividing surface.  \n\nThe transmission rate $k_{A\\rightarrow B}$ from cage $A$ to cage $B$ is then given by  \n\n$$\nk_{A\\rightarrow B}=\\kappa\\times\\sqrt{\\frac{k_{B}T}{2\\pi m}}\\times\\frac{\\mathrm{e}^{-\\beta F(q^{*})}}{\\int\\mathrm{e}^{-\\beta F(q)}\\mathrm{d}q},\n$$  \n\nwhere $\\beta=1/(k_{B}T),k_{i}$ $k_{B}$ is the Boltzmann constant, $T$ the temperature, $m$ the mass involved in the reaction coordinate and $F(q)$ the Helmholtz free energy as a function of $q$ . Calculating TST rate constants is therefore equivalent to calculating free energy differences. The exact rate can be recovered by running short MD trajectories from the dividing surface to compute a dynamical correction.[72] The extension to non-zero loading (or to a flexible framework) simply involves sampling these effects ‘in the background’.[73,74] In Figure 11 the free energy profile for methane in LTL is plotted for an average loading of three molecules per unit cell (a unit cell contains two channels). The barrier of this free energy profile is denoted as $q^{*}$ and corresponds to an entropic constriction of the channel. The dynamic correction is computed from many snapshots with one particle constrained to $q^{*}$ and $N-1$ particles free (the snapshots are easily sampled using MC). Each snapshot is used to start an MD path with initial velocities sampled from a Maxwell–Boltzmann distribution. The velocity of the barrier particle is pointing towards cage $B$ . For all of these snapshots, MD paths are simulated and the flux at the top of the barrier is computed. Figure 12 shows data for methane, ethane and propane in LTL-type zeolite with dcTST compared to MD (for this system both are feasible). It can be seen that the two methods give identical results. The dcTST method, however, is also applicable for slow diffusion $(\\ll10^{-12}\\mathrm{m}^{2}/s)$ that is (currently) impossible to compute with MD.  \n\n![](images/7278743449ba7e6388992d59a101c56ac5644b272ee86ea3abff6dda7ff6985e.jpg)  \nFigure 11. (Colour online) A typical snapshot of a tagged methane particle (green) in LTL-type zeolite restrained to the barrier $q^{\\bar{*}}$ surface at an average loading of three methane molecules per unit cell (there are two parallel channels per unit cell) at $30\\mathrm{{0}K}$ . Four unit cells each of $7.474\\mathring\\mathrm{A}$ in length are shown. The constrictions are caused by the 12-T-membered rings, which form free energy barriers impeding diffusion. The free energy profile in dimensionless units at this average loading is plotted in white, where the reaction coordinate is chosen parallel to the channel direction. If the free energy barriers are high enough, diffusion can be considered a hopping process from minimum to minimum $(q_{A},q_{B},q_{C}$ ; etc).  \n\nThe dcTST sampling is an example where it is convenient to be able to constrain MC moves. RASPA includes ways to constrain the movement of a component to a line, plane or sub-volume (box, cylinder, etc.). Any attempt to move a particle outside the sub-volume is rejected. Another useful feature is ‘blocking’ volumes that are large enough to contain a molecule, but where that volume is not accessible from the main channel. Yet another common example is a channel system where one would like to always have equal particles for each channel. This can be achieved by creating a different component for each channel, and only allow MC moves to a cylinder that encompasses that channel.  \n\n![](images/5b83543fa33c2fc79aae9f4b072cb7cd7d77dff863264b02b83663e7087058db.jpg)  \nFigure 12. (Colour online) Diffusion of methane $(C_{1})$ , ethane $(C_{2})$ and propane $(C_{3})$ at $300\\mathrm{K}$ as a function of loading in LTLtype zeolite computed by TST, dcTST and MD.  \n\n# 8. Material properties  \n\n# 8.1 Surface area, void fraction and pore-size distribution  \n\nSurface area is the most basic property of porous materials. Along with pore volume, surface area has become the main benchmark characterisation method for any porous material. The surface area is usually determined for experimental samples by measuring a nitrogen isotherm at $77\\mathrm{K}$ and then applying the Brunauer–Emmett–Teller (BET) model. Walton and Snurr [75] examined the consistency of the surface areas obtained from the BET model with those calculated geometrically from the crystal structure for several prototypical MOFs with varying pore sizes. Geometric surface areas can be calculated by using a simple MC integration technique in which a nitrogen probe $(3.681\\mathring{\\mathrm{A}})$ molecule is rolled along the surface of the framework.[26,76,77] Walton and Snurr provided compelling evidence for the importance of calculating the BET surface area from the proper region of the adsorption isotherm.[75] Commercial ‘BET’ instruments are typically set to automatically choose a fixed range for BET fitting. The operator must ensure that the range results in consistent BET model parameters.  \n\nThe pore-size distribution (PSD) can be calculated geometrically in RASPA using the method of Gelb and Gubbins [77,78]. For every point in the void volume, the largest sphere is found that encloses the point but does not overlap with any framework atoms. This yields the cumulative pore volume curve. Let $V_{\\mathrm{pore}}(r)$ be the volume of the void space ‘coverable’ by spheres of radius $r$ or smaller; a point $x$ is in $V_{\\mathrm{pore}}(r)$ only if we can construct a sphere of radius $r$ that overlaps $x$ and does not overlap any substrate atoms. This volume is equivalent to that enclosed by the pore’s ‘Connolly surface’. $V_{\\mathrm{pore}}(r)$ is a monotonically decreasing function of $r$ and is easily compared with the ‘cumulative pore volume’ curves often calculated in isotherm-based PSD methods. The derivative $-\\mathrm{d}V_{\\mathrm{pore}}(r)/\\mathrm{d}r$ is the fraction of volume coverable by spheres of radius $r$ but not by spheres of radius $r+\\mathrm{d}r$ and is a direct definition of the pore size distribution. The $V_{\\mathrm{pore}}(r)$ function can be calculated by a MC volume integration in RASPA.  \n\nKnowledge of the density of porous materials is critical for full characterisation of the adsorbents and for fixed-bed adsorber design studies. Talu and Myers [25] proposed a simulation methodology that mimics the experimental procedure. For consistency with experiment, the helium void fraction $\\xi$ is determined by probing the framework with a helium molecule using the Widom particle insertion method:  \n\n$$\n\\xi=\\int\\mathrm{e}^{-\\beta U}\\mathrm{d}\\mathbf{r}.\n$$  \n\nWidom insertion uses a probe particle that is inserted at random positions to measure the energy required for or obtained by insertion of the particle in the system. Usually a reference temperature of $25^{\\circ}\\mathrm{C}$ (298 K) is chosen for the determination of the helium void volume. Computationally, one can also use the $r\\longrightarrow0$ limit of the pore size distribution to evaluate the void fraction. The helium void fraction is needed to convert absolute loadings to excess values (or vice versa).  \n\n# 8.2 Thermal and mechanical properties  \n\nStructural flexibility is a well-known property of MOFs. [79] For example, MIL-53 exhibits breathing [6] and IRMOFs exhibit negative thermal expansion.[34] RASPA allows a wide variety of flexible models for the framework. Flexible models are needed to obtain properties such as thermal expansion of the framework itself. Thermal and mechanical transport properties are calculated either from equilibrium Green–Kubo relations or by setting up a small non-equilibrium flux across the system. Thermal expansion can be calculated from NpT MD simulations using a flexible framework.  \n\nFigure 13 plots the unit cell size of IRMOF-1 as a function of temperature.[34] The structure become smaller with increasing temperature, in contrast to most materials which expand when you heat them. The classical models, quantum mechanical results and experimental data are qualitatively consistent. The quantum results at zero Kelvin depend very much on the used level of theory and basis sets. The negative thermal expansion of IRMOFs is a direct result of the inherent structure of MOFs: linker molecules connected via metal corners. The wiggling of the linkers becomes larger with increasing temperature leading to a reduced ‘projected’ length. Viewed oppositely, when the temperature is lowered the linkers ‘stretch out’. Therefore, negative thermal expansion is very likely a generic property of many MOFs.  \n\n![](images/beb591aa4484e307ef450476f20354d2322593e4dfe7eed82841cf878deb1916.jpg)  \nFigure 13. (Colour online) Unit cell size of IRMOF-1 as a function of temperature; experimental data, quantum simulations and three classical models. For further details see text and Ref. [34].  \n\n# 9. Zero Kelvin modelling  \n\n# 9.1 Unit cells  \n\nTo study the shape and size of unit cells, one needs a model for the framework itself. Core-shell models are very suitable for minerals, metal-oxides and zeolites. The coreshell model introduces charged, massless shells around each core. For minimisation of core-shell models, the generalised Hessian matrix contains both the cores and the shells because the shells need to be minimised with respect to the cores, too. In the shell model, the short-range repulsion and van der Waals interactions are taken to act between the shell particles.  \n\nIn Table 1 we compare our results to those obtained with GULP [80,81] for the minimisation of various zeolites and minerals using Baker’s mode-following technique.[83] Using mode-following minimisation, the gradients on the cell and particles can be lowered arbitrarily close to zero and a true minimum energy is obtained (i.e. all positive eigenvalues of the Hessian matrix). The results of the RASPA and GULP codes are identical. The structures are minimised in space group P1 (no symmetry) with a cut-off of $12\\mathring{\\mathrm{A}}$ using the mode-following technique with full unit cell fluctuation, i.e. all cell lengths and angles are allowed to change. The GULP simulations are computed using GULP 3.1 and experimental data are taken from Schro¨der and  \n\nSauer [82]. The RASPA simulations were fully converged to forces smaller than $10^{-8}\\mathrm{K/\\mathring{A}}$ and $10^{-8}\\mathrm{K}/\\$ strain (1 degree Kelvin is $8.621738\\times10^{-5}\\mathrm{eV})$ .  \n\n# 9.2 Elastic constants  \n\nElastic constants express the degree to which a material possesses elasticity and mechanical stability (Born criteria). The elasticity tensor $C_{\\alpha\\beta\\mu\\nu}$ is the second derivative of the energy with respect to strain $\\eta$ and can be described in terms of fluctuations in the stress tensor $\\sigma$ [84]  \n\n$$\n\\begin{array}{c}{{C_{\\alpha\\beta\\mu\\nu}=\\Big\\langle C_{\\alpha\\beta\\mu\\nu}^{B}\\Big\\rangle}}\\\\ {{{}}}\\\\ {{-\\displaystyle\\frac{V}{k_{B}T}\\Big[\\Big\\langle\\sigma_{\\alpha\\beta}^{B}\\sigma_{\\mu\\nu}^{B}\\Big\\rangle-\\Big\\langle\\sigma_{\\alpha\\beta}^{B}\\Big\\rangle\\Big\\langle\\sigma_{\\mu\\nu}^{B}\\Big\\rangle\\Big]}}\\\\ {{{}}}\\\\ {{+\\rho k_{B}T\\Big(\\delta_{\\alpha\\mu}\\delta_{\\beta\\nu}+\\delta_{\\alpha\\nu}\\delta_{\\beta\\mu}\\Big),}}\\end{array}\n$$  \n\nwhere the first term on the right is the so-called Born term  \n\n$$\nC_{\\alpha\\beta\\mu\\nu}^{B}=\\frac{1}{V}\\frac{\\partial^{2}U}{\\partial\\eta_{\\alpha\\beta}\\partial\\eta_{\\mu\\nu}}\n$$  \n\nand the second and third terms are the stress-fluctuations term and ideal gas term, respectively. The $\\delta$ is the Kronecker’s delta, the function is 1 if the variables are equal, and 0 otherwise.  \n\nTable 1. Comparison of RASPA vs GULP for two core-shell models: the model of Catlow and Gale as provided in GULP [80,81] and Schro¨der and Sauer [82].   \n\n\n<html><body><table><tr><td rowspan=\"2\">Structure</td><td colspan=\"6\">RASPA</td><td colspan=\"6\">GULP</td></tr><tr><td>a[A]</td><td>b[A]</td><td>c[A]</td><td>α[]</td><td>β[°]</td><td>[]</td><td>a[A]</td><td>b [A]</td><td>c[A]</td><td>α[°]</td><td>β[]</td><td>[]</td></tr><tr><td>FAU (Catlow)</td><td>24.226</td><td>24.226</td><td>24.226</td><td>90</td><td>90</td><td>90</td><td>24.226</td><td>24.226</td><td>24.226</td><td>90</td><td>90</td><td>90</td></tr><tr><td>FAU (Schroder)</td><td>24.631</td><td>24.631</td><td>24.631</td><td>90</td><td>90</td><td>90</td><td>24.632</td><td>24.632</td><td>24.632</td><td>90</td><td>90</td><td>90</td></tr><tr><td>FAU (exp.)</td><td>24.26</td><td>24.26</td><td>24.26</td><td>90</td><td>90</td><td>90</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>MFI (Catlow)</td><td>19.979</td><td>19.739</td><td>13.320</td><td>90.814</td><td>90</td><td>90</td><td>19.980</td><td>19.740</td><td>13.320</td><td>90.813</td><td>90</td><td>90</td></tr><tr><td>MFI (Schroder)</td><td>20.425</td><td>20.205</td><td>13.634</td><td>90</td><td>90</td><td>90</td><td>20.425</td><td>20.205</td><td>13.634</td><td>90</td><td>90</td><td>90</td></tr><tr><td>MFI (exp.)</td><td>20.11</td><td>19.88</td><td>13.37</td><td>90.7</td><td>90</td><td>90</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CHA (Catlow)</td><td>9.198</td><td>9.198</td><td>9.198</td><td>94.752</td><td>94.752</td><td>94.752</td><td>9.198</td><td>9.198</td><td>9.198</td><td>94.751</td><td>94.751</td><td>94.751</td></tr><tr><td>CHA (Schroder)</td><td>9.335</td><td>9.335</td><td>9.335</td><td>94.524</td><td>94.524</td><td>94.524</td><td>9.335</td><td>9.335</td><td>9.335</td><td>94.524</td><td>94.524</td><td>94.524</td></tr><tr><td>CHA (exp.)</td><td>9.326</td><td>9.326</td><td>9.326</td><td>94.7</td><td>94.7</td><td>94.7</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>TON (Catlow)</td><td>13.814</td><td>17.387</td><td>5.002</td><td>90</td><td>90</td><td>90</td><td>13.815</td><td>17.388</td><td>5.002</td><td>90</td><td>90</td><td>90</td></tr><tr><td>TON (Schroder)</td><td>14.127</td><td>17.779</td><td>5.155</td><td>90</td><td>90</td><td>90</td><td>14.127</td><td>17.779</td><td>5.155</td><td>90</td><td>90</td><td>90</td></tr><tr><td>TON (exp.)</td><td>13.86</td><td>17.42</td><td>5.04</td><td>90</td><td>90</td><td>90</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SOD (Catlow)</td><td>8.767</td><td>8.767</td><td>8.767</td><td>90</td><td>90</td><td>90</td><td>8.767</td><td>8.767</td><td>8.767</td><td>90</td><td>90</td><td>90</td></tr><tr><td>SOD (Schroder)</td><td>8.951</td><td>8.951</td><td>8.951</td><td>90</td><td>90</td><td>90</td><td>8.951</td><td>8.951</td><td>8.951</td><td>90</td><td>90</td><td>90</td></tr><tr><td>SOD (exp.)</td><td>8.83</td><td>8.83</td><td>8.83</td><td>90</td><td>90</td><td>90</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Coesite (Catlow)</td><td>7.025</td><td>12.290</td><td>7.115</td><td>90</td><td>122.485</td><td>90</td><td>7.026</td><td>12.290</td><td>7.115</td><td>90</td><td>122.485</td><td>90</td></tr><tr><td>Coesite (Schroder)</td><td>7.187</td><td>12.538</td><td>7.256</td><td>90</td><td>122.803</td><td>90</td><td>7.187</td><td>12.538</td><td>7.256</td><td>90</td><td>122.803</td><td>90</td></tr><tr><td>Coesite (exp.)</td><td>7.14</td><td>12.37</td><td>7.17</td><td>90</td><td>120.3</td><td>90</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>α-quartz (Catlow)</td><td>4.835</td><td>4.835</td><td>5.346</td><td>90</td><td>90</td><td>120</td><td>4.836</td><td>4.836</td><td>5.346</td><td>90</td><td>90</td><td>120</td></tr><tr><td>α-quartz (Schroder)</td><td>4.988</td><td>4.988</td><td>5.506</td><td>90</td><td>90</td><td>120</td><td>4.988</td><td>4.988</td><td>5.506</td><td>90</td><td>90</td><td>120</td></tr><tr><td>α-quartz (exp.)</td><td>4.92</td><td>4.92</td><td>5.41</td><td>90</td><td>90</td><td>120</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table></body></html>  \n\nAt zero Kelvin, the elastic constants reduce to the Born term minus a ‘relaxation term’ [85]  \n\n$$\n\\begin{array}{r l}&{C_{\\alpha\\beta\\mu\\nu}=-\\frac{\\partial\\sigma_{\\alpha\\beta}^{B}}{\\partial\\eta_{\\mu\\nu}}\\biggr\\rvert_{h=0}}\\\\ &{\\qquad=\\underbrace{\\cfrac{1}{V}\\frac{\\partial^{2}U}{\\partial\\eta_{\\alpha\\beta}\\partial\\eta_{\\mu\\nu}}}_{\\mathrm{Borm~term}}-\\underbrace{\\cfrac{1}{V}\\frac{\\mathrm{d}\\sigma_{\\alpha\\beta}}{\\mathrm{d}r_{i\\lambda}}\\left(\\mathcal{H}^{-1}\\right)_{i\\lambda,j\\bar{\\xi}}\\frac{\\mathrm{d}\\sigma_{\\mu\\nu}}{\\mathrm{d}r_{i\\bar{\\xi}}}}_{\\mathrm{Relaxaition~term}}.}\\end{array}\n$$  \n\nNote that the derivative needs to be evaluated at constant zero gradient $\\pmb{h}=0$ , which is an algebraic relation between the coordinates at zero temperature. That is, the state before and after a strain is applied must be in a state of zero net force. When more than one particle is present in the system, this requires a ‘relaxation’ of the atoms relative to one another when the system is strained. [85] The zero temperature limit of the stress fluctuation term in Equation (8) is the relaxation term (and the ideal gas term vanishes in this limit). All expressions in Equation (10) are contained in the generalised Hessian matrix, which is the central quantity used in Baker’s minimisation scheme. The elastic constants at $0\\mathrm{K}$ can therefore be computed with very high accuracy. In Table 2 we show the elastic constants at $0\\mathrm{K}$ computed from RASPA and GULP for several zeolites and other silicates. The results are identical. Note that RASPA uses the upper triangular matrix for the simulation cell with the $a$ direction of the lattice always aligned with the $x$ -axis. This means that during the minimisation, the cell does not change orientation. This is convenient when computing elastic constants (which are directional) because the elastic constants are computed along the Cartesian axes (so the crystal should be aligned with these axes).  \n\n# 9.3 Approach angles  \n\nIn catalysis, insight into chirality transfer from catalyst to reactant can be gained by performing constrained minimisations of the reactant–catalyst complex. Because the mechanism of asymmetric induction for epoxidation of olefins by (salen)Mn catalysts is thought to involve steric interactions between the olefin and the catalyst, the direction of olefin approach has been studied by Oxford et al. [86] using hybrid MC simulations combined with classical optimisations. Four main directions of approach to the Mn-oxo moiety have been proposed in the literature (Figure 14(a)); see Ref. [86] and references therein. To examine the approach of 2,2-dimethyl-2H-chromene to the active site of $\\mathrm{(salen)Mn=O}$ , the potential energy surface for rotation around the dihedral angle of approach was mapped using constrained classical optimisations. In these optimisations using RASPA, the distance between the oxo ligand and $\\mathbf{C}_{1}$ atom was constrained to $2.0\\mathring\\mathrm{A}$ , and the angle defined by the manganese atom, oxoligand and $\\mathbf{C}_{1}$ atom was constrained to $122^{\\circ}$ (see Figure 14(b)) because this geometry is similar to that expected in the transition state.[87] Hard constraints were employed, using the $r^{2}$ -SHAKE, $\\cos^{2}\\cdot$ -SHAKE and the $\\varphi$ -SHAKE algorithms [88] for the bond, bend and torsion angle constraints, respectively. The minimisation method used guaranteed that the minimum found was a true minimum (all eigenvalues of the Hessian matrix are positive).  \n\nTable 2. Comparison of elastic constants at zero Kelvin in units of GPa as computed by GULP and RASPA.   \n\n\n<html><body><table><tr><td></td><td>MFI (RASPA)</td><td>MFI (GULP)</td><td>CHA (RASPA)</td><td>CHA (GULP)</td><td>Coesite (RASPA)</td><td>Coesite (GULP)</td><td>α-quartz (RASPA)</td><td>α-quartz (GULP)</td></tr><tr><td>C11</td><td>97.71</td><td>97.75</td><td>124.05</td><td>124.07</td><td>124.69</td><td>124.75</td><td>94.55</td><td>94.59</td></tr><tr><td>C22</td><td>88.99</td><td>89.03</td><td>122.24</td><td>122.26</td><td>210.93</td><td>210.95</td><td>94.55</td><td>94.59</td></tr><tr><td>C33</td><td>79.36</td><td>79.40</td><td>119.85</td><td>119.88</td><td>160.29</td><td>160.35</td><td>116.04</td><td>116.06</td></tr><tr><td>C44</td><td>28.65</td><td>28.67</td><td>17.51</td><td>17.52</td><td>28.28</td><td>28.32</td><td>49.97</td><td>50.00</td></tr><tr><td>C55</td><td>26.27</td><td>26.29</td><td>17.91</td><td>17.92</td><td>71.00</td><td>71.00</td><td>49.97</td><td>50.00</td></tr><tr><td>C66</td><td>23.09</td><td>23.10</td><td>18.32</td><td>18.32</td><td>43.55</td><td>43.58</td><td>38.06</td><td>38.08</td></tr><tr><td>C12</td><td>12.09</td><td>12.10</td><td>57.70</td><td>57.71</td><td>84.15</td><td>84.16</td><td>18.43</td><td>18.41</td></tr><tr><td>C13</td><td>26.49</td><td>26.49</td><td>56.82</td><td>56.84</td><td>98.09</td><td>98.09</td><td>19.67</td><td>19.67</td></tr><tr><td>C14</td><td>-2.15</td><td>-2.16</td><td>-5.53</td><td>-5.52</td><td>一</td><td>一</td><td>- 14.48</td><td>-14.49</td></tr><tr><td>C15</td><td>一</td><td>一</td><td>-5.53</td><td>-5.52</td><td>-34.12</td><td>-34.13</td><td>一</td><td>一</td></tr><tr><td>C16</td><td>一</td><td></td><td>-5.11</td><td>-5.10</td><td></td><td></td><td>一</td><td></td></tr><tr><td>C23</td><td>25.44</td><td>-25.45</td><td>56.05</td><td>56.06</td><td>45.24</td><td>45.25</td><td>19.67</td><td>19.67</td></tr><tr><td>C24</td><td>-7.93</td><td>-7.93</td><td>-6.32</td><td>-6.31</td><td>一</td><td>一</td><td>14.48</td><td>14.49</td></tr><tr><td>C25</td><td>一</td><td>一</td><td>-5.48</td><td>-5.47</td><td>9.73</td><td>9.73</td><td>一</td><td>一</td></tr><tr><td>C26</td><td></td><td>一</td><td>-7.24</td><td>-7.23</td><td>一</td><td>一</td><td>一</td><td>一</td></tr><tr><td>C34</td><td>-5.78</td><td>-5.78</td><td>-8.71</td><td>-8.70</td><td>一</td><td>一</td><td>一</td><td></td></tr><tr><td>C35</td><td>一</td><td>一</td><td>-8.10</td><td>-8.09</td><td>-62.03</td><td>-62.02</td><td>一</td><td>一</td></tr><tr><td>C36</td><td>一</td><td>一</td><td>5.31</td><td>-5.30</td><td>一</td><td>一</td><td>一</td><td>一</td></tr><tr><td>C45</td><td></td><td>一</td><td>- 2.73</td><td>-2.73</td><td>/</td><td>一</td><td>一</td><td>一</td></tr><tr><td>C46</td><td>一</td><td>一</td><td>- 2.69</td><td>- 2.69</td><td>0.63</td><td>0.65</td><td>一</td><td>一</td></tr><tr><td>C56</td><td>-0.78</td><td>-0.78</td><td>2.53</td><td>2.53</td><td>一</td><td>一</td><td>-14.48</td><td>-14.49</td></tr></table></body></html>  \n\nFigure 15 shows adsorption energies of 2,2-dimethyl2H-chromene on the homogeneous catalyst as a function of the approach angle. The approach angle $\\varphi$ was the dihedral angle defined by the midpoint of the oxygen atoms in the salen catalyst, the manganese atom, the oxo ligand and the $\\mathbf{C}_{1}$ atom of the reactant that would be forming a bond with the oxo ligand in the first transition state to the radical intermediate. The $\\mathbf{C}_{1}$ atom of chromene was chosen assuming that the phenyl group would stabilise the radical on the $\\mathbf{C}_{2}$ atom. The Si and Re enantiofaces of chromene are defined by the chirality of the $\\mathrm{C}_{1}$ atom in the reactant–catalyst complex. The reactant–catalyst complex was optimised at $10^{\\circ}$ intervals in $\\varphi$ . The simulations predict the Re enantioface to be the preferred enantioface, in agreement with experiments. The Re enantioface favours the approach from $\\varphi=-60^{\\circ}$ (approximately equivalent to the approach from direction C in Figure 14), while the Si enantioface prefers to approach from direction D, with a minimum in energy at $\\varphi=-10^{\\circ}$ .  \n\n![](images/6c0f4b7129f8b47af455605aa81abcb3986f5280092196676e94dcc165b84011.jpg)  \nFigure 14. (Colour online) Asymmetric induction for epoxidation of olefins by (salen)Mn catalysts. (a) Proposed directions of approach to the active Mn-oxo moiety of (salen)Mn. $\\varphi$ is the approach angle defined by the midpoint between the oxygen atoms in the salen ligand, the manganese atom, the oxo ligand and the carbon of the reactant forming a bond with the oxo ligand. (b) The bond-, bend- and dihedral constraints of the saddle point. The inset shows the two enantiofaces (Si and Re) of 2,2-dimethyl-2H-chromene. Figure courtesy of G.A.E. Oxford.  \n\n![](images/fe3be13a64e2cc952819f240518b33afcceafa618b53ac442e7500d21c5225cb.jpg)  \nFigure 15. (Colour online) Adsorption energies of 2,2- dimethyl-2H-chromene on the homogeneous salen catalyst as a function of the approach angle.  \n\n# 10. Visualisation  \n\nIt is difficult to determine the connectivity, shape and size of a channel/cage system just by examining the atomic positions of the framework. Early simulation work therefore used visualisations of energy contour plots and 3D density distributions, e.g. for benzene in silicalite,[90] to obtain siting information. The visualization toolkit (VTK) is an open-source, freely available software system for 3D computer graphics, image processing and visualisation.[91] RASPA has a stand-alone utility, written in $^{C++}$ , which visualises output files written by RASPA using VTK. RASPA produces 3D VTK files for visualising channel structures and 3D VTK files of the histograms of molecule positions during adsorption. For mixtures, a 3D histogram is produced for each component. This allows one to check and study ‘segregation’ of molecules.[92] Figures 3 and 11 have been made using RASPA and VTK.  \n\nFigure 16(a) shows the MFI-type zeolite. The orthorhombic unit cell has edge lengths $a=20.022\\mathring\\mathrm{A}$ , $b=19.899\\mathring{\\mathrm{A}}$ and $c=13.383\\mathring\\mathrm{A}$ . The visualisation shows two straight channels, two ‘zig–zag’ channels and four intersections per unit cell. The depicted surface is how a methane molecule would feel the adsorption surface. To visualise molecules inside the structure, the pore walls can be rendered transparent. RASPA generates 3D energy landscapes using the free energy obtained from the Widom insertion method. The simulation cell is divided up into, e.g. $150\\times150\\times150$ voxels. The adsorbate is randomly inserted millions of times and the voxels corresponding to the atom positions of the adsorbate are updated with the current Boltzmann weight. The resulting data-set has regions with value $\\langle\\mathrm{e}^{-\\beta U}\\rangle\\approx0$ , which correspond to overlap with the structure. The ratio of the non-zero values to the total number of voxels is the void fraction. Multiplying by the volume of the unit cell, we can compute the pore volume.  \n\nFigure 16(b) shows a snapshot of 2,3-dimethylbutane in MFI. To see the molecules themselves, the framework has to be either cut open or rendered transparent. The combination of the snapshot and the transparent framework allows for an analysis of adsorption sites, molecular positions and orientations, and molecule – molecule correlations. Snapshots are useful to detect differences in adsorption sites of the various species. For example, in this system the linear alkanes predominantly adsorb in the channels while the dibranched molecules adsorb first in the intersections.  \n\nSnapshots are very useful, but sometimes one needs to examine a large number of them to start to see a pattern. By keeping track of the atomic positions using 3D histograms the ‘density’ can be visualised. Figure 16(c) is the average of many snapshots. Therefore, the density is very convenient to obtain the siting information at the unit cell level. The picture is made using atomic positions (you could also use the centre of mass position) and, therefore, gives information on the average configuration (position and orientation). Using this type of approach we previously showed that average positions and occupations of the adsorption sites of argon and nitrogen in IRMOF-1 match well with experiments.[93]  \n\n![](images/8e6634efb96c76e9b5fe12121c41b3081e9b58203f7054540a867a75f8988cf0.jpg)  \nFigure 16. (Colour online) Energy landscape of MFI. The MFI unit cell has edge lengths $a=20.022\\mathring\\mathrm{A}$ , $b=19.899\\mathring{\\mathrm{A}}$ and $c=13.383\\mathring\\mathrm{A}$ , with cell angles $\\alpha=\\beta=\\gamma=90^{\\circ}$ . The MFI pore system (a) consists of straight channels running in the c-direction, which are connected via ‘zig–zag’ channels. About $29\\%$ of the structure is void. Colour code: oxygen (red), silicon (yellow). The snapshot (b) and density plot (c) are at $433\\mathrm{K}$ and $100\\mathrm{{kPa}}$ . Pictures adapted from Ref. [89].  \n\n# 11. Conclusions  \n\nWe have provided an overview of the algorithms that RASPA implements and showed examples of its application in computing coexistence properties, adsorption isotherms for single and multiple components, selfand collective diffusivities and reaction systems. RASPA is provided as source code under the GPL. The login information for the ‘git’-server can be obtained by emailing one of the authors of this manuscript. RASPA is provided without any kind of support or warranty. It should be viewed as an educational ‘research-code’ that could be useful for researchers working in the field.  \n\n# Acknowledgements  \n\nThe partition function values for the RxMC propene metathesis were computed by Sayee Prasaad Balaji; Diego A. G´omezGualdro´n provided Figure 7a; Ariana Torres Knoop provided Figure 3; R. Krishna provided the IAST computation in Figure 5. We would like to thank the following people for their help and input to improve the program and for the very helpful discussions about the algorithms: Sayee Prasaad Balaji, Youn-Sang Bae, Xiaoying Bao, Rocı´o Bueno Pe´rez, Nicholas C. Burtch, Tom Caremans, Ana Martı´n Calvo, Yamil Colon, Juan Manuel Castillo Sanchez, Allison Dickey, Tina Du¨ren, Titus van Erp, Denise Ford, Houston Frost, Rachel Getman, Pritha Ghosh, Elena Garcı´a Pe´rez, Gloria Oxford, Sudeep Punnathanam, Almudena Garcia Sanchez, Juan Jose Gutierrez Sevillano, John J. Low, Patrick Merkling, Patrick Ryan, Lev Sarkisov, Ben Sikora, Ariana Torres Knoop, Krista S. Walton, Chris Wilmer, Ozgur Yazaydin and Decai Yu. Very special thanks to Thijs Vlugt.  \n\n# Funding  \n\nThis material is supported by the Netherlands Research Council for Chemical Sciences (NWO/CW) through a VIDI grant (David Dubbeldam), by the European Research Council through an ERC Starting Grant [grant number ERC-StG-279520] (Sofia Calero), and by the U.S. National Science Foundation Grant [grant number DMR-1308799] (Randall Snurr).  \n\n# Disclosure statement  \n\nNo potential conflict of interest was reported by the authors.  \n\n# Notes  \n\n1. Email: scalero@upo.es   \n2. 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    "id": "10.1016_j.marpolbul.2016.01.006",
    "DOI": "10.1016/j.marpolbul.2016.01.006",
    "DOI Link": "http://dx.doi.org/10.1016/j.marpolbul.2016.01.006",
    "Relative Dir Path": "mds/10.1016_j.marpolbul.2016.01.006",
    "Article Title": "Synthetic fibers in atmospheric fallout: A source of microplastics in the environment?",
    "Authors": "Dris, R; Gasperi, J; Saad, M; Mirande, C; Tassin, B",
    "Source Title": "MARINE POLLUTION BULLETIN",
    "Abstract": "Sources, pathways and reservoirs of microplastics, plastic particles smaller than 5 mm, remain poorly documented in an urban context. While some studies pointed out wastewater treatment plants as a potential pathway of microplastics, none have focused on the atmospheric compartment. In this work, the atmospheric fallout of microplastics was investigated in two different urban and sub-urban sites. Microplastics were collected continuously with a stainless steel funnel. Samples were then filtered and observed with a stereomicroscope. Fibers accounted for almost all the microplastics collected. An atmospheric fallout between 2 and 355 particles/m(2)/day was highlighted. Registered fluxes were systematically higher at the urban than at the sub-urban site. Chemical characterization allowed to estimate at 29% the proportion of these fibers being all synthetic (made with petrochemicals), or a mixture of natural and synthetic material. Extrapolation using weight and volume estimates of the collected fibers, allowed a rough estimation showing that between 3 and 10 tons of fibers are deposited by atmospheric fallout at the scale of the Parisian agglomeration every year (2500 km(2)). These results could serve the scientific community working on the different sources of microplastic in both continental and marine environments. (C) 2016 Elsevier Ltd. All rights reserved.",
    "Times Cited, WoS Core": 1336,
    "Times Cited, All Databases": 1518,
    "Publication Year": 2016,
    "Research Areas": "Environmental Sciences & Ecology; Marine & Freshwater Biology",
    "UT (Unique WOS ID)": "WOS:000374198100046",
    "Markdown": "# Synthetic fibers in atmospheric fallout: A source of microplastics in the environment?  \n\nRachid Dris ⁎, Johnny Gasperi, Mohamed Saad, Cécile Mirande, Bruno Tassin  \n\nUniversité Paris-Est, LEESU (laboratoire eau environnement et systèmes urbains), 61 avenue du Général de Gaulle, 94010 Cedex Créteil, France  \n\n# a r t i c l e i n f o  \n\nArticle history:   \nReceived 2 October 2015   \nReceived in revised form 22 December 2015   \nAccepted 5 January 2016   \nAvailable online xxxx  \n\nKeywords: Microplastics Urban environment Atmospheric fallout Microplastic sources Synthetic fibers  \n\n# a b s t r a c t  \n\nSources, pathways and reservoirs of microplastics, plastic particles smaller than 5 mm, remain poorly documented in an urban context. While some studies pointed out wastewater treatment plants as a potential pathway of microplastics, none have focused on the atmospheric compartment. In this work, the atmospheric fallout of microplastics was investigated in two different urban and sub-urban sites. Microplastics were collected continuously with a stainless steel funnel. Samples were then filtered and observed with a stereomicroscope. Fibers accounted for almost all the microplastics collected. An atmospheric fallout between 2 and 355 particles $/\\mathrm{m}^{2}/$ day was highlighted. Registered fluxes were systematically higher at the urban than at the sub-urban site. Chemical characterization allowed to estimate at $29\\%$ the proportion of these fibers being all synthetic (made with petrochemicals), or a mixture of natural and synthetic material. Extrapolation using weight and volume estimates of the collected fibers, allowed a rough estimation showing that between 3 and 10 tons of fibers are deposited by atmospheric fallout at the scale of the Parisian agglomeration every year $(2500\\mathrm{km}^{2}.$ . These results could serve the scientific community working on the different sources of microplastic in both continental and marine environments.  \n\n$\\circledcirc$ 2016 Elsevier Ltd. All rights reserved.  \n\n# 1. Introduction  \n\nMicroplastics are a widespread particular contaminant originating from the breakdown of larger plastic debris (secondary) or directly manufactured on a millimetric or submilletric size (primary) (Cole et al., 2011). These plastics have been defined as particles with the largest dimension smaller than $5\\mathrm{mm}$ (Arthur et al., 2008); they cover a continuous spectrum of sizes and shapes including 1D-fibers, 2D-fragments and 3D-spheres.  \n\nGiven their size, these microparticles can be ingested by a wide range of species, either in marine (Anastasopoulou et al., 2013; Lusher et al., 2013; Thompson et al., 2004) or freshwater environments (Sanchez et al., 2014). These microplastics have negative effects on organisms and the possibility of their translocation, bioaccumulation and trophic accumulation is currently being debated (Wright et al., 2013).  \n\nWhile marine plastic pollution has been well documented, there has been limited focus on the continental contamination (Dris et al., 2015b; Wagner et al., 2014). Moreover, its sources, pathways and reservoirs in urban environments remain largely unknown. It is crucial to gather a better knowledge about these particles in the continental environment as rivers are said to be the main source of marine microplastics (Andrady, 2011). If it is very often cited that $80\\%$ of the fibers in the marine environment come from the continent, this estimation is not well documented and demonstrated.  \n\nSynthetic fibers are one of the forms in which microplastics can be found. They derive presumably from synthetic clothing or macroplastics. Different pathways are thought to be an important source of fibrous microplastics in the aquatic environment. It has been shown that laundry washing machines discharge large amounts of microplastics into wastewaters (reaching 1900 fibers in one wash (Browne et al., 2011)). During wastewater treatment, synthetic fibers are known to contaminate sewage sludge (Habib et al., 1998; Zubris and Richards, 2005). The sources and fate of microplastics in the various compartments of the urban environment are poorly documented (Dris et al., 2015a); this paper focuses on the atmospheric compartment and investigates the contribution of the atmospheric fallout as a potential vector of plastic pollution.  \n\n# 2. Materials and methods  \n\nTotal atmospheric fallout was collected on two sampling sites: one in a dense urban environment $(48^{\\circ}47^{\\prime}17.8^{\\prime\\prime}\\mathrm{N}$ , $2^{\\circ}26^{\\prime}36.2^{\\prime\\prime}\\mathrm{E}$ – Site 1 – Fig. 1) and one in a less dense sub-urban environment. $(48^{\\circ}50^{\\prime}27.8^{\\prime\\prime}\\mathrm{N}$ , $2^{\\circ}35^{\\prime}$ $15.3^{\\prime\\prime}\\mathrm{E}$ – Site 2 – Fig. 1). Site 1 was monitored over a period of one year (February 19th 2014 to March 12th 2015) and site 2 for a shorter period from October 3rd to 12th March 2015. Site 1 is localized in an area of 7900 inhabitants $\\ensuremath{{\\mathrm{km}}^{2}}$ while site 2 is characterized by a surrounding population of 3300 inhabitants $\\mathrm{\\km}^{2}$ (Insee — Régions, départements et villes de France, 2011).  \n\nThe sampling surface was $0.325~\\mathrm{m}^{2}$ allowing for total atmospheric fallout (dry and wet deposition) to be collected through a stainless steel funnel. A $20\\mathrm{L}$ glass bottle was placed at the bottom of the funnel in an opaque box to collect the water. The samples were collected at various frequencies during the monitoring period depending on cumulative rainfall leading to 24 samples at site 1 and 9 at site 2. When both sites were monitored, the collection of samples was carried out the same day for both sites in order to allow comparison. No interruption of the sampling occurred during the whole monitoring period in each site, providing a full view of the annual variability of the atmospheric fallout.  \n\n![](images/f8cd888a5200df69e7ffee1b6e51504e28eed32b16d0e198bfe9cd8e0556c23d.jpg)  \nFig. 1. Localization, sampling device and synthetic fibers for each site.  \n\nEach time the atmospheric fallout was collected, the funnel was rinsed with $3^{*}1\\mathrm{~L~}$ of reverse osmosed water in order to recover all particles adhering to the funnel. As commonly done in studies focusing on the pollutant fluxes in atmospheric fallout, preliminary tests demonstrated the efficiency of such rinses. Consecutive rinses with 1 L showed that in the fourth rinse, the number of microplastics is similar to the one in the laboratory blanks. After the rinsing step, samples were immediately covered until the processing step to avoid any contamination. Given the sampling period and the collecting surface area, the atmospheric fallout is expressed as a number of particles per square meter per day. The rainfall was recorded for both sites.  \n\nAll samples were filtered on quartz fiber GF/A Whatman filters $(1.6\\upmu\\mathrm{m})$ . To minimize post-sampling contamination from indoor air, samples were always covered. The filters and the vessel were heated to $500^{\\circ}\\mathrm{C}$ prior to their use. All laboratory procedures were performed wearing a cotton laboratory coat. Laboratory blanks were performed to verify that no microplastics are added to the samples during the laboratory procedures. Globally, blank results do not reveal any significant contamination in comparison to the levels found in samples (1 to 2 fibers per filter, representing between 0.5 and $5\\%$ of the fibers on the samples).  \n\nFilters were observed with a stereomicroscope (Leica MZ12). Previously used criteria were employed in order to identify synthetic fibers (Dris et al., 2015a; Hidalgo-Ruz et al., 2012; Norén, 2007). The accuracy of the method was estimated by comparing the counting of 3 different observers on the same filters. No difference $>5\\%$ in the total number of fibers was observed. In 11 of the collected samples, the length of the fibers was measured during the counting (the software “Histolab” coupled with the stereomicroscope). The observation size limit was defined to $50\\upmu\\mathrm{m}$ .  \n\nAtmospheric fallout in this study is presented as a number of total fibers. Chemical characterization was also performed. A subsample of ${\\mathfrak{n}}=24$ fibers was analyzed with Fourier Transform infrared (FT-IR) micro spectroscopy (Microscope LUMOS FT-IR — Brucker) coupled with an ATR (Attenuated Total Reflectance) accessory in order to characterize the proportion of synthetic and natural fibers and identify the predominant plastic polymers. The fibers were categorized according to the classification proposed by the international organization for standardization (ISO/TR 11827:2012 Textiles — Composition testing — Identification of fibers).  \n\n# 3. Results and discussion  \n\nBased on a long term monitoring (one year), our results show large amounts of fibers in the atmospheric fallout, which has not yet been reported in the literature. Throughout the year of monitoring (site 1), the atmospheric fallout ranged from 2 to 355 particles $/\\mathrm{m}^{2}.$ /day (Fig. 2) with an average atmospheric fallout of $110\\pm96$ particles $/\\mathrm{m}^{2}.$ /day (mean $\\pm$ SD), indicating a high annual variability. On Site 2 (6-month monitoring), the atmospheric fallout was around $53\\pm38$ particles $\\langle{\\bf m}^{2}$ /day $(\\mathrm{mean}\\pm\\mathrm{SD})$ ).  \n\nFig. 3 illustrates the proportion of fibers belonging to each size range considered for 11 samples. It can be seen that the smallest fibers (in the $200{-}400\\upmu\\mathrm{m}$ and $400{-}600\\upmu\\mathrm{m}$ size ranges) are predominant while fibers in the larger size ranges are rare. Few fibers have been found in the $50\\upmu\\mathrm{m}-200\\upmu\\mathrm{m}$ size range. Fibers smaller than $<50\\upmu\\mathrm{m}$ are also observed with the stereomicroscope, but since their nature can be hardly identified, they were not taken into account. The length of the fibers was measured during counting. The diameter of the fibers varies mainly between 7 and $15\\upmu\\mathrm{m}$ .  \n\nRainfall seems to be an important factor influencing the fallout flux. In fact, particularly low numbers of fibers were found during dry weather or low cumulated rainfall periods. For rainfall between 0 and $0.2\\ \\mathrm{mm/day}$ , an atmospheric fallout between 2 and 34 particles $/\\mathrm{m}^{2}$ /day was recorded. During rainy periods (from 2 to $5\\mathrm{mm/day},$ ), highly variable levels of atmospheric fallout were encountered ranging from 11 to 355 particles/ ${\\mathrm{m}}^{2}$ /day. No significant correlation between the levels of fibers in atmospheric fallout and the mean daily rainfall was highlighted. The same conclusion is obtained by considering the cumulative rainfall and the total amount of fibers collected. This indicates that if the absence of rain limits the microplastics atmospheric fallout, rainfall height is a significant factor, though not the only one, contributing to fallout variability; other mechanisms and temporal conditions also contribute to the fallout flux, but remain to be identified.  \n\nWhen the levels on both sites are compared side by side, it can be seen that through all the monitoring periods, the sub-urban site systematically showed fewer fibers then the urban one (Fig. 4). Statistical tests showed a significant difference between the atmospheric fallout on the urban and the sub-urban site (Wilcoxon matched pairs test, confidence level of $5\\%$ , $\\mathsf{p}_{\\mathrm{value}}=0.007$ ). One of the reasons that could explain this difference is the density of the surrounding population, which is considered as a proxy of the local activity. This is a pilot study as it contains only two sampling locations leading to a limited statistical power.  \n\nFourier Transform infrared (FT-IR) spectroscopy showed that half of the analyzed fibers are natural fibers $(50\\%)$ being mainly cotton or wool. The remaining fibers are man-made. For $21\\%$ of the total fibers, they are manufactured by transformation of natural polymers (rayon or acetate from cellulose). In other hand, $17\\%$ of the fibers correspond to purely synthetic fibers, mainly polyethylene-terephthalate and only one fiber of polyamide. The other $12\\%$ fibers are made with a mixture of different materials including purely synthetic materials fibers (mixture of polyethylene-terephthalate and polyurethane) and fibers being a mixture of natural and synthetic materials (cotton and polyamide). Therefore, petrochemicals are found in $29\\%$ of the analyzed fibers in atmospheric fallout.  \n\n![](images/0e165f22b2980d94b56f2832c740c604cee4a31494d7937f2b7996aacaa35f70.jpg)  \nFig. 3. Size distribution of synthetic fibers in 11 atmospheric fallout samples (site 1). Boxplots are plotted from bottom to top with: [Lower quartile $^{-1.5*}$ Interquartile range], [Lower quartile], [Median], [Upper quartile], and [Upper quartile $^{+1.5*}$ Interquartile range]. Outliers are displayed as isolated points.  \n\nGiven the lack of studies and data addressing the issue of plastic sources and fluxes in an urban area, it is hard to evaluate the importance of the atmospheric fallout as a source of microplastics. According to the average atmospheric flux of total fibers on each site (110 and 53 parti${\\mathrm{cles}}/{\\mathrm{m}}^{2}/{\\mathrm{day}})$ , the length of the fibers (Fig. 3), their approximated section $(80\\upmu\\mathrm{m}^{2}.$ ), and the mass of fibers in the atmospheric fallout per year may be assessed. Two densities were considered: 1 for the polyamide and $1.45~\\mathrm{g/cm^{3}}$ for the polyethylene-terephthalate, corresponding to two low and heavy plastic polymers widely used in the textile industry (Hidalgo-Ruz et al., 2012). At the scale of the Parisian agglomeration (area around $2500\\mathrm{km}^{2}.$ ), it was estimated that between 3 and 10 tons of synthetic fibers, including microplastics, could originate annually from the atmosphere.  \n\n# 4. Conclusions  \n\nThese results show a significant amount of fibers in atmospheric fallout, which leads to the hypothesis that the atmospheric compartment  \n\n![](images/25f0ad2d6aca7972a2664360e62e6b7f914ef914dd9e887480b1c96bbbc826d6.jpg)  \nFig. 2. Atmospheric fallout of microplastics on the site 1 in parallel with daily rainfall.  \n\nPlease cite this article as: Dris, R., et al., Synthetic fibers in atmospheric fallout: A source of microplastics in the environment?, Marine Pollution Bulletin (2016), http://dx.doi.org/10.1016/j.marpolbul.2016.01.006  \n\n![](images/6623a747523361b0d55e69b8e05ec369814b0c440e6fe2bb9b2f5216561eeaf3.jpg)  \nFig. 4. Atmospheric fallout of microplastics on an urban site (site 1) and a sub-urban site (site 2).  \n\nshould not be neglected as a potential source of microplastics, specially knowing that we estimated at $29\\%$ the amount of these fibers containing at least partially plastic polymers. These microplastics have different possible sources: synthetic fibers from clothes and houses, degradation of macroplastics, and landfills or waste incineration. The characterization indicates that the hypothesis of the clothing being the main source of these fibers is the more plausible. These fibers in the atmosphere, including microplastics, could be transported by wind to the aquatic environment (Free et al., 2014) or deposited on surfaces of cities or agrosystems. After deposition, they could impact terrestrial organisms or be transported into the aquatic systems through the runoff. More work is needed in order to investigate these atmospheric fibers and understands where they come from, where they end up and which mechanisms and factors lead to their transport and their fallout.  \n\nThe micro IR-TF spectroscopy analysis shows the presence of microplastic fibers in the atmospheric fallout. Nevertheless, some questions arise regarding the definition of microplastics in relation with their nature and the ISO classification. Only fibers made of petrochemicals are generally considered in the literature as microplastics (Song et al., 2015). Authors think that fibers made of a mixture of natural and synthetic materials should be also included in the identification of microplastics. Further discussions are also needed to identify if artificial fibers, which are manufactured by transformation of natural polymers, could be included in microplastics. In fact, these fibers might also be prevalent in marine and continental environments and could cause physical impacts on organisms. Moreover, harmful additives and dyes can be used when manufacturing these fibers.  \n\nFurthermore, this study could serve the implementation of the marine strategy framework directive (MSFD) (Galgani et al., 2013) aiming to achieve by 2020 the good environmental status for European marine waters. The indicator 10 of the MSFD related to marine liter with a focus on microplastics indicates that the solution for marine plastic pollution is to tackle the problem at its source. This work does not aim to provide accurate atmospheric fluxes for microplastics as a lot of uncertainties remain and more studies are needed, but serves to highlight the important role that this source could play in continental and marine environments and encourages scientists to take it into consideration in future works.  \n\n# Acknowledgments  \n\nWe address sincere thanks to the members of the LISA (Laboratoire Interuniversitaire des Systèmes Atmosphériques), especially Anne Chabas. The PhD of Rachid Dris is funded by the region Île-de-France Research Network on Sustainable Development (2013-02) (R2DS Ile-deFrance). We thank Kelsey Flanagan for lingual improvements on the manuscript. We thank the members of the ICMPE (Institut de Chimie et des Materiaux Paris-Est) especially Mohamed Guerrouache and Valérie Langlois. We also thank Ludovic Lemee and Gregory Candor from Brucker Optics.  \n\n# References  \n\nAnastasopoulou, A., Mytilineou, C., Smith, C.J., Papadopoulou, K.N., 2013. Plastic debris ingested by deep-water fish of the Ionian Sea (Eastern Mediterranean). Deep Sea Res. Part Oceanogr. Res. Pap. 74, 11–13. http://dx.doi.org/10.1016/j.dsr.2012.12.008.   \nAndrady, A.L., 2011. Microplastics in the marine environment. Mar. Pollut. Bull. 62, 1596–1605. http://dx.doi.org/10.1016/j.marpolbul.2011.05.030.   \nArthur, C., Baker, J., Bamford, H., 2008. Proceedings of the International Research. Presented at the Worshop on the Occurence, Effects and Fate of Microplastic Marine Debris. Sept 9–11 2008. NOAA Technical Memorandum NOS-OR&R-30.   \nBrowne, M.A., Crump, P., Niven, S.J., Teuten, E., Tonkin, A., Galloway, T., Thompson, R., 2011. Accumulation of microplastic on shorelines woldwide: sources and sinks. Environ. Sci. Technol. 45, 9175–9179. http://dx.doi.org/10.1021/es201811s.   \nCole, M., Lindeque, P., Halsband, C., Galloway, T.S., 2011. Microplastics as contaminants in the marine environment: a review. Mar. Pollut. Bull. 62, 2588–2597. http://dx.doi. org/10.1016/j.marpolbul.2011.09.025.   \nDris, R., Gasperi, J., Rocher, V., Saad, M., Renault, N., Tassin, B., 2015a. Microplastic contamination in an urban area: a case study in Greater Paris. Environ. Chem. 12, 592–599.   \nDris, R., Imhof, H., Sanchez, W., Gasperi, J., Galgani, F., Tassin, B., Laforsch, C., 2015b. Beyond the ocean: contamination of freshwater ecosystems with (micro-) plastic particles. Environ. Chem. 12, 539–550.   \nFree, C.M., Jensen, O.P., Mason, S.A., Eriksen, M., Williamson, N.J., Boldgiv, B., 2014. Highlevels of microplastic pollution in a large, remote, mountain lake. Mar. Pollut. Bull. 85, 156–163. http://dx.doi.org/10.1016/j.marpolbul.2014.06.001.   \nGalgani, F., Hanke, G., Werner, S., De Vrees, L., 2013. Marine litter within the European marine strategy framework directive. ICES J. Mar. Sci. 70, 1055–1064. http://dx.doi. org/10.1093/icesjms/fst122.   \nHabib, D., Locke, D.C., Cannone, L.J., 1998. Synthetic fibers as indicators of municipal sewage sludge, sludge products, and sewage treatment plant effluents. Water Air Soil Pollut. 103, 1–8.   \nHidalgo-Ruz, V., Gutow, L., Thompson, R.C., Thiel, M., 2012. Microplastics in the marine environment: a review of the methods used for identification and quantification. Environ. Sci. Technol. 46, 3060–3075. http://dx.doi.org/10.1021/es2031505.   \nInsee — Régions, départements et villes de France, 2011. [WWW Document]. URL http:// www.insee.fr/fr/themes/theme.asp?theme $\\mathrel{\\mathop:}=\\dot{}$ 1&sous_theme $^{\\cdot=2}$ (accessed 5.22.15).   \nLusher, A.L., McHugh, M., Thompson, R.C., 2013. Occurrence of microplastics in the gastrointestinal tract of pelagic and demersal fish from the English channel. Mar. Pollut. Bull. 67, 94–99. http://dx.doi.org/10.1016/j.marpolbul.2012.11.028.   \nNorén, F., 2007. Small plastic particles in Coastal Swedish waters — KIMO reports. http:// www.kimointernational.org/WebData/Files/Small%20plastic%20particles%20in% 20Swedish%20West%20Coast%20Waters.pdf.   \nSanchez, W., Bender, C., Porcher, J.-M., 2014. Wild gudgeons (Gobio gobio) from French rivers are contaminated by microplastics: preliminary study and first evidence. Environ. Res. 128, 98–100. http://dx.doi.org/10.1016/j.envres.2013.11.004.   \nSong, Y.K., Hong, S.H., Jang, M., Han, G.M., Rani, M., Lee, J., Shim, W.J., 2015. A comparison of microscopic and spectroscopic identification methods for analysis of microplastics in environmental samples. Mar. Pollut. Bull. http://dx.doi.org/10.1016/j.marpolbul. 2015.01.015.   \nThompson, R.C., Olsen, Y., Mitchell, R.P., Davis, A., Rowland, S.J., John, A.W., McGonigle, D., Russell, A.E., 2004. Lost at sea: where is all the plastic? Science 304, 838-838. http:// dx.doi.org/10.1126/science.1094559.   \nWagner, M., Scherer, C., Alvarez-Muñoz, D., Brennholt, N., Bourrain, X., Buchinger, S., Fries, E., Grosbois, C., Klasmeier, J., Marti, T., Rodriguez-Mozaz, S., Urbatzka, R., Vethaak, A.D., Winther-Nielsen, M., Reifferscheid, G., 2014. Microplastics in freshwater ecosystems: what we know and what we need to know. Environ. Sci. Eur. 26, 1–9. http://dx. doi.org/10.1186/s12302-014-0012-7.   \nWright, S.L., Thompson, R.C., Galloway, T.S., 2013. The physical impacts of microplastics on marine organisms: a review. Environ. Pollut. 178, 483–492. http://dx.doi.org/10. 1016/j.envpol.2013.02.031.   \nZubris, K.A.V., Richards, B.K., 2005. Synthetic fibers as an indicator of land application of sludge. Environ. Pollut. 138, 201–211. http://dx.doi.org/10.1016/j.envpol.2005.04.013.  ",
    "Target": null
  },
  {
    "id": "10.1038_npjcompumats.2016.28",
    "DOI": "10.1038/npjcompumats.2016.28",
    "DOI Link": "http://dx.doi.org/10.1038/npjcompumats.2016.28",
    "Relative Dir Path": "mds/10.1038_npjcompumats.2016.28",
    "Article Title": "A general-purpose machine learning framework for predicting properties of inorganic materials",
    "Authors": "Ward, L; Agrawal, A; Choudhary, A; Wolverton, C",
    "Source Title": "NPJ COMPUTATIONAL MATERIALS",
    "Abstract": "A very active area of materials research is to devise methods that use machine learning to automatically extract predictive models from existing materials data. While prior examples have demonstrated successful models for some applications, many more applications exist where machine learning can make a strong impact. To enable faster development of machine-learning-based models for such applications, we have created a framework capable of being applied to a broad range of materials data. Our method works by using a chemically diverse list of attributes, which we demonstrate are suitable for describing a wide variety of properties, and a novel method for partitioning the data set into groups of similar materials to boost the predictive accuracy. In this manuscript, we demonstrate how this new method can be used to predict diverse properties of crystalline and amorphous materials, such as band gap energy and glass-forming ability.",
    "Times Cited, WoS Core": 1121,
    "Times Cited, All Databases": 1266,
    "Publication Year": 2016,
    "Research Areas": "Chemistry; Materials Science",
    "UT (Unique WOS ID)": "WOS:000426821500024",
    "Markdown": "# ARTICLE OPEN A general-purpose machine learning framework for predicting properties of inorganic materials  \n\nLogan Ward1, Ankit Agrawal2, Alok Choudhary2 and Christopher Wolverton1  \n\nA very active area of materials research is to devise methods that use machine learning to automatically extract predictive models from existing materials data. While prior examples have demonstrated successful models for some applications, many more applications exist where machine learning can make a strong impact. To enable faster development of machine-learning-based models for such applications, we have created a framework capable of being applied to a broad range of materials data. Our method works by using a chemically diverse list of attributes, which we demonstrate are suitable for describing a wide variety of properties, and a novel method for partitioning the data set into groups of similar materials to boost the predictive accuracy. In this manuscript, we demonstrate how this new method can be used to predict diverse properties of crystalline and amorphous materials, such as band gap energy and glass-forming ability.  \n\nnpj Computational Materials (2016) 2, 16028; doi:10.1038/npjcompumats.2016.28; published online 26 August 2016  \n\n# INTRODUCTION  \n\nRational design of materials is the ultimate goal of modern materials science and engineering. As part of achieving that goal, there has been a large effort in the materials science community to compile extensive data sets of materials properties to provide scientists and engineers with ready access to the properties of known materials. Today, there are databases of crystal structures, superconducting critical temperatures (http://supercon.nims.go. jp/), physical properties of crystalline compounds2–5 and many other repositories containing useful materials data. Recently, it has been shown that these databases can also serve as resources for creating predictive models and design rules—the key tools of rational materials design.6–12 These databases have grown large enough that the discovery of such design rules and models is impractical to accomplish by relying simply on human intuition and knowledge about material behaviour. Rather than relying directly on intuition, machine learning offers the promise of being able to create accurate models quickly and automatically.  \n\nTo date, materials scientists have used machine learning to build predictive models for a handful of applications.13–27 For example, there are now models to predict the melting temperatures of binary inorganic compounds,21 the formation enthalpy crystalline compounds,14,15,28 which crystal structure is likely to form at a certain composition,5,16,29–31 band gap energies of certain classes of crystals32,33 and the mechanical properties of metal alloys.24,25 While these models demonstrate the promise of machine learning, they only cover a small fraction of the properties used in materials design and the data sets available for creating such models. For instance, no broadly-applicable, machine-learning-based models exist for band gap energy or glass-forming ability even though large-scale databases of these properties have existed for years.2,34  \n\nProvided the large differences between the approaches used in the literature, a systematic path forward to creating accurate machine learning models across a variety of new applications is not clear. While techniques in data analytics have advanced significantly, the development of routine methods for transforming raw materials data into the quantitative descriptions required for employing these algorithms is yet to emerge. In contrast, the chemoinformatics community benefits from a rich library of methods for describing molecular structures, which allow for standard approaches for deciding inputs into the models and, thereby, faster model development.35–37 What is missing are similar flexible frameworks for building predictive models of material properties.  \n\nIn this work, we present a general-purpose machine-learningbased framework for predicting the properties of materials based on their composition. In particular, we focus on the development of a set of attributes—which serve as an input to the machine learning model—that could be reused for a broad variety of materials problems. Provided a flexible set of inputs, creating a new material property model can be reduced to finding a machine learning algorithm that achieves optimal performance—a wellstudied problem in data science. In addition, we employ a novel partitioning scheme to enhance the accuracy of our predictions by first partitioning data into similar groups of materials and training separate models for each group. We show that this method can be used regardless of whether the materials are amorphous or crystalline, the data are from computational or experimental studies, or the property takes continuous or discrete values. In particular, we demonstrate the versatility of our technique by using it for two distinct applications: predicting novel solar cell materials using a database of density functional theory (DFT)- predicted properties of crystalline compounds and using experimental measurements of glass-forming ability to suggest new metallic glass alloys. Our vision is that this framework could be used as a basis for quickly creating models based on the data available in the materials databases and, thereby, initiate a major step forward in rational materials design.  \n\n# RESULTS AND DISCUSSION  \n\nThe results of this study are described in two major subsections. First, we will discuss the development of our method and the characterisation of the attribute set using data from the Open Quantum Materials Database (OQMD). Next, we will demonstrate the application of this method to two distinct material problems.  \n\nGeneral-purpose method to create materials property models Machine learning (ML) models for materials properties are constructed from three parts: training data, a set of attributes that describe each material, and a machine learning algorithm to map attributes to properties. For the purposes of creating a general-purpose method, we focused entirely on the attributes set because the method needs to be agnostic to the type of training data and because it is possible to utilise already-developed machine learning algorithms. Specifically, our objective is to develop a general set of attributes based on the composition that can be reused for a broad variety of problems.  \n\nThe goal in designing a set of attributes is to create a quantitative representation that both uniquely defines each material in a data set and relates to the essential physics and chemistry that influence the property of interest.14,17 As an example, the volume of a crystalline compound is expected to relate to the volume of the constituent elements. By including the mean volume of the constituent elements as an attribute, a machine learning algorithm could recognise the correlation between this value and the compound volume, and use it to create a predictive model. However, the mean volume of the constituent elements neither uniquely defines a composition nor perfectly describes the volumes of crystalline materials.38 Consequently, one must include additional attributes to create a suitable set for this problem. Potentially, one could include factors derived from the electronegativity of the compound to reflect the idea that bond distances are shorter in ionic compounds, or the variance in atomic radius to capture the effects of polydisperse packing. The power of machine learning is that it is not necessary to know which factors actually relate to the property and how before creating a model—those relationships are discovered automatically.  \n\nThe materials informatics literature is full of successful examples of attribute sets for a variety of properties.13–16,21,32,39 We observed that the majority of attribute sets were primarily based on statistics of the properties of constituent elements. As an example, Meredig et al.15 described a material based on the fraction of each element present and various intuitive factors, such as the maximum difference in electronegativity, when building models for the formation energy of ternary compounds. Ghiringhelli et al.14 used combinations of elemental properties such as atomic number and ionisation potential to study the differences in energy between zinc-blende and rocksalt phases. We also noticed that the important attributes varied significantly depending on material property. The best attribute for describing the difference in energy between zinc-blende and rocksalt phases was found to be related to the pseudopotential radii, ionisation potential and electron affinity of the constituent elements.14 In contrast, melting temperature was found to be related to atomic number, atomic mass and differences between atomic radii.21 From this, we conclude that a general-purpose attribute set should contain the statistics of a wide variety of elemental properties to be adaptable.  \n\nBuilding on existing strategies, we created an expansive set of attributes that can be used for materials with any number of constituent elements. As we will demonstrate, this set is broad enough to capture a sufficiently-diverse range of physical/ chemical properties to be used to create accurate models for many materials problems. In total, we use a set of 145 attributes, which are described in detail and compared against other attribute sets in the Supplementary Information, that fall into four distinct categories:  \n\n1. Stoichiometric attributes that depend only on the fractions of elements present and not what those elements actually are. These include the number of elements present in the compound and several $L^{p}$ norms of the fractions.   \n2. Elemental property statistics, which are defined as the mean, mean absolute deviation, range, minimum, maximum and mode of 22 different elemental properties. This category includes attributes such as the maximum row on periodic table, average atomic number and the range of atomic radii between all elements present in the material.   \n3. Electronic structure attributes, which are the average fraction of electrons from the $s,p,d$ and $f$ valence shells between all present elements. These are identical to the attributes used by Meredig et $a I$ .15   \n4. Ionic compound attributes that include whether it is possible to form an ionic compound assuming all elements are present in a single oxidation state, and two adaptations of the fractional ‘ionic character’ of a compound based on an electronegativitybased measure.40  \n\nFor the third ingredient, the machine learning algorithm, we evaluate many possible methods for each individual problem. Previous studies have used machine learning algorithms including partial least-squares regression,13,29 Least Absolute Shrinkage and Selection Operator (LASSO),14,33,41 decision trees,15,16 kernel ridge regression,17–19,42 Gaussian process regression19–21,43 and neural networks.22–24 Each method offers different advantages, such as speed or interpretability, which must be weighed carefully for a new application. We generally approach this problem by evaluating the performance of several algorithms to find one that has both reasonable computational requirements (i.e., can be run on available hardware in a few hours) and has low error rates in cross-validation—a process that is simplified by the availability of well-documented libraries of machine learning algorithms.44,45 We often find that ensembles of decision trees (e.g., rotation forests46) perform best with our attribute set. These algorithms also have the advantage of being quick to train, but are not easily interpretable by humans. Consequently, they are less suited for understanding the underlying mechanism behind a material property but, owing to their high predictive accuracy, excellent choices for the design of new materials.  \n\nWe also utilise a partitioning strategy that enables a significant increase in predictive accuracy for our ML models. By grouping the data set into chemically-similar segments and training a separate model on each subset, we boost the accuracy of our predictions by reducing the breadth of physical effects that each machine learning algorithm needs to capture. For example, the physical effects underlying the stability intermetallic compounds are likely to be different than those for ceramics. In this case, one could partition the data into compounds that contain only metallic elements and another including those that do not. As we demonstrate in the examples below, partitioning the data set can significantly increase the accuracy of predicted properties. Beyond using our knowledge about the physics behind a certain problem to select a partitioning strategy, we have also explored using an automated, unsupervised-learning-based strategy for determining distinct clusters of materials.47 Currently, we simply determine the partitioning strategy for each property model by searching through a large number of possible strategies and selecting the one that minimises the error rate in crossvalidation tests.  \n\n# Justification for large attribute set  \n\nThe main goal of our technique is to accelerate the creation of machine learning models by reducing or eliminating the need to develop a set of attributes for a particular problem. Our approach was to create a large attribute set, with the idea that it would contain a diverse enough library of descriptive factors such that it is likely to contain several that are well-suited for a new problem. To justify this approach, we evaluated changes in the performance of attributes for different properties and types of materials using data from the OQMD. As described in greater detail in the next section, the OQMD contains the DFT-predicted formation energy, band gap energy and volume of hundreds of thousands of crystalline compounds. The diversity and scale of the data in the OQMD make it ideal for studying changes in attribute performance using a single, uniform data set.  \n\nWe found that the attributes which model a material property best can vary significantly depending on the property and type of materials in the data set. To quantify the predictive ability of each attribute, we fit a quadratic polynomial using the attribute and measured the root mean squared error of the model. We found the attributes that best describe the formation energy of crystalline compounds are based on the electronegativity of the constituent elements (e.g., maximum and mode electronegativity). In contrast, the best-performing attributes for band gap energy are the fraction of electrons in the $p$ shell and the mean row in the periodic table of the constituent elements. In addition, the attributes that best describe the formation energy vary depending on the type of compounds. The formation energy of intermetallic compounds is best described by the variances in the melting temperature and number of $d$ electrons between constituent elements, whereas compounds that contain at least one nonmetal are best modelled by the mean ionic character (a quantity based on electronegativity difference between constituent elements). Taken together, the changes in which attributes are the most important between these examples further support the necessity of having a large variety of attributes available in a generalpurpose attribute set.  \n\nIt is worth noting that the 145 attributes described in this paper should not be considered the complete set for inorganic materials. The chemical informatics community has developed thousands of attributes for predicting the properties of molecules.35 What we present here is a step towards creating such a rich library of attributes for inorganic materials. While we do show in the examples considered in this work that this set of attributes is sufficient to create accurate models for two distinct properties, we expect that further research in materials informatics will add to the set presented here and be used to create models with even greater accuracy. Furthermore, many materials cannot be described simply by average composition. In these cases, we propose that the attribute set presented here can be extended with representations designed to capture additional features such as structure (e.g., Coulomb Matrix17 for atomic-scale structure) or processing history. We envision that it will be possible to construct a library of general-purpose representations designed to capture structure and other characteristics of a material, which would drastically simplify the development of new machine learning models.  \n\n# Example applications  \n\nIn the following sections, we detail two distinct applications for our novel material property prediction technique to demonstrate its versatility: predicting three physically distinct properties of crystalline compounds and identifying potential metallic glass alloys. In both cases, we use the same general framework, i.e., the same attributes and partitioning-based approach. In each case, we only needed to identify the most accurate machine learning algorithm and find an appropriate partitioning strategy. Through these examples, we discuss all aspects of creating machinelearning-based models to design a new material: assembling a training set to train the models, selecting a suitable algorithm, evaluating model accuracy and employing the model to predict new materials.  \n\nAccurate models for properties of crystalline compounds DFT is a ubiquitous tool for predicting the properties of crystalline compounds, but is fundamentally limited by the amount of computational time that DFT calculations require. In the past decade, DFT has been used to generate several databases containing the ${\\cal T}=0\\mathsf{K}$ energies and electronic properties of $\\sim10^{5}$ crystalline compounds, $\\lambda^{2-5,4\\bar{8}}$ which each required millions of hours of CPU time to construct. While these databases are indisputablyuseful tools, as evidenced by the many materials they have been used to design,3,49–54 machine-learning-based methods offer the promise of predictions at several orders of magnitude faster rates. In this example, we explore the use of data from the DFT calculation databases as training data for machine learning models that can be used rapidly to assess many more materials than what would be feasible to evaluate using DFT.  \n\nTraining data. We used data from the OQMD, which contains the properties of \\~ 300,000 crystalline compounds as calculated using DFT.2,3 We selected a subset of 228,676 compounds from OQMD that represents the lowest-energy compound at each unique composition to use as a training set. As a demonstration of the utility of our method, we developed models to predict the three physically distinct properties currently available through the OQMD: band gap energy, specific volume and formation energy.  \n\nMethod. To select an appropriate machine learning algorithm for this example, we evaluated the predictive ability of several algorithms using 10-fold cross-validation. This technique randomly splits the data set into 10 parts, and then trains a model on 9 partitions and attempts to predict the properties of the remaining set. This process is repeated using each of the 10 partitions as the test set, and the predictive ability of the model is assessed as the average performance of the model across all repetitions. As shown in Table 1, we found that creating an ensemble of reduced-error pruning decision trees using the random subspace technique had the lowest mean absolute error in cross-validation for these properties among the 10 ML algorithms we tested (of which, only 4 are listed for clarity).55 Models produced using this machine learning algorithm had the lowest mean absolute error in crossvalidation, and had excellent correlation coefficients of above 0.91 between the measured and predicted values for all three properties.  \n\nAs a simple test for how well our band gap model can be used for discovering new materials, we simulated a search for compounds with a band gap within a desired range. To evaluate the ability of our method to locate compounds that have band gap energies within the target range, we devised a test where a model was trained on $90\\%$ of the data set and then was tasked with selecting which 30 compounds in the remaining $10\\%$ were most likely to have a band gap energy in the desired range for solar cells: $0.9\\mathrm{-}1.7\\mathrm{eV.}^{56}$ For this test, we selected a subset of the OQMD that only includes compounds that were reported to be possible to be made experimentally in the ICSD (a total of 25,085 entries) so that only band gap energy, and not stability, needed to be considered.  \n\nFor this test, we compared three selection strategies for finding compounds with desirable band gap energies: randomly selecting nonmetal-containing compounds (i.e., without machine learning), using a single model trained on the entire training set to guide selection, and a model created using the partitioning approach introduced in this manuscript. As shown in Figure 1, randomly selecting a nonmetal-containing compound would result in just over $12\\%$ of the 30 selected compounds to be within the desired range of band gap energies. Using a single model trained on the entire data set, this figure dramatically improves to $\\sim46\\%$ of selected compounds having the desired property. We found the predictive ability of our model can be increased to $\\sim67\\%$ of predictions actually having the desired band gap energy by partitioning the data set into groups of similar compounds before training. Out of the 20 partitioning strategies we tested, we found the best composite model works by first partitioning the data set using a separate model trained to predict the expected range, but not the actual value, of the band gap energy (e.g., compounds predicted to have a band gap between 0 and $1.5\\mathsf{e V}$ are grouped together), and then on whether a compound contains a halogen, chalcogen or pnictogen. Complete details of the hierarchical model are available in the Supplementary Information. By partitioning the data into smaller subsets, each of the individual machine learning models only evaluates compounds with similar chemistries (e.g., halogen-containing compounds with a band gap expected to be between 0 and $1.5\\mathsf{e V}$ ), which we found enhances the overall accuracy of our model.  \n\n<html><body><table><tr><td rowspan=\"2\">Table 1.</td><td colspan=\"3\">Comparison of the ability ofseveral machine learning algorithms to predict propertiesof materials from the OQMD</td></tr><tr><td colspan=\"3\">Machine learning algorithm</td></tr><tr><td></td><td>Linear regresson Reduced-error pruning tree (REPTree) Rotation forest46+REPTreeRandom subspace5+REPTree</td><td></td><td></td></tr><tr><td colspan=\"2\">Property Volume (A² per atom) 1.22</td><td>0.816</td><td>0.563</td></tr><tr><td></td><td></td><td>0.126</td><td></td></tr><tr><td>Formation energy (eV per atom)</td><td></td><td>0.0701</td><td></td></tr><tr><td>Band gap energy (eV)</td><td></td><td></td><td></td></tr><tr><td colspan=\"4\">Abbreviations: DFT, density functional theory; OQMD, Open Quantum Materials Database.</td></tr><tr><td colspan=\"2\">compounds.</td><td>Datafffl</td><td></td></tr></table></body></html>  \n\n![](images/97d7c832015eaeeff476becfb9882bf7196d5e2d62f9036622bf747f0a0e3467.jpg)  \nFigure 1. Performance of three different strategies to locate compounds with a band gap energy within a desired range: randomly selecting nonmetal-containing compounds, and two strategies using the machine-learning-based method presented in this work. The first machine learning strategy used a single model trained on the computed band gap energies of 22,667 compounds from the ICSD. The second method a model created by first partitioning the data into groups of similar materials, and training a separate model on each subset. The number of materials that were actually found to have a band gap within the desired range after 30 guesses was over 5 times larger when using our machine learning approach than when randomly selecting compounds. Error bars represent the $95\\%$ confidence interval.  \n\n<html><body><table><tr><td>Table 2. Compositions and predicted band gap energies of materials predicted using machine learning to be candidates for solar cell applications</td></tr><tr><td>Composition Eg (eV)</td></tr><tr><td>ScHg4Cl7 1.26 VHgCl7 1.16 Mn6CCl8 1.28 Hf4S11Cl 1.11 VCu5Clg 1.19</td></tr><tr><td>Abbreviations: DFT, density functional theory; OQMD, open quantum materialsdatabase. Compositions represent the nominal compositions of novel ternary compounds predicted by using methods developed in ref. 15. Band gap energies were predicted using a machine learning model trained on DFT band gap energies from the OQMD² using methods described in this work.</td></tr></table></body></html>  \n\nOnce we established the reliability of our model, we used it to search for new compounds (i.e., those not yet in the OQMD) with a band gap energy within the desired range for solar cells: 0.9–1.7 eV. To gain the greatest predictive accuracy, we trained our band gap model on the entire OQMD data set. Then, we used this model to predict the band gap energy of compositions that were predicted by Meredig et $\\mathsf{\\bar{\\Pi}}_{a l.}\\mathsf{\\Pi}^{15}$ to be as-yet-undiscovered ternary compounds. Out of this list of 4,500 predicted compounds, we found that 223 are likely to have favourable band gap energies. A subset with the best stability criterion (as reported in ref. 15) and band gap energy closest to $1.3\\mathsf{e V}$ are shown in Table 2. As demonstrated in this example and a recent work by Sparks et al.,57 having access to several machine learning models for different properties can make it possible to rapidly screen materials based on many design criteria. Provided the wide range of applicability of the machine learning technique demonstrated in this work and the growing availability of material property data, it may soon be possible to screen for materials based on even more properties than those considered here using models constructed based on several different data sets.  \n\n# Locating novel metallic glass alloys  \n\nMetallic glasses possess a wide range of unique properties, such as high-wear resistance and soft magnetic behaviour, but are only possible to create at special compositions that are difficult to determine a priori.58 The metallic glass community commonly relies on empirical rules (e.g., systems that contain many elements of different sizes are more likely to form glasses59) and extensive experimentation to locate these special compositions.55 While searches based on empirical rules have certainly been successful (as evidenced by the large variety of known alloys,60) this conventional method is known to be slow and resourceintensive.61 Here, we show how machine learning could be used to accelerate the discovery of new alloys by using known experimental data sets to construct predictive models of glassforming ability.  \n\n![](images/f67bd6b0deb81b5074ebb44ec7c2568baf39fb4f0d3e11ef15f63890ca2bd8a8.jpg)  \nFigure 2. (a) Experimental measurements of metallic glass-forming ability in the Al–Ni–Zr ternary, as reported in ref. 34. Green circles (AM) mark compositions at which it is possible to create a fully-amorphous ribbon via melt spinning, blue squares (AC) mark compositions at which only a partially-amorphous ribbon can be formed, and red crosses (CR) mark compositions where it is not possible to form any appreciable amount of amorphous phase. (b) Predicted glass-forming ability from our machine learning model. Points are coloured based on relative likelihood of glass formation, where 1 is the mostly likely and 0 is the least. The model used to make these predictions was developed using the methods outlined in this work, and was not trained on any measurements from the Al–Ni–Zr ternary or any of its constituent binaries.  \n\nData. We used experimental measurements taken from ‘Nonequilibrium Phase Diagrams of Ternary Amorphous Alloys,’ a volume of the Landolt–Börnstein collection.32 This data set contains measurements of whether it is possible to form a glass using a variety of experimental techniques at thousands of compositions from hundreds of ternary phase diagrams. For our purposes, we selected 5,369 unique compositions where the ability to form an amorphous ribbon was assessed using melt spinning. In the event that multiple measurements for glassforming ability were taken at a single composition, we assume that it is possible to form a metallic glass if at least one measurement found it was possible to form a completely amorphous sample. After the described screening steps, $70.8\\%$ of the entries in the training data set correspond to metallic glasses.  \n\nMethod. We used the same set of 145 attributes as in the band gap example and ensembles of Random Forest classifiers62 created using the random subspace technique as the machine learning algorithm, which we found to be the most accurate algorithm for this problem. This model classifies the data into two categories (i.e., can and cannot form a metallic glass) and computes the relative likelihood that a new entry would be part of each category. For the purposes of validating the model, we assume any composition predicted to have a $>50\\%$ probability of glass formation to be a positive prediction of glass-forming ability. Using a single model trained on the entire data set, we were able to create a model with $90\\%$ accuracy in 10-fold cross-validation.  \n\nAs a test of the ability of our method to predict new alloys, we removed all entries that contained exclusively Al, Ni and Zr (i.e., all Al–Ni–Zr ternary compounds, and any binary formed by any two of those elements) from our training data set and then predicted the probability of an alloy being able to be formed into the amorphous state for the Al–Ni–Zr ternary system. As shown in Figure 2a, it is possible to form amorphous ribbons with melt spinning in one region along the Ni–Zr binary and in a second, Al-rich ternary region. Our model is able to accurately predict both the existence of these regions and their relative locations (see Figure 2b), which shows that models created using our method could serve to accurately locate favourable compositions in yet-unassessed alloy systems.  \n\nWe further validated the ability of our models to extrapolate to alloy systems not included in the training set by iteratively using each binary system as a test set. This procedure works by excluding all alloys that contain both of the elements in the binary, training a model on the remaining entries and then predicting the glass-forming ability of the alloys that were removed. For example, if the Al–Ni binary were being used as a test set, then $\\mathsf{A l}_{50}\\mathsf{N i}_{50}$ and $\\mathsf{A l}_{50}\\mathsf{N i}_{25}\\mathsf{F e}_{25}$ would be removed but $\\mathsf{A l}_{50}\\mathsf{F e}_{50}$ and $\\mathsf{A l}_{50}\\mathsf{F e}_{25}\\mathsf{Z r}_{25}$ would not. This process is then repeated for all 380 unique binaries in the data set. We measured that our model has an $80.2\\%$ classification accuracy over 15,318 test entries where $71\\%$ of entries were measured to be glasses—in contrast to the $90.1\\%$ measured in 10-fold cross-validation with a similar fraction of glasses in the test set. We also found that by training separate models for alloys that contain only metallic elements and those that contain a nonmetal/metalloid it is possible to slightly increase the prediction accuracy to $80.7\\%$ —a much smaller gain than that observed in the band gap example $(23\\%)$ . Overall, this exclusion test strongly establishes that our model is able to predict the glass-forming ability in alloy systems that are completely unassessed.  \n\nTo search for new candidate metallic glasses, we used our model to predict the probability of glass formation for all possible ternary alloys created at $2\\ \\mathsf{a t}\\%$ spacing by any combination of elements found in the training set. Considering that the data set included 51 elements, this space includes $\\sim24$ million candidate alloys, which required $\\sim6\\mathsf{h}$ to evaluate on eight $2.2G H z$ processors. To remove known alloys from our prediction results, we first removed all entries where the $L_{1}$ distance between the composition vector (i.e., $\\langle x_{\\mathsf{H}},x_{\\mathsf{H e}},x_{\\mathsf{L i}},\\hdots\\rangle)$ of the alloy and any amorphous alloy in the training set was $<30\\mathsf{a t\\%}$ . We then found the alloys with the highest predicted probability of glass formation in each binary and ternary. Eight alloys with the highest probability of glass formation are shown in Table 3. One top candidate, $Z\\mathrm{r}_{0.38}\\mathsf{C o}_{0.24}\\mathsf{C u}_{0.38},$ is particularly promising considering the existence of Zr-lean Zr–Co and $Z r-C u$ binary alloys and $Z r{-}A l{-}$ ${\\mathsf{C o}}{\\mathsf{-C u}}$ bulk metallic glasses.63 To make the ability to find new metallic glasses openly available to the materials science community, we have included all of the software and data necessary to use this model in the Supplementary Information and created an interactive, web-based tool(http://oqmd.org/static/ analytics/glass_search.html).  \n\n<html><body><table><tr><td>Table 3. Compositions of candidate metallic glass alloys predicted using a machine learning model trained on experimental measurements of glass-forming ability</td></tr><tr><td>Alloycomposition Zr0.38CO0.24CU0.38 Hfo.7Sio.16Nio.14 V0.16Nio.64B0.2 Hfo.48Zr0.16Nio.36 Zr0.46Cr0.36Nio.18 Zro.48Fe0.46Nio.06 Zro.5Fe0.38W0.12 Smo.22Fe0.54Bo.24</td></tr><tr><td>These alloys were predicted to have the highest probability being able to be formed into an amorphous ribbon via melting spinning out of 24 million candidates.</td></tr></table></body></html>  \n\n# CONCLUSIONS  \n\nIn this work, we introduced a general-purpose machine learning framework for predicting the properties of a wide variety of materials and demonstrated its broad applicability via illustration of two distinct materials problems: discovering new potential crystalline compounds for photovoltaic applications and identifying candidate metallic glass alloys. Our method works by using machine learning to generate models that predict the properties of a material as a function of a wide variety of attributes designed to approximate chemical effects. The accuracy of our models is further enhanced by partitioning the data set into groups of similar materials. In this manuscript, we show that this technique is capable of creating accurate models for properties as different as the electronic properties of crystalline compounds and glass formability of metallic alloys. Creating new models with our strategy requires only finding which machine learning algorithm maximises accuracy and testing different partitioning strategies, which are processes that could be eventually automated.64 We envision that the versatility of this method will make it useful for a large range of problems, and help enable the quicker deployment and wider-scale use machine learning in the design of new materials.  \n\n# MATERIALS AND METHODS  \n\nAll machine learning models were created using the Weka machine learning library.44 The Materials Agnostic Platform for Informatics and Exploration (Magpie) was used to compute attributes, perform the validation experiments and run searches for new materials. Both Weka and Magpie are available under open-source licenses. The software, training data sets and input files used in this work are provided in the Supplementary Information associated with this manuscript.  \n\n# ACKNOWLEDGEMENTS  \n\nThis work was performed under the following financial assistance award 70NANB14H012 from U.S. Department of Commerce, National Institute of Standards and Technology as part of the Center for Hierarchical Materials Design (CHiMaD). In addition, AA and AC were supported in part by the following grants: DARPA SIMPLEX award N66001-15-C-4036; NSF awards IIS-1343639 and CCF-1409601; DOE award DESC0007456; and AFOSR award FA9550-12-1-0458. LW was partially supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program.  \n\n# CONTRIBUTIONS  \n\nCW conceived the project, and jointly developed the method with LW, AA and AC. LW wrote all software and performed the necessary calculations with help and guidance from AA and AC. 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Mater. 13, 494–500 (2014).   \n62. Breiman, L. Random forests. Mach. Learn. 45, 5–32 (2001).   \n63. Wada, T., Zhang, T. & Inoue, A. Formation and high mechanical strength of bulk glassy alloys in Zr-Al-Co-Cu system. Mater. Trans. 44, 1839–1844 (2003).   \n64. Thornton, C., Hutter, F., Hoos, H. H. & Leyton-Brown, K. in Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. (ACM, New York, NY, 2013).  \n\nThis work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/ by/4.0/  \n\n$\\circledcirc$ The Author(s) 2016  ",
    "Target": null
  },
  {
    "id": "10.1039_c6ee01674j",
    "DOI": "10.1039/c6ee01674j",
    "DOI Link": "http://dx.doi.org/10.1039/c6ee01674j",
    "Relative Dir Path": "mds/10.1039_c6ee01674j",
    "Article Title": "Transition of lithium growth mechanisms in liquid electrolytes",
    "Authors": "Bai, P; Li, J; Brushett, FR; Bazant, MZ",
    "Source Title": "ENERGY & ENVIRONMENTAL SCIENCE",
    "Abstract": "Next-generation high-energy batteries will require a rechargeable lithium metal anode, but lithium dendrites tend to form during recharging, causing short-circuit risk and capacity loss, by mechanisms that still remain elusive. Here, we visualize lithium growth in a glass capillary cell and demonstrate a change of mechanism from root-growing mossy lithium to tip-growing dendritic lithium at the onset of electrolyte diffusion limitation. In sandwich cells, we further demonstrate that mossy lithium can be blocked by nulloporous ceramic separators, while dendritic lithium can easily penetrate nullopores and short the cell. Our results imply a fundamental design constraint for metal batteries (Sand's capacity''), which can be increased by using concentrated electrolytes with stiff, permeable, nulloporous separators for improved safety.",
    "Times Cited, WoS Core": 1211,
    "Times Cited, All Databases": 1330,
    "Publication Year": 2016,
    "Research Areas": "Chemistry; Energy & Fuels; Engineering; Environmental Sciences & Ecology",
    "UT (Unique WOS ID)": "WOS:000386336200026",
    "Markdown": "# Transition of lithium growth mechanisms in liquid electrolytes†  \n\nPeng Bai,\\*a Ju Li,bc Fikile R. Brushetta and Martin Z. Bazant\\*ade  \n\nReceived 10th June 2016, Accepted 8th August 2016  \n\nDOI: 10.1039/c6ee01674j www.rsc.org/ees  \n\nNext-generation high-energy batteries will require a rechargeable lithium metal anode, but lithium dendrites tend to form during recharging, causing short-circuit risk and capacity loss, by mechanisms that still remain elusive. Here, we visualize lithium growth in a glass capillary cell and demonstrate a change of mechanism from root-growing mossy lithium to tip-growing dendritic lithium at the onset of electrolyte diffusion limitation. In sandwich cells, we further demonstrate that mossy lithium can be blocked by nanoporous ceramic separators, while dendritic lithium can easily penetrate nanopores and short the cell. Our results imply a fundamental design constraint for metal batteries (‘‘Sand’s capacity’’), which can be increased by using concentrated electrolytes with stiff, permeable, nanoporous separators for improved safety.  \n\n# Broader context  \n\nConsumer electronic devices, portable power tools, and electric vehicles have been enabled, but also constrained, by the steady improvement of lithium-ion batteries. To develop batteries with higher energy density, such as $\\mathbf{Li-O}_{2}$ , Li–S, and other Li metal batteries using intercalation cathodes, lithium is believed to be the ideal anode material for its extremely high theoretical specific capacity $(3860~\\mathrm{\\mA~h~g^{-1}})$ , low density $\\left(0.59\\mathrm{~g~cm}^{-3}\\right)$ and the lowest negative electrochemical potential $_{-3.04\\mathrm{~V~}}$ vs. the standard hydrogen electrode). Unfortunately, lithium growth is unstable during battery recharging and leads to rough, mossy deposits, whose fresh surfaces consume the electrolyte to form solid–electrolyte interphase layers, resulting in high internal resistance, low Coulombic efficiency and short cycle life. Finger-like lithium dendrites can also short-circuit the cell by penetrating the porous separator, leading to catastrophic accidents. Controlling such hazardous instabilities requires accurately determining their mechanisms, which are more complex than the well-studied diffusion-limited growth of copper or zinc from aqueous solutions. Such fundamental understanding is critical for the success of the lithium metal anode and could provide guidance for the optimal design and operation of rechargeable lithium metal batteries.  \n\n# Introduction  \n\nThe lithium metal anode is a key component of future highenergy batteries, such as Li–S and Li– ${\\bf O}_{2}$ batteries,1 for economical and long-range electric vehicles.2 It also holds the promise to reduce the volume and weight of lithium-ion batteries by replacing the standard graphite anode, if lithium dendrites can be safely controlled during recharging to avoid internal shorts and life-threatening accidents.3 While it has been demonstrated that electrolyte additives,4–6 artificial solid electrolyte interphase (SEI) layers,7,8 and increasing the salt concentration in electrolytes,9,10 either alone or in combination, can improve the stability of lithium under small currents4–7 and low capacities,9,11 the challenge of suppressing dendrites at practical currents $\\left(>1\\mathrm{\\mA\\cm}^{-2}\\right)$ ) and areal capacities $(>1\\mathrm{\\mA}\\mathrm{\\h\\cm}^{-2})$ remains a major obstacle for the development of rechargeable lithium metal batteries.8,12 The time is ripe for a thorough investigation of lithium growth mechanisms under these conditions, in order to establish theoretical principles and design constraints for dendrite-free charging.  \n\nThe prevailing understanding of lithium growth instability is largely based on the simpler case of aqueous copper electrodeposition,13–17 where dendritic fractal patterns are telltale signs of long-range diffusion-limited growth.18–20 When a current is applied to recharge the battery, cations are consumed by reduction reaction, as anions are expelled by the electric field. In a binary electrolyte, the evolution of neutral salt concentration obeys an effective diffusion equation.21 For currents exceeding diffusion limitation, the salt concentration at the electrode surface decreases to zero at a characteristic time,22,23 and uniform electroplating becomes unstable.13,14,17 This characteristic time, $t_{\\mathrm{Sand}},$ was first derived by Sand in 1901,22 and is now known as ‘‘Sand’s time’’,23 after which the scarce supply of cations preferentially deposits onto surface protrusions, leading to a selfamplifying process of dendritic growth (i.e. tip growth mode) that propagates at the velocity of bulk anion electromigration, in order to preserve electroneutrality.13,17,21,24,25  \n\nAttempts to transfer this understanding from copper to lithium have been inconclusive. In lithium/polymer-electrolyte cells,26–28 the onset time for dendritic growth exhibits similar scaling with current as Sand’s time, but surprisingly, far below the diffusion-limited current.27 In lithium/liquid-electrolyte cells, decreasing the mobility and the transference number of anions by using modified separators can enhance the cycle life,29 albeit again at currents well below the diffusion-limited current. Ramified moss-like or ‘‘mossy’’ deposits have even been observed at a current density of $10~{\\upmu\\mathrm{A}}~{\\mathrm{cm}}^{-2}$ (ref. 30) and have been observed to grow from their roots,31 rather than their tips, in contrast to all existing growth models. Moreover, the microscopic morphology of serpentine lithium filaments observed in various electrolytes,3,5,9,32–35 over a range of length scales, do not resemble the branched, fractal structure of copper dendrites. These striking discrepancies between lithium and copper metal electrodeposition have lingered for decades without a clear explanation.  \n\nIn this study, we aim to determine the precise conditions for short-inducing dendritic lithium to form, in order to establish design constraints for safe rechargeable metal batteries. We choose one of the most successful electrolytes for lithium-ion batteries, $^{36}\\mathrm{LiPF}_{6}$ in the $1{:}1$ mixture of ethylene carbonate (EC) and dimethyl carbonate (DMC), and conduct two sets of experiments to investigate the mechanisms of lithium dendrite growth under various conditions. The first experiments with novel capillary cells reveal that the relatively dense ‘‘mossy’’ lithium growth is reaction-limited and changes to fractal ‘‘dendritic’’ lithium in response to electrolyte diffusion limitation. The second experiments using sandwich cells demonstrate that root-growing mossy lithium can be blocked by a nanoporous ceramic membrane, while tip-growing dendritic lithium can easily penetrate the nanopores and cause the internal short only at over-limiting currents. Our results suggest that optimizing and monitoring the intrinsic transport properties of the battery could eliminate the formation of dendritic lithium and the risk of internal shorts thereafter.  \n\n# Results  \n\n# Transition from mossy to dendritic lithium  \n\nTo better track the growth of lithium in situ, we fabricate a lithium|electrolyte|lithium symmetric cell in a special glass capillary, whose middle part is pulled thinner (Fig. 1a). A small piece of lithium metal is pushed into each end of the capillary until it lodges in the tapering part to seal the cell. Here, we are interested in the lithium deposition (reduction reaction), so the corresponding electrode should be called cathode. However, in order to be consistent with the convention of lithium metal anodes, we assume that the electrodeposition is a recharging process and the electrode is designated as an anode.  \n\nWhen a constant current is applied, moss-like lithium starts to deposit (Fig. 1c and d), and the salt concentration near the surface starts to decrease,37 as indicated by the gradually increasing voltage (Fig. 1b). After $\\sim40\\ \\mathrm{min}$ of polarization, the voltage starts to diverge upon salt depletion at the anode surface,38 and a wispy dendrite suddenly shoots out (at 2678s in Fig. 1e) in an obvious tip-growing manner, leaving behind stagnant mossy lithium. The dendrite’s fractal structure remains the same after two weeks of relaxation.  \n\nThe striking differences in morphology and dynamics imply two different mechanisms, switching from reaction-limited to transport-limited growth at the voltage spike. In the early stages of electrodeposition, mossy lithium mainly grows from its roots, as revealed by the movement of the tips, which barely change shape as they are pushed forward $^{\\mathrm{(ESI,\\dag}}$ Fig. S1 and Movies S1, S2). Root growth has also been observed by Yamaki et al.31 below the limiting current and attributed to internal stress release beneath the SEI layer on the lithium electrode. While growing into the open electrolyte, the mossy lithium also thickens, and the process has been described vividly as ‘‘rising dough’’.30 It is noteworthy that at the microscopic scale, the relatively dense moss-like structure is composed of whiskers, although the width of an individual whisker varies in different electrolytes.5,9,30,35 Such random surface growth is typical of reaction-limited deposition.19 Compared with copper electrodeposition,14 the key difference is that lithium, covered by SEI, develops whiskers and mossy structures, while copper, without SEI, forms whisker-free yet compact deposition before Sand’s time.39 Due to the insulating SEI that forms on individual lithium whiskers, mossy lithium is unable to transform into a uniform metallic film through a ripening process, even under mechanical pressure. At the voltage spike, sparse lithium dendrites grow explosively from their tips with the fractal morphology of diffusion-limited aggregation,19 also shared by copper dendrites,13,14 because electrodeposition is in the same universality class.40,41 The similarity between lithium and copper dendrites implies that both metals have similar surface tension, so the formation of dendritic lithium is correlated with the lack of lithium salt to form SEI different from that of mossy lithium, or very little SEI until the concentration relaxes after the initial burst of growth.  \n\nTo test the hypothesis of diffusion limitation, the experimental times to reach the voltage spike are used to calculate an apparent diffusion coefficient Dapp from Sand’s formula,22  \n\n$$\nt_{\\mathrm{Sand}}=\\pi D_{\\mathrm{app}}{\\frac{\\left(z_{\\mathrm{c}}c_{0}F\\right)^{2}}{4\\left(J t_{\\mathrm{a}}\\right)^{2}}}\n$$  \n\nwhere $z_{\\mathrm{c}}$ is the charge number of the cation $\\left(z_{\\mathrm{c}}\\ =\\ 1\\right.$ for $\\mathrm{Li}^{+}$ , $c_{0}$ is the bulk salt concentration, $F$ is the Faraday’s constant,  \n\n![](images/5fd1a7abe2d62ea02514ee180136e28c3026d063db0a083ffd3422c3f5c664a5.jpg)  \nFig. 1 In situ observations of lithium electrodeposition in a glass capillary filled with an electrolyte solution consisting of 1 M $\\mathsf{L i P F}_{6}$ in EC/DMC. (a) Photo of the capillary cell, whose middle part was pulled thinner for easier optical observation. (b) Voltage responses of the capillary cell at a deposition current density of $2.61\\mathsf{m A}\\mathsf{c m}^{-2}$ . $({\\mathsf{C}}-{\\mathsf{g}})$ In situ snapshots of the growth of lithium during the electrodeposition. Red arrow in (e) points to the emergence of dendritic lithium. Red dash line in (g) labels the clear morphological difference between the pre- and post-Sand’s time lithium deposits. (h) Theoretical interpretation of the growth mechanisms of lithium electrodeposition during concentration polarization.  \n\n$J$ is the current density, and $t_{\\mathrm{Li}}=0.38$ and $t_{\\mathrm{a}}=1-t_{\\mathrm{Li}}$ are the transference numbers of lithium cations and associated anions. For $c_{0}=1\\mathrm{~M~}$ , the calculated value, $D_{\\mathrm{app}}=1.0\\times10^{-6}~\\mathrm{cm}^{2}~\\mathrm{s}^{-1}$ , is consistent with reported values $3.0{-}3.5\\ \\times\\ 10^{-6}\\ \\mathrm{cm}^{2}\\ \\mathrm{s}^{-1}$ for small-current relaxation.42,43 As shown in Fig. 2a, the voltage spike at Sand’s time is consistently observed above the limiting current density, $J_{\\mathrm{lim}}=2z_{\\mathrm{c}}c_{0}F D_{\\mathrm{app}}(t_{\\mathrm{a}}L)^{-1}\\approx1\\mathrm{\\mA\\cm}^{-2}$ , where $L\\approx5\\:\\mathrm{mm}$ is the distance between the electrodes. With the aid of the in situ snapshots (Fig. 2b), we accurately measured the ‘‘experimental Sand’s time’’ for the onset of dendrites at each current density. The log–log plot can be fitted with a slope of $-1.40$ (Fig. 2c). Scaling exponents $>-2$ have also been reported (without explanation) for the short-circuiting time in other lithium cells.44 As shown in the $\\mathrm{ESI},\\dag$ the deviation observed here is attributable to convection by electro-osmotic flow in the depleted zone,45,46 although other effects, such as spatially varying porosity and/or deposit morphology, can also lead to different scaling laws for propagating diffusion layers in porous media.47  \n\nAs a new battery relevant metric, we convert Sand’s time into ‘‘Sand’s capacity’’ by multiplying with the current density. The plot of Sand’s capacity versus current density (Fig. 2d) provides a simple design constraint to avoid dendritic lithium. Interestingly, most state-of-the-art lithium metal anodes do not operate in the regime of dendritic lithium identified by the capillary cell, which is already much lower than that of the sandwich cells. Since the growth mechanism switches by diffusion limitation, absolute current densities cannot be meaningfully compared across different cells. It is the relative current density, with respect to the system-specific limiting current, that controls the transition from mossy to dendritic lithium.  \n\n# Blockage of mossy lithium  \n\nWe then apply the knowledge of growth mechanisms from the capillary cell to investigate the ability of a nanoporous separator to block mossy and dendritic lithium in a battery relevant sandwich cell. Since the smallest known whiskers in mossy lithium are $\\sim1\\upmu\\mathrm{m}$ thick,3 we construct the cell using an anodic aluminum oxide (AAO) membrane with submicron pores $(<200\\ \\mathrm{nm})$ , to see whether the mossy lithium can be blocked (Fig. 3a).  \n\n![](images/d28cf7bb5b48d44e293c36e6f70e2cf72021b9f87d5e2eb877a4679295344219.jpg)  \nFig. 2 Change of growth mechanism at Sand’s time during concentration polarization. (a) Voltage responses of capillary cells at various deposition current densities. (b) Representative optical images of lithium deposits demonstrating the clear change of morphologies at Sand’s time for various current densities. (c) Log–log plot of the experimental Sand’s times for various current densities. (d) Current-dependent Sand’s capacity with previous reports shown.  \n\nBased on the electrode separation, $300\\upmu\\mathrm{m}$ , the limiting current density for the sandwich cell is approximately $20\\mathrm{\\mA}\\mathrm{\\cm}^{-2}$ When an under-limiting current density of $10\\mathrm{\\mA}\\mathrm{\\cm}^{-2}$ is applied, the voltage stabilizes at $0.2{\\mathrm{~V~}}$ and lasts for a capacity well beyond the pure lithium limit that the compartment below the AAO can accommodate (Fig. 3a), indicating some deformation of the membrane. After dissembling the cell, an intact, free-standing AAO membrane is recovered with a dense lithium disk below it (Fig. S5, $\\mathrm{ESI\\dagger}$ ), which confirms the complete blockage of mossy lithium growth. Surprisingly, even when the AAO membrane is pressed in direct contact with the lithium metal anode, a significant amount of porous lithium can still be deposited below the AAO, as shown in Fig. 3c and d. The mossy lithium shown in the SEM images is clearly too bulky to penetrate AAO, and only forms a space-filling porous layer between the electrode and the separator, reducing the risk of short circuit below the limiting current. These results help explain why various ceramic membranes can prevent lithium short circuits,6,44,48,49 especially under normal conditions $\\left(<10\\mathrm{\\mA\\cm^{-2}}\\right)$ ), where only the dense root-growing mossy lithium is developed in the cell.  \n\nAs demonstrated in the capillary cell experiments, once Sand’s capacity is exceeded, dendritic lithium suddenly appears. When an over-limiting current density of $50\\mathrm{\\mA}\\mathrm{cm}^{-2}$ is applied to the sandwich cell, the voltage quickly increases and leads to a short circuit. As revealed by the SEM images, very thin lithium filaments can now be found among mossy deposits on the anode side (Fig. 3e and f). Clusters of granular deposits, smaller than the pores of AAO, are also clearly visible on the cathode side (Fig. $_{39}$ and h), which confirm that lithium penetration through the ceramic nanopores caused the short circuit. The stark difference between mossy and dendritic lithium deposits leads us to propose that the term ‘‘dendrite’’ be used more narrowly, only to describe a fractal, tip-growing deposit resulting from diffusion-limited growth, consistent with the well-studied copper and zinc dendrites. If lithium dendrite penetration in AAO were mainly opposed by surface tension, then the breakthrough voltage (where the overpotential exceeds the Young–Laplace pressure) would scale with the inverse of the pore size. The dendrite penetration may be further suppressed by modifying the surface charge of the nanopores,15 when over-limiting mass transfer is opposed by surface conduction.46  \n\n![](images/b5214a08431037459baf35e36655fd7bc123bd99b49db47295634efb0f6f9d73.jpg)  \nFig. 3 Lithium electrodeposition in sandwich cells. (a) Structure of the symmetric sandwich cell, where names of the electrodes follow the convention of lithium batteries, i.e. lithium deposits onto the anode during recharging. (b) Voltage responses of the sandwich cells, indicating the complete blockage of lithium deposits even beyond the theoretical capacity of the lower compartment at $10\\mathsf{m A}\\mathsf{c m}^{-2}$ , as well as the quick penetration of AAO and shortcircuiting of the cell at $50\\ m\\mathsf{A}\\mathsf{c m}^{-2}$ . (c and d) Scanning electron microscopy (SEM) images of the AAO/Li-deposit interface, revealing the blockage of bulky porous lithium formed in the under-limiting current conditions. (e and f) SEM images of the anode-facing side of AAO, displaying both bulky and needle-like lithium deposits formed in the over-limiting current conditions. (g and h) SEM images of the cathode-facing side of AAO, showing nanosized lithium deposits coming out of the nanopores of AAO. Inset: Magnification of the clusters of granular lithium deposits around the pores of AAO (appear as black dots).  \n\n# Sand’s capacity as the safety limit  \n\nOur results reveal why the risk of dendrites increases with aging, and how to mitigate it. According to the definition of Sand’s capacity,  \n\n$$\nC_{\\mathrm{Sand}}=J t_{\\mathrm{Sand}}=\\pi D_{\\mathrm{app}}{\\frac{(z_{\\mathrm{c}}c_{0}F)^{2}}{4J t_{\\mathrm{a}}^{2}}}\n$$  \n\ndilution of the electrolyte alone could significantly lower the safety limit, which is verified experimentally with our capillary cells by varying the salt concentration $c_{0}$ (Fig. 4). In practical cells, the cycling of mossy lithium consumes a large amount of the liquid electrolyte (salt and solvent) to form SEI layers.8,12,51,52 This lowers the amount of the dissolved lithium salt, and thereby the effective diffusivity $D_{\\mathrm{app}}$ , which not only results in higher impedance, but also steadily lowers the intrinsic Sand’s capacity. While the high impedance has been identified in a few experiments as a major cause of battery failure,8,9,12,51 dendritic lithium can still develop and short aged cells when using 1 M electrolyte,9 where the decrease of the intrinsic Sand’s capacity below the cycled capacity could be an explanation. Therefore, increasing Sand’s capacity by increasing the salt concentration in the electrolyte should be an effective method to improve the safety of rechargeable metal batteries. Interestingly, highly concentrated electrolytes have already enabled very high Coulombic efficiency,9,10,35 which is beneficial to longer cycle life.  \n\n# Dimensionless safety criterion  \n\nIn order to compare different systems and make general scaling predictions about the safety limit, we employ dimensional analysis. Following earlier definitions, $L$ is the distance between the two electrodes, then $L/2$ is a characteristic length scale for electrolyte diffusion with apparent diffusivity $D_{\\mathrm{app}}.$ i.e. the distance from the electrode (where salt depletion occurs) to a reservoir at concentration $c_{0}$ . Let $J_{\\mathrm{lim}}$ be the steady-state diffusion-limited current, and $C_{\\mathrm{Sand}}$ be the maximum (Sand’s) areal capacity for safe operation at a given current density $J.$ If these are the only important parameters, then, simply as a consequence of their physical units, Buckingham’s theorem53 states that there must exist a scaling relation $\\tilde{C}_{\\mathrm{Sand}}=f(\\tilde{J})$ between the dimensionless Sand’s capacity and the dimensionless applied current density,  \n\n$$\n\\tilde{C}_{\\mathrm{Sand}}=\\frac{4C_{\\mathrm{Sand}}D_{\\mathrm{app}}}{J L^{2}}\n$$  \n\n$$\n\\tilde{J}=\\frac{J}{J_{\\mathrm{lim}}}\n$$  \n\nSubstitution of eqn (2) and (4) into eqn (3), with the limiting current density for the dilute binary electrolyte, $J_{\\mathrm{lim}}=2z_{\\mathrm{c}}c_{0}F D_{\\mathrm{app}}(t_{\\mathrm{a}}L)^{-1}$ , yields the scaling function,  \n\n$$\n\\tilde{C}_{\\mathrm{Sand}}=f_{\\mathrm{dilute}}(\\tilde{J})=\\frac{\\pi}{4\\tilde{J}^{2}}\n$$  \n\n![](images/393eeebe5c294f6c9fa52661c9255a36263350fd3cd1810438eb6becfa926882.jpg)  \nFig. 4 Concentration-dependent Sand’s behavior. Experimental (a) Sand’s times and (b) Sand’s capacities for $0.5M$ and $2M$ electrolytes, with results of 1 M electrolyte from Fig. 2 as references.  \n\n![](images/79578fda5936ff787bf5754e6f31b97f07d201475a6935b89425360272428c02.jpg)  \nFig. 5 Linear and logarithmic (inset) plots of dimensionless Sand’s capacity versus dimensionless current density. Dashed line is the prediction of Sand’s formula for dilute electrolytes, while solid line is the best fit to the experimental data.  \n\nThe same scaling function governs the time, $t_{\\mathrm{Sand}}~=~C_{\\mathrm{Sand}}{\\cal J}^{-1},$ required to reach Sand’s capacity at constant current, scaled to the diffusion time: $\\tilde{t}_{\\mathrm{Sand}}=4t_{\\mathrm{Sand}}D_{\\mathrm{app}}L^{-2}=f(\\tilde{J}),$ which can also be derived by solving the ambipolar diffusion equation for transient overlimiting current density $\\tilde{J}>1\\left(\\mathrm{ESI}\\dagger\\right)$ .  \n\nThe scaling function will differ for concentrated electrolytes54 (with concentration-dependent diffusivities and coupled Stefan– Maxwell fluxes) in porous separators47 (with possibly variable porosity and tortuosity, surface conduction and electro-osmotic flows), but the trend should be the same as predicted by dilute solution theory.55 When the experimental data shown in Fig. 4b are nondimensionalized with corresponding $D_{\\mathrm{app}}$ (Table S3, ESI†) and plotted in Fig. 5 as $\\tilde{C}_{\\mathrm{Sand}}\\nu e r s u s\\tilde{J};$ a reasonable data collapse is observed, similar to the dilute solution prediction, eqn (5), but with a modified scaling function, $f(\\tilde{J})=0.265\\tilde{J}^{-1.274}$ , which is mainly attributable to electro-osmotic convection in the capillary cells (Fig. S4, ESI†).  \n\n# Discussion  \n\nOur results suggest that monitoring the capacity loss and transient responses to estimate transport properties and the associated Sand’s capacity could enable battery management systems to avoid dendrites by adjusting the applied current or cycled capacity windows in real time, which is particularly important for rechargeable lithium metal batteries cycling at high capacities. This prediction may seem at odds with the fact that some Li–S prototypes can be cycled at a very large specific capacity for hundreds of times without signs of internal shorts. There is no contradiction, however, after accounting for capacity differences. By multiplying the specific capacity with the small loading mass of the active sulfur,56 the converted areal capacity (in units of mA h $\\mathrm{cm}^{-2}$ ) that matters for the metal anode is actually smaller than those of mature lithium-ion batteries.56,57 For future rechargeable lithium metal batteries that possess a high specific energy with respect to the total mass, and operate at a truly large areal capacity, on-board diagnosis of the intrinsic Sand’s capacity of the battery to avoid dendritic lithium may become a practical solution for safe operation, before a robust chemistry that can completely suppress the continuous consumption of electrolyte (due to the growth of lithium whiskers) is developed.  \n\nCarbonate-base electrolytes, such as what we use in this work, are known to effect relatively thin lithium whiskers.32 Ether-based electrolytes, in contrast, allow lithium whiskers to grow much thicker.7–9,58,59 At a given areal capacity, thicker lithium whiskers create less surface area and therefore consume less lithium salt and solvent to develop SEI layers. In addition, fluorosulfonate species used in these electrolytes, such as lithium bis(trifluoromethane-sulphonyl)imide (LiTFSI) and lithium bis(fluorosulfonyl)imide (LiFSI), could undergo extensive reactions with lithium to form a robust LiF-rich SEI,52,59 which could also be facilitated by employing very high salt concentrations.9 However, whether the SEI layers formed in ether-based electrolytes will remain stable during cycling at larger areal capacities, and thereby retard the continuous consumption of electrolytes12,60 to retain high Coulombic efficiency and long cycle life is yet to be verified experimentally. Investigating the fundamental mechanisms alongside may help engineer better SEI in other high-voltage solvents,36,61 with which the standard graphite anode in lithiumion batteries may be replaced by ultrathin lithium metal anodes or simply removed to double the energy density. Of course, the chemistry of SEI does not override transport processes in electrolytes. Transitions from root-growing mossy lithium to tip-growing dendritic lithium also occur in ether-based electrolytes (Fig. S7 and Movie S6, S7, ESI†).  \n\nAn important implication of our study is the need for consistent terminology, not only to refer to the different lithium morphologies, but also to clarify the underlying mechanisms for rational battery design and engineering. Comparing various published work with ours, the thin needle-like lithium filaments that grow from their roots below the limiting current should be called ‘‘whiskers’’, which interweave with each other to form a ‘‘mossy’’ structure as the capacity increases. In contrast, the widely-used term ‘‘dendrites’’ should be reserved for the classical branched fractal structures that grow at their tips, which only occur at diffusion limitation and cannot revert to form a mossy structure. Although individual whiskers in the mossy structure may become thinner or disconnected over many deposition/ dissolution cycles, as long as the current density remains underlimiting, the root-growth mechanism will make penetrating ceramic nanopores as difficult as threading a needle. With further investigations of SEI formation on mossy lithium and its interaction with ceramic separators during cycling, an ultimate safe solution should be possible.  \n\n# Conclusions  \n\nWhile the failure mechanisms in practical batteries with opaque separators are still challenging to investigate in situ,62 our capillary cells provide a simple and effective means to explore the hidden physics. We have demonstrated that lithium growth in liquid electrolytes follows two different mechanisms, depending on the applied current and capacity. Below Sand’s capacity, reaction-limited mossy lithium mainly grows from the roots and cannot penetrate hard ceramic nanopores in a sandwich cell. Above Sand’s capacity, transport-limited dendritic lithium grows at the tips and can easily cross the separator to short the cell. Our results suggest maximizing Sand’s capacity by increasing the salt concentration in the electrolyte. Electrolyte degradation should also be monitored to prevent dendrites by keeping the cycled capacity below Sand’s capacity. Ceramic separators with pores smaller than mossy lithium whiskers could replace conventional polyolefin separators with flexible large pores to enhance safety and cycle life, and the effect could be further reinforced with lithium salts and solvents that favor thicker columnar deposits. To the broader field of electrodeposition, our results clarify the physical connections between lithium and copper/zinc dendrites formed in liquid electrolytes. Mechanisms and mathematical models of copper/zinc dendrite growths cannot be and should not be applied to explain either the development or the suppression of lithium whiskers. Future theoretical investigations should take into account the dynamics of SEI formation during both the root-growth and tip-growth processes of lithium electrodeposition.  \n\n# Methods  \n\n# Materials  \n\nThe battery grade electrolyte $(1\\mathrm{M}\\mathrm{LiPF}_{6}$ in ethylene carbonate/ dimethyl carbonate with a volume ratio of $\\boldsymbol{1}:\\boldsymbol{1}$ ), ethylene carbonate (EC, anhydrous, $99\\%$ ), dimethyl carbonate (DMC, anhydrous, $\\geq99\\%$ ), and Whatman AAO membranes (pore size $100\\ \\mathrm{nm}$ , thickness $60~{\\upmu\\mathrm{m}}$ , diameter $13\\ \\mathrm{mm}$ ) were purchased from Sigma-Aldrich, and used as received. Lithium bis(trifluoromethane-sulphonyl)imide (LiTFSI), 1,3-dioxolane (DOL) and 1,2-dimethoxyethane (DME) were purchased from BASF Corporation. Lithium bis(fluorosulfonyl)imide (LiFSI) was purchased from Oakwood Products Inc. Copper wires, stainless steel wires, and polyvinylidene fluoride (PVDF) sheets were purchased from McMaster-Carr. The glass capillaries were purchased from Narishige Co., Ltd. Lithium chips $(99.9\\%$ , thickness $250~{\\upmu\\mathrm{m}}.$ diameter $15.6~\\mathrm{mm}$ ) were purchased from MTI Corporation.  \n\n# Cell fabrication and electrochemical testing  \n\nThe glass capillaries were pulled 7 mm longer with a vertical type micropipette puller (PC-10, Narishige Co., Ltd). The pulled capillary was bonded onto a glass slide with silicone and then transferred into the Argon-filled glovebox. To avoid gas bubbles, the electrolyte was filled in only by the capillary effect. Then, a small piece of lithium metal was pushed into each end of the cell by a metal wire to clog at the tapering part of the capillary to seal the cell. Separation between the lithium electrodes is around $5~\\mathrm{mm}$ for all cells. Sandwich cells were constructed in the split test cells purchased from MTI Corporation. A piece of lithium chip was first gently pressed onto the bottom part of the cell and covered by a customized PVDF washer punched off from the PVDF sheet. Several drops of electrolyte were dispensed on the surface of lithium, which also immerse the PVDF washer. A piece of AAO was then carefully placed on top of the PVDF washer and covered by another piece of PVDF washer. A few more drops of electrolyte were dispensed on AAO until the second PVDF washer is immersed. Finally the second lithium chip was stacked on top of the second PVDF, and then covered by a stainless steel disk. The whole cell was assembled together with the upper part (spring-loaded) of the test cell. Electrochemical tests were conducted with an Arbin battery tester (BT 2043, Arbin Instruments). In situ images were captured by an optical microscope (MU500, AmScope). All experiments were performed at room temperature in an Argon-filled glovebox (Vigor Tech USA) with water and oxygen content less than 1 ppm.  \n\n# SEM characterization  \n\nAAO separators with lithium deposits harvested from sandwich cells were washed with DMC for three times, then fixed onto the  \n\nSEM sample holders with carbon adhesive and sealed in an air-tight box before moving out of the Argon-filled glovebox. The residual DMC on the samples helps protect the lithium from the ambient atmosphere when transferring them into the chamber of the Analytical Scanning Electron Microscope (JEOL, 6010LA), which usually takes less than 10 s before the vacuum evacuation.  \n\n# Acknowledgements  \n\nThis work is supported by Robert Bosch LLC through the MIT Energy Initiative (MITei). P. B. thanks Mr William DiNatale in the Institute for Soldier Nanotechnologies at MIT for providing the access to the micropipette puller. J. L. acknowledges support by NSF DMR-1410636. M. Z. B. acknowledges support from the Global Climate and Energy Project at Stanford University and by the US Department of Energy, Basic Energy Sciences through the SUNCAT Center for Interface Science and Catalysis. The authors thank Dr Sarah Stewart for helpful discussions.  \n\n# Notes and references  \n\n1 P. G. Bruce, S. A. Freunberger, L. J. Hardwick and J. M. Tarascon, Nat. Mater., 2012, 11, 19–29.   \n2 J. M. Tarascon and M. Armand, Nature, 2001, 414, 359–367.   \n3 W. Xu, J. Wang, F. Ding, X. Chen, E. Nasybulin, Y. Zhang and J.-G. Zhang, Energy Environ. Sci., 2014, 7, 513–537.   \n4 F. Ding, W. Xu, G. L. Graff, J. Zhang, M. L. Sushko, X. L. Chen, Y. Y. Shao, M. H. Engelhard, Z. M. Nie, J. Xiao, X. J. Liu, P. V. Sushko, J. Liu and J. G. Zhang, J. Am. Chem. Soc., 2013, 135, 4450–4456.   \n5 W. Y. Li, H. B. Yao, K. Yan, G. Y. Zheng, Z. Liang, Y. M. Chiang and Y. Cui, Nat. Commun., 2015, 6, 7436.   \n6 Y. Y. Lu, Z. Y. Tu and L. A. Archer, Nat. Mater., 2014, 13, 961–969.   \n7 G. Y. 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    "Target": null
  },
  {
    "id": "10.1002_aenm.201501590",
    "DOI": "10.1002/aenm.201501590",
    "DOI Link": "http://dx.doi.org/10.1002/aenm.201501590",
    "Relative Dir Path": "mds/10.1002_aenm.201501590",
    "Article Title": "Electrochemical Stability of Li10GeP2S12 and Li7La3Zr2O12 Solid Electrolytes",
    "Authors": "Han, FD; Zhu, YZ; He, XF; Mo, YF; Wang, CS",
    "Source Title": "ADVANCED ENERGY MATERIALS",
    "Abstract": "The electrochemical stability window of solid electrolyte is overestimated by the conventional experimental method using a Li/electrolyte/inert metal semiblocking electrode because of the limited contact area between solid electrolyte and inert metal. Since the battery is cycled in the overestimated stability window, the decomposition of the solid electrolyte at the interfaces occurs but has been ignored as a cause for high interfacial resistances in previous studies, limiting the performance improvement of the bulk-type solid-state battery despite the decades of research efforts. Thus, there is an urgent need to identify the intrinsic stability window of the solid electrolyte. The thermodynamic electrochemical stability window of solid electrolytes is calculated using first principles computation methods, and an experimental method is developed to measure the intrinsic electrochemical stability window of solid electrolytes using a Li/electrolyte/electrolyte-carbon cell. The most promising solid electrolytes, Li10GeP2S12 and cubic Li-garnet Li7La3Zr2O12, are chosen as the model materials for sulfide and oxide solid electrolytes, respectively. The results provide valuable insights to address the most challenging problems of the interfacial stability and resistance in high-performance solid-state batteries.",
    "Times Cited, WoS Core": 911,
    "Times Cited, All Databases": 988,
    "Publication Year": 2016,
    "Research Areas": "Chemistry; Energy & Fuels; Materials Science; Physics",
    "UT (Unique WOS ID)": "WOS:000374703900002",
    "Markdown": "# Electrochemical Stability of Li10GeP2S12 and Li7La3Zr2O12 Solid Electrolytes  \n\nFudong Han, Yizhou Zhu, Xingfeng He, Yifei Mo,\\* and Chunsheng Wang\\*  \n\nThe electrochemical stability window of solid electrolyte is overestimated by the conventional experimental method using a Li/electrolyte/inert metal semiblocking electrode because of the limited contact area between solid electrolyte and inert metal. Since the battery is cycled in the overestimated stability window, the decomposition of the solid electrolyte at the interfaces occurs but has been ignored as a cause for high interfacial resistances in previous studies, limiting the performance improvement of the bulk-type solid-state battery despite the decades of research efforts. Thus, there is an urgent need to identify the intrinsic stability window of the solid electrolyte. The thermodynamic electrochemical stability window of solid electrolytes is calculated using first principles computation methods, and an experimental method is developed to measure the intrinsic electrochemical stability window of solid electrolytes using a Li/electrolyte/electrolyte-carbon cell. The most promising solid electrolytes, $\\mathsf{L i}_{10}\\mathsf{G e P}_{2}\\mathsf{S}_{12}$ and cubic Li-garnet $\\begin{array}{r}{\\mathsf{L i}_{7}\\mathsf{L a}_{3}\\mathsf{Z r}_{2}\\mathsf{O}_{12},}\\end{array}$ are chosen as the model materials for sulfide and oxide solid electrolytes, respectively. The results provide valuable insights to address the most challenging problems of the interfacial stability and resistance in high-performance solid-state batteries.  \n\n# 1. Introduction  \n\nThe safety issue of Li-ion batteries has resulted in fire incidences for electric vehicles and airplanes. The use of flammable organic electrolytes in commercial Li-ion batteries is often blamed. Replacing the organic electrolyte with inorganic, ceramic solid electrolytes, which are intrinsically nonflammable, to assemble all-solid-state Li-ion batteries has the promise to ultimately resolve the safety issue of Li-ion batteries. Similar to an organic liquid electrolyte, a solid electrolyte has also to satisfy three critical requirements: (1) high Li ionic conductivity of $>10^{-3}\\ \\mathrm{S}\\ \\mathrm{cm}^{-1}$ and low electronic conductivity, (2) wide electrochemical stability window, and (3) chemical compatibility with the anode and cathode. In the past few years, major advances have been achieved in increasing the Li ionic conductivity of the solid electrolytes. The state-of-the-art solid electrolyte materials, such as Li-garnet $\\mathrm{Li}_{7}\\mathrm{La}_{3}\\mathrm{Zr}_{2}\\mathrm{O}_{12}$ (LLZO) and $\\mathrm{Li}_{10}\\mathrm{GeP}_{2}\\mathrm{S}_{12}$ (LGPS) have achieved an ionic conductivity of $10^{-3}$ to $10^{-2}\\ \\mathrm{S}\\ \\mathrm{cm}^{-1}$ 1,2  which are comparable to commercial organic liquid electrolytes. The high ionic conductivity in solid electrolytes has ignited the research of all-solid-state Li-ion batteries. After achieving adequate Li ionic conductivity in the solid electrolyte materials, current research efforts turned to enhancing the electrochemical stability of the solid electrolytes and chemical compatibility between the solid electrolytes and electrodes, so that Li metal anode and high voltage cathode materials can achieve higher energy density in all-solid-state Li-ion batteries.  \n\nTo enable the highest voltage output of the solid-state battery by coupling a lithium metal anode with a high voltage cathode material, a very wide electrochemical stability window (0.0–5.0 V) is desired for an ideal solid electrolyte. The electrochemical stability window of solid electrolyte was typically obtained by applying the linear polarization on the Li/solid electrolyte/ inert metal (e.g., Pt) semiblocking electrode. Tested by this method, very wide electrochemical stability windows of 0.0 to $5.0\\mathrm{V}$ were reported for both LGPS and LLZO. 2,3  However, the electrochemical performances of the bulk-type all-solid-state battery batteries assembled with these solid electrolytes 2,4  are far worse than the liquid-electrolyte based batteries even though the solid electrolyte has a comparable ionic conductivity to the liquid electrolyte. The high interfacial resistance is often blamed as the main limiting factor for the performance of the solid state battery. 5  The origin of the interfacial resistance, though still not fully understood, is often attributed to the poor physical interfacial contact, the formation of space charge layers, 6 and/or the formation of interphase layers due to the chemical reactions between the electrolyte and electrode. 7  Although a variety of interfacial processing techniques, such as dynamic pressing, 8  nanosizing, 9  cosintering, 10  screen printing, 11  surface coatings 12,13  have been attempted to engineer the interfaces between the electrodes and electrolytes, the performances of the solid-state battery are still much lower than the liquidelectrolyte based batteries. The limited electrochemical stability of the solid electrolyte is rarely thought to be an issue, since the batteries are cycled within the “wide” stability window of electrolytes measured using the semiblocking electrode. 14  \n\nHowever, recent studies have challenged the claimed stability of the solid electrolyte materials. For example, LiPON, a solid electrolyte demonstrated to be compatible with Li metal anode, has recently been shown to decompose against Li metal. 15  In addition, our first principles computational and experimental study demonstrated the reversible reduction and oxidization of the LGPS solid electrolyte materials at $_{0-1.7\\mathrm{~V~}}$ and $2{-}2.5\\mathrm{~V},$ respectively, 5  which indicated a true electrochemical window of the LGPS significantly narrower than the $0.0{-}5.0\\mathrm{~V~}$ window obtained using the semiblocking electrode. 5,16  These results suggest that the electrochemical window measurements based on the semiblocking electrodes significantly overestimated the true electrochemical window governed by the intrinsic thermodynamics of the material. The overestimated electrochemical stability of solid electrolytes is caused by the slow kinetics of the decomposition reactions due to the small contact area between LGPS and current collectors. 5  However, in the bulk-type allsolid-state battery, a large amount of carbon and solid electrolyte are mixed together with the active materials to form the composite electrode. 9,10  As a result, the reduction or oxidation kinetics of the solid electrolyte in the composite electrode is significantly accelerated because of the significantly-increased contact area between the solid electrolyte and electronic conductive additives. The electrochemical stability window of the electrolyte in the carbon-electrolyte-active material electrode composite cannot be properly captured by the semiblock electrodes, which may be only more suitable for the cell configurations in thin-film solid-state batteries. 17  Therefore, a proper cell design is needed to evaluate the electrochemical window of the solid electrolyte in the bulk-type all-solid-state batteries.  \n\nMore importantly, the limited stability of the solid electrolyte materials, though still neglected by battery community, has significantly restricted the performance of all-solid-state Li-ion batteries. At the cycling voltages beyond the stability window of the solid electrolyte, the decomposition products of the solid electrolyte would form as an interphase at the interfaces between solid electrolyte and electronic conductive additives. Depending on the properties of the decomposition products, the interphase may lead to an increase in interfacial resistances and a decrease in the performance of the bulk-type solid-state battery. Unfortunately, the interfacial resistance arising from the decomposition of solid electrolytes has been ignored so far due to the overestimated stability window from the semiblocking electrode measurements. The intrinsic (true) electrochemical stability window of solid electrolytes is critical in understanding the origins of high interfacial resistance in the bulk-type solid-state Li-ion batteries. However, only few theoretical studies have examined the electrochemical stability of solid electrolytes, and no existing experimental technique can measure the true stability window of the solid electrolytes.  \n\nIn this study, we challenge the claimed stability of the solid electrolyte materials and the use of semiblock electrode design for evaluating the electrochemical window for solid electrolyte materials. The most promising solid electrolytes, $\\mathrm{Li}_{10}\\mathrm{GeP}_{2}\\mathrm{S}_{12}$ and cubic Li-garnet $\\mathrm{Li}_{7}\\mathrm{La}_{3}\\mathrm{Zr}_{2}\\mathrm{O}_{12}$ were chosen as the model materials for sulfide and oxide solid electrolytes, respectively.  \n\nFirst principles calculations were performed to obtain the intrinsic thermodynamic electrochemical stability windows. A new Li/electrolyte/electrolyte-carbon cell was proposed to replace current Li/electrolyte/Pt semiblocking electrode for the measurement of the true electrochemical stability window of solid electrolytes. The first principles computation and experimental results are in good agreement, indicating that both of these solid electrolyte materials have narrower electrochemical window than what was previously claimed. The understanding of the intrinsic thermodynamics about the solid electrolyte materials at different voltages during the battery cycling provides invaluable guidance for the development of the bulk-type all-solid-state battery.  \n\n# 2. Results  \n\n# 2.1. Electrochemical Stability of Li10GeP2S12  \n\nLithium sulfide-based solid electrolytes exhibit high ionic conductivity, low grain boundary resistance, and the excellent mechanical property, which allows forming a good interfacial contact with the electrode by cold-pressing without high temperature sintering. 18,19  In this study, $\\mathrm{Li}_{10}\\mathrm{GeP}_{2}\\mathrm{S}_{12}$ (LGPS) is chosen as a typical example of sulfide electrolytes. LGPS was reported to have the highest room-temperature ionic conductivity $({\\approx}10^{-2}\\ \\mathrm{S}\\ \\mathrm{cm}^{-1})^{[2]}$ among all solid electrolyte materials and a wide “apparent” electrochemical stability window of $0.0{-}5.0\\mathrm{V}$ determined by cyclic voltammetry of a Li/LGPS/Pt semiblock electrode. 2  \n\nHowever, the first principles computation using Li grand potential phase diagram demonstrated that the intrinsic stability window is much narrower than 0.0–5.0 V. 16  The Li grand potential phase diagram identifies the phase equilibria at different potentials and the most thermodynamically favorable reactions at the given potential, assuming the full thermodynamic equilibrium and no kinetic limitation in the reaction and transportation. The same computation scheme has been used in the calculations of voltages and reaction energies in the lithiation/delithiation of battery materials. Figure 1 shows the calculated voltage profile and phase equilibria of LGPS upon lithiation and delithiation, confirming that LGPS has a much narrower electrochemical window than $5.0~\\mathrm{V}.^{[2]}$ The reduction of the LGPS starts at 1.71 V, where LGPS is lithiated and turns into $\\mathrm{Li}_{4}\\mathrm{GeS}_{4}$ P, and $\\mathrm{Li}_{2}\\mathrm{S}$ . With further decrease of the potential, there are multiple thermodynamic voltage plateaus corresponding to the Li-P and Li-Ge alloying processes upon lithiation. Our calculations predicted the reduction products of LGPS to be $\\mathrm{Li}_{2}\\mathrm{S}$ , $\\mathrm{Li}_{15}\\mathrm{Ge}_{4}$ $\\mathrm{Li}_{3}\\mathrm{P}$ at $0\\mathrm{V},$ which have been confirmed by the experimental results. 5  On the other hand, the oxidization of the LGPS to $\\mathrm{Li}_{3}\\mathrm{PS}_{4}$ S, and ${\\mathrm{GeS}}_{2}$ starts at only $2.14~\\mathrm{V},$ and the formed $\\mathrm{Li}_{3}\\mathrm{PS}_{4}$ is further oxidized into S and $\\mathrm{P}_{2}\\mathrm{S}_{5}$ at $2.31\\mathrm{V}.$ In summary, our calculation results have shown that the LGPS has a limited electrochemical stability window from 1.7 to $2.1\\mathrm{V}.$  \n\nCyclic voltammetry (CV) was used to experimentally evaluate the electrochemical stability of LGPS. Using the conventional Li/LGPS/Pt semiblocking electrode (voltage range: $-0.6$ to 5.0 V), the decomposition current within the voltage window of 0.0 to $5.0\\mathrm{~V~}$ cannot be observed from the CV of LGPS. 2,5  \n\n![](images/654214ffbe1f6379b5d8551f1b365cc3ab46a4e67334f0d7c88548d85f68835f.jpg)  \nFigure 1. The first principles calculation results of the voltage profile and phase equilibria of LGPS solid electrolyte upon lithiation and delithiation.  \n\nThe “wide” electrochemical stability window of $0.0{-}5.0\\mathrm{~V~}$ is because the decomposition current is very small and is underestimated by the huge Li deposition/dissolution peaks. 20  To avoid the huge Li deposition/dissolution peaks, the conventional Li/LGPS/Pt semiblocking electrode was scanned within restricted voltage windows $(0.0{-}2.5\\mathrm{V}$ and 2.5–4.0 V). As shown in Figure S1 (Supporting Information), apparent current due to the decomposition of LGPS could be clearly observed in the linear scan of the Li/LGPS/Pt although the reaction current is still very low due to the limited interfacial contact between LGPS and Pt in the Li/LGPS/Pt cell. In this regard, we propose a novel experimental method to measure the electrochemical stability window of LGPS using a Li/LGPS/LGPS-C/Pt cell. A large amount of carbon (graphite, KS-4) was mixed into LGPS (weight ratio of LGPS to carbon is 75:25) to form the electrode. The increased contact between LGPS and carbon would significantly improve the kinetics of the decomposition reaction due to the facile electron transport as well as the significantly increased active area for charge-transfer reaction. Thus, the intrinsic stability window of LGPS is expected to be obtained from the CV scan of the Li/LGPS/LGPS-C/Pt cell. Since the electrochemical decomposition and the lithiation/delithiation of the LGPS are essentially the same process but described from two different perspectives, the reversible decomposition of LGPS electrolyte had been demonstrated using the same $\\operatorname{Li}/$ LGPS/LGPS-C/Pt cell in Figure S2 in the Supporting Information 5  of our previous work. The result indicates that the reduction of LGPS starts at $1.7\\mathrm{~V~}$ while the oxidation of LGPS starts at $2.1~\\mathrm{V}.$ This electrochemical behavior agrees very well with the computational results, and both computational and experimental results indicate the true electrochemical stability window of 1.7 to $2.1\\mathrm{~V~}$ for LGPS. Additionally, the oxidation of S at high potentials and the formation of Li-Ge and Li-P alloys at the low potentials were also confirmed by the X-ray photoelectron spectrum results. 5  The main function of carbon in the LGPS-C composite is to increase the electronic conductivity of LGPS so that the decomposition kinetics could be improved. In this regard, carbon is not the only option for the electronic-conductive additive. To exclude the potential interactions between carbon and LGPS, we replaced carbon with the inert metal powder (Pt black), i.e., $25\\mathrm{wt\\%}$ Pt black and mixed Pt with LGPS to form the LGPS-Pt composite electrode. The CV curves of the  \n\nLi/LGPS/LGPS-Pt/Pt cell are shown in Figure S2 (Supporting Information). Both the oxidation and reduction peaks could be observed at similar voltages in the CV curves of the Li/LGPS/ LGPS-C/Pt cell. The result implies that the redox peaks in Li/ LGPS/LGPS-C/Pt cell is not induced by the reaction between carbon and LGPS but the decomposition of LGPS itself. These results demonstrated that the thermodynamic electrochemical stability window of LGPS can be accurately calculated using our computation scheme, and that Li/LGPS/LGPS-C/Pt cell can be used to measure the true electrochemical stability of LGPS.  \n\nTherefore, our proposed method of measuring the electrochemical stability of the electrolyte in Li/electrolyte/electrolyteC cell is demonstrated to obtain the “true” electrochemical stability window based on the intrinsic thermodynamics of the solid electrolyte. The Li/electrolyte/electrolyte-C cell provides improved kinetics from large and continuous physical contacts between solid electrolyte and carbon to facilitate the thermodynamically favorable decomposition reactions of the solid electrolyte. The kinetics of these reactions is limited in the semiblocking electrode, which yields overestimated electrochemical stability. Moreover, the use of the Li/electrolyte/electrolyte-C cell mimics the cell configuration in the bulk-type solid-state battery and represents the real microstructural architectures in the solid-state electrode composite, where carbon and solid electrolyte are mixed with the active material. It should be noted that in this work we mainly focused on the thermodynamic electrochemical stability of the solid electrolyte materials. The degree (extent) of the decomposition of a solid electrolyte depends on the kinetics of decomposition reaction. The particle size of solid electrolyte, the electronic and ionic conductivities of electrolytecarbon composite, the electronic and ionic conductivities of decomposition products, and the applied current (or CV scan rate) all change the reaction kinetics, thus the degree of decomposition of solid electrolyte. The passivation from the electronic insulating decomposition products may also prevent further decomposition of solid electrolyte. These indicate that the decomposition of the solid electrolyte in the real all-solid-state cell may not be as severe as that in the Li/electrolyte/electrolyte-C cell because of the low content of carbon in the electrode composite. However, even a slight amount of decomposition of the solid electrolyte may cause a huge interfacial resistance in the real cell, which was always ignored and will be discussed in detail in Section 3. Therefore, the measurements based on the Li/electrolyte/electrolyte-C cell could help to understand the electrochemical interfacial behavior of the solid electrolyte in the real bulk-type solid-state battery.  \n\nIn addition, we calculated the electrochemical stability of other sulfide electrolytes, such as $\\mathrm{Li}_{3.25}\\mathrm{Ge}_{0.25}\\mathrm{P}_{0.75}\\mathrm{S}_{4}$ $\\mathrm{Li}_{3}\\mathrm{PS}_{4}$ $\\mathrm{Li}_{4}\\mathrm{GeS}_{4}$ $\\mathrm{Li}_{6}\\mathrm{PS}_{5}\\mathrm{Cl}$ , and $\\mathrm{Li_{7}P_{2}S_{8}I}$ , using the same computation scheme. 21 The thermodynamically intrinsic electrochemical stability windows and the decomposition phase equilibria beyond their stability window are very similar to those of LGPS. The cathodic limit is around $1.6{-}1.7\\mathrm{V}$ for the reduction of Ge or $\\mathrm{\\DeltaP}$ contained in the sulfide electrolytes, and the anodic limit is usually around $2.1{-}2.3\\mathrm{~V~}$ corresponding to the oxidization of S. Doping halogen elements, such as Cl and I, into the materials increases the the potential to fully delithiate the materials. 22–24  The results indicate that the narrow electrochemical stability window is originated from the reduction of P/Ge and the oxidization of S.  \n\n# 2.2. Electrochemical Stability of $\\ L i_{7}\\ L a_{3}Z r_{2}O_{72}$  \n\nDespite the high ionic conductivity, most of the sulfide electrolytes are sensitive to moisture and/or oxygen in the ambient environment. The oxide-based solid electrolytes, which have better stability in air, therefore attract a lot of interests. In particular, cubic Li-stuffed garnet (i.e., $\\mathrm{Li}_{7}\\mathrm{La}_{3}\\mathrm{Zr}_{2}\\mathrm{O}_{12}\\big)$ ) reported with a wide electrochemical stability window of $0.0{-}6.0\\ \\mathrm{V}^{[3,25]}$ and a high ionic conductivity of $10^{-4}–10^{-3}\\mathrm{~S~cm^{-1},}^{!}$ 1  is considered as one of the most promising oxide solid electrolytes. In this section, the same research methodology was applied to study the electrochemical stability window of lithium oxide-based solid electrolyte, especially LLZO.  \n\nThe voltage profile of LLZO upon lithiation/delithiation and the detailed phase equilibria of LLZO at different voltages were calculated using the first principles method (Figure 2). The results show that the thermodynamic electrochemical stability window of LLZO is also smaller than the reported value of $0.0{-}6.0\\ \\mathrm{V}.^{[3]}$ The oxidation decomposition of LLZO occurs at as low as $2.91\\mathrm{~V~}$ to form $\\mathrm{Li}_{2}\\mathrm O_{2}$ $\\mathrm{Li}_{6}\\mathrm{Zr}_{2}\\mathrm{O}_{7}$ and $\\mathrm{La}_{2}\\mathrm{O}_{3}$ As the voltage increases above $3.3~\\mathrm{V},$ $\\mathrm{O}_{2}$ is generated from the oxidation of $\\mathrm{Li}_{2}\\mathrm O_{2}$ (Figure 2). At below $0.05\\mathrm{~V},$ LLZO is lithiated and reduced into $\\mathrm{Li}_{2}\\mathrm O$ , $\\mathrm{Zr}_{3}\\mathrm{O}$ , and $\\mathrm{La}_{2}\\mathrm{O}_{3}$ , and $\\mathrm{Zr}_{3}\\mathrm{O}$ may be further reduced into $\\mathrm{zr}$ metal at below 0.004 V (Figure 2). The thermodynamic results based on the energetics of DFT calculations indicate LLZO is not thermodynamically stable against Li metal. However, the reduction potential of LLZO (0.05 V) is very close to Li metal deposition potential (0 V), the thermodynamic driving force for the reduction is very small. Since these values of energy and voltage (0.004 V) for the further reduction of $\\mathrm{Zr}_{3}\\mathrm{O}$ are as small as the potential errors of typical DFT calculations and the approximations in our calculation scheme, the exact potential to reduce $\\mathrm{Zr}_{3}\\mathrm{O}$ into $\\mathrm{zr}$ may be below or above $0\\ \\mathrm{V}.$ However, if the potential is significantly lower than $0\\mathrm{~V~},$ the formation of $\\mathrm{zr}$ would be thermodynamically favorable. In addition, we also evaluated the electrochemical stability of the garnet phases doped by the cation dopants, such as Ta, Nb, and Al (Tables S1–S3, Supporting Information), which are commonly applied to stabilize the cubic phase of LLZO and to increase the Li ionic conductivity. The calculations indicate that a small amount of dopants, such as Ta, Al, Nb, which may be reduced at a slightly higher reduction potential, does not have a large effect on the reduction/oxidation of the host elements in LLZO (Tables S1–S3, Supporting Information). At $0.0\\mathrm{V},$ the doped cations Ta and Nb are reduced into metallic states, and Al is reduced into $\\mathrm{Zr{\\cdot}A l}$ alloys. Considering the low amount of dopants in LLZO, the effects of dopants on the stability window are small. Given the low reduction potentials for the garnet reduction, the good stability of the garnet LLZO may be explained by the formation of surface passivation after decomposition, such as $\\mathrm{Li}_{2}\\mathrm O$ , $\\mathrm{La}_{2}\\mathrm{O}_{3}$ and other oxides. In addition, the formation of $\\mathrm{Li}_{2}\\mathrm{CO}_{3}$ surface layers due to reaction of LLZO with the $\\mathrm{CO}_{2}$ in the air may also help passivating the LLZO. 26,27 These results may explain why LLZO was widely observed to be stable with Li at room temperature in many studies. 25,28  \n\n![](images/ce24e1b309474bc51d1f9d0a1acba9170076a2fcef1452c1e0d9cb24951c4f0e.jpg)  \nFigure 2. The first principles calculation results of the voltage profile of LLZO solid electrolyte upon lithiation and delithiation.  \n\nThe CV of the Li/LLZO/LLZO-C/Pt cell was used to measure the electrochemical stability window of LLZO, which was doped with a small amount of Ta to stabilize the cubic phase of LLZO. 29  To increase the contact area between the LLZO and carbon, the as-obtained LLZO powder was ground using a high-energy ball mill to reduce its particle size below $1\\upmu\\mathrm{m}$ , and then a thin-layer of carbon was coated on LLZO, as shown in Figure S3 (Supporting Information). The carbon-coated LLZO was then mixed with carbon black (weight ratio is 40:60) to make the LLZO-C composite electrode. All these processes were done in an Ar atmosphere to protect the LLZO from the slow reaction with the $_{\\mathrm{H}_{2}\\mathrm{O}/\\mathrm{CO}_{2}}$ in air. 26,27  The XRD test (Figure S4, Supporting Information) confirmed that the LLZO structure remained after grinding and carbon-coating processing. The same $\\ensuremath{\\mathrm{Zr}}3d$ spectra of LLZO before and after carbon coating (Figure S5, Supporting Information) indicates that LLZO is stable upon high-temperature carbonization process and no apparent carbothermal reduction of $\\mathrm{zr}$ could be observed. The thermodynamic oxidation stability of the LLZO was examined by the CV scan of the Li/LLZO/LLZO-C cell within the voltage range of $2.6\\mathrm{-}10.0\\ \\mathrm{V}.$ As can be observed from Figure 3  the apparent oxidation of LLZO starts at about $4.0\\mathrm{V},$ which is much lower than the reported value of $6.0{\\mathrm{V}}.$ The subsequent cathodic scan indicates the oxidation reaction is not reversible, and no oxidation peak can be observed in the second cycle. The maximum current of ${\\approx}5~{\\upmu}\\mathrm{A}$ in Figure 3 indicates that only a small amount of LLZO was oxidized. It should also be noted that the small oxidation current could also come from the insufficient ionic conduction in the LLZO-C composite because of the large amount of carbon additives as well as the large grain boundary resistance between LLZO particles. A larger current would be expected if a continuous ionic pathway through LLZO was formed in the LLZO-C composite (e.g., from co-sintering of LLZO solid electrolyte and LLZO-C electrode. 10 ) The higher voltage (4.0 V) compared with the calculation result (2.91 V) can be explained by the large over-potential for the oxidation of LLZO. It should be noted that the CV scan of Li/LLZO/ LLZO-C/Pt cell was tested in an Ar-filled glovebox and similar results were obtained when graphite was used as the electronic conductive additive, excluding the oxidation of carbon additives if LLZO is stable. Since the reduction potential of the LLZO at $0.05\\mathrm{~V~}$ is very close to the voltage of Li metal, it is difficult to distinguish the reduction of LLZO from the Li deposition in the CV scan and to quantify the reduction potential of LLZO.  \n\nX-ray photoelectron spectroscopy (XPS) was used to identify the reduction and oxidation products of LLZO beyond its stability window. In order to increase the yields of decomposition products for characterization, the LLZO-C composite electrode was cycled against Li metal in a liquid electrolyte, which provided faster reaction kinetics. A 5-V class liquid electrolyte, $1\\mathrm{~M~LiPF}_{6}$ in a mixed solvent of FEC, FEMC, and HFE (volume ratio is 2:6:2), was used to minimize the oxidation from the liquid electrolyte. The Li/LLZO half-cells were charged to $4.5~\\mathrm{V}$ or discharged to $0\\mathrm{~V~}$ at a current density of $10\\mathrm{\\mA\\g^{-1}}$ and were then maintained at the voltages for $^{72\\mathrm{~h~}}$ . The charge and discharge curves of the LLZO-C composite electrodes are provided in the Supporting Information (Figure S6, Supporting Information). However, it is impossible to conclude the decomposition of the LLZO simply from the charge/discharge curve of the LLZO-C electrode in Figure S7 (Supporting Information) because carbon in the LLZO-C electrode will also reacts with lithium and solid-electrolyte-interphase (SEI) is also formed on carbon.  \n\n![](images/9fbe9c7f045b8600c1f9dd61874ff83f07bc73d28fd12b9ca5adc4839e3d1698.jpg)  \nFigure 3. Cyclic voltammetry of Li/LLZO/LLZO-C/Pt cell within the voltage range of $2.6\\mathrm{-}10.0\\ V.$ .  \n\nTherefore, XPS was used to characterize the decomposition of LLZO. Figure 4a shows the XPS survey of the fresh and charged LLZO electrodes. The atomic percentages of the O and $\\mathrm{zr}$ derived from the survey (Table 1) indicates the atomic ratio of O to $\\mathrm{zr}$ decreases from 7.6:1 to 4.9:1 after LLZO was charged to $4.5~\\mathrm{V}.$ It should be noted that 2h $\\mathrm{Ar^{+}}$ sputtering was performed on the surface of the charged LLZO before collecting the atomic concentrations of O and $\\mathrm{zr}$ in order to completely remove the surface layers caused by the decomposition of the liquid electrolyte at a high potential. The complete removal of SEI after sputtering is confirmed by the XPS spectra of C 1s of the LLZO-C samples upon different sputtering times (Figure S7, Supporting Information). Multiple peaks above 284.6 eV (carbon black) could be observed for the charged LLZO before sputtering, indicating that several carbon-containing species are present at the surface. These carbon-containing species are most likely attributed to the decomposition products of the liquid electrolyte. However, after $^\\textrm{\\scriptsize1h}$ sputtering, only one peak at $284.6\\ \\mathrm{eV}$ corresponding to the carbon black in the LLZO-C electrode could be observed in the sample, which means that all the SEI species were removed. One more hour sputtering was performed in order to completely remove the surface layer. This result confirms that $\\mathrm{O}_{2}$ was released from the charged LLZO, which is consistent with our computation result (Figure 2). No obvious binding energy shift can be observed for Li, La, Zr, O elements after the LLZO was charged. On the other hand, Figure 4b compares the high-resolution spectra of $\\textstyle\\operatorname{Zr}3d$ of LLZO electrodes at the fresh and discharged states. All $\\textstyle\\operatorname{Zr}3d$ spectra exhibit a doublet with a fixed difference of $2.43\\ \\mathrm{eV}$ because of the spinsplit coupling between $3d_{5/2}$ and $3d_{3/2}$ For the fresh LLZO electrode, two different chemical environments of the $\\mathrm{zr}$ can be observed. The main peak of Zr $3d_{5/2}$ located at the $181.8\\mathrm{eV}$ corresponds to the $\\mathrm{zr}$ in the cubic garnet, 26  while the side peak of the $\\mathrm{Zr~}3d_{5/2}$ at the $179.7\\ \\mathrm{eV}$ may be ascribed to the oxide impurities (e.g., $\\mathrm{La}_{2}\\mathrm{Zr}_{2}\\mathrm{O}_{7}$ ) at the surface of the sample though the amount of the impurities is too small to be detected in the XRD test. 30,31  Both $\\mathrm{zr}$ peaks remained with the increased relative intensity of the side peak at $179.7\\ \\mathrm{eV},$ after the LLZO was discharged to $0\\mathrm{V}.$ In addition, another peak at a lower binding energy of $178.2\\ \\mathrm{eV}$ also appears for the discharged LLZO. It is known that various $\\mathrm{zr}$ suboxides exist and their binding energy will shift to a lower value as the oxidation state of $\\mathrm{zr}$ decreases. 32  The increase in the relative intensity of side peak at $179.7\\ \\mathrm{eV}$ as well as the appearance of a new peak at a lower binding energy $(178.2\\ \\mathrm{eV},$ ascribed to $\\mathrm{Zr}_{3}\\mathrm{O}$ herein) confirmed the reduction of $\\mathrm{zr}$ in the discharged LLZO, which agrees with the calculation result. It should be noted that the main peak at $181.8\\ \\mathrm{eV}$ of $\\mathrm{zr}$ still remained after the LLZO was discharged to $0\\mathrm{V},$ indicating that only the surface of the LLZO was reduced and most of LLZO was still stable. Nevertheless, our results demonstrated that the electrochemical stability window of garnet is not as wide as previously reported, and the reduction of $\\mathrm{zr}$ and the oxidation of O contained in LLZO occur beyond the stability window of LLZO.  \n\n![](images/70eea5dd12c6b0c44f8c7e82e362cc2ed22c0fcc580ca87f3a32e21281e56ddc.jpg)  \nFigure 4. a) The XPS survey spectrum of the fresh and charged LLZO. The atomic percentage of O and $Z\\boldsymbol{\\mathsf{r}}$ in the sample is obtained from the area of $\\textsf{O}$ 1s and $Z r3d$ peak, respectively. b) High resolution $Z r3d$ core XPS spectra of fresh and discharged LLZO. The curve fits were obtained using fixed spin splits $(3d_{3/2}-3d_{5/2}=2.43\\ \\mathrm{eV})$ .  \n\nTable 1. XPS analysis-derived O and Zr elements atomic concerntrations.   \n\n\n<html><body><table><tr><td>Samples</td><td>O content [at%]</td><td>Zr content [at%]</td><td>O/Zr ratio</td></tr><tr><td>Fresh</td><td>6.64</td><td>0.87</td><td>7.6:1</td></tr><tr><td>Charged to 4.5 V</td><td>2.12</td><td>0.43</td><td>4.9:1</td></tr></table></body></html>  \n\nOur computation and experimental results provide a new mechanism for the short-circuiting across the Li/LLZO/Li cell during Li striping/plating test at a high overpotential. 33–35 It was reported that the Li dendrite growth across the LLZO electrolyte layer is responsible for the short circuiting of $\\mathrm{Li}/$ LLZO/Li electrolyte cell. However, the growth of soft, ductile Li dendrite through the hard, dense layer of the LLZO is not understood. Here, we propose an alternative mechanism on the basis of the reduction of LLZO at very large overpotentials. As a result of the cation reduction, the formation of metallic states at the interfaces of the Li-LLZO and of the LLZO grain boundaries facilitates the electronic conduction at these interfaces. The electronic conduction would facilitate the deposition of Li in the materials from the Li electrode or the Li ions in the garnet materials. In addition, the coloration of the LLZO surface from tan white to gray black was observed after LLZO was immersed in molten Li $(300~^{\\circ}\\mathrm{C})$ for $168\\mathrm{~h~}$ . 36  We believe that the coloration is related to the reduction of $\\mathrm{zr}$ and/or the dopant (Al) in LLZO. The undetected oxidation change of $\\mathrm{zr}$ in their XPS result 36  may be caused by the re-oxidation of the top-surface of the sample stored in dry room, since the surface of $\\mathrm{zr}$ is very sensitive to oxygen and will be gradually oxidized to $\\mathrm{ZrO}_{2}$ after long time exposure of air. 32  \n\nThe thermodynamic electrochemical stability windows and the decomposition phase equilibria at different voltages of other common oxide solid electrolytes were also calculated. 21 The oxide solid electrolytes generally have a wider stability window than sulfides. The stability window of oxide solid electrolyte varies significantly from one material to another. Ligarnet LLZO has the lowest cathodic limit of $0.05\\mathrm{V},$ suggesting the best resistance to reduction. The NASICON-type materials, $\\mathrm{Li}_{1.3}\\mathrm{Al}_{0.3}\\mathrm{Ti}_{1.7}(\\mathrm{PO}_{4})_{3}$ (LATP) and $\\mathrm{Li}_{1.5}\\mathrm{Al}_{0.5}\\mathrm{Ge}_{1.5}(\\mathrm{PO}_{4})_{3}$ (LAGP), have the highest reduction potential of 1.7 and $2.2~\\mathrm{V},$ respectively, and also have the highest oxidization potential of ${\\approx}4.2~\\mathrm{V}.$ We found that the anodic limit of the electrolyte is related with the oxidation of the O in the compounds. The reduction of Ge,  \n\nTi, P, Zn, and Al elements contained in the solid electrolytes is generally responsible for the cathodic limit.  \n\n# 3. Discussions  \n\nOur first principles computation and experimental results indicate most solid electrolytes, especially sulfides, have an intrinsically narrower electrochemical stability window than the “apparent” window obtained from the linear scan of semiblocking electrode. No solid electrolyte is thermodynamically stable over the wide range from 0.0 to $5.0~\\mathrm{V}.$ Therefore, most electrolytes are not stable within the cycling voltage range of typical Li-ion battery cells based on the lithium anode and $\\mathrm{LiCoO}_{2}$ cathode. The main problem for operating the solid electrolyte beyond the limited thermodynamic stability window is the formation of new interphases due to the decomposition at the active material-electrolyte and carbon-electrolyte interfaces. The decomposition interphases, which likely have poorer Li ion conductivity than the solid electrolyte, would impede the Li transport between the solid electrolyte and the active materials and would increase the interfacial resistance. Therefore, the performance of the bulk-type solid-state battery is greatly affected, depending on the properties of the decomposition interphases, such as ionic conductivity, electronic conductivity, and electrochemical reversibility.  \n\nThe most desired properties of the interphases are electrochemically irreversible, highly ionic conducting but electronic insulating. The interphase with such properties is essentially the SEI, which kinetically inhibit further decompositions of solid electrolyte and extend the electrochemical window. The formation of the SEI layer is similar to that on the graphite electrode in the commercialized lithium ion battery, which enabled the liquid electrolyte to be used beyond its stability window. 14  For example, the decomposition products of $\\mathrm{Li}_{2}\\mathrm O$ $\\mathrm{Li}_{3}\\mathrm{N}$ , and $\\mathrm{Li}_{3}\\mathrm{P}$ formed at the reduction and lithiation of LiPON serve as an excellent SEI, 15  enabling its stability with Li metal for extremely long charge/discharge cycles. 17  In addition, $\\mathrm{Li}_{3}\\mathrm{N}$ and $\\mathrm{Li}_{3}\\mathrm{P}$ are good Li ionic conductor materials, which lower the interfacial resistance. 37,38  However, it is more likely to have the interphase with lower Li ionic conductivity than the original electrolyte, causing high interfacial resistance at the interface. Even worse, the interphase would be highly detrimental if the decomposition products have sufficient electronic conductivity. In this case, the decomposition of the solid electrolyte would continue into the bulk of the solid electrolyte, eventually causing short circuiting of the battery. For example, the wellknown reduction of the LLTO is due to the high electronic conductivity of LLTO after the reduction of Ti at low potentials. 39 The formation of metals or metal alloys at reduction, which is typical for the solid electrolytes containing certain cations, such as Ge, Ti, Zn, and Al, prevents the formation of SEI layers. For such solid electrolyte materials, an artificial SEI layer is required to be inserted at the electrode/electrolyte interface to passivate the solid electrolyte and to suppress the decomposition of the solid electrolyte beyond its stability window.  \n\nIn addition, it is highly undesired to have reversible or partially reversible decomposition reactions during lithiation/ delithiation, which make the electrolyte essentially an active  \n\n# www.MaterialsViews.com  \n\nelectrode. 39,40  The decomposition of the electrolyte at the interfaces would reduce the electrolyte content in the electrode composite, and the repeated volume changes during the cycling may lead to the poor physical contacts at the interfaces of the electrolyte. For example, the oxidation products of sulfide electrolytes at high voltages contain S, which is a well-known cathode material in Li-S batteries. The lithiation/delithiation of S at the interfaces of LGPS-cathode and LGPS-carbon interfaces generates a large volume change of up to $180\\%$ at the interface. 41  In addition, the changes of electronic and ionic conductivities in the interphase upon lithiation/delithiation would also affect the interfacial resistances and performance of the solid state batteries during cycling. The EIS test of the Li/LGPS/LGPS-C cells at different voltages (Figures S8 and S9, Supporting Information) confirmed that oxidation and reduction decomposition of LGPS will increase the interfacial resistance of the cells.  \n\nTo avoid the undesirable decompositions of the solid electrolyte, one strategy is to limit the voltage of the battery to suppress the formation of detrimental decomposition products. For example, the decomposition of LLZO will be intrinsically avoided if we use Li-In alloy as an anode (0.6 V) and S as a cathode (2.3 V). In addition, Li-In anode is widely used for the sulfide solid electrolytes, because the Li-In alloying potential higher than the reduction of Ge suppresses the Li-Ge alloying and further decompositions in the sulfide solid electrolytes. However, the use of Li-In anode significantly decreases the capacity and voltage of the battery. Another strategy to extend the stability of the solid electrolyte is to apply the coating layers at the electrolyte–electrode interfaces, since the choice of the materials is very limited to simultaneously satisfy all battery criteria (e.g., voltage, capacity, and chemical compatibility). 21 For example, the artificial coating layer, such as $\\mathrm{Li}_{4}\\mathrm{Ti}_{5}\\mathrm{O}_{12}$ and ${\\mathrm{LiNb}}O_{3}$ has been applied at the interface between the sulfide solid electrolyte and cathode materials. 12,42  These coating layers are found to suppress the interfacial mutual diffusion and to reduce the interfacial resistance. In addition, the formation of $\\mathrm{Li}_{2}\\mathrm{CO}_{3}$ on the surface of LLZO after exposing to air 26,27  can be considered as a SEI, which protects the reduction of LLZO against Li. At the anode side, Polyplus has applied the coating layers to stabilize the LATP materials against Li metal anode. 43 The above examples suggest the formation and coating of the SEI-like layers is an effective strategy to extend the stability window of the solid electrolyte and to improve the performance of all-solid-state batteries.  \n\nOn the basis of our new understanding, we provide specific recommendations for the engineering of sulfides and oxides solid electrolyte materials in the all-solid-state batteries. Since LGPS has a limited electrochemical stability window with a reduction potential of $1.7\\mathrm{V}$ and an oxidization potential of $2.1\\mathrm{V},$ the anode materials, such as In, with the lithiation potential higher than Li-Ge alloying is recommended for LGPS electrolyte to avoid the formation of highly electronic conductive Li-Ge alloys. The problems of the LGPS solid electrolyte at the cathode side is that the oxidation products, $\\mathrm{P}_{2}\\mathrm{S}_{5}$ S, and ${\\mathrm{GeS}}_{2}$ are neither electronic nor ionic conductive, and that the oxidation product S is electrochemically reversible if mixed with carbon. Therefore, applying an artificial SEI layer is recommended at the interface between the high voltage cathode and LGPS to provide good battery performance. LLZO has a wider electrochemical window than LGPS. In particular, LLZO holds great promises for the application with lithium metal anode, because the stability of LLZO against Li metal can be easily circumvented by kinetic protections, given the very small thermodynamic driving force for the reduction of LLZO at 0 V. Such kinetic protections should be able to sustain large current densities and high temperatures, which would facilitate the Li reduction of LLZO, during the operation of the LLZO-based batteries. At the cathode side, the stability of the LLZO may not be an issue as the oxidation products consisting electronic insulating $\\mathrm{La}_{2}\\mathrm{Zr}_{2}\\mathrm{O}_{7}$ and $\\mathrm{La}_{2}\\mathrm{O}_{3}$ can provide good passivation. However, these decomposition phases are poor ionic conductors, which give rise to high interfacial resistance. Therefore, the application of coating layers is also recommended between LLZO and the cathode to reduce interfacial resistance. The introduction of Nb oxides at cathode interfaces is recently demonstrated to effectively reduce the interfacial resistance. 42,44  \n\nIn addition to the electrochemical decomposition of the solid electrolyte itself at the interfaces with electronically conductive additives (e.g., carbon), the presence of the active material in a real all-solid-state cell may also induce the electrochemical decomposition of the solid electrolyte. Therefore, the effect of the active material on the thermodynamic electrochemical stability of solid electrolytes should also be considered. Given the conventional understanding about that the high interfacial resistance caused by the chemical incompatibility between solid electrolyte and active material during high-temperature sintering process and/or room-temperature charge/discharge processes, our work provides unprecedented insight for the understanding of the interfacial resistances in all-solid-state lithium ion batteries.  \n\n# 4. Conclusion  \n\nIn summary, the thermodynamic stability windows and decomposition phase equilibria of LGPS and LLZO were calculated using the first principles computation method. A Li/electrolyte/ electrolyte-carbon cell was proposed to replace current Li/electrolyte/Pt semiblocking electrode to obtain the intrinsic stability window of the solid electrolytes. The reduction and oxidation of both LGPS and LLZO are confirmed by the new CV scans and the XPS results. The results indicate that both solid electrolytes have significantly narrower electrochemical window than previously reported apparent window based on the semiblocking electrode. Therefore, the high interfacial resistances arising from the decomposition of solid electrolyte should be addressed by stabilizing the solid electrolyte. Extending the stability window of the solid electrolytes through the spontaneous formation or artificial application of SEI layers is the key to good performance of the bulk-type all-solid-state lithium ion batteries.  \n\n# 5. Experimental Section  \n\nSynthesis: Polycrystalline $\\mathsf{L i}_{10}\\mathsf{G e P}_{2}\\mathsf{S}_{12}$ powder was prepared with the same method reported elsewhere. 2  A Ta-doped cubic garnet compound with the composition of $\\mathsf{L i}_{6.75}\\mathsf{L a}_{3}\\mathsf{Z r}_{1.75}\\mathsf{T a}_{0.25}\\mathsf{O}_{12}$ was prepared through solid state reaction. Starting materials of $L i O H\\cdot H_{2}O$ $99.995\\%$ , Sigma  \n\n# www.MaterialsViews.com  \n\nAldrich), $L a(O H)_{3}$ $(99.9\\%$ , Sigma Aldrich), $Z\\mathsf{r O}_{2}$ $(99.99\\%$ , Sigma Aldrich), $\\mathsf{T a}_{2}\\mathsf{O}_{5}$ $(99.99\\%$ , Sigma Aldrich), were weighed and mixed based on the stoichiometric ratio. $10\\%$ excess $\\mathsf{L i O H}.\\mathsf{H}_{2}\\mathsf{O}$ was used to compensate the Li loss during high-temperature calcinations and sintering. The mixture was ball-milled (PM 100, Retsch) in 2-propanol for $24\\mathrm{~h~}$ with zirconia balls in a zirconia vial, and then dried, heated in air at $950^{\\circ}\\mathsf C$ for $\\mathsf{12h}$ . The ball-milling and heating were repeated once to enhance purity. The collected powder samples were pressed into pellets under isostatic pressure $(720\\ M\\mathsf{P a})$ . The pellet was fully covered with powder with the same composition and sintered in air at $1230^{\\circ}\\mathsf C$ for $\\mathsf{l}2\\mathsf{h}$ in a ${\\mathsf{M g O}}$ crucible. The residual powder samples were transferred to the Ar-filled glovebox to protect its slow reaction with the ${\\sf H}_{2}{\\sf O}/{\\sf C}{\\sf O}_{2}$ in air. For the preparation of the carbon-coated LLZO particles, the as-prepared LLZO powder was ground using a high-energy vibrating mill (SPEX SamplePrep\\* 8000M Mixer/Mill) for $\\texttt{l h}$ (to reduce its particle size), dispersed into a solution of polyvinylpyrrlidone ( $70\\mathrm{\\ut\\%}$ in ethanol), and then vigorously stirred for $30~\\mathrm{min}$ . The product was then dried and sintered at $700^{\\circ}\\mathsf C$ for $\\rceil\\mathfrak{h}$ in argon flow to enable carbon coating.  \n\nCharacterization: Powder X-ray diffraction patterns were obtained with a D8 Advance with LynxEye and SolX (Bruker AXS, WI, USA) using $\\mathsf{C u}$ $\\mathsf{K}\\alpha$ radiation. The morphologies of the sample were examined using a Hitachi a SU-70 field-emission scanning electron microscope and JEOL 2100F field emission transmission electron microscope (TEM). The surface chemistry of the samples was examined by X-ray photoelectron spectroscopy (XPS) using a Kratos Axis 165 spectrometer. To prepare the sample for XPS test, LLZO electrodes were charged or discharged to a certain voltage in a liquid electrolyte using a Swagelok cell, and held at that voltage for $24\\mathrm{~h~}$ . The electrodes were then taken out from the cell, and rinsed by dimethyl carbonate (DMC) inside the glove box for three times. All samples were dried under vacuum overnight, placed in a sealed bag, and then transferred into the XPS chamber under inert conditions in a nitrogen-filled glove bag. $\\mathsf{A r}^{+}$ sputtering was performed for $2\\ h$ ( $0.5\\mathrm{~h~}$ per step) until the carbon and/or SEI layer on the surface of the LLZO electrodes are removed. XPS data were collected using a monochromated Al $\\mathsf{K}\\alpha$ X-ray source $(7486.7\\ \\mathrm{eV})$ . The working pressure of the chamber was lower than $6.6\\times10^{-9}$ Pa. All reported binding energy values are calibrated to the C 1s peak at $284.8\\ \\mathrm{eV}.$  \n\nElectrochemistry: $\\boldsymbol{120}\\:\\mathrm{~mg}$ LGPS powder was pressed into a pellet (diameter $13m m$ ; thickness $2\\mathsf{m m}$ ) under isostatic pressure $(\\mathsf{l}20\\mathsf{M P a})$ in an Ar atmosphere. It was then sputtered with Pt on one side and attached with Li metal on the other side to make the Li/LGPS/Pt cell. To make the Li/LGPS/LGPS-C cell, $10\\mathrm{\\mg\\LGPS{\\cdot}C}$ powder (LGPS: graphite is 75:25 in weight) was put on the top of 120 LGPS powder and then cold-pressed together under $360~\\mathsf{M P a}$ , while Li metal was attached on the other side of LGPS pellet. The cyclic voltammograms of the Li/ LGPS/Pt and Li/LGPS/LGPS-C cells were measured with a scan rate of $0.7\\ m\\vee s^{-1}$ . The LLZO electrodes were prepared by mixing the carboncoated LLZO and carbon black (weight ratio of carbon-coated LLZO to carbon is 40:60) by hand-grinding in the mortar, and mixing with $10\\mathrm{~wt\\%~}$ polyvinylidene fluoride (PVDF) and n-methylpyrrolidinone (NMP) to make the electrode slurry. The electrodes were prepared by casting the electrode slurry onto copper or aluminum foils and dried at $120^{\\circ}\\mathsf C$ overnight. The loading of the active material on each electrode is about $\\textsf{l m g}$ . The charge/discharge tests of the LLZO electrodes were carried out in Swagelok cells using Li metal as the counter electrode and 1M $\\mathsf{L i P F}_{6}$ in a mixed solvent of FEC, HFE, and FEMC (volume ratio is 2:6:2) as the liquid electrolyte. To make the Li/LLZO/LLZO-C cell for the electrochemical stability window test, the LLZO electrode slurry was coated on the top surface of LLZO pellet, dried at ${\\mathsf{120}}\\ {\\mathsf{^{o}C}}$ overnight, and then sputtered with Pt to improve the electrical contact. After that, Li metal was attached on the other side of the pellet and cured at $200^{\\circ}\\mathsf{C}$ to enhance the interfacial contact between Li and LLZO. The cyclic voltammogram of the Li/LLZO/LLZO-C cell was tested with a scan rate of $0.01\\ m\\vee\\mathsf{s}^{-1}$ . The charge/discharge behavior was tested using an Arbin BT2000 workstation at room temperature. The cyclic voltammetry measurements were carried on an electrochemistry workstation (Solartron 1287/1260).  \n\nFirst Principles Computation Methods: All density functional theory (DFT) calculations in the work were performed using the Vienna Ab initio Simulation Package (VASP) within the projector augmented-wave approach, and the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) functional was used. The DFT parameters were consistent with the parameters used in Materials Project (MP). 45  The crystal structures of LGPS and LLZO were obtained from the ICSD database and ordered using pymatgen if the material has disordering sites. The electrochemical stability of the solid electrolyte materials was studied using the grand potential phase diagrams constructed using pymatgen, 46  which identify the phase equilibria of the material in equilibrium with an opening Li reservoir of Li chemical potential $\\mu_{\\mathrm{Li}}$ . As in the previous studies, 16,47  the applied potential $\\phi$ was considered in the Li chemical potential $\\mu_{\\mathrm{Li}}$ as  \n\n$$\n\\mu_{\\mathrm{Li}}(\\phi)=\\mu_{\\mathrm{Li}}^{0}-e\\phi\n$$  \n\nwhere $\\mu_{\\mathrm{Li}}^{0}$ is the chemical potential of Li metal, and the potential $\\phi$ is referenced to Li metal.  \n\n# Supporting Information  \n\nSupporting Information is available from the Wiley Online Library or from the author.  \n\n# Acknowledgements  \n\nF.H. and Y.Z. contributed equally to this work. The authors thank Dr. Karen J. Gaskell at the Surface Analysis Center of University of Maryland for the help on the XPS data analysis. The authors thank Dr. Kang Xu at the U.S. Army Research Laboratory for providing the high-voltage liquid electrolyte. C.W. and F.H. thank the support from National Science Foundation under Award No. 1235719 and Army Research Office (Program Manager: Dr. Robert Mantz), under Award No. W911NF1510187. Y.M. and Y.Z. thank the support from U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, under Award No. DE-EE0006860, and the computational facilities from the University of Maryland supercomputing resources and from the Extreme Science and Engineering Discovery Environment (XSEDE) supported by National Science Foundation Award No. TG-DMR130142.  \n\nReceived: August 9, 2015   \nRevised: December 21, 2015   \nPublished online: January 21, 2016  \n\n[10] A. Aboulaich, R. Bouchet, G. Delaizir, V. Seznec, L. Tortet, M. Morcrette, P. Rozier, J. M. Tarascon, V. Viallet, M. Dolle, Adv. Energy Mater. 2011, 1, 179.   \n[11] S. Ohta, S. Komagata, J. Seki, T. Saeki, S. Morishita, T. Asaoka, J. Power Sources 2013, 238, 53.   \n[12] N. Ohta, K. Takada, L. Q. Zhang, R. Z. Ma, M. Osada, T. Sasaki, Adv. Mater. 2006, 18, 2226.   \n[13] R. B. Cervera, N. Suzuki, T. Ohnishi, M. Osada, K. Mitsuishi, T. Kambara, K. Takada, Energy Environ. Sci. 2014, 7, 662.   \n[14] J. B. 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    "Target": null
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    "id": "10.1088_1361-648X_aa8f79",
    "DOI": "10.1088/1361-648X/aa8f79",
    "DOI Link": "http://dx.doi.org/10.1088/1361-648X/aa8f79",
    "Relative Dir Path": "mds/10.1088_1361-648X_aa8f79",
    "Article Title": "Advanced capabilities for materials modelling with QUANTUM ESPRESSO",
    "Authors": "Giannozzi, P; Andreussi, O; Brumme, T; Bunau, O; Nardelli, MB; Calandra, M; Car, R; Cavazzoni, C; Ceresoli, D; Cococcioni, M; Colonna, N; Carnimeo, I; Dal Corso, A; de Gironcoli, S; Delugas, P; DiStasio, RA; Ferretti, A; Floris, A; Fratesi, G; Fugallo, G; Gebauer, R; Gerstmann, U; Giustino, F; Gorni, T; Jia, J; Kawamura, M; Ko, HY; Kokalj, A; Küçükbenli, E; Lazzeri, M; Marsili, M; Marzari, N; Mauri, F; Nguyen, NL; Nguyen, HV; Otero-de-la-Roza, A; Paulatto, L; Poncé, S; Rocca, D; Sabatini, R; Santra, B; Schlipf, M; Seitsonen, AP; Smogunov, A; Timrov, I; Thonhauser, T; Umari, P; Vast, N; Wu, X; Baroni, S",
    "Source Title": "JOURNAL OF PHYSICS-CONDENSED MATTER",
    "Abstract": "QUANTUM ESPRESSO is an integrated suite of open-source computer codes for quantum simulations of materials using state-of-the-art electronic-structure techniques, based on density-functional theory, density-functional perturbation theory, and many-body perturbation theory, within the plane-wave pseudopotential and projector-augmented-wave approaches. QUANTUM ESPRESSO owes its popularity to the wide variety of properties and processes it allows to simulate, to its performance on an increasingly broad array of hardware architectures, and to a community of researchers that rely on its capabilities as a core open-source development platform to implement their ideas. In this paper we describe recent extensions and improvements, covering new methodologies and property calculators, improved parallelization, code modularization, and extended interoperability both within the distribution and with external software.",
    "Times Cited, WoS Core": 6474,
    "Times Cited, All Databases": 6784,
    "Publication Year": 2017,
    "Research Areas": "Physics",
    "UT (Unique WOS ID)": "WOS:000413705400001",
    "Markdown": "PAPER  \n\n# Related content  \n\n# Advanced capabilities for materials modelling with Quantum ESPRESSO  \n\n- QUANTUM ESPRESSO: a modular and open-source software project for quantumsimulations of materials Paolo Giannozzi, Stefano Baroni, Nicola Bonini et al.  \n\nTo cite this article: P Giannozzi et al 2017 J. Phys.: Condens. Matter 29 465901  \n\n- Electronic structure calculations with GPAW: a real-space implementation of the projectoraugmented-wave method J Enkovaara, C Rostgaard, J J Mortensen et al.  \n\nView the article online for updates and enhancements.  \n\n- exciting: a full-potential all-electron package implementing density-functional theory and many-body perturbation theory Andris Gulans, Stefan Kontur, Christian Meisenbichler et al.  \n\n# Advanced capabilities for materials modelling with Quantum ESPRESSO  \n\nP Giannozzi $^1\\textcircled{\\circ}$ , O Andreussi2,9, T Brumme3, O Bunau4, M Buongiorno Nardelli5, M Calandra4, R Car6, C Cavazzoni7, D Ceresoli8, M Cococcioni9, N Colonna9, I Carnimeo1, A Dal Corso10,32, S de Gironcoli10,32, P Delugas10, R A DiStasio Jr 11, A Ferretti12, A Floris13, G Fratesi $14\\textcircled{\\mathbb{P}}$ , G Fugallo15, R Gebauer16, U Gerstmann17, F Giustino18, T Gorni4,10, J Jia11, M Kawamura $19_{\\textcircled{\\mathbb{D}}}$ , H-Y $\\mathsf{K o}^{6}$ , A Kokalj20, E Küçükbenli10, M Lazzeri4, M Marsili21, N Marzari9, F Mauri22, N L Nguyen9, H-V Nguyen23, A Otero-de-la-Roza24, L Paulatto4, S Poncé18, D Rocca25,26, R Sabatini27, B Santra $6\\textcircled{\\textcircled{\\circ}}$ , M Schlipf18, A P Seitsonen28,29, A Smogunov30, I Timrov9 , T Thonhauser31, P Umari21,32, N Vast33, X Wu34 and S Baroni $10_{\\textcircled{\\textcircled{10}}}$  \n\n1  Department of Mathematics, Computer Science, and Physics, University of Udine, via delle Scienze   \n206, I-33100 Udine, Italy   \n2  Institute of Computational Sciences, Università della Svizzera Italiana, Lugano, Switzerland   \n3  Wilhelm-Ostwald-Institute of Physical and Theoretical Chemistry, Leipzig University, Linnéstr. 2,   \nD-04103 Leipzig, Germany   \n4  IMPMC, UMR CNRS 7590, Sorbonne Universités-UPMC University Paris 06, MNHN, IRD, 4 Place   \nJussieu, F-75005 Paris, France   \n5  Department of Physics and Department of Chemistry, University of North Texas, Denton, TX,   \nUnited States of America   \n6  Department of Chemistry, Princeton University, Princeton, NJ 08544, United States of America   \n7  CINECA—Via Magnanelli 6/3, I-40033 Casalecchio di Reno, Bologna, Italy   \n8  Institute of Molecular Science and Technologies (ISTM), National Research Council (CNR), I-20133   \nMilano, Italy   \n9  Theory and Simulation of Materials (THEOS), and National Centre for Computational Design   \nand Discovery of Novel Materials (MARVEL), Ecole Polytechnique Fédérale de Lausanne, CH-1015   \nLausanne, Switzerland   \n10  SISSA-Scuola Internazionale Superiore di Studi Avanzati, via Bonomea 265, I-34136 Trieste, Italy   \n11  Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14853,   \nUnited States of America   \n12  CNR Istituto Nanoscienze, I-42125 Modena, Italy   \n13  School of Mathematics and Physics, College of Science, University of Lincoln, United Kingdom   \n14  Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy   \n15  ETSF, Laboratoire des Solides Irradiés, Ecole Polytechnique, F-91128 Palaiseau cedex, France   \n16  The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34151   \nTrieste, Italy   \n17  Department Physik, Universität Paderborn, D-33098 Paderborn, Germany   \n18  Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom   \n19  The Institute for Solid State Physics, Kashiwa, Japan   \n20  Department of Physical and Organic Chemistry, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana,   \nSlovenia   \n21  Dipartimento di Fisica e Astronomia, Università di Padova, via Marzolo 8, I-35131 Padova, Italy   \n22  Dipartimento di Fisica, Università di Roma La Sapienza, Piazzale Aldo Moro 5, I-00185 Roma, Italy   \n23  Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Hanoi, Vietnam   \n24  Department of Chemistry, University of British Columbia, Okanagan, Kelowna BC V1V 1V7, Canad   \n25  Université de Lorraine, $\\dot{\\mathrm{CRM}}^{2}$ , UMR 7036, F-54506 Vandoeuvre-lès-Nancy, France   \n26  CNRS, $\\mathbf{CRM}^{2}$ , UMR 7036, F-54506 Vandoeuvre-lès-Nancy, France   \n27  Orionis Biosciences, Newton, MA 02466, United States of America   \n28  Institut für Chimie, Universität Zurich, CH-8057 Zürich, Switzerland   \n29  Département de Chimie, École Normale Supérieure, F-75005 Paris, France   \n30  SPEC, CEA, CNRS, Université Paris-Saclay, F-91191 Gif-Sur-Yvette, France   \n31  Department of Physics, Wake Forest University, Winston-Salem, NC 27109, United States of America  \n\n32  CNR-IOM DEMOCRITOS, Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche, Italy 33  Laboratoire des Solides Irradiés, École Polytechnique, CEA-DRF-IRAMIS, CNRS UMR 7642, Université Paris-Saclay, F-91120 Palaiseau, France 34  Department of Physics, Temple University, Philadelphia, PA 19122-1801, United States of America  \n\nE-mail: paolo.giannozzi@uniud.it  \n\nReceived 5 July 2017, revised 23 September 2017   \nAccepted for publication 27 September 2017   \nPublished 24 October 2017  \n\n![](images/36cb9e159752d0b39df0e591b4daa3813eccfa0ffe5193a55a1a5118300f3f09.jpg)  \n\n# Abstract  \n\nQuantum ESPRESSO is an integrated suite of open-source computer codes for quantum simulations of materials using state-of-the-art electronic-structure techniques, based on density-functional theory, density-functional perturbation theory, and many-body perturbation theory, within the plane-wave pseudopotential and projector-augmented-wave approaches. Quantum ESPRESSO owes its popularity to the wide variety of properties and processes it allows to simulate, to its performance on an increasingly broad array of hardware architectures, and to a community of researchers that rely on its capabilities as a core open-source development platform to implement their ideas. In this paper we describe recent extensions and improvements, covering new methodologies and property calculators, improved parallelization, code modularization, and extended interoperability both within the distribution and with external software.  \n\nKeywords: density-functional theory, density-functional perturbation theory, many-body perturbation theory, first-principles simulations  \n\n(Some figures may appear in colour only in the online journal)  \n\n# 1.  Introduction  \n\nNumerical simulations based on density-functional theory (DFT) [1, 2] have become a powerful and widely used tool for the study of materials properties. Many such simulations are based upon the ‘plane-wave pseudopotential method’, often using ultrasoft pseudopotentials [3] or the projector augmented wave method (PAW) [4] (in the following, all of these modern developments will be referred to under the generic name of ‘pseudopotentials’). An important role in the diffusion of DFT-based techniques has been played by the availability of robust and efficient software implementations [5], as is the case for Quantum ESPRESSO, which is an open-source software distribution—i.e. an integrated suite of codes—for electronic-structure calculations based on DFT or many-body perturbation theory, and using plane-wave basis sets and pseudo­potentials [6].  \n\nThe core philosophy of Quantum ESPRESSO can be summarized in four keywords: openness, modularity, efficiency, and innovation. The distribution is based on two core packages, PWscf and CP, performing self-consistent and molecular-dynamics calculations respectively, and on additional packages for more advanced calculations. Among these we quote in particular: PHonon, for linear-response calculations of vibrational properties; PostProc, for data analysis and postprocessing; atomic, for pseudopotential generation; XSpectra, for the calculation of $\\mathbf{\\boldsymbol{x}}$ -ray absorption spectra; GIPAW, for nuclear magnetic resonance and electron paramagn­etic resonance calculations.  \n\nIn this paper we describe and document the novel or improved capabilities of Quantum ESPRESSO up to and including version 6.2. We do not cover features already present in v.4.1 and described in [6], to which we refer for further details. The list of enhancements includes theoretical and methodological extensions but also performance enhancements for current parallel machines and modularization and extended interoperability with other software.  \n\nAmong the theoretical and methodological extensions, we mention in particular:  \n\n•\tFast implementations of exact (Fock) exchange for hybrid functionals [7, 42–44]; implementation of non-local van der Waals functionals [9] and of explicit corrections for van der Waals interactions [10–13]; improvement and extensions of Hubbard-corrected functionals [14, 15].   \n•\tExcited-state calculations within time-dependent densityfunctional and many-body perturbation theories.   \n•\tRelativistic extension of the PAW formalism, including spin–orbit interactions in density-functional theory [16, 17].   \n•\tContinuum embedding environments (dielectric solvation models, electronic enthalpy, electronic surface tension, periodic boundary corrections) via the Environ module [18, 19] and its time-dependent generalization [20].  \n\nSeveral new packages, implementing the calculation of new properties, have been added to Quantum ESPRESSO. We quote in particular:  \n\n•\tturboTDDFT [21–24] and turboEELS [25, 26], for excited-state calculations within time-dependent DFT (TDDFT), without computing virtual orbitals, also interfaced with the Environ module (see above).   \n•\tQE-GIPAW, replacing the old GIPAW package, for nuclear magnetic resonance and electron paramagnetic resonance calculations.   \n•\tEPW, for electron–phonon calculations using Wannierfunction interpolation [27].   \n•\tGWL and SternheimerGW for quasi-particle and excited-state calculations within many-body perturbation theory, without computing any virtual orbitals, using the Lanczos bi-orthogonalization [28, 29] and multi-shift conjugate-gradient methods [30], respectively.   \n•\tthermo_pw, for computing thermodynamical properties in the quasi-harmonic approximation, also featuring an advanced master-slave distributed computing scheme, applicable to generic high-throughput calculations [31].   \n• $\\mathtt{d}3\\mathtt{q}$ and thermal2, for the calculation of anharmonic 3-body interatomic force constants, phonon-phonon interaction and thermal transport [32, 33].  \n\nImproved parallelization is crucial to enhance performance and to fully exploit the power of modern parallel architectures. A careful removal of memory bottlenecks and of scalar sections  of code is a pre-requisite for better and extending scaling. Significant improvements have been achieved, in par­ ticular for hybrid functionals [34, 35].  \n\nComplementary to this, a complete pseudopotential library, pslibrary, including fully-relativistic pseudopotentials, has been generated [36, 37].A curation effort [38] on all the pseudo­ potential libraries available for Quantum ESPRESSO has led to the identification of optimal pseudopotentials for efficiency or for accuracy in the calculations, the latter delivering an agreement comparable to any of the best all-electron codes [5]. Finally, a significant effort has been dedicated to modularization and to enhanced interoperability with other software. The structure of the distribution has been revised, the code base has been re-organized, the format of data files re-designed in line with modern standards. As notable examples of interoperability with other software, we mention in particular the interfaces with the LAMMPS molecular dynamics (MD) code [39] used as molecular-mechanics ‘engine’ in the Quantum ESPRESSO implementation of the QM–MM methodology [40], and with the i PI MD driver [41], also featuring path-integral MD.  \n\nAll advances and extensions that have not been documented elsewhere are described in the next sections. For more details on new packages we refer to the respective references.  \n\nThe paper is organized as follows. Section  2 contains a description of new theoretical and methodological developments and of new packages distributed together with Quantum ESPRESSO. Section  3 contains a description of improvements of parallelization, updated information on the philosophy and general organization of Quantum ESPRESSO, notably in the field of modularization and interoperability. Section 4 contains an outlook of future directions and our conclusions.  \n\n# 2. Theoretical, algorithmic, and methodological extensions  \n\nIn the following, CGS units are used, unless noted otherwise.  \n\n# 2.1.  Advanced functionals  \n\n2.1.1.  Advanced implementation of exact (Fock) exchange and hybrid functionals.  Hybrid functionals are already the de facto standard in quantum chemistry and are quickly gaining popularity in the condensed-matter physics and computational materials science communities. Hybrid functionals reduce the self-interaction error that plagues lower-rung exchange-correlation functionals, thus achieving more accurate and reliable predictive capabilities. This is of particular importance in the calculation of orbital energies, which are an essential ingredient in the treatment of band alignment and charge transfer in heterogeneous systems, as well as the input for higher-level electronic-structure calculations based on many-body perturbation theory. However, the widespread use of hybrid functionals is hampered by the often prohibitive computational requirements of the exact-exchange (Fock) contribution, especially when working with a plane-wave basis set. The basic ingredient here is the action $(\\hat{V}_{x}\\phi_{i})(\\mathbf{r})$ of the Fock operator $\\hat{V}_{x}$ onto a (single-particle) electronic state $\\phi_{i}$ , requiring a sum over all occupied Kohn–Sham (KS) states $\\{\\psi_{j}\\}$ . For spinunpolarized systems, one has:  \n\n$$\n(\\hat{V}_{x}\\phi_{i})(\\mathbf{r})=-e^{2}\\sum_{j}\\psi_{j}(\\mathbf{r})\\int\\mathrm{d}\\mathbf{r}^{\\prime}\\frac{\\psi_{j}^{*}(\\mathbf{r}^{\\prime})\\phi_{i}(\\mathbf{r}^{\\prime})}{|\\mathbf{r}-\\mathbf{r}^{\\prime}|},\n$$  \n\nwhere $-e$ is the charge of the electron. In the original algorithm [6] implemented in PWscf, self-consistency is achieved via a double loop: in the inner one the $\\psi^{\\dagger}$ s entering the definition of the Fock operator in equation  (1) are kept fixed, while the outer one cycles until the Fock operator converges to within a given threshold. In the inner loop, the integrals appearing in equation (1):  \n\n$$\nv_{i j}(\\mathbf{r})=\\int{\\mathrm{d}\\mathbf{r}^{\\prime}}\\frac{\\rho_{i j}(\\mathbf{r}^{\\prime})}{|\\mathbf{r}-\\mathbf{r}^{\\prime}|},\\qquad\\rho_{i j}(\\mathbf{r})=\\psi_{i}^{*}(\\mathbf{r})\\phi_{j}(\\mathbf{r}),\n$$  \n\nare computed by solving the Poisson equation  in reciprocal space using fast Fourier transforms (FFT). This algorithm is straightforward but slow, requiring $\\mathcal{O}\\big((N_{b}N_{k})^{2}\\big)$ FFTs, where $N_{b}$ is the number of electronic states (‘bands’ in solid-state parlance) and $N_{k}$ the number of $\\mathbf{k}$ points in the Brillouin zone (BZ). While feasible in relatively small cells, this unfavorable scaling with the system size makes calculations with hybrid functionals challenging if the unit cell contains more than a few dozen atoms.  \n\nTo enable exact-exchange calculations in the condensed phase, various ideas have been conceived and implemented in recent Quantum ESPRESSO versions. Code improvements aimed at either optimizing or better parallelizing the standard algorithm are described in section  3.1. In this section we describe two important algorithmic developments in Quantum ESPRESSO, both entailing a significant reduction in the computational effort: the adaptively compressed exchange (ACE) concept [7] and a linear-scaling $(\\mathcal{O}(N_{b}))$ framework for performing hybrid-functional ab initio molecular dynamics using maximally localized Wannier functions (MLWF) [42–44].  \n\n2.1.1.1. Adaptively compressed exchange.  The simple formal derivation of ACE allows for a robust implementation, which applies straightforwardly both to isolated or aperiodic systems $\\mathrm{{T-}}$ only sampling of the BZ, that is, $\\mathbf{k}=0$ ) and to periodic ones (requiring sums over a grid of $\\mathbf{k}$ points in the BZ); to norm conserving and ultrasoft pseudopotentials or PAW; to spin-unpolarized or polarized cases or to non-collinear magnetization. Furthermore, ACE is compatible with, and takes advantage of, all available parallelization levels implemented in Quantum ESPRESSO: over plane waves, over $\\mathbf{k}$ points, and over bands.  \n\nWith ACE, the action of the exchange operator is rewritten as  \n\n$$\n|\\hat{V}_{x}\\phi_{i}\\rangle\\simeq\\sum_{j m}|\\xi_{j}\\rangle(M^{-1})_{j m}\\langle\\xi_{m}|\\phi_{i}\\rangle,\n$$  \n\nwhere $|\\xi_{i}\\rangle=\\hat{V}_{x}|\\psi_{i}\\rangle$ and $M_{j m}=\\langle\\psi_{j}|\\xi_{m}\\rangle$ . At self-consistency, ACE becomes exact for $\\phi_{i}$ ’s in the occupied manifold of KS states. It is straightforward to implement ACE in the doubleloop structure of PWscf. The new algorithm is significantly faster while not introducing any loss of accuracy at conv­ ergence. Benchmark tests on a single processor show a $3\\times$ to $4\\times$ speedup for typical calculations in molecules, up to $6\\times$ in extended systems [45].  \n\nAn additional speedup may be achieved by using a reduced FFT cutoff in the solution of Poisson equations. In equation  (1), the exact FFT algorithm requires a FFT grid containing $G$ -vectors up to a modulus $G_{\\mathrm{max}}=2G_{c}$ , where $G_{c}$ is the largest modulus of $G$ -vectors in the plane-wave basis used to expand $\\psi_{i}$ and $\\phi_{j}.$ , or, in terms of kinetic energy cutoff, up to a cutoff $E_{x}=4E_{c}$ , where $E_{c}$ is the plane-wave cutoff. The presence of a $1/{G}^{2}$ factor in the reciprocal space expression suggests, and experience confirms, that this condition can be relaxed to $E_{x}\\sim2E_{c}$ with little loss of precision, down to $E_{x}=E_{c}$ at the price of increasing somewhat this loss [46]. The kinetic-energy cutoff for Fock-exchange computations can be tuned by specifying the keyword ecutfock in input.  \n\nHybrid functionals have also been extended to the case of ultrasoft pseudopotentials and to PAW, following the method of [47]. A large number of integrals involving augmentation charges $q_{l m}$ are needed in this case, thus offsetting the advantage of a smaller plane-wave basis set. Better performances are obtained by exploiting the localization of the $q_{l m}$ and computing the related terms in real space, at the price of small aliasing errors.  \n\nThese improvements allow to significantly speed up a calcul­ation, or to execute it on a larger number of processors, thus extending the reach of calculations with hybrid functionals. The bottleneck represented by the sum over bands and by the FFT in equation (1) is however still present: ACE just reduces the number of such expensive calculations, but does not eliminate them. In order to achieve a real breakthrough, one has to get rid of delocalized bands and FFTs, moving to a representation of the electronic structure in terms of localized orbitals. Work along this line using the selected column density matrix localization scheme [48, 49] is ongoing. In the next section we describe a different approach, implemented in the CP code, based on maximally localized Wannier functions (MLWF).  \n\n2.1.1.2. Ab initio molecular dynamics using maximally localized Wannier functions.  The CP code can now perform highly efficient hybrid-functional ab initio MD using MLWFs [50] $\\big\\{\\overline{{\\varphi}}_{i}\\big\\}$ to represent the occupied space, instead of the canonical KS orbitals $\\{\\psi_{i}\\}$ , which are typically delocalized over the entire simulation cell. The MLWF localization procedure can be written as a unitary transformation, $\\begin{array}{r}{\\overline{{\\varphi}}_{i}({\\bf r})=\\sum_{j}U_{i j}\\psi_{j}({\\bf r}).}\\end{array}$ where $U_{i j}$ is computed at each MD time step by minimizing the total spread of the orbitals via a second-order damped dynamics scheme, starting with the converged $U_{i j}$ from the previous time step as initial guesses [51].  \n\nThe natural sparsity of the exchange interaction provided by a localized representation of the occupied orbitals (at least in systems with a finite band gap) is efficiently exploited during the evaluation of exact-exchange based applications (e.g. hybrid DFT functionals). This is accomplished by computing each of the required pair-exchange potentials $\\overline{{v}}_{i j}(\\mathbf{r})$ (corresponding to a given localized pair-density $\\overline{{\\rho}}_{i j}(\\mathbf{r}))$ through the numerical solution of the Poisson equation:  \n\n$$\n\\nabla^{2}\\overline{{v}}_{i j}(\\mathbf{r})=-4\\pi\\overline{{\\rho}}_{i j}(\\mathbf{r}),\\qquad\\overline{{\\rho}}_{i j}(\\mathbf{r})=\\overline{{\\varphi}}_{i}^{*}(\\mathbf{r})\\overline{{\\varphi}}_{j}(\\mathbf{r})\n$$  \n\nusing finite differences on the real-space grid. Discretizing the Laplacian operator $(\\nabla^{2})$ using a 19-point central-difference stencil (with an associated $\\mathcal{O}(h^{6})$ accuracy in the grid spacing $h$ ), the resulting sparse linear system of equations is solved using the conjugate-gradient technique subject to the boundary conditions imposed by a multipolar expansion of $\\overline{{v}}_{i j}(\\mathbf{r})$ :  \n\n$$\n\\overline{{v}}_{i j}(\\mathbf{r})=4\\pi\\sum_{l m}\\frac{Q_{l m}}{2l+1}\\frac{Y_{l m}(\\theta,\\phi)}{r^{l+1}},\\qquadQ_{l m}=\\int\\mathrm{d}\\mathbf{r}Y_{l m}^{*}(\\theta,\\phi)r^{l}\\overline{{\\rho}}_{i j}(\\mathbf{r})\n$$  \n\nin which the $Q_{l m}$ are the multipoles describing $\\overline{{\\rho}}_{i j}(\\mathbf{r})$ [42–44].  \n\nSince $\\overline{{v}}_{i j}(\\mathbf{r})$ only needs to be evaluated for overlapping pairs of MLWFs, the number of Poisson equations that need to be solved is substantially decreased from $\\mathcal{O}(N_{b}^{2})$ to $\\mathcal{O}(N_{b})$ . In addition, $\\overline{{v}}_{i j}(\\mathbf{r})$ only needs to be solved on a subset of the real-space grid (that is in general of fixed size) that encompasses the overlap between a given pair of MLWFs. This further reduces the overall computational effort required to evaluate exact-exchange related quantities and results in a linear-scaling $(\\mathcal{O}(N_{b}))$ algorithm. As such, this framework for performing exact-exchange calculations is most efficient for non-metallic systems (i.e. systems with a finite band gap) in which the occupied KS orbitals can be efficiently localized.  \n\nThe MLWF representation not only yields the exactexchange energy $E_{\\mathrm{xx}}$ ,  \n\n$$\nE_{\\mathrm{xx}}=-e^{2}\\sum_{i j}\\int\\mathrm{d}\\mathbf{r}\\overline{{\\rho}}_{i j}(\\mathbf{r})\\overline{{v}}_{i j}(\\mathbf{r}),\n$$  \n\nat a significantly reduced computational cost, but it also provides an amenable way of computing the exactexchange contributions to the (MLWF) wavefunction forces, $\\begin{array}{r}{\\overline{{D}}_{x x}^{i}({\\bf r})=e^{2}\\sum_{j}\\overline{{v}}_{i j}({\\bf r})\\overline{{\\varphi}}_{j}({\\bf r}).}\\end{array}$ , which serve as the central quanti­ ties in Car–Parrinello MD simulations [53]. Moreover, the exact-exchange contributions to the stress tensor are readily available, thereby providing a general code base which enables hybrid DFT based simulations in the NVE, NVT, and NPT ensembles for simulation cells of any shape [44]. We note in passing that applications of the current implementation of this MLWF-based exact-exchange algorithm are limited to $\\Gamma$ -point calculations employing norm-conserving pseudo-potentials.  \n\nThe MLWF-based exact-exchange algorithm in CP employs a hybrid MPI/OpenMP parallelization strategy that has been extensively optimized for use on large-scale massively-parallel (super-) computer architectures. The required set of Poisson equations—each one treated as an independent task—are distributed across a large number of MPI ranks/ processes using a task distribution scheme designed to minimize the communication and to balance computational workload. Performance profiling demonstrates excellent scaling up to 30 720 cores (for the $\\alpha$ -glycine molecular crystal, see figure 1) and up to 65 536 cores (for $\\left(\\mathrm{H}_{2}\\mathrm{O}\\right)_{256}$ , see [43]) on Mira (BG/Q) with extremely promising efficiency. In fact, this algorithm has already been successfully applied to the study of long-time MD simulations of large-scale condensed-phase systems such as $\\mathrm{(H}_{2}\\mathrm{O})_{128}$ [43, 52]. For more details on the performance and implementation of this exact-exchange algorithm, we refer the reader to [44].  \n\n2.1.2.  Dispersion interactions.  Dispersion, or van der Waals, interactions arise from dynamical correlations among charge fluctuations occurring in widely separated regions of space. The resulting attraction is a non-local correlation effect that cannot be reliably captured by any local (such as local density approximation, LDA) or semi-local (generalized gradient approximation, GGA) functional of the electron density [54]. Such interactions can be either accounted for by a truly non-local exchange-correlation (XC) functional, or modeled by effective interactions amongst atoms, whose parameters are either computed from first principles or estimated semiempirically. In Quantum ESPRESSO both approaches are implemented. Non-local XC functionals are activated by selecting them in the input_dft variable, while explicit interactions are turned on with the vdw_corr option. From the latter group, DFT-D2 [10], Tkatchenko–Scheffler [11], and exchange-hole dipole moment models [12, 13] are cur­ rently implemented (DFT-D3 [55] and the many-body dispersion (MBD) [56–58] approaches are already available in a development version).  \n\n2.1.2.1. Non-local van der Waals density functionals.  A fully non-local correlation functional able to account for van der Waals interactions for general geometries was first developed in 2004 and named vdW-DF [59]. Its development is firmly rooted in many-body theory, where the so-called adiabatic connection fluctuation-dissipation theorem (ACFD)  \n\n[60] provides a formally exact expression for the XC energy through a coupling constant integration over the response function—see section 2.1.4. A detailed review of the vdW-DF formalism is provided in [9]. The overall XC energy given by the ACFD theorem—as a functional of the electron density $n$ —is then split in vdW-DF into a GGA-type XC part $E_{\\mathrm{xc}}^{0}[n]$ and a truly non-local correlation part $E_{\\mathrm{c}}^{\\mathrm{nl}}[n].$ i.e.  \n\n$$\nE_{\\mathrm{xc}}[n]=E_{\\mathrm{xc}}^{0}[n]+E_{\\mathrm{c}}^{\\mathrm{nl}}[n],\n$$  \n\nwhere the non-local part is responsible for the van der Waals forces. Through a second-order expansion in the plasmonresponse expression used to approximate the response function, the non-local part turns into a computationally tractable form involving a universal kernel $\\Phi(\\mathbf{r},\\mathbf{r}^{\\prime})$ ,  \n\n$$\nE_{\\mathrm{c}}^{\\mathrm{nl}}[n]=\\frac{1}{2}\\int\\mathrm{d}\\mathbf{r}\\mathrm{d}\\mathbf{r}^{\\prime}n(\\mathbf{r})\\Phi(\\mathbf{r},\\mathbf{r}^{\\prime})n(\\mathbf{r}^{\\prime}).\n$$  \n\nThe kernel $\\Phi(\\mathbf{r},\\mathbf{r}^{\\prime})$ depends on $\\mathbf{r}$ and $\\mathbf{r}^{\\prime}$ only through $q_{0}(\\mathbf{r})|\\mathbf{r}-\\mathbf{r}^{\\prime}|$ and $q_{0}(\\mathbf{r}^{\\prime})|\\mathbf{r}-\\mathbf{r}^{\\prime}|$ , where $q_{0}(\\mathbf{r})$ is a function of $n(\\mathbf{r})$ and $\\nabla n(\\mathbf{r})$ . As such, the kernel can be pre-calculated, tabulated, and stored in some external file. To make the scheme self-consistent, the XC potential $V_{\\mathrm{c}}^{\\mathrm{nl}}({\\bf r})={\\delta E_{\\mathrm{c}}^{\\mathrm{nl}}[n]}/{\\delta n({\\bf r})}$ also needs to be computed [61]. The evaluation of $E_{\\mathrm{c}}^{\\mathrm{nl}}[n]$ in equation  (8) is computationally expensive. In addition, the evaluation of the corresponding potential $V_{\\mathrm{c}}^{\\mathrm{nl}}(\\mathbf{r})$ requires one spatial integral for each point r. A significant speedup can be achieved by writing the kernel in terms of splines [62]  \n\n$$\n\\begin{array}{l}{\\displaystyle\\Phi(\\mathbf{r},\\mathbf{r}^{\\prime})=\\Phi\\big(q_{0}(\\mathbf{r}),q_{0}(\\mathbf{r}^{\\prime}),|\\mathbf{r}-\\mathbf{r}^{\\prime}|\\big)}\\\\ {\\approx\\displaystyle\\sum_{\\alpha\\beta}\\Phi\\big(q_{\\alpha},q_{\\beta},|\\mathbf{r}-\\mathbf{r}^{\\prime}|\\big)p_{\\alpha}\\big(q_{0}(\\mathbf{r})\\big)p_{\\beta}\\big(q_{0}(\\mathbf{r}^{\\prime})\\big),}\\end{array}\n$$  \n\nwhere $q_{\\alpha}$ are fixed values and $p_{\\alpha}$ are cubic splines. Equation (8) then becomes a convolution that can be simplified to  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\cal E}_{\\mathrm{c}}^{\\mathrm{nl}}[n]=\\frac{1}{2}\\sum_{\\alpha\\beta}\\int\\mathrm{d}{\\bf r}\\mathrm{d}{\\bf r}^{\\prime}\\theta_{\\alpha}({\\bf r})\\Phi_{\\alpha\\beta}(|{\\bf r}-{\\bf r}^{\\prime}|)\\theta_{\\beta}({\\bf r}^{\\prime})}}\\\\ {{\\displaystyle~=\\frac{1}{2}\\sum_{\\alpha\\beta}\\int\\mathrm{d}{\\bf k}\\theta_{\\alpha}^{*}({\\bf k})\\Phi_{\\alpha\\beta}(k)\\theta_{\\beta}({\\bf k}).}}\\end{array}\n$$  \n\nHere $\\theta_{\\alpha}(\\mathbf{r})=n(\\mathbf{r})p_{\\alpha}{\\big(}q_{0}(\\mathbf{r}){\\big)}$ and $\\theta_{\\alpha}(\\mathbf{k})$ is its Fourier transform. Accordingly, $\\Phi_{\\alpha\\beta}(k)$ is the Fourier transform of the original kernel $\\Phi_{\\alpha\\beta}(r)=\\Phi(q_{\\alpha},q_{\\beta},|\\mathbf{r}-\\mathbf{r}^{\\prime}|)$ . Thus, two spatial integrals are replaced by one integral over Fourier transformed quantities, resulting in a considerable speedup. This approach also provides a convenient evaluation for $V_{\\mathrm{c}}^{\\mathrm{nl}}(\\mathbf{r})$ .  \n\nThe vdW-DF functional was implemented in Quantum ESPRESSO version 4.3, following equation  (10). As a result, in large systems, compute times in vdW-DF calculations are only insignificantly longer than for standard GGA functionals. The implementation uses a tabulation of the Fourier transformed kernel $\\Phi_{\\alpha\\beta}(k)$ from equation  (10) that is computed by an auxiliary code, generate_vdW_kernel_table.x, and stored in the external file vdW_kernel_table. The file then has to be placed either in the directory where the calculation is run or in the directory where the corresponding pseudopotentials reside. The formalism for vdW-DF stress was derived and implemented in [63]. The proper spin extension of vdW-DF, termed svdW-DF [64], was implemented in Quantum ESPRESSO version 5.2.1.  \n\n![](images/6dc32e909093b4944a9db836051e88f3160276bbccbc1f63b86a61cc3d50882d.jpg)  \nFigure 1.  Strong (left) and weak (right) scaling plots on Mira (BG/Q) for hybrid-DFT simulations of the $\\alpha$ -glycine molecular crystal polymorph using the linear-scaling exact-exchange algorithm in CP. In these plots, unit cells containing 16–64 glycine molecules (160–640 atoms, 240–960 bands) were considered as a function of $z$ , the number of MPI ranks per band $(z=0.5-2)$ . On Mira, 30 720 cores (1920 MPI ranks $\\times16$ OpenMP threads/rank $\\times1$ core/OpenMP thread) were utilized for the largest system (gly064, $z=2$ ), retaining over $88\\%$ (strong scaling) and $80\\%$ (weak scaling) of the ideal efficiencies (dashed lines). Deviations from ideal scaling are primarily due to the FFT (which scales non-linearly) required to provide the MLWFs in real space.  \n\nAlthough the ACFD theorem provides guidelines for the total XC functional in equation  (7), in practice $E_{\\mathrm{xc}}^{0}[n]$ is approximated by simple GGA-type functional forms. This has been used to improve vdW-DF—and correct the often too large binding separations found in its original form—by optimizing the exchange contribution to $E_{\\mathrm{xc}}^{0}[n]$ . The naming convention for the resulting variants is that the extension should describe the exchange functional used. In this context, the functionals vdW-DF-C09 [65], vdW-DF-obk8 [66], vdW-DF-ob86 [67], and vdW-DF-cx [68] have been developed and implemented in Quantum ESPRESSO. While all of these variants use the same kernel to evaluate $E_{\\mathrm{c}}^{\\mathrm{nl}}[n].$ , advances have also been made in slightly adjusting the kernel form, which is referred to and implemented as vdW-DF2 [69]. A corresponding variant, i.e. vdW-DF2-b86r [70], is also implemented. Note that vdWDF2 uses the same kernel file as vdW-DF.  \n\nThe functional VV10 [71] is related to vdW-DF, but adheres to fewer exact constraints and follows a very different design philosophy. It is implemented in Quantum ESPRESSO in a form called rVV10 [72] and uses a different kernel and kernel file that can be generated by running the auxiliary code generate_vdW_kernel_table.x.  \n\n2.1.2.2. Interatomic pairwise dispersion corrections.  An alternative approach to accounting for dispersion forces is to add to the XC energy $E_{\\mathrm{xc}}^{0}$ a dispersion energy, $E_{\\mathrm{disp}}$ , written as a damped asymptotic pairwise expression:  \n\n$$\nE_{\\mathrm{xc}}=E_{\\mathrm{xc}}^{0}+E_{\\mathrm{disp}},\\qquadE_{\\mathrm{disp}}=-{\\frac{1}{2}}\\sum_{n=6,8,10}\\sum_{I\\ne J}{\\frac{C_{I J}^{(n)}f_{n}(R_{I J})}{R_{I J}^{n}}}\n$$  \n\nwhere $I$ and $J$ run over atoms, $R_{I J}=|\\mathbf{R}_{I}-\\mathbf{R}_{J}|$ is the interatomic distance between atoms $I$ and $J$ , and $f_{n}(R)$ is a suitable damping function. The interatomic dispersion coefficients $C_{I J}^{(n)}$ can be derived from fits, as in DFT-D2 [10], or calculated non-empirically, as in the Tkatchenko–Scheffler (TS-vdW) [11] and exchange-hole dipole moment (XDM) models [12, 13].  \n\nIn XDM, the $C_{I J}^{(n)}$ coefficients are calculated assuming that dispersion interactions arise from the electrostatic attraction between the electron-plus-exchange-hole distributions on different atoms [12, 13]. In this way, XDM retains the simplicity of a pairwise dispersion correction, like in DFT-D2, but derives the $C_{I J}^{(n)}$ coefficients from the electronic properties of the system under study. The damping functions $f_{n}$ in equation  (11) suppress the dispersion interaction at short distances, and serve the purpose of making the link between the short-range correlation (provided by the XC functional) and the long-range dispersion energy, as well as mitigating erroneous behavior from the exchange functional in the representation of intermolecular repulsion [13]. The damping functions contain two adjustable parameters, available online [73] for a number of popular density functionals. Although any functional for which damping parameters are available can be used, the functionals showing best performance when combined with XDM appear to be B86bPBE [74, 75] and PW86PBE [75, 76], thanks to their accurate modeling of Pauli repulsion [13]. Both functionals have been implemented in Quantum ESPRESSO since version 5.0.  \n\nIn the canonical XDM implementation, recently included in Quantum ESPRESSO and described in detail elsewhere [77], the dispersion coefficients are calculated from the electron density, its derivatives, and the kinetic energy density, and assigned to the different atoms in the system using a Hirshfeld atomic partition scheme [78]. This means that XDM is effectively a meta-GGA functional of the dispersion energy whose evaluation cost is small relative to the rest of the self-consistent calculation. Despite the conceptual and computational simplicity of XDM, and because the dispersion coefficients depend upon the atomic environment in a physically meaningful way, the XDM dispersion correction offers good performance in the calculation of diverse properties, such as lattice energies, crystal geometries, and surface adsorption energies. XDM is especially good for modeling organic crystals and organic/inorganic interfaces. For a recent review, see [13].  \n\nThe XDM dispersion calculation is turned on by specifying vdw_corr $\\scriptstyle\\gamma=\\prime\\ \\_{\\mathrm{XClm}}\\prime$ and optionally selecting appropriate damping function parameters (with the xdm_a1 and xdm_a2 keywords). Because the reconstructed all-electron densities are required during self-consistency, XDM can be used only in combination with a PAW approach. The XDM contribution to forces and stress is not entirely consistent with the energies because the current implementation neglects the change in the dispersion coefficients. Work is ongoing to remove this limitation, as well as to make XDM available for Car–Parrinello MD, in future Quantum ESPRESSO releases.  \n\nIn the TS-vdW approach $(\\mathrm{vdw\\_corr}\\mathrm{r}\\mathrm{=}^{\\prime}\\mathrm{t}\\mathrm{s}\\mathrm{-}\\mathrm{vdw}^{\\prime})$ , all vdW parameters (which include the atomic dipole polarizabilities, $\\alpha_{I}$ , vdW radii, $R_{I}^{0}$ , and interatomic $C_{I J}^{(6)}$ dispersion coefficients) are functionals of the electron density and computed using the Hirshfeld partitioning scheme [78] to account for the unique chemical environment surrounding each atom. This approach is firmly based on a fluctuating quantum harmonic oscillator (QHO) model and results in highly accurate $C_{I J}^{(6)}$ coefficients with an associated error of approximately $5.5\\%$ [11]. The TS-vdW approach requires a single empirical range-separation parameter based on the underlying XC functional and is recommended in conjunction with non-empirical DFT functionals such as PBE and PBE0. For a recent review of the TS-vdW approach and several other vdW/dispersion corrections, please see [79].  \n\nThe implementation of the density-dependent TS-vdW correction in Quantum ESPRESSO is fully self-consistent [80] and currently available for use with norm-conserving pseudo-potentials. An efficient linear-scaling implementation of the TS-vdW contribution to the ionic forces and stress tensor allows for Born–Oppenheimer and Car–Parrinello MD simulations at the DFT $+$ TS-vdW level of theory; this approach has already been successfully employed in longtime MD simulations of large-scale condensed-phase systems such as $\\mathrm{(H}_{2}\\mathrm{O})_{128}$ [43, 52]. We note in passing that the Quantum ESPRESSO implementation of the TS-vdW correction also includes analytical derivatives of the Hirshfeld weights, thereby completely reflecting the change in all TS-vdW parameters during geometry/cell optimizations and MD simulations.  \n\n2.1.3.  Hubbard-corrected functionals: DFT+U.  Most approximate XC functionals used in modern DFT codes fail quite spectacularly on systems with atoms whose ground-state electronic structure features partially occupied, strongly localized orbitals (typically of the $d$ or $f$ kind), that suffer from strong self-interaction effects and a poor description of electronic correlations. In these circumstances, $\\tt D F T+U$ is often, although not always, an efficient remedy. This method is based on the addition to the DFT energy functional $E_{\\mathrm{DFT}}$ of a correction $E_{U}$ , shaped on a Hubbard model Hamiltonian: $E_{\\mathrm{DFT+}U}=E_{\\mathrm{DFT}}+E_{U}$ . The original implementation in  \n\nQuantum ESPRESSO, extensively described in [81, 82], is based on the simplest rotationally invariant formulation of $E_{U}$ , due to Dudarev and coworkers [83]:  \n\n$$\nE_{U}=\\frac{1}{2}\\sum_{I}U^{I}\\sum_{m,\\sigma}\\left\\{n_{m m}^{I\\sigma}-\\sum_{m^{\\prime}}n_{m m^{\\prime}}^{I\\sigma}n_{m^{\\prime}m}^{I\\sigma}\\right\\},\n$$  \n\nwhere  \n\n$$\nn_{m m^{\\prime}}^{I\\sigma}=\\sum_{\\mathbf{k},\\nu}f_{\\mathbf{k}\\nu}^{\\sigma}\\langle\\psi_{\\mathbf{k}\\nu}^{\\sigma}\\vert\\phi_{m}^{I}\\rangle\\langle\\phi_{m^{\\prime}}^{I}\\vert\\psi_{\\mathbf{k}\\nu}^{\\sigma}\\rangle,\n$$  \n\n$|\\psi_{{\\bf k}\\nu}^{\\sigma}\\rangle$ is the valence electronic wave function for state $\\mathbf{k}\\nu$ of spin $\\sigma$ , $f_{{\\bf k}\\nu}^{\\sigma}$ is the corresponding occupation number, $\\vert\\phi_{m}^{I}\\rangle$ is the chosen Hubbard manifold of atomic orbitals, centered on atomic site $I$ , that may be orthogonalized or not. The presence of the Hubbard functional results in extra terms in energy derivatives such as forces, stresses, elastic constants, or forceconstant (dynamical) matrices. For instance, the additional term in forces is  \n\n$$\n\\mathbf{F}_{I\\alpha}^{U}=-\\frac{1}{2}\\sum_{I,m,m^{\\prime},\\sigma}U^{I}\\left(\\delta_{m m^{\\prime}}-2n_{m m^{\\prime}}^{I\\sigma}\\right)\\frac{\\partial n_{m m^{\\prime}}^{I\\sigma}}{\\partial R_{I\\alpha}}\n$$  \n\nwhere $R_{I\\alpha}$ is the $\\alpha$ component of position for atom $I$ in the unit cell,  \n\n$$\n\\begin{array}{c}{\\displaystyle\\frac{\\partial n_{m m^{\\prime}}^{I\\sigma}}{\\partial R_{I\\alpha}}=\\sum_{\\mathbf{k},\\nu}f_{\\mathbf{k}\\nu}^{\\sigma}\\left(\\left\\langle\\psi_{\\mathbf{k}\\nu}^{\\sigma}\\left\\vert\\frac{\\partial\\phi_{m}^{I}}{\\partial R_{I\\alpha}}\\right\\rangle\\langle\\phi_{m^{\\prime}}^{I}\\vert\\psi_{\\mathbf{k}\\nu}^{\\sigma}\\rangle\\right.\\right.}\\\\ {\\displaystyle\\left.\\left.+\\langle\\psi_{\\mathbf{k}\\nu}^{\\sigma}\\vert\\phi_{m}^{I}\\rangle\\left\\langle\\frac{\\partial\\phi_{m^{\\prime}}^{I}}{\\partial R_{I\\alpha}}\\right\\vert\\psi_{\\mathbf{k}\\nu}^{\\sigma}\\right\\rangle\\right).}\\end{array}\n$$  \n\n2.1.3.1. Recent extensions of the formulation.  As a correction to the total energy, the Hubbard functional naturally contributes an extra term to the total potential that enters the KS equations. An alternative formulation [14] of the $\\mathrm{DFT+U}$ method, recently introduced and implemented in Quantum ESPRESSO for transport calculations, eliminates the need of extra terms in the potential by incorporating the Hubbard correction directly into the (PAW) pseudopotentials through a renormalization of the coefficients of their nonlocal terms.  \n\nA simple extension to the Dudarev functional, $\\mathrm{\\DeltaDFT+U+J0}$ , was proposed in [15] and used to capture the insulating ground state of $\\mathrm{CuO}$ . In $\\mathtt{C u O}$ the localization of holes on the $d$ states of $\\mathrm{cu}$ and the consequent on-set of a magnetic ground state can only be stabilized against a competing tendency to hybridize with oxygen $p$ states when on-site exchange interactions are precisely accounted for. A simplified functional, depending upon the on-site (screened) Coulomb interaction $U$ and the Hund’s coupling $J$ , can be obtained from the full second-quantization formulation of the electronic interaction potential by keeping only on-site terms that describe the interaction between up to two orbitals and by approximating on-site effective interactions with the (orbital-independent) atomic averages of Coulomb and exchange terms:  \n\n$$\nE_{U+J}=\\sum_{I,\\sigma}\\frac{U^{I}-J^{I}}{2}\\operatorname{Tr}\\Big[\\mathbf{n}^{I\\sigma}\\left(\\mathbf{1}-\\mathbf{n}^{I\\sigma}\\right)\\Big]+\\sum_{I,\\sigma}\\frac{J^{I}}{2}\\operatorname{Tr}\\Big[\\mathbf{n}^{I\\sigma}\\mathbf{n}^{I-\\sigma}\\Big].\n$$  \n\nThe on-site exchange coupling $J^{I}$ not only reduces the effective Coulomb repulsion between like-spin electrons as in the simpler Dudarev functional (first term of the right-hand side), but also contributes a second term that acts as an extra penalty for the simultaneous presence of anti-aligned spins on the same atomic site and further stabilizes ferromagnetic ground states.  \n\nThe fully rotationally invariant scheme of Liechtenstein et al [84], generalized to non-collinear magnetism and twocomponent spinor wave-functions, is also implemented in the current version of Quantum ESPRESSO. The corrective energy term for each correlated atom can be quite generally written as:  \n\n$$\n\\begin{array}{r l}{\\displaystyle}&{E_{U}^{\\mathrm{full}}=\\frac{1}{2}\\sum_{\\alpha\\beta\\gamma\\delta}U_{\\alpha\\beta\\gamma\\delta}\\langle c_{\\alpha}^{\\dagger}c_{\\beta}^{\\dagger}c_{\\delta}c_{\\gamma}\\rangle_{\\mathrm{DFT}}}\\\\ {\\displaystyle}&{=\\frac{1}{2}\\sum_{\\alpha\\beta\\gamma\\delta}\\big(U_{\\alpha\\beta\\gamma\\delta}-U_{\\alpha\\beta\\delta\\gamma}\\big)n_{\\alpha\\gamma}n_{\\beta\\delta},}\\end{array}\n$$  \n\nwhere the average is taken over the DFT Slater determinant, $U_{\\alpha\\beta\\gamma\\delta}$ are Coulomb integrals, and some set of orthonormal spin-space atomic functions, $\\{\\alpha\\}$ , is used to calculate the occupation matrix, $n_{\\alpha\\beta}$ , via equation (13). These basis functions could be spinor wave functions of total angular momentum $j=l\\pm1/2$ , originated from spherical harmonics of orbital momentum $l$ , which is a natural choice in the presence of spin–orbit coupling. Another choice, adopted in our implementation, is to use the standard basis of separable atomic functions, $R_{l}(r)Y_{l m}(\\theta,\\phi)\\chi(\\sigma)$ , where $\\chi(\\sigma)$ are spin up/down projectors and the radial function, $R_{l}(r)$ , is an eigenfunction of the pseudo-atom. In the presence of spin–orbit coupling, the radial function is constructed by averaging between the two radial functions $R_{l\\pm1/2}$ . These radial functions are read from the file containing the pseudopotential, in this case a fully-relativistic one. In this conventional basis, the corrective functional takes the form:  \n\n$$\nE_{U}^{\\mathrm{{full}}}=\\frac{1}{2}\\sum_{i j k l,\\sigma\\sigma^{\\prime}}U_{i j k l}n_{i k}^{\\sigma\\sigma}n_{j l}^{\\sigma^{\\prime}\\sigma^{\\prime}}-\\frac{1}{2}\\sum_{i j k l,\\sigma\\sigma^{\\prime}}U_{i j l k}n_{i k}^{\\sigma\\sigma^{\\prime}}n_{j l}^{\\sigma^{\\prime}\\sigma},\n$$  \n\nwhere $\\{i j k l\\}$ run over azimuthal quantum number $m$ . The second term contains a spin-flip contribution if $\\sigma^{\\prime}\\neq\\sigma$ . For collinear magnetism, when $n_{i j}^{\\sigma\\sigma^{\\prime}}=\\delta_{\\sigma\\sigma^{\\prime}}n_{i j}^{\\sigma}$ , the present form­ ulation reduces to the scheme of Liechtenstein et al [84]. All Coulomb integrals, $U_{i j k l}$ , can be parameterized by few input parameters such as $U$ (s-shell); $U$ and $J$ $\\overset{\\cdot}{p}$ -shell); $U,J$ and $B$ ( $\\mathit{\\Pi}_{M}$ -shell); $U,J,E_{2}$ , and $E_{3}$ $f$ -shell), and so on. We note that if all parameters but $U$ are set to zero, the Dudarev functional is recovered.  \n\n2.1.3.2. Calculation of Hubbard parameters.  The Hubbard corrective functional $E_{U}$ depends linearly upon the effective on-site interactions, $U^{I}$ . Therefore, using a proper value for these interaction parameters is crucial to obtain quantitatively reliable results from $\\mathrm{\\DeltaDFT+U}$ calculations. The Quantum ESPRESSO implementation of $\\tt D F T+U$ has also been the basis to develop a method for the calcul­ation of $U$ [81], based on linear-response theory. This method is fully ab initio and provides values of the effective interactions that are consistent with the system and with the ground state that the Hubbard functional aims at correcting. A comparative analysis of this method with other approaches proposed in the literature to compute the Hubbard interactions has been initiated in [82] and will be further refined in forthcoming publications by the same authors.  \n\nWithin linear-response theory, the Hubbard interactions are the elements of an effective interaction matrix, computed as the difference between bare and screened inverse susceptibilities [81]:  \n\n$$\nU^{I}=\\big(\\chi_{0}^{-1}-\\chi^{-1}\\big)_{I I}.\n$$  \n\nIn equation  (19) the susceptibilities $\\chi$ and $\\chi_{0}$ measure the response of atomic occupations to shifts in the potential acting on the states of single atoms in the system. In particular, $\\chi$ is defined as $\\begin{array}{r}{\\chi_{I J}=\\sum_{m\\sigma}\\left(d n_{m m}^{I\\sigma}/d\\alpha^{J}\\right)}\\end{array}$ and is evaluated at self consistency, while $\\chi_{0}$ has a similar definition but is computed before the self-consistent re-adjustment of the Hartree and XC potentials. In the current implementation these susceptibilities are computed from a series of self-consistent total energy calculations (varying the strength $\\alpha$ of the perturbing potential over a range of values) performed on supercells of sufficient size for the perturbations to be isolated from their periodic replicas. While easy to implement, this approach is quite cumbersome to use, requiring multiple calculations, expensive convergence tests of the resulting parameters and complex post-processing tools.  \n\nThese difficulties can be overcome by using density-functional perturbation theory (DFpT) to automatize the calcul­ ation of the Hubbard parameters. The basic idea is to recast the entries of the susceptibility matrices into sums over the BZ:  \n\n$$\n\\frac{\\mathrm{d}n_{m m^{\\prime}}^{I\\sigma}}{\\mathrm{d}\\alpha^{J}}=\\frac{1}{N_{\\mathbf{q}}}\\sum_{\\mathbf{q}}^{N_{\\mathbf{q}}}\\mathrm{e}^{\\mathrm{i}\\mathbf{q}\\cdot\\left(\\mathbf{R}_{l}-\\mathbf{R}_{l^{\\prime}}\\right)}\\Delta_{\\mathbf{q}}^{s^{\\prime}}n_{m m^{\\prime}}^{s\\sigma},\n$$  \n\nwhere $I\\equiv(l,s)$ and $J\\equiv(l^{\\prime},s^{\\prime})$ , $l$ and $l^{\\prime}$ label unit cells, $s$ and $s^{\\prime}$ label atoms in the unit cell, $\\mathbf{R}_{l}$ and $\\mathbf{R}_{l^{\\prime}}$ are Bravais lattice vectors, and $\\Delta_{\\mathbf{q}}^{s^{\\prime}}n_{m m^{\\prime}}^{s\\sigma}$ represent the (lattice-periodic) response of atomic occupations to monochromatic perturbations constructed by modulating the shift to the potential of all the periodic replicas of a given atom by a wave-vector $\\mathbf{q}$ . This quantity is evaluated within DFpT (see section  2.2), using linear-response routines contained in LR_Modules (see section  3.4.3). This approach eliminates the need for supercell calculations in periodic systems (along with the cubic scaling of their computational cost) and automatizes complex postprocessing operations needed to extract $U$ from the output of calculations. The use of DFpT also offers the perspective to directly evaluate inverse susceptibilities, thus avoiding the matrix inversions of equation  (19), and to calculate the Hubbard parameters for closed-shell systems, a notorious problem for schemes based on perturbations to the potential. Full details about this implementation will be provided in a forthcoming publication [85] and the corresponding code will be made available in one of the next Quantum ESPRESSO releases.  \n\n2.1.4.  Adiabatic-connection fluctuation-dissipation theory.  In the quest for better approximations for the unknown XC energy functional in KS-DFT, the approach based on the adiabatic connection fluctuation-dissipation (ACFD) theorem [60] has received considerable interest in recent years. This is largely due to some attractive features: (i) a formally exact expression for the XC energy in term of density linear response functions can be derived providing a promising way for a systematic improvement of the XC functional; (ii) the method treats the exchange energy exactly, thus canceling out the spurious selfinteraction error present in the Hartree energy; (iii) the correlation energy is fully non local and automatically includes long-range van der Waals interactions (see section 2.1.2.1).  \n\nWithin the ACFD framework a formally exact expression for the XC energy $E_{\\mathrm{xc}}$ of an electronic system can be derived:  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\cal E}_{\\mathrm{xc}}=-\\frac{\\hbar}{2\\pi}\\int_{0}^{1}\\mathrm{d}\\lambda\\int\\mathrm{d}{\\bf r}\\mathrm{d}{\\bf r}^{\\prime}\\frac{e^{2}}{\\left|{\\bf r}-{\\bf r}^{\\prime}\\right|}}\\ ~}\\\\ {{\\displaystyle~\\times\\left[\\int_{0}^{\\infty}\\chi_{\\lambda}({\\bf r},{\\bf r}^{\\prime},\\mathrm{i}u)\\mathrm{d}u+\\delta({\\bf r}-{\\bf r}^{\\prime})n({\\bf r})\\right]},}\\end{array}\n$$  \n\nwhere $\\hbar=h/2\\pi$ and $h$ is the Planck constant, $\\chi_{\\lambda}(\\mathbf{r},\\mathbf{r}^{\\prime};\\mathrm{i}u)$ is the density response function at imaginary frequency iu of a system whose electrons interact via a scaled Coulomb interaction, i.e. $\\lambda e^{2}/|\\mathbf{r}-\\mathbf{r}^{\\prime}|,$ and are subject to a local potential such that the electronic density $n(\\mathbf{r})$ is independent of $\\lambda$ , and is thus equal to the ground-state density of the fully interacting system ( $\\langle\\lambda=1$ ). The XC energy, equation (21), can be further separated into a KS exact-exchange energy $E_{\\mathrm{xx}}$ , equation (6), and a correlation energy $E_{\\mathrm{c}}$ . The former is routinely evaluated as in any hybrid functional calculation (see section 2.1.1). Using a matrix notation, the latter can be expressed in a compact form in terms of the Coulomb interaction, $v_{c}=e^{2}/|\\mathbf{r}-\\mathbf{r}^{\\prime}|$ , and of the density response functions:  \n\n$$\nE_{\\mathrm{c}}=-\\frac{\\hbar}{2\\pi}\\int_{0}^{1}\\mathrm{d}\\lambda\\int_{0}^{\\infty}\\mathrm{d}u\\mathrm{tr}\\big[v_{c}[\\chi_{\\lambda}(\\mathrm{i}u)-\\chi_{0}(\\mathrm{i}u)]\\big].\n$$  \n\nFor $\\lambda>0$ , $\\chi_{\\lambda}$ can be related to the noninteracting density response function $\\chi_{0}$ via a Dyson equation  obtained from TDDFT:  \n\n$$\n\\chi_{\\lambda}(\\mathbf{i}u)=\\chi_{0}(\\mathbf{i}u)+\\chi_{0}(\\mathbf{i}u)\\left[\\lambda v_{c}+f_{\\mathrm{xc}}^{\\lambda}(\\mathbf{i}u)\\right]\\chi_{\\lambda}(\\mathbf{i}u).\n$$  \n\nThe exact expression of the XC kernel $f_{\\mathrm{xc}}$ is unknown, and in practical applications one needs to approximate it. In the ACFDT package, the random phase approximation (RPA), obtained by setting $f_{\\mathrm{xc}}^{\\lambda}=0$ , and the RPA plus exact-exchange kernel (RPAx), obtained by setting $f_{\\mathrm{xc}}^{\\lambda}=\\lambda f_{\\mathrm{x}}$ , are implemented. The evaluation of the RPA and RPAx correlation energies is based on an eigenvalue decomposition of the noninteracting response functions and of its first-order correction in the limit of vanishing electron-electron interaction [86–88].  \n\nSince only a small number of these eigenvalues are relevant for the calculation of the correlation energy, an efficient iterative scheme can be used to compute the low-lying modes of the RPA/RPAx density response functions.  \n\nThe basic operation required for the eigenvalue decomposition is a number of loosely coupled DFpT calculations for different imaginary frequencies and trial potentials. Although the global scaling of the iterative approach is the same as for implementations based on the evaluation of the full response matrices $(N^{4})$ , the number of operation involved is 100 to 1000 times smaller [87], thus largely reducing the global scaling pre-factor. Moreover, the calculation can be parallelized very efficiently by distributing different trial potentials on different processors or groups of processors.  \n\nIn addition, the local EXX and RPA-correlation potentials can be computed through an optimized effective potential (OEP) scheme fully compatible with the eigenvalue decomposition strategy adopted for the evaluation of the EXX/RPA energy. Iterating the energy and the OEP calculations and using an effective mixing scheme to update the KS potential, a self-consistent minimization of the EXX/RPA functional can be achieved [89].  \n\n# 2.2.  Linear response and excited states without virtual orbitals  \n\nOne of the key features of modern DFT implementations is that they do not require the calculation of virtual (unoccupied) orbitals. This idea, first pioneered by Car and Parrinello in their landmark 1985 paper [53] and later adopted by many groups world-wide, found its way in the computation of excited-state properties with the advent of density-functional perturbation theory (DFpT) [90–93]. DFpT is designed to deal with static perturbations and its use is therefore restricted to those excitations that can be described in the Born–Oppenheimer approx­ imation, such as lattice vibrations. The main idea underlying DFpT is to represent the linear response of KS orbitals to an external perturbation as generic orbitals satisfying an orthogonality constraint with respect to the occupied-state manifold and a self-consistent Sternheimer equation  [94, 95], rather than as linear combinations of virtual orbitals (which would require the computation of all, or a large number, of them).  \n\nSubstantial progress has been made over the past decade, allowing one to extend DFpT to the dynamical regime, and thus simulate sizable portions of the optical and loss spectra of complex molecular and extended systems, without making any explicit reference to their virtual states. Although the Sternheimer approach can be easily extended to timedependent perturbations [96–98], its use is hampered in practice by the fact that a different Sternheimer equation has to be solved for each different value of the frequency of the perturbation. When the perturbation acting on the system vanishes, the frequency-dependent Sternheimer equation  becomes a non-Hermitian eigenvalue equation, whose eigenvalues are the excitation energies of the system. In the TDDFT community, this equation is known as the Casida equation [99, 100], which is the immediate translation to the DFT parlance of the timedependent Hartree–Fock equation  [101]. This approach to excited states is optimal in those cases where one is interested in a few excitations only, but can hardly be extended to continuous spectra, such as those arising in extended systems or above the ionization threshold of even finite ones. In those cases where extended portions of a continuous spectrum is required, a new method has been developed, based on the Lanczos bi-orthogonalization algorithm, and dubbed the Liouville– Lanczos approach to time-dependent density-functional perturbation theory (TDDFpT). This method allows one to reuse intermediate products of an iterative process, essentially identical to that used for static perturbations, to build dynamical response functions from which spectral properties can be computed for a whole wide spectral range at once [21, 22]. A similar approach to linear optical spectroscopy was proposed later, based on the multi-shift conjugate gradient algorithm [102], instead of Lanczos. This powerful idea has been generalized to the solution of the Bethe–Salpeter equation, which is formally very similar to the eigenvalue equations arising in TDDFpT [103–105], and to the computation of the polarization propagator and self-energy operator appearing in the $G W$ equations  [28, 29, 106]. It is presently exploited in several components of the Quantum ESPRESSO distribution, as well as in other advanced implementations of many-body perturbation theory [106].  \n\n2.2.1.  Static perturbations and vibrational spectroscopy.  The computation of vibrational properties in extended systems is one of the traditional fields of application of DFpT, as thoroughly described, e.g. in [93]. The latest releases of Quantum ESPRESSO feature the linear-response implementation of several new functionals in the van der Waals and DFT+U families. Explicit expressions of the XC kernel, implementation details, and a thorough benchmark are reported in [107] for the first case. As for the latter, DFpT $+\\mathrm{U}$ has been implemented for both the Dudarev [83] and $\\mathrm{\\DeltaDFT+U+J0}$ functionals [15], allowing one to account for electronic localization effects acting selectively on specific phonon modes at arbitrary wave-vectors, thus substantially improving the description of the vibrational properties of strongly correlated systems with respect to ‘standard’ LDA/GGA functionals. The current implementation allows for both norm-conserving and ultrasoft pseudopotentials, insulators and metals alike, also including the spin-polarized case. The implementation of $\\mathrm{\\DeltaDFpT+U}$ requires two main additional ingredients with respect to standard DFpT [108]. First, the dynamical matrix contains an additional term coming from the second derivative of the Hubbard term $E_{U}$ with respect to the atomic positions (denoted $\\lambda$ or $\\mu_{\\mathrm{-}}$ ), namely:  \n\n$$\n\\begin{array}{l}{{\\displaystyle\\Delta^{\\mu}(\\partial^{\\lambda}E_{U})=\\sum_{I\\sigma m m^{\\prime}}U^{I}\\left[\\frac{\\delta_{m m^{\\prime}}}{2}-n_{m m^{\\prime}}^{I\\sigma}\\right]\\Delta^{\\mu}\\left(\\partial^{\\lambda}n_{m m^{\\prime}}^{I\\sigma}\\right)}}\\\\ {{-\\sum_{I\\sigma m m^{\\prime}}U^{I}\\Delta^{\\mu}n_{m m^{\\prime}}^{I\\sigma}\\partial^{\\lambda}n_{m m^{\\prime}}^{I\\sigma},}}\\end{array}\n$$  \n\nwhere the notations are the same as in equation  (12). The symbols $\\partial$ and $\\Delta$ indicate, respectively, a bare derivative (leaving the KS wavefunctions unperturbed) and a total derivative (including also linear-response contributions). Second, in order to obtain a consistent electronic density response to the atomic displacements from the $\\mathrm{\\DeltaDFT+U}$ ground state, the perturbed KS potential $\\Delta V_{S C F}$ in the Sternheimer equation is augmented with the perturbed Hubbard potential $\\Delta^{\\lambda}V_{U}$ :  \n\n$$\n\\begin{array}{r}{\\begin{array}{r}{\\Delta^{\\lambda}V_{U}=\\displaystyle\\sum_{I\\sigma m m^{\\prime}}U^{I}\\left[\\frac{\\delta_{m m^{\\prime}}}{2}-n_{m m^{\\prime}}^{I\\sigma}\\right]\\times\\left[|\\Delta^{\\lambda}\\phi_{m^{\\prime}}^{I}\\rangle\\langle\\phi_{m}^{I}|+|\\phi_{m^{\\prime}}^{I}\\rangle\\langle\\Delta^{\\lambda}\\phi_{m}^{I}|\\right]}\\\\ {-\\displaystyle\\sum_{I\\sigma m m^{\\prime}}U^{I}\\Delta^{\\lambda}n_{m m^{\\prime}}^{I\\sigma}|\\phi_{m^{\\prime}}^{I}\\rangle\\langle\\phi_{m}^{I}|,}\\end{array}}\\end{array}\n$$  \n\nwhere the notations are the same as in equation  (13). The unperturbed Hamiltonian in the Sternheimer equation  is the $\\tt D F T+U$ Hamiltonian (including the Hubbard potential $V_{U,}$ . More implementation details will be given in a forthcoming publication [109].  \n\nApplications of $\\mathrm{\\DeltaDFpT+U}$ include the calculation of the vibrational spectra of transition-metal monoxides like MnO and NiO [108], investigations of properties of materials of geophysical interest like goethite [110, 111], iron-bearing [112, 113] and aluminum-bearing bridgmanite [114]. These results feature a significantly better agreement with experiment of the predictions of various lattice-dynamical properties, including the LO-TO and magnetically-induced TO splittings, as compared with standard LDA/GGA calculations.  \n\n2.2.2.  Dynamic perturbations: optical, electron energy loss, and magnetic spectroscopies.  Electronic excitations can be described in terms of the dynamical charge- and spin-density susceptibilities, which are accessible to TDDFT [115, 116]. In the linear regime the TDDFT equations can be solved using first-order perturbation theory. The time Fourier transform of the charge-density response, $\\tilde{n}^{\\prime}(\\mathbf{r},\\omega)$ , is determined by the projection over the unoccupied-state manifold of the Fourier transforms of the first-order corrections to the one-electron orbitals, $\\tilde{\\psi}_{{\\bf k}\\nu}^{\\prime}({\\bf r},\\omega)$ , [21–24, 117]. For each band index $\\mathbf{k}\\nu$ , two response orbitals $x_{\\mathbf{k}\\nu}$ and $y_{\\mathbf{k}\\nu}$ can be defined as  \n\n$$\nx_{\\mathbf{k}\\nu}(\\mathbf{r})=\\frac{1}{2}\\hat{Q}\\left(\\tilde{\\psi}_{\\mathbf{k}\\nu}^{\\prime}(\\mathbf{r},\\omega)+\\tilde{\\psi}_{-\\mathbf{k}\\nu}^{\\prime*}(\\mathbf{r},-\\omega)\\right)\n$$  \n\n$$\ny_{\\mathbf{k}\\nu}(\\mathbf{r})=\\frac{1}{2}\\hat{Q}\\left(\\tilde{\\psi}_{\\mathbf{k}\\nu}(\\mathbf{r},\\omega)-\\tilde{\\psi}_{-\\mathbf{k}\\nu}^{\\prime*}(\\mathbf{r},-\\omega)\\right),\n$$  \n\nwhere $\\hat{\\boldsymbol{Q}}$ is the projector on the unoccupied-state manifold. The response orbitals $x_{\\mathbf{k}\\nu}$ and $y_{\\mathbf{k}\\nu}$ can be collected in socalled batches $X=\\left\\{x_{\\mathbf{k}\\nu}\\right\\}$ and $Y=\\{y_{{\\bf k}\\nu}\\}$ , which uniquely determine the response density matrix. In a similar way, the perturbing potential $\\hat{V}^{\\prime}$ can be represented by the batch $Z=\\{z_{{\\bf k}\\nu}\\}=\\{\\hat{Q}\\hat{V}^{\\prime}\\psi_{{\\bf k}\\nu}\\}$ . Using these definitions, the linearresponse equations of TDDFpT take the simple form:  \n\n$$\n\\left(\\hbar\\omega-\\hat{\\mathcal{L}}\\right)\\cdot\\binom{X}{Y}=\\binom{0}{Z},\\qquad\\hat{\\mathcal{L}}=\\left(\\begin{array}{l l}{0}&{\\hat{D}}\\\\ {\\hat{D}+\\hat{K}}&{0}\\end{array}\\right),\n$$  \n\nwhere the super-operators $\\hat{D}$ and $\\hat{K}$ , which enter the definition of the Liouvillian super-operator, $\\hat{\\mathcal{L}}$ , are defined in terms of the unperturbed Hamiltonian and of the perturbed Hartreeplus-XC potential [21–24, 117]. This implies that a Liouvillian build costs roughly twice as much as a single iteration in time-independent DFpT. It is important to note that $\\hat{D}$ and $\\hat{K}$ , and therefore $\\hat{\\mathcal{L}}$ , do not depend on the frequency $\\omega$ . For this reason, when in equation  (28) the vector on the right-hand side, $(0,Z)^{\\top}$ , is set to zero, a linear eigenvalue equation  is obtained (Casida’s equation).  \n\nThe quantum Liouville equation  (28) can be seen as the equation  for the response density matrix operator $\\hat{\\rho}^{\\prime}(\\omega)$ , namely $(\\hbar\\omega-\\hat{\\mathcal{L}})\\cdot\\hat{\\rho}^{\\prime}(\\omega)=[\\hat{V}^{\\prime},\\hat{\\rho}^{\\circ}],$ , where $[\\cdot,\\cdot]$ is the commutator and $\\hat{\\rho}^{\\circ}$ is the ground-state density matrix operator. With this at hand, we can define a generalized susceptibility $\\chi_{A V}(\\omega)$ , which characterizes the response of an arbitrary one-electron Hermitian operator $\\hat{\\boldsymbol A}$ to the external perturbation $\\hat{V}^{\\prime}$ as  \n\n$$\n\\chi_{A V}(\\omega)=\\mathrm{Tr}\\left[\\hat{A}\\hat{\\rho}^{\\prime}(\\omega)\\right]=\\left\\langle\\hat{A}\\ \\right|(\\hbar\\omega-\\hat{\\mathcal{L}})^{-1}\\cdot[\\hat{V}^{\\prime},\\hat{\\rho}^{0}]\\right\\rangle,\n$$  \n\nwhere $\\langle\\cdot|\\cdot\\rangle$ denotes a scalar product in operator space. For instance, when both $\\hat{\\boldsymbol A}$ and $\\hat{V^{\\prime}}$ are one of the three Cartesian components of the dipole (position) operator, equation  (29) gives the dipole polarizability of the system, describing optical absorption spectroscopy [21, 22]; setting $\\hat{\\boldsymbol A}$ and $\\hat{V}^{\\prime}$ to one of the space Fourier components of the electron charge-density operator would correspond to the simulation of electron energy loss or inelastic $\\mathbf{\\boldsymbol{x}}$ -ray scattering spectroscopies, giving access to plasmon and exciton excitations in extended systems [25, 26]; two different Cartesian components of the Fourier transform of the spin polarization would give access to spectroscopies probing magnetic excitations (e.g. inelastic neutron scattering or spin-polarized electron energy loss) [118], and so on. When dealing with macroscopic electric fields, the dipole operator in periodic boundary conditions is treated using the standard DFpT prescription, as explained in [119, 120].  \n\nThe Quantum ESPRESSO distribution contains several codes to solve the Casida’s equation or to directly compute generalized susceptibilities according to equation  (29) and by solving equation (28) using different approaches for different pairs of $\\hat{A}/\\hat{V}^{\\prime}$ , corresponding to different spectroscopies. In particular, equation (28) can be solved iteratively using the Lanczos recursion algorithm, which allows one to avoid computationally expensive inversion of the Liouvillian. The basic principle of how matrix elements of the resolvent of an operator can be calculated using a Lanczos recursion chain has been worked out by Haydock et al [121, 122] for the case of Hermitian operators and diagonal matrix elements. The quantity of interest can be written as  \n\n$$\ng_{v}(\\omega)=\\left\\langle v\\left|(\\hbar\\omega-\\hat{L})^{-1}v\\right\\rangle.\n$$  \n\nA chain of vectors is defined by  \n\n$$\n\\begin{array}{r l}&{\\left|q_{0}\\right\\rangle=0}\\\\ &{\\left|q_{1}\\right\\rangle=\\left|v\\right\\rangle}\\\\ &{\\quad\\alpha_{n}=\\left\\langle q_{n}\\right|\\hat{L}q_{n}\\right\\rangle}\\\\ &{\\quad\\beta_{n+1}\\left|q_{n+1}\\right\\rangle=\\left(\\hat{L}-\\alpha_{n}\\right)\\left|q_{n}\\right\\rangle-\\beta_{n}\\left|q_{n-1}\\right\\rangle,}\\end{array}\n$$  \n\nwhere $\\beta_{n+1}$ is given by the condition $\\langle q_{n+1}|q_{n+1}\\rangle=1$ . The vectors $\\left|q_{n}\\right\\rangle$ created by this recursive chain are orthonormal.  \n\nFurthermore, the operator $\\hat{L}$ , written in the basis of these vectors, is tridiagonal. If one limits the chain to the $M$ first vectors $\\vert q_{0}\\rangle,\\vert q_{1}\\rangle,\\cdot\\cdot\\cdot,\\vert q_{M}\\rangle$ , then the resulting representation of $\\hat{L}$ is a $M\\times M$ square matrix $T_{M}$ which reads  \n\n$$\nT_{M}=\\left(\\begin{array}{c c c c c c}{{\\alpha_{1}}}&{{\\beta_{2}}}&{{0}}&{{\\cdots}}&{{0}}\\\\ {{}}&{{}}&{{}}&{{}}&{{}}\\\\ {{\\beta_{2}}}&{{\\alpha_{2}}}&{{\\beta_{3}}}&{{\\ddots}}&{{}}&{{\\vdots}}\\\\ {{}}&{{}}&{{}}&{{\\ddots}}&{{\\ddots}}&{{}}\\\\ {{0}}&{{\\beta_{3}}}&{{\\ddots}}&{{\\ddots}}&{{}}&{{0}}\\\\ {{\\vdots}}&{{\\ddots}}&{{\\ddots}}&{{\\ddots}}&{{\\alpha_{M-1}}}&{{\\beta_{M}}}\\\\ {{}}&{{}}&{{\\cdots}}&{{0}}&{{\\beta_{M}}}&{{\\alpha_{M}}}\\end{array}\\right).\n$$  \n\nUsing such a truncated representation of $\\hat{L}$ , the resolvent in equation (30) can be approximated as  \n\n$$\ng_{v}(\\omega)\\approx\\left\\langle v\\left|\\left(\\hbar\\omega-T_{M}\\right)^{-1}v\\right\\rangle.\n$$  \n\nThanks to the tridiagonal form of $T_{M}.$ the approximate resolvent can finally be written as a continued fraction  \n\n$$\ng_{v}(\\omega)\\approx\\frac{1}{\\hbar\\omega-\\alpha_{1}+\\frac{\\beta_{2}^{2}}{\\hbar\\omega-\\alpha_{2}+...}}.\n$$  \n\nNote that performing the recursion (31)–(34) is the computational bottleneck of this algorithm, while evaluating the continued fraction in equation (37) is very fast. The recursion being independent of the frequency $\\omega$ , a single recursion chain yields information about any desired number of frequencies, at negligible additional computational cost. It is also important to note that at any stage of the recursion chain, only three vectors need to be kept in memory, namely $|q_{n-1}\\rangle,|q_{n}\\rangle$ and $\\left|q_{n+1}\\right>$ . This is a considerable advantage with respect to the direct calculation of $N$ eigenvectors of $\\bar{L}$ where all $N$ vectors need to be kept in memory in order to enforce orthogonality.  \n\nThe Liouvillian $\\hat{\\mathcal{L}}$ in equation (28) is not a Hermitian operator. For this reason, the Lanczos algorithm presented above cannot be directly applied to the calculation of the generalized susceptibility (29). There are two distinct algorithms that can be applied in the non-Hermitian case. The non-Hermitian Lanczos bi-orthogonalization algorithm [22, 23] amounts to recursively applying the operator $\\hat{\\mathcal{L}}$ and its Hermitian conjugate $\\hat{\\mathcal{L}}^{\\dagger}$ to two Lanczos vectors $\\left|v_{n}\\right\\rangle$ and $\\left|w_{n}\\right\\rangle$ . In this way, a pair of bi-orthogonal basis sets is created. The operator $\\hat{\\mathcal{L}}$ can then be represented in this basis as a tridiagonal matrix, similarly to the Hermitian case, equation (35). The Liouvillian $\\hat{\\mathcal{L}}$ of TDDFpT belongs to a special class of non-Hermitian operators which are called pseudo-Hermitian [24, 123]. For such operators, there exists a second recursive algorithm to compute the resolvent— pseudo-Hermitian Lanczos algorithm— which is numerically more stable and requires only half the number of operations per Lanczos step [24, 123]. Both algorithms have been implemented in Quantum ESPRESSO. Because of its speed and numerical stability, the use of the pseudo-Hermitian method is recommended.  \n\nThis methodology has also been extended—presently only in the case of absorption spectroscopy—to employ hybrid functionals [24, 103, 104] (see section  2.1.1). In this case the calculation requires the evaluation of the linear response of the non-local Fock potential, which is readily available from the response density matrix, represented by the batches of response orbitals. The corresponding hybrid-functional Liouvillian features additional terms with respect to the definition in equation  (28), but presents a similar structure and similar mathematical properties. Accordingly, semi-local and hybrid-functional TDDFpT employ the same numerical algorithms in practical calculations.  \n\n2.2.2.1.  Optical absorption spectroscopy.  The turbo_ lanczos.x [23, 24] and turbo_davidson.x [24] codes are designed to simulate the optical response of molecules and clusters. turbo_lanczos.x computes the dynamical dipole polarizability (see equation (29)) of finite systems over extended frequency ranges without ever computing any eigenpairs of the Casida equation. This goal is achieved by applying a recursive non-Hermitian or pseudo-Hermitian Lanczos algorithm. The two flavours of the Lanczos algorithm implemented in turbo_lanczos.x are particularly suited in those cases where one is interested in the spectrum over a wide frequency range comprising a large number of individual excitations. In turbo_davidson.x a Davidson-like algorithm [124] is used to selectively compute a few eigenvalues and eigenvectors of $\\hat{\\mathcal{L}}$ . This is useful when very few low-lying excited states are needed and/or when the excitation eigenvector is explicitly needed, e.g. to compute ionic forces on excited potential energy surfaces, a feature that will be implemented in one of the forthcoming releases. Both turbo_lanczos.x and turbo_davidson.x are interfaced with the Environ module [18], to simulate the absorption spectra of complex molecules in solution using the self-consistent continuum solvation model [20] (see section 2.5.1).  \n\n2.2.2.2.  Electron energy loss spectroscopy.  The turbo_ eels.x code [26] computes the response of extended systems to an incoming beam of electrons or $\\mathrm{\\Delta}X$ rays, aimed at simulating electron energy loss (EEL) or inelastic $\\mathbf{\\boldsymbol{x}}$ -ray scattering (IXS) spectroscopies, sensitive to collective chargefluctuation excitations, such as plasmons. Similarly to the description of vibrational modes in a lattice by the PHonon package, here the perturbation can be represented as a sum of monochromatic components corresponding to different momenta, q, and energy transferred from the incoming electrons to electrons of the sample. The quantum Liouville equation  (28) in the batch representation can be formulated for individual $\\mathbf{q}$ components of the perturbation, which can be solved independently [25]. The recursive Lanczos algorithm is used to solve iteratively the quantum Liouville equation, much like in the case of absorption spectroscopy. The entire EEL/IXS spectrum is obtained in an arbitrarily wide energy range (up to the core-loss region) with only one Lanczos chain. Such a numerical algorithm allows one to compute directly the diagonal element of the charge-density susceptibility, see equation (29), by avoiding computationally expensive matrix inversions and multiplications characteristic of standard methods based on the solution of the Dyson equation [125].  \n\nThe current version of turbo_eels.x allows to explicitly account for spin–orbit coupling effects [126].  \n\n2.2.2.3. Magnetic spectroscopy.  The response of the system to a magnetic perturbation is described by a spin-density susceptibility matrix, see equation (29), labeled by the Cartesian components of the perturbing magnetic field and magnetization response, whose poles characterize spin-wave (magnon) and Stoner excitations. The methodology implemented in turbo_eels.x to deal with charge-density fluctuations has been generalized to spin-density fluctuations so as to deal with magnetic (spin-polarized neutron and electron) spectroscopies in extended systems. In the spin-polarized formulation of TDDFpT the time-dependent KS wave functions are two-component spinors $\\{\\psi_{{\\bf k}\\nu}^{\\sigma}({\\bf r},t)\\}$ ( $\\overrightharpoon{\\boldsymbol{\\sigma}}$ is the spin index), which satisfy a time-dependent Pauli-type KS equations  and describe a time-dependent spin-charge-density, $\\begin{array}{r}{n_{\\sigma\\sigma^{\\prime}}(\\mathbf{r},t)=\\sum_{\\mathbf{k}\\nu}\\psi_{\\mathbf{k}\\nu}^{\\sigma*}(\\mathbf{r},t)\\psi_{\\mathbf{k}\\nu}^{\\sigma^{\\prime}}(\\mathbf{r},t)}\\end{array}$ . Instead of using the latter quantity it is convenient to change variables and use the charge density $\\begin{array}{r}{n(\\mathbf{r},t)=\\sum_{\\sigma}n_{\\sigma\\sigma}(\\mathbf{r},t)}\\end{array}$ and the spin density $\\begin{array}{r}{\\mathbf{m}(\\mathbf{r},t)=\\mu_{\\mathrm{B}}\\sum_{\\sigma\\sigma^{\\prime}}\\pmb{\\sigma}_{\\sigma\\sigma^{\\prime}}n_{\\sigma^{\\prime}\\sigma}(\\mathbf{r},t)}\\end{array}$ , where $\\mu_{\\mathrm{B}}$ is the Bohr magneton and $\\sigma$ is the vector of Pauli matrices. In the linearresponse regime, the charge- and spin-density response $n^{\\prime}(\\mathbf{r},t)$ and $\\mathbf{m}^{\\prime}(\\mathbf{r},t)$ are coupled via the scalar and magnetic XC response potentials $V_{\\mathrm{xc}}^{\\prime}(\\mathbf{r},t)$ and $\\mathbf{B}_{\\mathrm{xc}}^{\\prime}(\\mathbf{r},t)$ , which are treated on a par with the Hartree response potential $V_{\\mathrm{H}}^{\\prime}(\\mathbf{r},t)$ depending only on $n^{\\prime}(\\mathbf{r},t)$ , and which all enter the linear-response time-dependent Pauli-type KS equations. The lack of timereversal symmetry in the ground state means that the TDDFpT equations have to be generalized to treat KS spinors at $\\mathbf{k}$ and $-\\mathbf{k}$ and various combinations with the $\\mathbf{q}$ vector. Moreover, this also implies that no rotation of batches is possible, as in equations (26) and (27), and a generalization of the Lanczos algorithm to complex arithmetics is required. At variance with the cumbersome Dyson’s equation approach, which requires the separate calculation and coupling of charge-charge, spinspin, and charge-spin independent-electron polarizabilities, in our approach the coupling between spin and charge fluctuations is naturally accounted for via Lanczos chains for the spinor KS response orbitals. The current implementation supports general non-collinear spin-density distributions, which allows us to account for spin–orbit interaction and magnetic anisotropy. All the details about the present formalism will be given in a forthcoming publication [118] and the corre­ sponding code will be made available in one of the next Quantum ESPRESSO releases.  \n\n# 2.2.3.  Many-body perturbation theory.  \n\nMany-body perturbation theory refers to a set of computational methods, based on quantum field theory, that are designed to calculate electronic and optical excitations beyond standard DFT [125]. The most popular among such methods are the $G W$ approximation and the Bethe–Salpeter equation  (BSE) approach. The former is intended to calculate accurate quasiparticle excitations, e.g. ionization energies and electron affinities in molecules, band structures in solids, and accurate band gaps in semiconductor and insulators. The latter is employed to study optical excitations by including electron–hole interactions.  \n\nIn the GW method the XC potential of DFT is corrected using a many-body self-energy consisting of the product of the electron Green’s function $G$ and the screened Coulomb interaction W [127, 128], which represents the lowest-order term in the diagrammatic expansion of the exact electron self-energy. In the vast majority of $G W$ implementations, the evaluation of $G$ and $W$ requires the calculation of both occupied and unoccupied KS eigenstates. The convergence of the resulting self-energy correction with respect to the number of unoccupied states is rather slow, and in many cases it constitutes the main bottleneck in the calculations. During the past decade there has been a growing interest in alternative techniques which only require the calculation of occupied electronic states, and several computational strategies have been developed [29, 129–131]. The common denominator to all these strategies is that they rely on linear-response DFpT and the Sternheimer equation, as in the PHonon package.  \n\nIn Quantum ESPRESSO the GW approximation is realized based on a DFpT representation of response and selfenergy operators, thus avoiding any explicit reference to unoccupied states. There are two different implementations: the GWL (GW-Lanczos) package [28, 29] and the SternheimerGW package [30]. The former focuses on efficient GW calcul­ ations for large systems (including disordered solids, liquids, and interfaces), and also supports the calculations of optical spectra via the Bethe–Salpeter approach [105]. The latter focuses on high-accuracy calculations of band structures, frequency-dependent self-energies, and quasi-particle spectral functions for crystalline solids. In addition to these, the WEST code [106], not part of the Quantum ESPRESSO distribution, relies on Quantum ESPRESSO as an external library to perform similar tasks and to achieve similar goals.  \n\n2.2.3.1. GWL.  The GWL package consists of four different codes. The pw4gww.x code reads the KS wave-functions and charge density previously calculated by PWscf and prepares a set of data which are used by code gww.x to perform the actual $G W$ calculation. While pw4gww.x uses the plane-wave representation of orbitals and charges and the same Quantum ESPRESSO environment as all other linear response codes, gww.x does not rely on any specific representation of the orbitals. Its parallelization strategy is based on the distribution of frequencies. Only a few basic routines, such as the MPI drivers, are common with the rest of Quantum ESPRESSO.  \n\nGWL supports three different basis sets for representing polarisability operators: (i) plane wave-basis set, defined by an energy cutoff; (ii) the basis set formed by the most important eigenvectors (i.e. corresponding to the highest eigenvalues) of the actual irreducible polarisability operator at zero frequency calculated through linear response; (iii) the basis set formed by the most important eigenvectors of an approximated polarisability operator. The last choice permits the control of the balance between accuracy and dimension of the basis. The GW scheme requires the calculation of products in real space of KS orbitals with vectors of the polarisability basis. These are represented in GWL through dedicated additional basis sets of reduced dimensions.  \n\nGWL supports only the $\\Gamma\\cdot$ point sampling of the BZ and considers only real wave-functions. However, ordinary $\\mathbf{k}$ -point sampling of the BZ can be used for the long-range part of the (symmetrized) dielectric matrix. These terms are calculated by the head.x code. In this way reliable calcul­ ations for extended materials can be performed using quite small simulation cells (with cell edges of the order of 20 Bohr). Self-consistency is implemented in GWL, although limited to the quasi-particle energies; the so-called vertex term, arising in the diagrammatic expansion of the self-energy, is not yet implemented.  \n\nUsually ordinary $G W$ calculations for transition elements require the explicit inclusion of semicore orbitals in the valence manifold, resulting in a significantly higher computational cost. To cope with this issue, an approximate treatment of semicore orbitals has been introduced in GWL as described in [132]. In addition to collinear spin polarization, GWL provides a fully relativistic non collinear implementation relying on the scalar relativistic calculation of the screened Coulomb interactions [133].  \n\nThe bse.x code of the GWL package performs BSE calcul­ ations and permits to evaluate either the entire frequencydependent complex dielectric function through the Lanczos algorithm or a discrete set of excited states and their energies through a conjugate gradient minimization. In contrast to ordinary implementations, bse.x scales as $N^{3}$ instead of $N^{4}$ with respect to the system size $N$ (e.g. the number of atoms) thanks to the use of maximally localized Wannier functions for representing the valence manifold [105]. The bse.x code, apart from reading the screened Coulomb interaction at zero frequency from a gww.x calculation, works as a separate code and uses the Quantum ESPRESSO environ­ ment. Therefore it could be easily be interfaced with other GW codes.  \n\n2.2.3.2. SternheimerGW.  The SternheimerGW package calculates the frequency-dependent GW self-energy and the corresponding quasiparticle corrections at arbitrary $\\mathbf{k}$ -points in the BZ. This feature enables accurate calculations of band structures and effective masses without resorting to interpolation. The availability of the complete $G W$ self-energy (as opposed to the quasiparticle shifts) makes it possible to calculate spectral functions, for example including plasmon satellites [134]. The spectral function can be directly compared to angle-resolved photoelectron spectroscopy (ARPES) experiments. In SternheimerGW the screened Coulomb interaction W is evaluated for wave-vectors in the irreducible BZ by exploiting crystal symmetries. Calculations of $G$ and $W$ for multiple frequencies $\\omega$ rely on the use of multishift linear system solvers that construct solutions for all frequencies from the solution of a single linear system [131, 135]. This method is closely related to the Lanczos approach. The convolution in the frequency domain required to obtain the self energy from $G$ and $W$ can be performed either on the real axis or the imaginary axis. Padé functions are employed to perform approximate analytic continuations from the imaginary to the real frequency axis; the standard Godby-Needs plasmon pole model is also available to compare with literature results. The stability and portability of the SternheimerGW package are verified via a test-suite and a Buildbot test-farm (see section 3.6).  \n\n# 2.3.  Other spectroscopies  \n\n2.3.1.  QE-GIPAW: nuclear magnetic and electron paramagn­ etic resonance.  The QE-GIPAW package allows for the calcul­ation of various physical parameters measured in nuclear magnetic resonance (NMR) and electron paramagn­ etic resonance (EPR) spectroscopies. These encompass (i) NMR chemical shift tensors and magnetic susceptibility, (ii) electric field gradient (EFG) tensors, (iii) EPR $\\mathrm{\\bf{g}}$ -tensor, and (iv) hyperfine coupling tensor.  \n\nIn QE-GIPAW, the NMR and EPR parameters are obtained from the orbital linear response to an external uniform magn­ etic field. The response depends critically upon the exact shape of the electronic wavefunctions near the nuclei. Thus, all-electron wavefunctions are reconstructed from the pseudo-wavefunctions in a gauge- and translationally invariant way using the gauge-including projector augmented-wave (GIPAW) method [136]. The description of a uniform magnetic field within periodic boundary conditions is achieved by the longwavelength limit $(q\\ll1)$ of a sinusoidally modulated field in real space. In practice, for each $\\mathbf{k}$ point, we calculate the first order change of the wavefunctions at $\\mathbf{k}+\\mathbf{q}$ , where q runs over a star of 6 points. The magnetic susceptibility and the induced orbital currents are then evaluated by finite differences, in the limit of small $q$ . The induced magnetic field at the nucleus, which is the central quantity in NMR, is obtained from the induced current via the Biot–Savart law. In QE-GIPAW, the NMR orbital chemical shifts and magnetic susceptibility can be calculated both for insulators [34] and for metals [137] (the additional contribution for metals coming from the spinpolarization of valence electrons, namely the Knight shift, can also be computed but it is not yet ready for production at the time of writing). Similarly to the NMR chemical shift, the EPR g-tensor is calculated as the cross product of the induced current with the spin–orbit operator [138].  \n\nFor the quantities defined in zero magnetic field, namely the EFG, Mössbauer and relativistic hyperfine tensors, the usual PAW reconstruction of the wavefunctions is sufficient and these are computed as described in [139, 140]. The hyperfine Fermi contact term, proportional to the spin density evaluated at the nuclear coordinates, however requires the relaxation of the core electrons in response to the magnetization of valence electrons. We implemented the core relaxation in perturbation theory, according to [141]. Basically we compute the spherically averaged PAW spin density around each atom. Then we compute the change of the XC potential, $\\Delta V_{\\mathrm{XC}}$ , on a radial grid, and compute in perturbation theory the core radial wavefunction, both for spin up and spin down. This provides an extra contribution to the Fermi contact, in most cases opposite in sign to and as large as that of valence electrons.  \n\nBy combining the quadrupole coupling constants derived from EFG tensors and hyperfine splittings, electron nuclear double resonance (ENDOR) frequencies can be calculated. Applications highlighting all these features of the QE-GIPAW package can be found in [142]. These quantities are also needed to compute NMR shifts in paramagnetic systems, like novel cathode materials for Li batteries [143]. Previously restricted to norm-conserving pseudopotentials only, all features are now applicable using any kind of pseudization scheme and to PAW, following the theory described in [144]. The use of smooth pseudopotentials allows for the calcul­ation of chemical shifts in systems with several hundreds of atoms [145].  \n\nThe starting point of all QE-GIPAW calculations is a previous calculation of KS orbitals via PWscf. Hence, much like other linear response routines, the QE-GIPAW code uses many subroutines of PWscf and of the linear response module. As usually done in linear response methods, the response of the unoccupied states is calculated using the completeness relation between occupied and unoccupied manifolds [146]. As a result, for insulating as well as metallic systems, the linear response of the system is efficiently obtained without the need to include virtual orbitals.  \n\nAs an alternative to linear response method, the theory of orbital magnetization via Berry curvature [147, 148] can be used to calculate the NMR [149] and EPR parameters [150]. Specifically, it can be shown that the variation of the orbital magnetization $M^{\\mathrm{{orb}}}$ with respect to spin flip is directly $\\mathrm{g}$ einsstohr:e $\\begin{array}{r}{g_{\\mu\\nu}=g_{e}-\\frac{2}{\\alpha S}\\mathbf{e}_{\\mu}\\cdot\\mathbf{M}^{\\mathrm{orb}}(\\mathbf{e}_{\\nu})}\\end{array}$ ,hewthoetrael $g_{e}=2.002319$ $\\alpha$ $S$ spin, e are Cartesian unit vectors, provided that the spin–orbit interaction is explicitly considered in the Hamiltonian. This converse method of calculating the g-tensor has been implemented in an older version of QE-GIPAW. It is especially useful in critical cases where linear response is not appropriate, e.g. systems with quasi-degenerate HOMO-LUMO levels. A demonstration of this method applied to delocalized conduction band electrons can be found in [151].  \n\nThe converse method will be shortly ported into the current QE-GIPAW and we will explore the possibility of computing in a converse way the Knight shift as the response to a small nuclear magnetic dipole. The present version of the code allows for parameter-free calculations of g-tensors, hyperfine splittings, and ENDOR frequencies also for systems with total spin $S>1/2$ . Such triplet or even higher-spin states give rise to additional spin-spin interactions, that can be calculated within the magnetic dipole-dipole interaction approximation. This interaction results in a fine structure which can be measured in zero magnetic field. This so-called zero-field splitting is being implemented following the methodology described in [152].  \n\n2.3.2.  XSpectra: L2,3 x-ray absorption edges.  The XSpectra code [153, 154] has been extended to the calcul­ ation of $\\mathbf{\\boldsymbol{X}}$ -ray absorption spectra at the $L_{2,3}$ -edges [155]. The XSpectra code uses the self-consistent charge density produced by PWscf and acts as a post processing tool [153, 154, 156]. The spectra are calculated for the $L_{2}$ edge, while the $L_{3}$ edge is obtained by multiplying by two (single-particle statistical branching ratio) the $L_{2}$ edge spectrum and by shifting it by the value of the spin–orbit splitting of the $2p_{1/2}$ core levels of the absorbing atom. The latter can be taken either from a DFT relativistic all-electron calculation on the isolated atom, or from experiments.  \n\nIn practice, the $L_{3}$ edge is obtained from the $L_{2}$ with the spectra_correction.x tool. Such tool contains a table  of experimental $2p$ spin–orbit splittings for all the elements. In addition to computing $L_{3}$ edges, spectra_correction.x allows one to remove states from the spectrum below a certain energy, and to convolute the calculated spectrum with more elaborate broadenings. These operations can be applied to any edge.  \n\nTo evaluate the x-ray absorption spectrum for a system containing various atoms of the same species but in different chemical environments, one has to sum the contribution by each atom. This could be the case, for example of an organic molecule containing various C atoms in inequivalent sites. Such individual contributions can be computed separately by XSpectra, and the tool molecularnexafs.x allows one to perform their weighted sum taking into account the proper energy reference (initial and final state effects) [157, 158]. One should in fact notice that the reference for initial state effects will depend upon the environment (e.g. the vacuum level for gas phase molecules, or the Fermi level for molecules adsorbed on a metal).  \n\n# 2.4.  Other lattice-dynamical and thermal properties  \n\n2.4.1.  thermo_pw: thermal properties from the quasi-harmonic approximation.  thermo_pw [31] is a collection of codes aimed at computing various thermodynamical quantities in the quasi-harmonic approximation. The key ingredient is the vibrational contribution, $F_{\\mathrm{ph}}$ , to the Helmholtz free energy at temperature $T$ :  \n\n$$\nF_{\\mathrm{ph}}=k_{\\mathrm{B}}T\\sum_{\\mathbf{q},\\nu}\\ln\\left[2\\sinh\\left(\\frac{\\hbar\\omega_{\\mathbf{q}\\nu}}{2k_{\\mathrm{B}}T}\\right)\\right],\n$$  \n\nwhere $\\omega_{\\mathbf{q},\\nu}$ are phonon frequencies at wave-vector ${\\bf q},k_{\\mathrm{B}}$ is the Boltzmann constant. thermo_pw works by calling Quantum ESPRESSO routines from PWscf and PHonon, that perform one of the following tasks: (i) compute the DFT total energy and possibly the stress for a given crystal structure; (ii) compute for the same system the electronic band structure along a specified path; (iii) compute for the same system phonon frequencies at specified wave-vectors. Using such quantities, thermo_pw can calculate numerically the derivatives of the free energy with respect to the external parameters (e.g. different volumes). Several calls to such routines, with slightly different geometries, are typically needed in a run. All such tasks can be independently performed on different groups of processors (called images).  \n\nWhen the tasks carried out by different images require approximately the same time, or when the amount of numer­ ical work needed to accomplish each task is easy to estimate $a$ priori, it would be possible to statically assign tasks to images at the beginning of the run so that images do not need to communicate during the run. However, such conditions are seldom met in thermo_pw and therefore it would be impossible to obtain a good load balancing between images. thermo_pw takes advantage of an engine that controls these different tasks in an asynchronous way, dynamically assigning tasks to the images at run time.  \n\nAt the core of thermo_pw there is a module mp_asyn, based on MPI routines, that allows for asynchronous communication between different images. One of the images is the ‘master’ and assigns tasks to the other images (the ‘slaves’) as soon as they communicate that they have accomplished the previously assigned task. The master image also accomplishes some tasks but once in a while, with negligible overhead, it checks if there is an image available to do some work; if so, it assigns to it the next task to do. The code stops when the master recognizes that all the tasks have been done and communicates this information to the slaves. The routines of this communication module are quite independent of the thermo_pw variables and in principle can be used in conjunction with other codes to set up complex workflows to be executed in a massively parallel environment. It is assumed that each processor of each image reads the same input and that the only information that the image needs to synchronize with the other images is which tasks to do. The design of thermo_pw makes it easily extensible to the calculation of new properties in an incremental way.  \n\n2.4.2.  thermal2: phonon–phonon interaction and thermal transport.  Phonon–phonon interaction (ph–ph) plays a role in different physical phenomena: phonon lifetime (and its inverse, the linewidth), phonon-driven thermal transport in insulators or semi-metals, thermal expansion of mat­erials. Ph–ph is possible because the harmonic Hamiltonian of ionic motion, of which phonons are stationary states, is only approximate. At first order in perturbation theory we have the third derivative of the total energy with respect to three phonon perturbations, which we compute ab initio. This calcul­ ation is performed by the d3q code via the $2n+1$ theorem [32, 159]. The d3q code is an extension of the old D3 code, which only allowed the calculation of zone-centered phonon lifetimes and of thermal expansion. The current version can compute the three-phonon matrix element of arbitrary wavevectors $D^{(3)}(\\mathbf{q}_{1},\\mathbf{q}_{2},\\mathbf{q}_{3})=\\partial^{3}E/\\partial u_{\\mathbf{q}_{1}}\\partial u_{\\mathbf{q}_{2}}\\partial u_{\\mathbf{q}_{3}}.$ where $u$ are the phonon displacement patterns, the momentum conservation rule imposes ${\\bf q}_{1}+{\\bf q}_{2}+{\\bf q}_{3}=0$ . The current version of the code can treat any kind of crystal geometry, metals and insulators, both local density and gradient-corrected functionals, and multi-projector norm-conserving pseudopotentials. Ultrasoft pseudopotentials, PAW, spin polarization and noncollinear magnetization are not implemented. Higher order derivative of effective charges [160] are not implemented.  \n\nThe ph–ph matrix elements, computed from linear response, can be transformed, via a generalized Fourier transform, to the real-space three-body force constants which could be computed in a supercell by finite difference derivation:  \n\n$$\nD^{(3)}(\\P_{1},\\P_{2},\\P_{3})=\\sum_{\\mathbf{R}^{\\prime},\\mathbf{R}^{\\prime\\prime}}\\mathrm{e}^{-2\\mathrm{i}\\pi(\\mathbf{R}^{\\prime}\\cdot\\mathbf{q}_{2}+\\mathbf{R}^{\\prime\\prime}\\cdot\\mathbf{q}_{3})}F^{(3)}(\\mathbf{0},\\mathbf{R}^{\\prime},\\mathbf{R}^{\\prime\\prime}),\n$$  \n\nwhere $F^{3}({\\bf0},{\\bf R}^{\\prime},{\\bf R}^{\\prime\\prime})=\\partial^{3}E/\\partial\\tau\\partial(\\tau^{\\prime}+{\\bf R}^{\\prime})\\partial(\\tau^{\\prime\\prime}+{\\bf R}^{\\prime\\prime})$ is the derivative of the total energy w.r.t. nuclear positions of ions with basis $\\tau,\\tau^{\\prime}$ , $\\tau^{\\prime\\prime}$ from the unit cells identified by direct lattice vectors 0 (the origin), $\\mathbf{R}^{\\prime}$ and $\\mathbf{R}^{\\prime\\prime}$ . The sum over $\\ensuremath{\\mathbf{R}}^{\\prime}$ and $\\mathbf{R}^{\\prime\\prime}$ runs, in principle, over all unit cells, however the terms of the sum quickly decay as the size of the triangle $\\mathbf{0}-\\mathbf{R}^{\\prime}-\\mathbf{R}^{\\prime\\prime}$ increases. The real-space finite-difference calculation, performed by some external softwares [168], has some advantages: it is easier to implement and it can readily include all the capabilities of the self-consistent code; on the other hand it is much more computationally expensive than the linearresponse method we use, its cost scaling with the cube of the supercell volume, or the $9_{\\mathrm{th}}$ power of the number of side units of an isotropic system. We use the real-space formalism to apply the sum rule corresponding to translational invariance to the matrix elements. This is done with an iterative method that alternatively enforces the sum rule on the first matrix index and restores the invariance on the order of the derivations. We also use the real-space force constants to Fourierinterpolate the ph–ph matrices on a finer grid, assuming that the contrib­ution from triangles $\\mathbf{0}-\\mathbf{R}^{\\prime}-\\mathbf{R}^{\\prime\\prime}$ which we have not computed is zero; it is important in this stage to consider the periodicity of the system.  \n\nFrom many-body theory we get the first-order phonon linewidth [161] $(\\gamma_{\\nu})$ of mode $\\nu$ at $\\mathbf{q}$ , which is a sum over all the possible $N_{\\mathbf{q}}$ ’s final and initial states $(\\mathbf{q}^{\\prime},\\nu^{\\prime},\\nu^{\\prime\\prime})$ with conservation of energy $(\\hbar\\omega)$ and momentum $(\\mathbf{q}^{\\prime\\prime}=-\\mathbf{q}-\\mathbf{q}^{\\prime})$ , BoseEinstein occupations $(n(\\mathbf{q},\\nu)=(\\exp(\\hbar\\omega_{\\mathbf{q},\\nu}/k_{\\mathrm{B}}T)-1)^{-1})$ and an amplitude $V^{(3)}$ , proportional to the ${\\cal D}^{(3)}$ matrix element but renormalized with phonon energies and ion masses:  \n\n$$\n\\begin{array}{l}{{\\displaystyle\\gamma_{\\mathbf{q},\\nu}=\\frac{\\pi}{\\hbar^{2}N_{\\mathbf{q}}}\\sum_{\\mathbf{q}^{\\prime},\\nu^{\\prime},\\nu^{\\prime\\prime}}\\left\\vert V^{(3)}(\\mathbf{q}\\nu,\\mathbf{q}^{\\prime}\\nu^{\\prime},\\mathbf{q}^{\\prime\\prime}\\nu^{\\prime\\prime})\\right\\vert}}\\\\ {{\\displaystyle\\qquad\\times\\left[(1+n_{\\mathbf{q}^{\\prime},\\nu^{\\prime}}+n_{\\mathbf{q}^{\\prime\\prime},\\nu^{\\prime\\prime}})\\delta(\\omega_{\\mathbf{q},\\nu}-\\omega_{\\mathbf{q}^{\\prime},\\nu^{\\prime}}-\\omega_{\\mathbf{q}^{\\prime\\prime},\\nu^{\\prime\\prime}})\\right.}}\\\\ {{\\displaystyle\\qquad\\left.+2(n_{\\mathbf{q}^{\\prime},\\nu^{\\prime}}-n_{\\mathbf{q}^{\\prime\\prime},\\nu^{\\prime\\prime}})\\delta(\\omega_{\\mathbf{q},\\nu}+\\omega_{\\mathbf{q}^{\\prime},\\nu^{\\prime}}-\\omega_{\\mathbf{q}^{\\prime\\prime},\\nu^{\\prime\\prime}})\\right].}}\\end{array}\n$$  \n\nThis sum is computed in the thermal2 suite, which is bundled with d3q. A similar expression can be written for the phonon scattering probability which appears in the Boltzmann transport equation. In order to properly converge the integral of the Dirac delta function, we express it as finite-width Gaussian function and use an interpolation grid. This equation can be solved either exactly or in the single mode approximation (SMA) [162]. The SMA is a good tool at temperatures comparable to or larger than the Debye temperature, but is known to be inadequate at low temperatures [163, 164] or in the case of 2D materials [165–167]. The exact solution is computed using a variational form, minimized via a preconditioned conjugate gradient algorithm, which is guaranteed to converge, usually in less than 10 iterations [33].  \n\naddition to using our force constants from DFpT, the code supports importing 3-body force constants computed via finite differences with the thirdorder.py code [168]. Parallelization is implemented with both MPI (with great scalability up to thousand of CPUs) and OpenMP (optimal for memory reduction).  \n\n2.4.3.  EPW: electron–phonon coefficients from Wannier i­nterpolation.  The electron–phonon-Wannier (EPW) package is designed to calculate electron–phonon coupling using an ultra-fine sampling of the BZ by means of Wannier interpolation. EPW employs the relation between the electron–phonon matrix elements in the Bloch representation $g_{m n\\nu}(\\mathbf{k,q})$ , and in the Wannier representation, $g_{i j\\kappa\\alpha}(\\mathbf{R},\\mathbf{R}^{\\prime})$ [169],  \n\n$$\ng_{m n}(\\mathbf{k},\\mathbf{q})=\\sum_{\\mathbf{R},\\mathbf{R}^{\\prime}}\\mathrm{e}^{\\mathrm{i}\\left(\\mathbf{k}\\cdot\\mathbf{R}+\\mathbf{q}\\cdot\\mathbf{R}^{\\prime}\\right)}\\sum_{i j\\kappa\\alpha}U_{m i\\mathbf{k}+\\mathbf{q}}g_{i j\\kappa\\alpha}(\\mathbf{R},\\mathbf{R}^{\\prime})U_{j n\\mathbf{k}}^{\\dagger}u_{\\kappa\\alpha,\\mathbf{q}\\nu},\n$$  \n\nin order to interpolate from coarse $\\mathbf{k}$ -point and $\\mathbf{q}$ -point grids into dense meshes. In the above expression $\\mathbf{k}$ and $\\mathbf{q}$ represent the electron and phonon wave-vector, respectively, the indices $m,n$ and $i,j$ refer to Bloch states and Wannier states, respectively, and $\\mathbf{R},\\mathbf{R}^{\\prime}$ are direct lattice vectors. The matrices $U_{m i\\mathbf{k}}$ are unitary transformations and the vector $u_{\\kappa\\alpha,\\mathbf{q}\\nu}$ is the displacements of the atom $\\kappa$ along the Cartesian direction $\\alpha$ for the phonon of wavevector $\\mathbf{q}$ and branch $\\nu$ . The interpolation is performed with ab initio accuracy by relying on the localization of maximally-localized Wannier functions [170]. During its execution EPW invokes the Wannier90 software [171] in library mode in order to determine the matrices $U_{m i\\mathbf{k}}$ on the coarse $\\mathbf{k}$ -point grid.  \n\nEPW can be used to compute the following physical properties: the electron and phonon linewidths arising from electron–phonon interactions; the scattering rates of electrons by phonons; the total, averaged electron–phonon coupling strength; the electrical resistivity of metals, see figure 2(b); the critical temperature of electron–phonon superconductors; the anisotropic superconducting gap within the Eliashberg theory, see figure  2(c); the Eliashberg spectral function, transport spectral function, see figure 2(d) and the nesting function. The calculation of carrier mobilities using the Boltzmann transport equation in semiconductors is under development.  \n\nThe epw.x code exploits crystal symmetry operations (including time reversal) in order to limit the number of phonon calculations to be performed using PHonon to the irreducible wedge of the BZ. The code supports calculations of electron–phonon couplings in the presence of spin–orbit coupling. The current version does not support spin-polarized calculations, ultrasoft pseudopotentials nor the PAW method. As shown in figure 2(a), epw.x scales reasonably up to 2000 cores using MPI. A test farm (see section 3.6) was set up to ensure portability of the code on many architecture and compilers. Detailed information about the EPW package can be found in [27].  \n\nOn top of intrinsic ph–ph events, the thermal2 codes can also treat isotopic disorder and substitution effects and finite transverse dimension using the Casimir formalism. In  \n\n2.4.4.  Non-perturbative approaches to vibrational spectroscopies.  Although DFpT is in many ways the state of the art in the simulation of vibrational spectroscopies in extended systems, and in fact one of defining features of Quantum ESPRESSO, it is sometimes convenient to compute lattice-dynamical properties, the response to macroscopic electric fields, or combinations thereof (such as e.g. the infrared or Raman activities), using non-perturbative methods. This is so because DFpT requires the design of dedicated codes, which have to be updated and maintained separately, and which therefore not always follow the pace of the implementation of new features, methods, and functionals (such as e.g. DFT $+\\mathrm{U}$ , vdW-DF, hybrid functionals, or ACBN0 [172]) in their ground-state counterparts. Such a non-perturbative approach is followed in the FD package, which implements the ‘frozen-phonon’ method for the computation of phonons and vibrational spectra: the interatomic Force Constants (IFCs) and electronic dielectric constant are computed as finite differences of forces and polarizations, with respect to finite atomic displacements or external electric fields, respectively [173, 174]. IFC’s are computed in two steps: first, code fd.x generates the symmetry-independent displacements in an appropriate supercell; after the calculations for the various displacements are completed, code fd_ifc.x reads the forces and generates IFC’s. These are further processed in matdyn.x, where non-analytical long-ranged dipolar terms are subtracted out from the IFCs following the recipe of [175]. The calculation of dielectric tensor and of the Born effective charges proceeds from the evaluation of the electronic susceptibility following the method proposed by Umari and Pasquarello [174], where the introduction of a non local energy functional $E_{\\mathrm{tot}}^{\\pmb{\\mathscr{E}}}[\\psi]=E^{0}[\\psi]-\\pmb{\\mathscr{E}}\\cdot(\\mathbf{P}^{\\mathrm{ion}}+\\mathbf{P}^{\\mathrm{el}}[\\psi])$ allows to compute the electronic structure for periodic systems under finite homogeneous electric fields. $E^{0}$ is the ground state total energy in the absence of external electric fields; $\\mathbf{P}^{\\mathrm{ion}}$ is the usual ionic polarization, and $\\mathbf{P}^{\\mathrm{el}}$ is given as a Berry phase of the manifold of the occupied bands [176]. The high-frequency dielectric tensor $\\epsilon^{\\infty}$ is then computed as $\\epsilon_{i j}^{\\infty}=\\delta_{i,j}+4\\pi\\chi_{i j}$ , while Born effective-charge tensors $Z_{I,i j}^{*}$ are obtained as the polarization induced along the direction $i$ by a unit displacement of the Ith atom in the $j$ direction; alternatively, as the force induced on atom $I$ by an applied electric field, $\\varepsilon$ .  \n\n![](images/89817fd2d07675c5c860ec9ef67ddd85b14de9b9622748f2827ba09d83216060.jpg)  \nFigure 2.  Examples of calculations that can be performed using EPW. (a) Parallel scaling of EPW on ARCHER Cray XC30. This example corresponds to the calculation of electron–phonon couplings for wurtzite GaN. The parallelization is performed over $\\mathbf{k}$ -points using MPI. (b) Calculated temperature-dependent resistivity of Pb by including/neglecting spin–orbit coupling. (c) Calculated superconducting gap function of $\\mathbf{MgB}_{2}$ , color-coded on the Fermi surface. (d) Eliashberg spectral function $\\alpha^{2}{\\cal F}$ and transport spectral function $\\alpha^{2}F_{\\mathrm{tr}}$ of $\\mathrm{Pb}$ . (b)–(d) Reprinted from [27], Copyright 2016, with permission from Elsevier.  \n\nThe calculation of the Raman spectra proceeds along similar lines. Within the finite-field approach, the Raman tensor is evaluated in terms of finite differences of atomic forces in the presence of two electric fields [177]. In practice, the tensor $\\chi_{i j I k}^{(1)}$ is obtained from a set of calculations combining finite electric fields along different Cartesian directions. $\\chi_{i j I k}^{(1)}$ is then symmetrized to recover the full symmetry of the structure under study.  \n\n# 2.5.  Multi-scale modeling  \n\n2.5.1.  Environ: self-consistent continuum solvation embedding model.  Continuum models are among the most popular multiscale approaches to treat solvation effects in the quantumchemistry community [178]. In this class of models, the degrees of freedom of solvent molecules are effectively integrated out and their statistically-averaged effects on the solute are mimicked by those of a continuous medium surrounding a cavity in which the solute is thought to dwell. The most important interaction usually handled with continuum models is the electrostatic one, in which the solvent is described as a dielectric continuum characterized by its experimental dielectric permittivity.  \n\nFollowing the original work of Fattebert and Gygi [179] , a new class of continuum models was designed, in which a smooth transition from the QM-solute region to the continuumenvironment region of space is introduced and defined in terms of the electronic density of the solute. The corresponding free energy functional is optimized using a fully variational approach, leading to a generalized Poisson equation  that is solved via a multi-grid solver [179]. This approach, ideally suited for plane-wave basis sets and tailored for MD simulations, has been featured in the Quantum ESPRESSO distribution since v. 4.1. This approach was recently revised [18], by defining an optimally smooth QM/continuum transition, reformulated in terms of iterative solvers [180] and extended to handle in a compact and effective way non-electrostatic interactions [18]. The resulting self-consistent continuum solvation (SCCS) model, based on a very limited number of physically justified parameters, allows one to reproduce experimental solvation energies for aqueous solutions of neutral [18] and charged [181] species with accuracies comparable to or higher than state-of-the-art quantum-chemistry packages.  \n\nThe SCCS model involves different embedding terms, each representing a specific interaction with an external continuum environment and contributing to the total energy, KS potential, and interatomic forces of the embedded QM system. Every contribution may depend explicitly on the ionic (rigid schemes) and/or electronic (self-consistent or soft schemes) degrees of freedom of the embedded system. All the different terms are collected in the stand-alone Environ module [182]. The present discussion refers to release 0.2 of Environ, which is compatible with Quantum ESPRESSO starting from versions 5.1. The module requires a separate input file with the specifications of the environment interactions to be included and of the numerical parameters required to compute their effects. Fully parameterized and tuned SCCS environ­ ments, e.g. corresponding to water solutions for neutral and charged species, are directly available to the users. Otherwise individual embedding terms can be switched on and tuned to the specific physical conditions of the required environ­ ment. Namely, the following terms are currently featured in Environ:  \n\n•\tSmooth continuum dielectric, with the associated generalized Poisson problem solved via a direct iterative approach or through a preconditioned conjugate gradient algorithm [180].   \n•\tElectronic enthalpy functional, introducing an energy term proportional to the quantum-volume of the system and able to describe finite systems under the effect of an applied external pressure [183].   \n•\tElectronic cavitation functional, introducing an energy term proportional to the quantum-surface able to describe free energies of cavitation and other surface-related interaction terms [184].   \n•\tParabolic corrections for periodic boundary conditions in aperiodic and partially periodic (slab) systems [19, 185].   \n•\tFixed dielectric regions, allowing for the modelling of complex inhomogenous dielectric environments.  \n\n•\tFixed Gaussian-smoothed distributions of charges, allowing for a simplified modelling of countercharge distributions, e.g. in electrochemical setups.  \n\nDifferent packages of the Quantum ESPRESSO distribution have been interfaced with the Environ module, including PWscf, CP, PWneb, and turboTDDFT, the latter featuring a linear-response implementation of the SCCS model (see section  2.2.2). Moreover, continuum environ­ ment effects are fully compatible with the main features of Quantum ESPRESSO, and in particular, with reciprocal space integration and smearing for metallic systems, with both norm-conserving and ultrasoft pseudopotentials and PAW, with all XC functionals.  \n\n2.5.2.  QM–MM.  QM–MM was implemented in v.5.0.2 using the method documented in [40]. Such methodology accounts for both mechanical and electrostatic coupling between the QM (quantum-mechanical) and MM (molecular-mechanics) regions, but not for bonding interactions (i.e. bonds between the QM and MM regions). In practice, we need to run two different codes, Quantum ESPRESSO for the QM region and a classical force-field code for the MM region, that communicate atomic positions, forces, electrostatic potentials.  \n\nLAMMPS [39] is the software chosen to deal with the classical (MM) degrees of freedom. This is a well-known and well-maintained package, released under an opensource license that allows redistribution together with Quantum ESPRESSO. The communications between the QM and MM regions use a ‘shared memory’ approach: the MM code runs on a master node, communicates directly via the memory with the QM code, which is typically running on a massively parallel machine. Such approach has some advantages: the MM part is typically much faster than the QM one and can be run in serial execution, wasting no time on the HPC machine; there is a clear and neat separation between the two codes, and very small code changes in either codes are needed. It has however also a few drawbacks, namely: the serial computation of the MM part may become a bottleneck if the MM region contains many atoms; direct access to memory is often restricted for security reasons on HPC machines.  \n\nAn alternative approach has been implemented in v.5.4. A single (parallel) executable runs both the MM and the QM codes. The two codes exchange data and communicate via MPI. This approach is less elegant than the previous one and requires a little bit more coding, but its implementation is quite straightforward thanks also to the changes in the logic of parallelization mentioned in section 3.4. The coupling of the two codes has required some modifications also to the qmmm library inside LAMMPS and to the related fix qmmm (a ‘fix’ in LAMMPS is any operation that is applied to the system during the MD run). In particular, electrostatic coupling has been introduced, following the approach described in [186]. The great advantage of this approach is that its performance on HPC machines is as good as the separate performances of the QM and MM codes. Since LAMMPS is very well parallelized, this is a significant advantage if the MM region contains many atoms. Moreover, it can be run without restrictions on any parallel machine. This new QM–MM implementation is an integral part of the Quantum ESPRESSO distribution and will soon be included into LAMMPS as well (the ‘fix’ is currently under testing) and it is straightforward to compile and execute it.  \n\n# 2.6.  Miscellaneous feature enhancements and additions  \n\n2.6.1.  Fully relativistic projector augmented-wave method.  By applying the PAW formalism to the equations  of relativistic spin density functional theory [187, 188], it is possible to obtain the fully relativistic PAW equations  for four-comp­ onent spinor pseudo-wavefunctions [16]. In this formalism the pseudo-wavefunctions can be written in terms of large $|\\tilde{\\Psi}_{i,\\sigma}^{A}\\rangle$ and small $|\\tilde{\\Psi}_{i,\\sigma}^{B}\\rangle$ components, both two-component spinors (the index $\\sigma$ runs over the two spin components). The latter is of order $\\textstyle{\\frac{v}{c}}$ of the former, where $\\nu$ is of the order of the velocity of the electron and $c$ is the speed of light. These equations can be simplified introducing errors of the order of the transferability error of the pseudopotential or of order $1/c^{2}$ , depending on which is the largest. In the final equations only the large components of the pseudo-wavefunctions appear. The non relativistic kinetic energy $\\ensuremath{\\mathbf{p}}^{2}/2m$ ( $\\dot{m}$ is the electron mass) acts on the large component of the pseudo-wavefunctions $|\\tilde{\\Psi}_{i,\\sigma}^{A}\\rangle$ in the mesh defined by the FFT grid and the same kinetic energy is used to calculate the expectation values of the Hamiltonian between partial pseudo-waves $\\big|\\Phi_{n,\\sigma}^{I,P S,A}\\big\\rangle$ . The Dirac kinetic energy is used instead to calculate the expectation values of the Hamiltonian between all-electron partial waves $\\vert\\Phi_{n,\\eta}^{I,A E}\\rangle$ (η is a four-component index). In this manner, relativistic effects are hidden in the coefficients of the non-local pseudopotential. The equations are formally very similar to the equations of the scalar-relativistic case:  \n\n$$\n\\begin{array}{r l r}{\\lefteqn{\\sum_{\\sigma_{2}}\\left[\\frac{{\\bf p}^{2}}{2m}\\delta^{\\sigma_{1},\\sigma_{2}}+\\sum_{\\eta_{1},\\eta_{2}}\\int\\mathrm{d}{\\bf r}\\tilde{V}_{\\mathrm{LOC}}^{\\eta_{1},\\eta_{2}}({\\bf r})\\tilde{K}({\\bf r})_{\\sigma_{1},\\sigma_{2}}^{\\eta_{1},\\eta_{2}}-\\varepsilon_{i}S^{\\sigma_{1},\\sigma_{2}}\\right.}}\\\\ &{}&{\\left.+\\sum_{I,m n}(D_{I,m n}^{1}-\\tilde{D}_{I,m n}^{1})|\\beta_{m,\\sigma_{1}}^{I,A}\\rangle\\langle\\beta_{n,\\sigma_{2}}^{I,A}|\\right]|\\tilde{\\Psi}_{i,\\sigma_{2}}^{A}\\rangle=0,}\\end{array}\n$$  \n\nwhere $D_{I,m n}^{1}$ and ${\\tilde{D}}_{I,m n}^{1}$ are calculated inside the PAW spheres:  \n\n$$\nD_{I,m n}^{1}=\\sum_{\\eta_{1},\\eta_{2}}\\langle\\Phi_{m,\\eta_{1}}^{I,A E}|T_{D}^{\\eta_{1},\\eta_{2}}+V_{\\mathrm{LOC}}^{I,\\eta_{1},\\eta_{2}}|\\Phi_{n,\\eta_{2}}^{I,A E}\\rangle,\n$$  \n\n$$\n\\begin{array}{l}{{\\displaystyle\\tilde{D}_{I,m n}^{1}=\\sum_{\\sigma_{1},\\sigma_{2}}\\langle\\Phi_{m,\\sigma_{1}}^{I,P S,{A}}\\vert\\frac{\\mathbf{p}^{2}}{2m}\\delta^{\\sigma_{1},\\sigma_{2}}+\\tilde{V}_{\\mathrm{LOC}}^{I,\\sigma_{1},\\sigma_{2}}\\vert\\Phi_{n,\\sigma_{2}}^{I,P S,{A}}\\rangle}}\\\\ {{\\displaystyle\\qquad+\\sum_{\\eta_{1},\\eta_{2}}\\int_{\\Omega_{I}}\\mathrm{d}\\mathbf{r}\\hat{Q}_{m n,\\eta_{1},\\eta_{2}}^{I}(\\mathbf{r})\\tilde{V}_{\\mathrm{LOC}}^{I,\\eta_{1},\\eta_{2}}(\\mathbf{r}).}}\\end{array}\n$$  \n\nHere $T_{D}$ is the Dirac kinetic energy:  \n\n$$\nT_{D}=c{\\pmb\\alpha}\\cdot{\\bf p}+(\\beta-{\\bf1}_{4\\times4})m c^{2},\n$$  \n\nwritten in terms of the $4\\times4$ Hermitian matrices $\\alpha$ and $\\beta$ and $V_{\\mathrm{LOC}}^{\\eta_{1},\\eta_{2}}$ is the sum of the local, Hartree, and XC potential $(V_{\\mathrm{eff}})$ together, in magnetic systems, with the contribution of the XC magnetic field:  \n\n$V_{\\mathrm{LOC}}^{\\eta_{1},\\eta_{2}}(\\mathbf{r})=V_{\\mathrm{eff}}(\\mathbf{r})\\delta^{\\eta_{1},\\eta_{2}}-\\mu_{\\mathrm{B}}\\mathbf{B}_{\\mathrm{xc}}(\\mathbf{r})\\cdot(\\boldsymbol{\\beta}\\pmb{\\Sigma})^{\\eta_{1},\\eta_{2}}$ . We refer to [16] for a detailed definition of the partial waves $\\vert\\Phi_{n,\\eta}^{I,A E}\\rangle$ , $\\big|\\Phi_{n,\\sigma}^{I,P S,A}\\big\\rangle$ dapnrdo $\\vert\\beta_{m,\\sigma}^{I,A}\\rangle$ , of tthe oavuerglmapenmtatiroixn iaonds $\\hat{Q}_{m n,\\eta_{1},\\eta_{2}}^{I}(\\mathbf{r})$ $\\tilde{K}(\\mathbf{r})_{\\sigma_{1},\\sigma_{2}}^{\\eta_{1},\\eta_{2}}$ $S^{\\sigma_{1},\\sigma_{2}}$ for their rewriting in terms of projector functions that contain only spherical harmonics. Solving these equations  it is possible to include spin–orbit coupling effects in electronic structure calculations. In Quantum ESPRESSO these equations  are used when input variables noncolin and lspinorb are both.TRUE.and the PAW data sets are fully relativistic, as those available with the pslibrary project.  \n\n2.6.2.  Electronic and structural properties in field-effect ­configuration.  Since Quantum ESPRESSO v.6.0 it is possible to compute the electronic structure under a fieldeffect transistor (FET) setup in periodic boundary conditions [189]. In physical FETs, a voltage is applied to a gate electrode, accumulating charges at the interface between the gate di­electric and a semiconducting system (see figure  3). The gate electrode is simulated with a charged plate, henceforth referred to as the gate. Since the interaction of this charged plate with its periodic image generates a spurious nonphysical electric field, a dipolar correction, equivalent to two planes of opposite charge, is added [190], canceling out the field on the left side of the gate. In order to prevent electrons from spilling towards the gate for large electron doping [191], a potential barrier can be added to the electrostatic potential, mimicking the effect of the gate dielectric.  \n\nThe setup for a system in FET configuration is shown in figure 3. The gate has a charge $n_{\\mathrm{dop}}A$ and the system has opposite charge. Here $n_{\\mathrm{dop}}$ is the number of doping electrons per unit area (i.e. negative for hole doping), $A$ is the area of the unit cell parallel to the surface. In practice the gate is represented by an external potential  \n\n$$\nV_{\\mathrm{gate}}(\\mathbf{r})=-2\\pi n_{\\mathrm{dop}}\\left(-|\\overline{{\\mathbf{z}}}|+\\frac{\\overline{{\\mathbf{z}}}^{2}}{L}+\\frac{L}{6}\\right).\n$$  \n\nHere ${\\overline{{\\mathsf{z}}}}=z-z_{\\mathrm{gate}}$ with $\\overline{{\\mathbf{z}}}\\in\\left[-L/2;L/2\\right]$ measures the distance from the gate (see figure 3). The dipole of the charged system plus the gate is canceled by an electric dipole generated by two planes of opposite charge [190, 192, 193], placed at $z_{\\mathrm{dip}}-d_{\\mathrm{dip}}/2$ and $z_{\\mathrm{dip}}+d_{\\mathrm{dip}}/2$ , in the vacuum region next to the gate ( $\\ensuremath{V_{\\mathrm{dip}}}$ in figure 3). Additionally one may include a potential barrier to avoid charge spilling towards the gate, or as a substitute for the gate dielectric. $V_{b}({\\bf r})$ is a periodic function of $z$ defined on the interval $z\\in[0,L]$ as equal to a constant $V_{b}$ for $z_{1}<z<z_{2}$ and zero elsewhere. Figure 3 shows the resulting total potential (black line). The following additional variables are needed: $z_{\\mathrm{gate}}$ , z1, $z_{2}$ , and $V_{0}$ . In the code these variables are named zgate, block_1, block_2, and block_height, respectively. The dipole corrections and the gate are activated by the options ${\\tt d i p f i e l d}=$ .true. and gate $:=$ .true. In order to enable the potential barrier and the relaxation of the system towards it, the new input parameters block and relaxz, respectively, have to be set to.true.More details about the implementation can be found in [189].  \n\n![](images/969f2f579d27bd2d45b5dd6281c27bb52eb17f69fedb3491a7e3c6a5f17b744f.jpg)  \nFigure 3.  Schematic picture of the planar averaged KS potential (without the exchange-correlation potential) for periodically repeated, charged slabs. The uppermost panel shows a sketch of a gated system. The different parts of the total KS potential are shown with different color: red—gate, $V_{\\mathrm{gate}}$ , green—dipole, $V_{\\mathrm{dip}}$ , blue— potential barrier, $V_{b}$ . The position of the gate is indicated by $z_{\\mathrm{gate}}$ . The black line shows the sum of $V_{\\mathrm{gate}}$ , $V_{\\mathrm{dip}}$ , $V_{b}$ , of the ionic potential $V_{\\mathrm{per}}^{i}$ , and of the Hartree potential $V_{H}$ . The length of the unit cell along $\\hat{z}$ is given by $L$ .  \n\n2.6.3.  Cold restart of Car–Parrinello molecular dynamics.  In the standard Lagrangian formulation of ab initio molecular dynamics [53], the coefficients of KS molecular orbitals over a given basis set (i.e. their Fourier coefficients, in the case of plane waves) are treated as classical degrees of freedom obeying Newton’s equations of motion that derive from a suitably defined extended Lagrangian. This Lagrangian is obtained from the Born–Oppenheimer total energy by augmenting it with a fictitious electronic kinetic-energy term and relaxing the constraint that the molecular orbitals stay at each instant of the trajectory in their instantaneous KS ground state. The idea is that, by choosing a suitably small fictitious electronic mass, the thermalization time of the electronic degrees of freedom can be made much longer than the typical simulation times, so that if the system is prepared in its electronic KS ground state at the start of the simulation, the electronic dynamics would follow almost adiabatically the nuclear one all over the simulation, thus effectively mimicking a bona fide Born– Oppenheimer dynamics.  \n\nWhile in Car–Parrinello MD both the physical nuclear and fictitious electronic velocities are determined by the equations of motion on a par, the question still remains as to how choose them at the start of the simulation. Initial nuclear velocities are dictated by physical considerations (e.g. thermal equilibrium) or may be taken from a previously interrupted MD run. Electronic velocities (i.e. the time derivatives of the KS molecular orbitals), instead, are not available when the simulation is started from scratch and are not independent of the physical nuclear ones, but are determined by the adiabatic time evolution of the system. Moreover, the projection over the occupied-state manifold of the electronic velocities, $\\dot{\\psi}_{v}^{\\parallel}\\dot{=}\\hat{P}\\dot{\\psi}_{v}$ is ill-defined because the KS ground-state solution is defined modulo a unitary transformation within this manifold. This means that the starting electronic velocities may not be simply obtained as finite differences of KS orbitals at times $t=0$ and $t=\\Delta t$ . Here and in the following $\\hat{P}$ indicates the projector over the occupied-state manifold, and $\\hat{\\boldsymbol Q}=1-\\hat{\\boldsymbol P}$ its complement (i.e. the projector over the virtual-orbital manifold).  \n\nThe component of the electronic velocities over the virtual-state manifold, $\\dot{\\psi}_{v}^{\\perp}\\dot{=}\\hat{Q}\\dot{\\psi}_{v}$ , is instead well defined and can be formally written using standard first-order perturbation theory:  \n\n$$\n\\dot{\\psi}_{v}^{\\perp}({\\bf r})=\\sum_{c}\\psi_{c}({\\bf r})\\frac{\\langle\\psi_{c}|\\dot{V}_{\\mathrm{KS}}|\\psi_{v}\\rangle}{\\epsilon_{v}-\\epsilon_{c}},\n$$  \n\nwhere $\\nu$ and $c$ indicate occupied (valence) and virtual (conduction) states, respectively, $\\epsilon_{n}$ the corresponding orbital energies, and $\\dot{V}_{\\mathrm{KS}}$ is the time derivative of the KS potential, $V_{\\mathrm{KS}}$ . $\\dot{V}_{\\mathrm{KS}}$ is the linear response of $V_{\\mathrm{KS}}$ to the perturbation in the external potential determined by an infinitesimal displacement of the nuclei MD trajectory: $\\begin{array}{r}{\\dot{V}_{\\mathrm{ext}}(\\mathbf{r})=\\sum_{\\mathbf{R}}\\frac{\\partial v_{\\mathbf{R}}(\\mathbf{r}-\\mathbf{R})}{\\partial\\mathbf{R}}\\cdot\\dot{\\mathbf{R}}}\\end{array}$ $v_{\\mathbf{R}}(\\mathbf{r}-\\mathbf{R})$ at position $\\mathbf{R}$ and $\\dot{\\bf R}$ its velocity. Electronic velocities can conveniently be initialized to the values given by equation (47), which are those that minimize their norm and, hence, the initial electronic temperature, which is defined as the sum of the squared norms of the electronic velocities.  \n\nWhile this could in principle be done using density-functional perturbation theory [90, 93], it is more convenient to compute them numerically, following the procedure described below. At $t=0$ the KS molecular orbitals are initialized from a ground-state computation, performed with whatever method is available or preferred (standard SCF calculation or global optimization, such as e.g. with conjugate gradients [194]). The KS molecular orbitals that would result from a perfectly adiabatic propagation at $t=\\Delta t$ are then determined from a second ground-state computation, performed after half a ‘velocity-Verlet’ MD step, i.e. at nuclear positions $\\mathbf{R}(\\Delta t)=\\mathbf{R}(0)+\\dot{\\mathbf{R}}(0)\\Delta t$ . The initial velocities are then obtained from the relation:  \n\n$$\n\\dot{\\psi}_{v}^{\\perp}=\\hat{\\dot{P}}\\psi_{v},\n$$  \n\nwhich is obtained by simply differentiating the definition of occupied-state projector, $\\hat{P}\\psi_{v}=\\psi_{v}$ . The right-hand side of equation (48) is finally easily computed by subtracting from each KS orbital at time $t=0$ , its component over the occupied-state manifold at $t=\\Delta t$ and dividing by $\\Delta t$ .  \n\n2.6.4.  Optimized tetrahedron method.  The integration over $\\mathbf{k}$ -points in the BZ is a crucial step in the calculation of the electronic structure of a periodic system, affecting not only the ground state but linear response as well. This is especially true for metallic systems where the integrand is discontinuous at the Fermi level. Even more problematic is the integration of Dirac delta functions, such as those appearing in the density of states (DOS), partial DOS and in the electron–phonon coupling constant.  \n\nQuantum ESPRESSO has always implemented a variety of ‘smearing’ methods, in which the delta function is replaced by a function of finite width (e.g. a Gaussian function, or more sophisticated choices). It has also always implemented the linear tetrahedron method [195] with the correction proposed by Blöchl [196], in which the BZ is divided into tetrahedra and the integration is performed analytically by linear interpolation of KS eigenvalues in each tetrahedron. Such method is however limited in its convenience and range of applicability: in fact the linear interpolation systematically overestimates convex functions, thus making the convergence against the number of $\\mathbf{k}$ -points slow. The linear tetrahedron method was thus mostly restricted to the calculation of DOS and partial DOS.  \n\nSince Quantum ESPRESSO v.6.1, the optimized tetrahedron method [197] is implemented. Such method overcomes the drawback of the linear tetrahedron method using an interpolation that accounts for the curvature of the interpolated function. The optimized tetrahedron method has better conv­ ergence properties and an extended range of applicability: in addition to the calculation of the ground-state charge density, DOS and partial DOS, it can be used in linear-response calculation of phonons and of the electron–phonon coupling constant.  \n\n2.6.5.  Wyckoff positions.  In Quantum ESPRESSO the crystal geometry is traditionally specified by a Bravais lattice index (called ibrav), by the crystal parameters (celldm, or a, b, c, cosab, cosac, cosbc) describing the unit cell, and by the positions of all atoms in the unit cell, in crystal or Cartesian axis.  \n\nSince v.5.1.1, it is possible to specify the crystal geometry in crystallographic style [198], according to the notations of the International Tables  of Crystallography (ITA) [199]. A complete description of the crystal structure is obtained by specifying the space-group number according to the ITA and the positions of symmetry-inequivalent atoms only in the unit cell. The latter can be provided either in the crystal axis of the conventional cell, or as Wyckoff positions: a set of special positions, listed in the ITA for each space group, that can be fully specified by a number of parameters, none to three depending upon the site symmetry. Table  1 reports a few examples of accepted syntax. The code generates the symmetry operations for the specified space group and applies them to inequivalent atoms, thus finding all atoms in the unit cell.  \n\nTable 1.  Examples of valid syntax for Wyckoff positions. $C$ is the element name, followed by the Wyckoff label of the site (number of equivalent atoms followed by a letter identifying the site), followed by the site-dependent parameters needed to fully specify the atomic positions.   \n\n\n<html><body><table><tr><td colspan=\"4\">ATOMIC_POSITIONS sg</td></tr><tr><td>C</td><td>la</td><td></td><td></td></tr><tr><td>C</td><td>8g</td><td>X</td><td></td></tr><tr><td>C</td><td>24m</td><td>X</td><td>y</td></tr><tr><td>C</td><td>48n</td><td>X</td><td>y</td></tr><tr><td>C</td><td></td><td></td><td></td></tr></table></body></html>  \n\nFor some crystal systems there are alternate descriptions in the ITA, so additional input parameters may be needed to select the desired one. For the monoclinic system the $\\mathrm{{^\\circc}}$ -unique’ orientation is the default and bunique $\\c=$ .TRUE.must be specified in input if the ‘b-unique’ orientation is desired. For some space groups there are two possible choices of the origin. The origin appearing first in the ITA is chosen by default, unless origin_choice $:=2$ is specified in input. Finally, for trigonal space groups the atomic coordinates can be referred to the rhombohedral or to the hexagonal Bravais lattices. The default is the rhombohedral lattice, so rhombohedral $\\c=$ .FALSE. must be specified in input to use the hexagonal lattice.  \n\nA final comment for centered Bravais lattices: in the crystallographic literature, the conventional unit cell is usually assumed. Quantum ESPRESSO however assumes the primitive unit cell, having a smaller volume and a smaller number of atoms, and discards atoms outside the primitive cell. Auxiliary code supercell.x, available in thermo_pw (see section  2.4.1), prints all atoms in the conventional cell when necessary.  \n\n# 3.  Parallelization, modularization, interoperability and stability  \n\n# 3.1.  New parallelization levels  \n\nThe basic modules of Quantum ESPRESSO are characterized by a hierarchy of parallelization levels, described in [6]. Processors are divided into groups, labeled by a MPI communicator. Each group of processors distributes a specific subset of computations. The growing diffusion of HPC machines based on nodes with many cores (32 and more) makes however pure MPI parallelization not always ideal: running one MPI process per core has a high overhead, limiting performances. It is often convenient to use mixed MPI-OpenMP parallelization, in which a small number of MPI processes per node use OpenMP threads, either explicitly (i.e. with compiler directives) or implicitly (i.e. via calls to OpenMP-aware library). Explicit OpenMP parallelization, originally confined to computationally intensive FFTs, has been extended to many more parts of the code.  \n\nOne of the challenges presented by a massively parallel machine is to get rid of both memory and CPU time bottlenecks, caused respectively by arrays that are not distributed across processors and by non-parallelized sections  of code. It is especially important to distribute all arrays and parallelize all computations whose size/complexity increases with the dimensions of the unit cell or of the basis set. Nonparallelized computations hamper ‘weak’ scalability, that is, parallel performance while increasing both the system size and the amount of computational resources, while non-distributed arrays may become an unavoidable RAM bottleneck with increasing problem size. ‘Strong’ scalability (that is, at fixed problem size and increasing number of CPUs) is even more elusive than weak scalability in electronic-structure calcul­ations, requiring, in addition to systematic distribution of computations, to keep to the minimum the ratio between time spent in communications and in computation, and to have a nearly perfect load balancing. In order to achieve strong scalability, the key is to add more parallelization levels and to use algorithms that permit to overlap communications and computations.  \n\nFor what concerns memory, notable offenders are arrays of scalar products between KS states $\\psi_{i}$ : $O_{i j}=\\langle\\psi_{i}|\\widehat{O}|\\psi_{j}\\rangle$ , where $\\hat{o}$ can be either the Hamiltonian or an overlap m\u001fatrix; and sca\u001flar products between KS states and pseudopotential projectors $\\beta$ , $p_{i j}=\\langle\\psi_{i}|\\beta_{j}\\rangle$ . The size of such arrays grows as the square of the size of the cell. Almost all of them are now distributed across processors of the ‘linear-algebra group’, that is, the group of processors taking care of linear-algebra operations on matrices. The most expensive of such operations are subspace diagonalization (used in PWscf in the iterative diagonalization) and iterative orthonormalization (used by CP). In both cases, a parallel dense-matrix diagonalization on distributed matrix is needed. In addition to ScaLAPACK, Quantum ESPRESSO can now take advantage of newer ELPA libraries [200], leading to significant performance improvements.  \n\nThe array containing the plane-wave representation, $c_{\\mathbf{k},n}(\\mathbf{G})$ , of KS orbitals is typically the largest array, or one of the largest. While plane waves are already distributed across processors of the ‘plane-wave group’ as defined in [6], it is now possible to distribute KS orbitals as well. Such a parallelization level is located between the $\\mathbf{k}$ -point and the plane-wave parallelization levels. The corresponding MPI communicator defines a subgroup of the $\\mathbf{\\hat{\\mu}}_{\\mathbf{k}}$ -point group’ of processors and is called ‘band group communicator’. In the CP package, band parallelization is implemented for almost all available calcul­ ations. Its usefulness is better appreciated in simulations of large cells—several hundreds of atoms and more—where the number of processors required by memory distribution would be too large to get good scalability from plane-wave parallelization only.  \n\nIn PWscf, band parallelization is implemented for calcul­ ations using hybrid functionals. The standard algorithm to compute Hartree–Fock exchange in a plane-wave basis set (see section 2.1.1) contains a double loop on bands that is by far the heaviest part of computation. A first form of parallelization, described in [34], was implemented in v.5.0. In the latest version, this has been superseded by parallelization of pairs of bands, [35]. Such algorithm is compatible with the ‘task-group’ parallelization level (that is: over KS states in the calculation of $V\\psi_{i}$ products) described in [6].  \n\nIn addition to the above-mentioned groups, that are globally defined and in principle usable in all routines, there are a few additional parallelization levels that are local to specific routines. Their goal is to reduce the amount of non-parallel computations that may become significant for many-atom systems. An example is the calculation of $\\tt D F T+U$ (section 2.1.3) terms in energy and forces, equations  (12) and (14) respectively. In all these expressions, the calculation of the scalar products between valence and atomic wave functions is in principle the most expensive step: for $N_{b}$ bands and $N_{p w}$ plane waves, $\\mathcal{O}(N_{p w}N_{b})$ floating-point operations are required (typically, $N_{p w}\\gg N_{b})$ . The calculation of these terms is however easily and effectively parallelized, using standard matrix-matrix multiplication routines and summing over MPI processes with a mpi_reduce operation on the plane-wave group. The sum over $\\mathbf{k}$ -points can be parallelized on the $\\mathbf{k}$ -point group. The remaining sums over band indices $\\nu$ and Hubbard orbitals $I,m$ may however require a significant amount of non-parallelized computation if the number of atoms with a Hubbard $U$ term is not small. The sum over band indices is thus parallelized by simply distributing bands over the plane-wave group. This is a convenient choice because all processors of the plane-wave group are available once the scalar products are calculated. The addition of band parallelization speeds up the computation of such terms by a significant factor. This is especially important for Car–Parrinello dynamics, requiring the calculation of forces at each time step, when a sizable number of Hubbard manifolds is present.  \n\n# 3.2.  Aspects of interoperability  \n\nOne of the original goals of Quantum ESPRESSO was to assemble different pieces of rather similar software into an integrated software suite. The choice was made to focus on the following four aspects: input data formats, output data files, installation mechanism, and a common base of code. While work on the first three aspects is basically completed, it is still ongoing on the fourth. It was however realized that a different form of integration—interoperability, i.e. the possibility to run Quantum ESPRESSO along with other software—was more useful to the community of users than tight integration. There are several reasons for this, all rooted in new or recent trends in computational materials science. We mention in par­ ticular the usefulness of interoperability for  \n\n1.\texcited-states calculations using many-body perturbation theory, at various levels of sophistication: GW, TDDFT, BSE (e.g. yambo [201], SaX [202], or BerkeleyGW [203]);   \n2.\tcalculations using quantum Monte Carlo methods;   \n3.\tconfiguration-space sampling, using such algorithms as nudged elastic band (NEB), genetic/evolutionary algorithms, meta-dynamics;   \n4.\tinclusion of quantum effects on nuclei via path-integral Monte Carlo;   \n5.\tmulti-scale simulations, requiring different theoretical approaches, each valid in a given range of time and length scale, to be used together;   \n6.\thigh-throughput, or ‘exhaustive’, calculations (e.g. AiiDA [204, 205] and $\\operatorname{AFLOW}\\pi$ [206]) requiring automated submission, analysis, retrieval of a large number of jobs;   \n7.\t‘steering’, i.e. controlling the computation in real time using either a graphical user interface (GUI) or an interface in a high-level interpreted language (e.g. python).  \n\nIt is in principle possible, and done in some cases, to implement all of the above into Quantum ESPRESSO, but this is not always the best practice. A better option is to use Quantum ESPRESSO in conjunction with external software performing other tasks.  \n\nCases 1 and 2 mentioned above typically use as starting step the self-consistent solution of KS equations, so that what is needed is the possibility for external software to read data files produced by the main Quantum ESPRESSO codes, notably the self-consistent code PWscf and the molecular dynamics code CP.  \n\nCases 3 and 4 typically require many self-consistent calcul­ ations at different atomic configurations, so that what is needed is the possibility to use the main Quantum ESPRESSO codes as ‘computational engine’, i.e. to call PWscf or CP from an external software, using atomic configurations supplied by the calling code.  \n\nThe paradigmatic case 5 is QM–MM (section 2.5.2), requiring an exchange of data, notably: atomic positions, forces, and some information on the electrostatic potential, between Quantum ESPRESSO and the MM code—typically a classical MD code.  \n\nCase 6 requires easy access to output data from one simulation, and easy on-the-fly generation of input data files as well. This is also needed for case 7, which however may also require a finer-grained control over computations performed by Quantum ESPRESSO routines: in the most sophisticated scenario, the GUI or python interface should be able to perform specific operations ‘on the fly’, not just running an entire self-consistent calculation. This scenario relies upon the existence of a set of application programming interfaces (API’s) for calls to basic computational tasks.  \n\n# 3.3.  Input/Output and data file format  \n\nOn modern machines, characterized by fast CPU’s and large RAM’s, disk input/output (I/O) may become a bottleneck and should be kept to a strict minimum. Since v.5.3 both PWscf and CP do not perform by default any I/O at run time, except for the ordinary text output (printout), for checkpointing if required or needed, and for saving data at the end of the run. The same is being gradually extended to all codes. In the following, we discuss the case of the final data writing.  \n\nThe original organization of output data files (or more exactly, of the output data directory) was based on a formatted ‘head’ file, with a XML-like syntax, containing general information on the run, and on binary data files containing the KS orbitals and the charge density. We consider the basic idea of such approach still valid, but some improvements were needed. On one hand, the original head file format had a number of small issues—inconsistencies, missing pieces of relevant information—and used a non-standard syntax, lacking a XML ‘schema’ for validation. On the other hand, data files suffered from the lack of portability of Fortran binary files and had to be transformed into text files, sometimes very large ones, in order to become usable on a different machine.  \n\n3.3.1.  XML files with schema.  Since v.6.0, the ‘head’ file is a true XML file using a consistent syntax, described by a XML schema, that can be easily parsed with standard XML tools. It also contains complete information on the run, including all data needed to reproduce the results, and on the correct execution and exit status. This aspect is very useful for highthroughput applications, for databasing of results and for verification and validation purposes.  \n\nThe XML file contains an input section  and can thus be used as input file, alternative to the still existing text-based input. It supersedes the previous XML-based input, introduced several years ago, that had a non-standard syntax, different from and incompatible with the one of the original head file. Implementing a different input is made easy by the clear separation existing between the reading and initialization phases: input data is read, stored in a separate module, copied to internal variables.  \n\nThe current XML file can be easily parsed and generated using standard XML tools and is especially valuable in conjunction with GUI’s. The schema can be found at the URL: www.quantum-espresso.org/ns/qes/qes-1.0.xsd.  \n\n3.3.2.  Large-record data file format.  Although not as I/Obound as other kinds of calculations, electronic-structure simulations may produce a sizable amount of data, either intermediate or needed for further processing. The largest array typically contains the plane-wave representation of KS orbitals; other sizable arrays contain the charge and spin density, either in reciprocal or in real space. In parallel execution using MPI, large arrays are distributed across processors, so one has two possibilities: let each MPI process write its own slice of the data (‘distributed’ I/O), or collect the entire array on a single processor before writing it (‘collected’ I/O). In distributed I/O, coding is straightforward and efficient, minimizing file size and achieving some sort of I/O parallelization. A global file system, accessible to all MPI processes, is needed. The data is spread into many files that are directly usable only by a code using exactly the same distribution of arrays, that is, exactly the same kind of parallelization. In collected I/O, the coding is less straightforward. In order to ensure portability, some reference ordering, independent upon the number of processors and the details of the parallelization, must be provided. For large simulations, memory usage and communication pattern must be carefully optimized when a distributed array is collected into a large array on a single processor.  \n\nIn the original I/O format, KS orbitals were saved in reciprocal space, in either distributed or collected format. For the latter, a reproducible ordering of plane waves (including the ordering within shells of plane waves with the same module), independent upon parallelization details and machine-independent, ensures data portability. Charge and spin density were instead saved in real space and in collected format. In the new I/O scheme, available since v.6.0, the output directory is simplified, containing only the XML data file, one file per $\\mathbf{k}$ -point with KS orbitals, one file for the charge and spin density. Both files are in collected format and both quantities are stored in reciprocal space. In addition to Fortran binary, it is possible to write data files in HDF5 format [207]. HDF5 offers the possibility to write structured record and portability across architectures, without significant loss in performances; it has an excellent support and is the standard for I/O in other fields of scientific computing. Distributed I/O is kept only for checkpointing or as a last-resort alternative.  \n\nIn spite of its advantages, such a solution has still a bottleneck in large-scale computations on massively parallel machines: a single processor must read and write large files. Only in the case of parallelization over $\\mathbf{k}$ -points is I/O parallelized in a straightforward way. More general solutions to implement parallel I/O using parallel extensions of HDF5 are currently under examination in view of enabling Quantum ESPRESSO towards ‘exascale’ computing (that is: towards $\\mathcal{O}(10^{18})$ floating-point operations per second).  \n\n# 3.4.  Organization of the distribution  \n\nCodes contained in Quantum ESPRESSO have evolved from a small set of original codes, born with rather restricted goals, into a much larger distribution via continuous additions and extensions. Such a process—presumably common to most if not all scientific software projects—can easily lead to uncoordinated growth and to bad decisions that negatively affect maintainability.  \n\n# 3.4.1.  Package re-organization and modularization.  In order to make the distribution easier to maintain, extend and debug, the distribution has been split into  \n\na.\tbase distribution, containing common libraries, tools and utilities, core packages PWscf, CP, PostProc, plus some commonly used additional packages, currently: atomic, PWgui, PWneb, PHonon, XSpectra, turboTDDFT, turboEELS, GWL, EPW; b.\texternal packages such as SaX [202], yambo [201], Wannier90 [171], WanT [208, 209], that are automatically downloaded and installed on demand.  \n\nThe directory structure now explicitly reflects the structure of Quantum ESPRESSO as a ‘federation’ of packages rather than a monolithic one: a common base distribution plus additional packages, each of which fully contained into a subdirectory.  \n\nIn the reorganization process, the implementation of the NEB algorithm was completely rewritten, following the paradigm sketched in section 3.2. PWneb is now a separate package that implements the NEB algorithm, using PWscf as the computational engine. The separation between the NEB algorithm and the self-consistency algorithm is quite complete: PWneb could be adapted to work in conjunction with a different computational engine with a minor effort.  \n\nThe implementation of meta-dynamics has also been reconsidered. Given the existence of a very sophisticated and well-maintained package [210] Plumed for all kinds of metadynamics calculations, the PWscf and CP packages have been adapted to work in conjunction with Plumed v.1.x, removing the old internal meta-dynamics code. In order to activate meta-dynamics, a patching process is needed, in which a few specific ‘hook’ routines are modified so that they call routines from Plumed.  \n\n3.4.2.  Modular parallelism.  The logic of parallelism has also evolved towards a more modular approach. It is now possible to have all Quantum ESPRESSO routines working inside a MPI communicator, passed as argument to an initialization routine. This allows in particular the calling code to have its own parallelization level, invisible to Quantum ESPRESSO routines; the latter can thus perform independent calculations, to be subsequently processed by the calling code. For instance: the ‘image’ parallelization level, used by NEB calculations, is now entirely managed by PWneb and no longer in the called PWscf routines. Such a feature is very useful for coupling external codes to Quantum ESPRESSO routines. To this end, a general-purpose library for calling PWscf or CP from external codes (either Fortran or $_{\\mathrm{C/C++}}$ using the Fortran 2003 ISO C binding standard) is provided in the directory COUPLE/.  \n\n3.4.3.  Reorganization of linear-response codes.  All linearresponse codes described in sections  2.2 and 2.1.4 share as basic computational step the self-consistent solution of linear systems $A x=b$ for different perturbations $b$ , where the operator $A$ is derived from the KS Hamiltonian $H$ and the linearresponse potential. Both the perturbations and the methods of solution differ by subtle details, leading to a plethora of routines, customized to solve slightly different versions of the same problem. Ideally, one should be able to solve any linearresponse problem by using a suitable library of existing code. To this end, a major restructuring of linear-response codes has been started. Several routines have been unified, generalized and extended. They have been collected into the same subdirectory, LR_Modules, that will be the container of ‘generic’ linear-response routines. Linear-response-related packages now contain only code that is specific to a given perturbation or property calculation.  \n\n# 3.5.  Quantum ESPRESSO and scripting languages  \n\nA desirable feature of electronic-structure codes is the ability to be called from a high-level interpreted scripting language. Among the various alternatives, python has emerged in the last years due to its simple and powerful syntax and to the availability of numerical extensions (NumPy). Since v.6.0, an interface between PWscf and the path integral MD driver i-PI [41] is available and distributed together with  \n\n![](images/cdada522bfbec079887ab060c8a351916be890606e54e7bc1ad84bba5580c7bd.jpg)  \nFigure 4.  A simple AiiDA directed acyclic graph for a Quantum ESPRESSO calculation using PWscf (square), with all the input nodes (data, circles; code executable, diamond) and all the output nodes that the daemon creates and connects automatically.  \n\nQuantum ESPRESSO. Various implementations of an interface between Quantum ESPRESSO codes and the atomic simulation environment (ASE) [211] are also available. In the following we briefly highlight the integration of Quantum ESPRESSO with AiiDA, the pwtk toolkit for PWscf, and the QE-emacs-modes package for userfriendly editing of Quantum ESPRESSO with the Emacs editor [212].  \n\n3.5.1.  AiiDA: a python materials’ informatics infrastructure.  AiiDA [204] is a comprehensive python infrastructure aimed at accelerating, simplifying, and organizing major efforts in computational science, and in particular computational materials science, with a close integration with the Quantum ESPRESSO distribution. AiiDA is structured around the four pillars of the ADES model (Automation, Data, Environment, and Sharing, [204])), and provides a practical and efficient implementation of all four. In particular, it aims at relieving the work of a computational scientist from the tedious and error-prone tasks of running, overseeing, and storing hundreds or more of calculations daily (Automation pillar), while ensuring that strict protocols are in place to store these calculations in an appropriately structured database that preserves the provenance of all computational steps (Data pillar). This way, the effort of a computational scientist can become focused on developing, curating, or exploiting complex workflows (Environment pillar) that calculate in a robust manner e.g. the desired materials properties of a given input structure, recording expertise in reproducible sequences that can be progressively perfected, while being able to share freely both the workflows and the data generated with public or private common repositories (Sharing). AiiDA is built using an agnostic structure that allows to interface it with any given code—through plugins and a plugin repository— or with different queuing systems, transports to remote HPC resources, and property calculators. In addition, it allows to use arbitrary object-relational mappers (ORMs) as backends (currently, Django and SQLAlchemy are supported). These ORMs map the AiiDA objects (‘Codes’, ‘Calculations’ and ‘Data’) onto python classes, and lead to the representation of calculations through Directed Acyclic Graphs (DAGs) connecting all objects with directional arrows; this ensures both provenance and reproducibility of a calculation. As an example, in figure 4 we present a simple DAG representing a PWscf calculation on ${\\bf B a T i O}_{3}$ .  \n\n3.5.2.  Pwtk: a toolkit for PWscf.  The pwtk, standing for PWscfToolKit, is a Tcl scripting interface for PWscf set of programs contained in the Quantum ESPRESSO distribution. It aims at providing a flexible and productive framework. The basic philosophy of pwtk is to lower the learning curve by using syntax that closely matches the input syntax of Quantum ESPRESSO. Pwtk features include: (i) assignment of default values of input variables on a project basis, (ii) reassignment of input variables on the fly, (iii) stacking of input data, (iv) math-parser, (v) extensible and hierarchical configuration (global, project-based, local), (vi) data retrieval functions (i.e. either loading the data from pre-existing input files or retrieving the data from output files), and (vii) a few predefined higher-level tasks, that consist of several seamlessly integrated calculations. Pwtk allows to easily automate large number of calculations and to glue together different computational tasks, where output data of preceding calcul­ ations serve as input for subsequent calculations. Pwtk and related documentation can be downloaded from http://pwtk. quantum-espresso.org.  \n\n![](images/f64a19b568f22fb8d5cd2963d81e35e3975400dfc40d652901dab06b27fd736e.jpg)  \nFigure 5.  (a) pw.x input file opened in Emacs with pw-mode highlighting the following elements: namelists and their variables (blue and brown), cards and their options (purple and green), comments (red), string and logical variable values (burgundy and cyan, respectively). A mistyped variable (i.e. ibrv instead of ibrav) is not highlighted. (b) An excerpt from the INPUT_PW.html file, which describes the pw.x input file syntax. Both the QE-emacs-modes and the INPUT_PW.html are automatically generated from the Quantum ESPRESSO’s internal definition of the input file syntax.  \n\n3.5.3.  QE-emacs-modes.  The QE-emacs-modes package is an open-source collection of Emacs major-modes for making the editing of Quantum ESPRESSO input files easier and more comfortable with Emacs. The package provides syntax highlighting (see figure  5(a)), auto-indentation, auto-completion, and a few utility commands, such as $\\mathtt{M}\\mathrm{-}\\mathtt{x}$ prog−insert template that inserts a respective input file template for the prog program (e.g. pw, neb, pp, projwfc, dos, bands). The QE-emacs-modes are aware of all namelists, variables, cards, and options that are explicitly documented in the INPUT_PROG.html files, which describe the respective input file syntax (see figure  5(b)), where PROG stands for the uppercase name of a given program of Quantum ESPRESSO. The reason for this is that both INPUT_PROG.html files and QE-emacs-modes are automatically generated by the internal helpdoc utility of Quantum ESPRESSO.  \n\n# 3.6.  Continuous integration and testing  \n\nThe modularization of Quantum ESPRESSO reduces the extent of code duplication, thus improving code maintainability, but it also creates interdependencies between the modules so that changes to one part of the code may impact other parts. In order to monitor and mitigate these side effects we developed a test-suite for non-regression testing. Its purpose is to increase code stability by identifying and correcting those changes that break established functionalities. The test-suite relies on a modified version of python script testcode [213].  \n\n![](images/1d0231085ac008188e408b2b595a44ff0fa476d962aab45ba96ee94b09f03cf0.jpg)  \nFigure 6.  Layout of the Quantum ESPRESSO test-suite. The program testcode runs Quantum ESPRESSO executables, extracts numerical values from the output files, and compares the results with reference data. If the difference between these data exceeds a specified threshold, testcode issues an error indicating that a recent commit might have introduced a bug in parts of the code.  \n\nThe layout of the test-suite is illustrated in figure  6. The suite is invoked via a Makefile that accepts several options to run sequential or parallel tests or to test one particular feature of the code. The test-suite runs the various executables of Quantum ESPRESSO, extracts the numerical data of interest, compares them to reference data, and decides whether the test is successful using specified thresholds. At the moment, the test-suite contains 181 tests for PW, 14 for PH, 17 for CP, 43 for EPW, and 6 for TDDFpT covering $43\\%$ , $30\\%$ , $29\\%$ , $63\\%$ and $25\\%$ of the blocks, respectively. Moreover, $60\\%$ , $44\\%$ , $47\\%$ , $76\\%$ and $32\\%$ of the subroutines in each of these codes are tested, respectively.  \n\nThe test-suite also contains the logic to automatically create reference data by running the relevant executables and storing the output in a benchmark file. These benchmarks are updated only when new tests are added or bugfixes modify the previous behavior.  \n\nThe test-suite enables automatic testing of the code using several Buildbot test farms. The test farms monitor the code repository continuously, and trigger daily builds in the night after every new commit. Several compilers (Intel, GFortran, PGI) are tested both in serial and in parallel (openmpi, mpich, Intel mpi and mvapich2) execution with different mathematical libraries (LAPACK, BLAS, ScaLAPACK, FFTW3, MKL, OpenBlas). More information can be found at testfarm.quantum-espresso.org.  \n\nThe official mirror of the development version of Quantum ESPRESSO (https://github.com/QEF/q-e) employs a subset of the test-suite to run TravisCI. This tool rapidly identifies erroneous commits and can be used to assist code review during a pull request.  \n\ncodes makes a rewrite for new architectures a challenging choice, and a risky one given the fast evolution of computer architectures.  \n\nWe think that the main directions followed until now in the development of Quantum ESPRESSO are still valid, not only for new methodologies, but also for adapting to new computer architectures and future ‘exascale’ machines. Namely, we will continue pushing towards code reusability by removing duplicated code and/or replacing it with routines performing well-defined tasks, by identifying the time-intensive sections of the code for machine-dependent optimization, by having documented APIs with predictable behavior and with limited dependency upon global variables, and we will continue to optimize performance and reliability. Finally, we will push towards extended interoperability with other software, also in view of its usefulness for data exchange and for cross-verification, or to satisfy the needs of high-throughput calculations.  \n\nStill, the investment in the development and maintenance of state-of-the-art scientific software has historically lagged behind compared to the investment in the applications that use such software, and one wonders is this the correct or even forward-looking approach given the strategic importance of such tools, their impact, their powerful contribution to open science, and their full and complete availability to the entire community. In all of this, the future of materials simulations appear ever more bright [214], and the usefulness and relevance of such tools to accelerating invention and discovery in science and technology is reflected in its massive uptake by the community at large.  \n\n# 4.  Outlook and conclusions  \n\nThis paper describes the core methodological developments and extensions of Quantum ESPRESSO that have become available, or are about to be released, after [6] appeared. The main goal of Quantum ESPRESSO to provide an efficient and extensible framework to perform simulations with well-established approaches and to develop new methods remains firm, and it has nurtured an ever growing community of developers and contributors.  \n\nAchieving such a goal, however, becomes increasingly challenging. On one hand, computational methods become ever more complex and sophisticated, making it harder not only to implement them on a computer but also to verify the correctness of the implementation (for a much needed initial effort on verification of electronic-structure codes based on DFT, see [5]). On the other hand, exploiting the current technological innovations in computer hardware can require massive changes to software and even algorithms. This is especially true for the case of ‘accelerated’ architectures (GPUs and the like), whose exceptional performance can translate to actual calculations only after heavy restructuring and optimization. The complexity of existing  \n\n# Acknowledgments  \n\nThis work has been partially funded by the European Union through the MaX Centre of Excellence (Grant No. 676598) and by the Quantum ESPRESSO Foundation. SdG acknowledges support from the EU Centre of Excellence E CAM (Grant No. 676531). OA, MC, NC, NM, NLN, and IT acknowledge support from the SNSF National Centre of Competence in Research MARVEL, and from the PASC Platform for Advanced Scientific Computing. TT acknowledges support from NSF Grant No. DMR-1145968. SP, MS, and FG are supported by the Leverhulme Trust (Grant RL-2012-001). MBN acknowledges support by DOD-ONR (N00014-13-1-0635, N00014-11-1-0136, N00014-15-1- 2863) and the Texas Advanced Computing Center at the University of Texas, Austin. RD acknowledges partial support from Cornell University through start-up funding and the Cornell Center for Materials Research (CCMR) with funding from the NSF MRSEC program (DMR-1120296). MK acknowledges support by Building of Consortia for the Development of Human Resources in Science and Technology from the MEXT of Japan. AK acknowledges support from the Slovenian Research Agency (Grant No. P2-0393). This research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-06CH11357. This research also used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231.  \n\n# ORCID iDs  \n\nP Giannozzi $\\textcircled{1}$ https://orcid.org/0000-0002-9635-3227   \nG Fratesi $\\circledcirc$ https://orcid.org/0000-0003-1077-7596   \nM Kawamura $\\circledcirc$ https://orcid.org/0000-0003-3261-1968   \nH-Y Ko $\\textcircled{1}$ https://orcid.org/0000-0003-1619-6514   \nB Santra $\\circledcirc$ https://orcid.org/0000-0003-3609-2106   \nI Timrov $\\textcircled{1}$ https://orcid.org/0000-0002-6531-9966   \nS Baroni $\\textcircled{1}$ https://orcid.org/0000-0002-3508-6663  \n\n# References  \n\n[1] Hohenberg P and Kohn W 1964 Phys. 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    "Target": null
  },
  {
    "id": "10.1038_NCHEM.2695",
    "DOI": "10.1038/NCHEM.2695",
    "DOI Link": "http://dx.doi.org/10.1038/NCHEM.2695",
    "Relative Dir Path": "mds/10.1038_NCHEM.2695",
    "Article Title": "Activating lattice oxygen redox reactions in metal oxides to catalyse oxygen evolution",
    "Authors": "Grimaud, A; Diaz-Morales, O; Han, BH; Hong, WT; Lee, YL; Giordano, L; Stoerzinger, KA; Koper, MTM; Shao-Horn, Y",
    "Source Title": "NATURE CHEMISTRY",
    "Abstract": "Understanding how materials that catalyse the oxygen evolution reaction (OER) function is essential for the development of efficient energy-storage technologies. The traditional understanding of the OER mechanism on metal oxides involves four concerted proton-electron transfer steps on metal-ion centres at their surface and product oxygen molecules derived from water. Here, using in situ O-18 isotope labelling mass spectrometry, we provide direct experimental evidence that the O-2 generated during the OER on some highly active oxides can come from lattice oxygen. The oxides capable of lattice-oxygen oxidation also exhibit pH-dependent OER activity on the reversible hydrogen electrode scale, indicating non-concerted proton-electron transfers in the OER mechanism. Based on our experimental data and density functional theory calculations, we discuss mechanisms that are fundamentally different from the conventional scheme and show that increasing the covalency of metal-oxygen bonds is critical to trigger lattice-oxygen oxidation and enable non-concerted proton-electron transfers during OER.",
    "Times Cited, WoS Core": 1718,
    "Times Cited, All Databases": 1786,
    "Publication Year": 2017,
    "Research Areas": "Chemistry",
    "UT (Unique WOS ID)": "WOS:000399785500013",
    "Markdown": "# Activating lattice oxygen redox reactions in metal oxides to catalyse oxygen evolution  \n\nAlexis Grimaud1†‡, Oscar Diaz-Morales2‡, Binghong Han3‡, Wesley T. Hong3, Yueh-Lin Lee1,4, Livia Giordano4,5, Kelsey A. Stoerzinger3, Marc T. M. Koper2 and Yang Shao-Horn1,3,4\\*  \n\nUnderstanding how materials that catalyse the oxygen evolution reaction (OER) function is essential for the development of efficient energy-storage technologies. The traditional understanding of the OER mechanism on metal oxides involves four concerted proton–electron transfer steps on metal-ion centres at their surface and product oxygen molecules derived from water. Here, using in situ $\\mathfrak{r}\\mathfrak{s}_{\\pmb{0}}$ isotope labelling mass spectrometry, we provide direct experimental evidence that the $\\bullet_{2}$ generated during the OER on some highly active oxides can come from lattice oxygen. The oxides capable of latticeoxygen oxidation also exhibit pH-dependent OER activity on the reversible hydrogen electrode scale, indicating nonconcerted proton–electron transfers in the OER mechanism. Based on our experimental data and density functional theory calculations, we discuss mechanisms that are fundamentally different from the conventional scheme and show that increasing the covalency of metal–oxygen bonds is critical to trigger lattice-oxygen oxidation and enable non-concerted proton–electron transfers during OER.  \n\nctivating the anionic redox chemistry of oxygen in metal oxides, on the surface and in bulk, can provide exciting opportunities for the design of materials in clean-energy and environmental applications. Such chemistry, which involves the formation of bulk peroxo-like $(\\mathrm{O}_{2})^{n-}$ species1, has been shown to be important in applications such as catalysing the oxygen evolution reaction (OER) in photoelectrochemical water splitting2–6, regenerative fuel cells7,8, rechargeable metal–air batteries and storing energy in lithium-ion battery materials9,10. In this Article we demonstrate that enabling the oxidation of lattice oxygen in highly covalent metal oxides during the OER can enhance OER activity and trigger non-concerted proton– electron transfer.  \n\nThe current understanding of the OER mechanism on metal oxides11,12, which is largely drawn from studies on metal surfaces13, involves four concerted proton–electron transfer steps on surface metal-ion centres, yielding $\\mathrm{\\pH}$ -independent activity on the reversible hydrogen electrode (RHE) scale14. With this mechanism, high OER activities can be obtained by optimizing the binding strength of reaction intermediates on surfaces (neither too strong nor too weak)11,12, and this is supported by experimental findings that the OER activities of oxides in basic solution (the $\\mathrm{O}_{2}/\\mathrm{H}_{2}\\mathrm{O}$ couple: $4\\mathrm{OH}^{-}\\rightarrow\\mathrm{O}_{2}+2\\mathrm{H}_{2}\\mathrm{O}+4\\mathrm{e}^{-})$ correlate with oxide electronic structure parameters such as the estimated $e_{\\mathrm{g}}$ occupancy of surface transitionmetal ions15. However, this conventional mechanism—centred on the redox chemistry of the metal cation—has been challenged by a few observations in recent years. First, some highly active oxides exhibit pH-dependent OER activity on the RHE scale16–18, suggesting that non-concerted proton–electron transfers participate in catalysing the $\\mathrm{OER^{14}}$ . Second, changing the bulk oxide electronic structure, for example, increasing the oxygen $2p$ -band centre (defined relative to the Fermi level)19,20 and metal–oxygen hybridization15,21, has been correlated with enhanced OER activities in perovskites, highlighting the role of bulk electronic structure in catalysing OER kinetics19,22,23. Third, mass spectrometry measurements of oxides stable in bulk during OER, such as $\\mathrm{NiCo}_{2}\\mathrm{O}_{4}$ in basic solution24, $\\mathrm{IrO}_{2}$ (ref. 25) and ${\\mathrm{RuO}}_{2}$ (ref. 26,27) in acid, and Co-Pi (ref. 28) in neutral solutions, reveal that the evolved oxygen molecules can come not only from water molecules but also from the oxide29–31.  \n\nIn this study, we provide, for the first time, direct experimental evidence for the involvement of lattice oxygen redox chemistry in the OER mechanisms within the perovskite family. On-line electrochemical mass spectrometry (OLEMS) measurements of $^{18}\\mathrm{O}$ -labelled perovskites reveal that the oxidation of lattice oxygen occurs during the OER for highly covalent oxides such as $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ , for which up to 37 monolayers of oxides $({\\sim}14~\\mathrm{nm})$ can be involved during the OER process, but not for the less covalent $\\mathrm{LaCoO}_{3}$ . Moreover, highly covalent oxides exhibit $\\mathrm{\\pH}$ -dependent OER activities (on the RHE scale), whereas $\\mathrm{LaCoO}_{3}$ shows the $\\mathrm{\\pH}$ -independent OER activity expected from the conventional OER mechanism. In light of these findings, we discuss potential OER mechanisms that are fundamentally different from the conventional scheme, which involve redox reactions of lattice oxygen and non-concerted proton–electron transfer steps to explain the lattice oxygen oxidation and the $\\mathrm{\\tt{pH}}$ -dependent OER activity observed with highly covalent and active catalysts.  \n\n# Results and discussion  \n\nEvidence of lattice oxygen oxidation during OER. Here, we compare the participation of lattice oxygen oxidation in the OER among $\\mathrm{LaCoO}_{3}$ , $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ (that is, double perovskite $\\mathrm{PrBaCo}_{2}\\mathrm{O}_{5+\\delta})$ and $\\mathrm{SrCoO}_{3-\\delta}$ . It is of particular interest to examine the role of metal–oxygen covalency on the participation of lattice oxygen oxidation in the OER. Density functional theory (DFT) studies19,20 have shown that substituting trivalent ions such as ${\\mathrm{La}}^{3+}$ with divalent ions such as $\\operatorname{Sr}^{2+}$ on the A-site of the perovskite structure $(\\mathrm{ABO}_{3})$ moves the Fermi level closer to the computed O $2p$ -band centre, which is accompanied by a reduced energy gap between the metal $3d$ and O $2p$ -band centres (Fig. 1a), in agreement with work by Cheng and co-authors32. As the Fermi level moves down in energy and closer to the O $2p$ states from $\\mathrm{LaCoO}_{3}$ to $\\mathrm{SrCoO}_{3}$ , the antibonding states below the Fermi level exhibit greater oxygen character, indicative of greater covalency of the metal–oxygen bond33. Similar hybridization has been reported for layered metal chalcogenides34, perovskite-like cuprates35 and layered ${\\mathrm{Li}}_{x}{\\mathrm{CoO}}_{2}$ $(x=\\sim\\bar{0})^{36}$ . The oxidation of lattice oxygen in the perovskites becomes thermodynamically favourable when O $2p$ states at the perovskite Fermi level lie above the redox energy of the $\\mathrm{O}_{2}/\\mathrm{H}_{2}\\mathrm{O}$ couple, as shown in Fig. 1b. Lattice oxygen oxidation is fundamentally different from oxygen intercalation (oxygen vacancy filling via $\\mathrm{ACoO}_{3-\\delta}+2\\delta\\mathrm{OH}^{-}\\rightarrow\\mathrm{ACoO}_{3}+\\delta\\mathrm{H}_{2}\\mathrm{O}+2\\delta\\mathrm{e}^{-})$ upon the oxidation of oxygen-deficient perovskites in an alkaline electrolyte37–39, which is driven by the energy difference between the $\\mathrm{H}_{2}\\mathrm{O}/\\mathrm{H}_{2}$ redox couple and the $\\mathrm{Co}\\quad3d$ states at the Fermi level of oxides (Fig. 1b).  \n\n![](images/2f29f8f2eb2f1bc329607bacfb44e67ab1a4a8af22c50095bb6f4aad9f613990.jpg)  \nFigure 1 | Electronic structures of Co-containing perovskite oxides. a, Difference between the Co $3d\\cdot$ -band centre and the $\\textsf{O}2p$ -band centre versus the O $2p$ -band centre relative to the Fermi level for stoichiometric ${\\mathsf{A C o O}}_{3}$ perovskites. Details of the computational approaches are provided in the Supplementary Information. b, Schematic rigid band diagrams of $\\mathsf{L a C o O}_{3}$ and $\\mathsf{S r C o O}_{3}$ . The position of the $0_{2}/\\mathsf{H}_{2}\\mathsf{O}$ redox couple at pH 14 $\\langle40\\mathsf{H}^{-}\\to0_{2}+2\\mathsf{H}_{2}\\mathsf{O}+4\\mathsf{e}^{-}\\rangle$ is 1.23 V versus RHE, as shown schematically on the right. The relationship between voltages under the RHE and standard hydrogen electrode (SHE) scale is $\\varepsilon_{\\mathsf{R H E}}=\\varepsilon_{\\mathsf{S H E}}+59\\mathsf{m V}\\times\\mathsf{p H}.$  \n\nOLEMS was used to detect the participation of lattice oxygen oxidation in the OER catalysed by these Co-based perovskites with different metal–oxygen covalency. $\\mathrm{LaCoO}_{3}$ , $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ were prepared by conventional solid-state synthesis (see Methods). These oxides are stable in bulk during OER, in contrast to $\\mathrm{Ba_{0.5}S r_{0.5}C o_{0.8}F e_{0.2}O_{3-\\delta}}$ , which becomes amorphous during $\\mathrm{OER^{40,41}}$ . In this work, we assert that oxides are not stable in bulk during OER, when considerable amorphization and loss of metal ions for oxide particles following OER are clearly discernable using transmission electron microscopy and associated spectroscopy, as reported previously40,41. This definition is different from the thermodynamic instability of oxides described in previous work42, which indicates that oxides are formally unstable when interfacing with aqueous electrolytes free of metal cations. Here, the oxide particles were dispersed on a gold disk electrode and labelled with $^{18}\\mathrm{O}$ by potentiostatic holding at $1.6\\mathrm{V}$ versus Au in $\\mathrm{H}_{2}^{\\ 18}\\mathrm{O}$ -labelled 0.1 M KOH solution for $10\\mathrm{min}$ (see Methods and Supplementary Fig. 19). After rinsing with $^{16}\\mathrm{O}$ water to remove $\\mathrm{H}_{2}^{\\mathrm{~18}}\\mathrm{\\dot{O}}$ , OLEMS measurements were performed on these oxide electrodes in a $0.1\\mathrm{{M}}$ KOH solution of $\\mathrm{~{~H~}}_{2}^{16}\\mathrm{{O}}$ using cyclic voltammetry (CV). The OER activities of these oxides on  \n\nAu during OLEMS (Supplementary Fig. 1) were found to be similar to those measured by rotating disk electrode (RDE) measurements on glassy carbon (Supplementary Fig. 2), which are also consistent with those reported previously19,40,41. Oxygen gas of different molecular weights generated during OER was measured in situ by mass spectroscopy, where the signal for mass-to-charge ratio $m/z=32$ represents $^{32}\\mathrm{O}_{2}$ $\\overset{(16}{\\mathrm{O}}^{16}\\mathrm{O})$ , $m/z=34$ represents $^{34}\\mathrm{O}_{2}$ $^{\\cdot16}\\mathrm{O}^{18}\\mathrm{O})$ and $m/z=36$ represents $^{36}\\mathrm{O}_{2}$ $^{\\prime18}\\mathrm{O}^{18}\\mathrm{O})$ . The signal of $m/z=34$ was normalized by the signal of $m/z=32$ to account for natural isotopic abundance $(\\sim0.2\\%)$ . This normalized signal of $m/z=34$ to $m/z=32$ and the mass signal of $m/z=36$ collected from the OER in the first cycle are shown in Fig. 2a and b, respectively. As the signals of $m/z=32$ detected by OLEMS include both molecular oxygen formed during the OER and oxygen from ambient air, the ratio of OER current involving $m/z=34$ and $m/z=36$ to that with $m/z=32$ could not be quantified in this study.  \n\nNeither normalized $m/z=34$ (Fig. 2a) nor $m/z=36$ (Fig. 2b) was detected during the OER for $\\mathrm{LaCoO}_{3}$ , indicating there was no oxygen from the $\\mathrm{LaCoO}_{3}$ lattice in the molecular oxygen evolved. For $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , the normalized $m/z=34$ rose slightly above the natural abundance of $^{18}\\mathrm{O}$ (Fig. 2a), and small signals of $m/z=36$ were detected at $1.6\\mathrm{V}$ versus RHE and above (Fig. 2b), indicating successful $^{18}\\mathrm{O}$ labelling and the involvement of lattice oxygen in the OER. Of significance, lattice oxygen oxidation during the OER measured from $\\mathrm{SrCoO}_{3-\\delta}$ was more pronounced than that for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}.$ , as both the normalized $m/z=34$ and $m/z=36$ signals detected from $\\mathrm{SrCoO}_{3-\\delta}$ were greater, with lower onset voltages $_{1.5\\mathrm{V}}$ versus RHE and greater), as shown in Fig. 2. The detection of $^{16}\\mathrm{O}^{18}\\mathrm{O}$ during the OER (one oxygen from the electrolyte and the other from the oxide lattice) as well as $^{18}\\mathrm{O}^{18}\\mathrm{O}$ (two oxygens from the oxide lattice) requires the oxidation of lattice oxygen from metal oxides and the formation of oxygen vacancies during the OER. The release of $^{16}\\mathrm{O}^{16}\\mathrm{O}$ , $^{16}\\mathrm{O}^{18}\\mathrm{O}$ and $^{18}\\mathrm{O}^{18}\\mathrm{O}$ for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ , which are stable in bulk during OER, differs from those for catalysts that are amorphized in bulk under OER, such as $\\mathrm{Ba_{0.5}S r_{0.5}C o_{0.8}F e_{0.2}O_{3-\\delta}}$ (refs 40,41), where $^{16}\\mathrm{O}^{16}\\mathrm{O}$ and $^{18}\\mathrm{O}^{18}\\mathrm{O}$ are released independently of the potential in the negative sweep (Supplementary Figs 1 and 3). The amount of $^{18}\\mathrm{O}$ detected during the OER was used to estimate the number of monolayers, or thickness normal to the (001) surface, in the cubic perovskite structure involved in the OER for $\\mathrm{LaCoO}_{3}$ , $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ (Supplementary Table 3 and Supplementary Fig. 4). The minimum oxide thickness involved during OER was estimated to be ${\\sim}0.5\\ \\mathrm{nm}$ for $\\mathrm{LaCoO}_{3}$ , indicating a surface OER process. In contrast, the minimum thicknesses estimated for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ were ${\\sim}0.7$ , 1.5 and $14\\mathrm{nm}$ , respectively, demonstrating the participation of oxygen from the oxides in the OER. Having considerable oxygen originating from oxides participating in the OER is not surprising, considering that the bulk diffusion of oxygen ions at room temperature can be large (for example, $\\tilde{D}_{O}\\approx\\mathrm{i}0^{-11}\\mathrm{cm}^{2}s^{-1}$ for $\\mathrm{SrCoO}_{3-\\delta})^{38}$ . The physical origin of the oxidation of lattice oxygen can be attributed to a shift of the Fermi level deeper into the O $2p$ band. When the $\\mathrm{O}_{2}/\\mathrm{H}_{2}\\mathrm{O}$ redox potential aligns with the energy corresponding to the O $2p$ states of the oxide, holes are created in the oxygen band, which can allow oxidized oxygen ions to form $(\\mathrm{O}_{2})^{\\dot{n}-}$ species, analogous to anionic redox of chalcogenides34 and/or molecular oxygen33 (leaving oxygen vacancies behind). The more $\\mathrm{O}_{2}$ generated during OER, the greater the value for $^{18}\\mathrm{O}/^{16}\\mathrm{O}$ (Supplementary Fig. 5), indicating that the amount of lattice oxygen detected during the OER process depends on the position of the O $2p$ states relative to the Fermi level and correlates with the OER activity for this family of oxides.  \n\n![](images/e6c889a1b7de9a0fe57079127f4caa03f59edb8a991d8d5f78e0f03ee1712394.jpg)  \nFigure 2 | Direct evidence of lattice oxygen oxidation involved in the OER of $^{18}0$ -labelled perovskites. Data were measured in 0.1 M KOH by OLEMS at a scan rate of $2{\\mathsf{m}}{\\mathsf{V}}{\\mathsf{s}}^{-1}$ (no gas bubbling). a, $^{34}\\mathrm{O}_{2}/{}^{32}\\mathrm{O}_{2}$ ratios, where the straight lines correspond to the natural abundance of $^{18}\\mathrm{O}$ of $0.2\\%$ . The arrows indicate forward and backward scans. b, $^{36}\\mathrm{O}_{2}$ signal. All data were taken from the first cycle (data from the second cycle are provided in Supplementary Fig. 1). $^{34}\\mathsf{O}_{2}$ but not $^{36}\\mathsf{O}_{2}$ was detected in the cathodic sweep at potentials where there is no OER activity. The origin of this is not understood. We speculate that chemical processes triggered by OER with lattice oxygen participation might be responsible for the evolution of $^{34}\\mathsf{O}_{2}$ in the cathodic sweep, but further studies are needed.  \n\nThe participation of lattice oxygen oxidation in the OER for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ highlights a fundamental mechanistic shift from the conventional OER mechanism involving four concerted proton–electron-transfer steps on surface metal ions11,12. The lattice oxygen oxidation found in the OER in this study is distinct from the participation of active vacancy sites in the $\\dot{\\mathrm{OER}}^{43}$ or the oxygen intercalation reaction in oxygendeficient oxides37–39, which largely involve the redox of the cation and do not form $_{\\mathrm{O-O}}$ bonds involving metal oxide lattice oxygen. As previous work shows that $\\mathrm{La}_{0.8}\\mathrm{Sr}_{0.2}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{La}_{0.6}\\mathrm{Sr}_{0.4}\\mathrm{CoO}_{3-\\delta}$ exhibit pH-dependent OER activity on the RHE scale16,17, we investigate in the following whether the participation of lattice oxygen oxidation in the OER is associated with non-concerted proton–electron transfer processes14, giving rise to $\\mathrm{\\pH}$ -dependent OER activity.  \n\npH-dependent OER kinetics. Oxides that exhibited the oxidation of lattice oxygen during the OER (that is, $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta};$ Fig. 2) were found to have pH-dependent OER kinetics on the RHE scale (Fig. 3 and  \n\n![](images/11a293f12df2f94c0ca302cd1e725442157941e8a44d73e41dea6a457176418a.jpg)  \nFigure 3 | pH-dependent OER activity on the RHE scale. a, CV measurements from $\\mathsf{O}_{2}$ -saturated 0.03 M KOH $\\cdot\\mathsf{p H}12.5)$ to 1 M KOH (pH 14) recorded at $\\mathsf{10}\\mathsf{m}\\mathsf{V}\\mathsf{s}^{-1}$ . ${\\big\\vert}\\mathbf{b},$ Specific OER activity (current normalized by oxide BET surface area) at $\\ensuremath{\\uparrow.55\\mathrm{V}}$ versus RHE after $i R$ correction as a function of pH. The nominal oxide loading is $0.25\\mathrm{mg}_{\\mathrm{oxide}}\\mathrm{cm}_{\\mathrm{disk}}^{-2}.$ Error bars represent standard deviation of three measurement results.  \n\nSupplementary Figs 6,7,8), while $\\mathrm{LaCoO}_{3}$ without any lattice oxygen oxidation exhibited pH-independent OER kinetics. RHE was used as the reference electrode to ensure that the OER overpotential with respect to the equilibrium $\\mathrm{O}_{2}/\\mathrm{H}_{2}\\mathrm{O}$ redox potential remained identical across different values of $\\mathrm{\\pH}$ (ref. 17). The pH dependence of OER activity on the RHE scale indicates the presence of non-concerted proton–electron transfer steps during the OER, where the rate-limiting step is either a proton transfer step or preceded by acid/base equilibrium14,17. OER currents from CV measurements were found to increase with increasing $\\mathrm{pH}$ for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ , but remained unchanged for $\\mathrm{LaCoO}_{3}$ , as shown in Fig. 3a. The intrinsic OER activities of $\\mathrm{LaCoO}_{3}$ , $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ , as estimated by either the surface-area-normalized current at $1.55\\mathrm{V}$ versus RHE (Fig. 3b) or the OER potential at $0.2\\ \\mathrm{mA}\\ \\mathrm{cm}_{\\mathrm{oxide}}^{-2}$ (Supplementary Fig. 6) from CV and galvanostatic measurements, are compared as a function of $\\mathsf{p H}$ . The specific OER activities of these oxides, with error bars, are presented in Supplementary Figs 7 and 8. Greater metal–oxygen covalency (Fig. 1) and lattice oxygen oxidation (Fig. 2) correlates with increasing OER activity, which is most evident at $\\mathrm{pH}14$ (Fig. 3b). Significantly, $\\mathrm{SrCoO}_{3-\\delta}$ was found to have OER activity approximately three times greater than that of $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ (ref. 19), one of the most active and stable oxides previously reported for the OER.  \n\nBoth CV and galvanostatic measurements show that the intrinsic OER activity of $\\mathrm{LaCoO}_{3}$ is independent of $\\mathrm{\\pH}$ , while those for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ are pHdependent (increasing the pH from 12.5 to 14 led to greater intrinsic OER activities by one order of magnitude). Interestingly, the specific OER currents measured from CV can be consistently higher than those from galvanostatic measurements. This difference can be attributed to the contribution from oxygen intercalation following the oxidation of ${\\mathrm{Co}}^{3+}$ ions (as evidenced by the redox peak at ${\\sim}1.2\\mathrm{V}$ versus RHE in Supplementary Fig. 9) in oxygen-deficient perovskites37–39 to the OER current measured from CV (measured over minutes), which is negligible in galvanostatic measurements (with much lower rates, measured over hours). This argument is further supported by the fact that the difference in the OER activity between CV and galvanostatic measurements increases with increasing charge associated with oxygen vacancy filling of the oxides before OER (Fig. 3 and Supplementary Fig. 10). Therefore, galvanostatic measurements provide a more accurate measure of OER activity than CV.  \n\nOER mechanisms with lattice oxygen oxidation. The conventional OER mechanism11,12 on surface metal sites (Fig. 4a and Supplementary Fig. 11) catalyses OER on the oxide surface, which cannot evolve more than ${\\sim}0.4~\\mathrm{nm}$ of $^{18}\\mathrm{O}$ during OER after $^{18}\\mathrm{O}$ -labelling. The observed $^{18}\\mathrm{O}$ released in the OER for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ can be attributed to oxide lattice participation greater than ${\\sim}0.4\\ \\mathrm{nm}$ , which cannot be explained by the traditional OER mechanism. In addition, this traditional mechanism with concerted proton–electron transfer cannot explain the pH-dependent OER activity on the above oxides. Therefore, mechanisms that are fundamentally different from the conventional scheme11,12 are needed to explain the involvement of lattice oxygen oxidation $\\binom{34}{\\phantom{^{34}\\mathrm{O}_{2}}}$ and/or $^{36}\\mathrm{O}_{2}\\qquad$ detected) in the OER for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , and $\\mathrm{SrCoO}_{3-\\delta},$ as well as their $\\mathrm{pH}$ -dependent OER kinetics. Conversely, given the absence of lattice oxygen oxidation and $\\mathrm{\\pH}$ -independent OER kinetics observed for $\\mathrm{LaCoO}_{3}$ , the OER kinetics on $\\mathrm{LaCoO}_{3}$ can be explained by the conventional concerted proton–electron transfer mechanism on surface cobalt sites (Fig. 4a)11,12. Considering the cobalt sites of the (001) $\\mathrm{CoO}_{2}$ surface, the computed free energies of concerted proton–electron transfer steps show that the $_{\\mathrm{O-O}}$ bond formation (step 2 in Fig. 4a) is rate-limiting for $\\mathrm{LaCoO}_{3}$ , in agreement with previous works11,44. Increasing metal–oxygen covalency from $\\mathrm{LaCoO}_{3}$ to $\\mathsf{S r C o O}_{3}$ was found to promote OER kinetics on the metal sites in this concerted mechanism (Fig. 4b), as indicated by the lowered potential needed for all steps becoming thermodynamically downhill.  \n\n![](images/213c37e49f54a221311323f5f16672e33b618fbdaced1d430f4d4c378bc4a625.jpg)  \nFigure 4 | OER mechanisms with concerted and non-concerted proton–electron transfer. a, Conventional OER mechanism involving concerted proton– electron transfers on surface metal sites12,58,59, with oxygen from the electrolyte in blue and from the oxide lattice in red. c,e, Possible OER mechanisms involving concerted proton–electron transfer on surface oxygen sites to yield $^{16}\\mathrm{O}^{18}\\mathrm{O}\\ (^{34}\\mathrm{O}_{2})$ and $^{18}\\mathrm{O}^{18}\\mathrm{O}$ $^{(36}\\mathsf{O}_{2})_{\\cdot}$ , respectively. b,d, Computed free energies (ΔG) of coupled proton–electron transfer OER steps on the metal $(\\pmb{\\ b})$ and oxygen $(\\pmb{\\mathsf{d}})$ sites of the (001) $M O_{2}$ surface to form $^{32}\\mathsf{O}_{2}$ , respectively, where all the steps are thermodynamically downhill. f, The computed $\\Delta G$ of (001) $M O_{2}$ shows that the formation of the $_{0-0}$ bond is energetically favourable for surface oxygen sites of $\\mathsf{S r C o O}_{3}$ but not for $\\mathsf{L a C o O}_{3}$ . g,h, Possible non-concerted proton–electron transfer OER mechanisms that evolve $^{34}\\mathsf{O}_{2}\\left(\\pmb{\\mathsf{g}}\\right)$ and $^{36}{\\mathsf O}_{2}({\\mathsf h})$ with pH-dependent OER activity, with electron-transfer steps in yellow, proton-transfer steps in green and charged intermediates accommodated by metal ion valence changes.  \n\nWe next discuss reaction schemes involving the oxidation of lattice oxygen and concerted proton–electron transfers to explain the detection of $^{34}\\mathrm{O}_{2}$ and $^{36}\\mathrm{O}_{2}$ , which will be further modified to include decoupled proton–electron transfers to explain the observed pH-dependent OER activities. The formation of $^{34}\\mathrm{O}_{2}$ can be explained by charge transfer steps on surface oxygen sites. One likely mechanism is shown in Fig. 4c, though other possibilities may exist. The first three steps on surface oxygen sites are concerted proton–electron transfers analogous to those on metal sites in Fig. 4a; these are followed by a chemical step to produce molecular $^{34}\\mathrm{O}_{2}\\:(^{16}\\mathrm{O}^{18}\\mathrm{O})$ and an oxygen vacancy, and a subsequent concerted proton–electron transfer step to regenerate the O–M–OH surface. Increasing metal–oxygen covalency from $\\mathrm{LaCoO}_{3}$ to $\\mathsf{S r C o O}_{3}$ promotes OER activity for the surface oxygen sites, as is evident from the lower computed potential necessary for all reaction steps to become thermodynamically favourable (Fig. 4d). The computed thermodynamic OER potential on surface oxygen sites of $\\mathsf{S r C o O}_{3}$ is comparable to that on surface Co sites (Fig. 4b), suggesting that both can be active for OER. In contrast, surface Co sites (Fig. 4b) of $\\mathrm{LaCoO}_{3}$ are more active than surface oxygen sites, suggesting that surface Co sites govern the OER activity. Note that the ratelimiting step of OER on $\\mathrm{LaCoO}_{3}$ was found to be different from that for $\\mathrm{SrCoO}_{3}$ , where $_{\\mathrm{O-O}}$ bond formation on surface metal and oxygen sites (step 2 in Fig. $^{4\\mathrm{a},\\mathrm{c}}$ ) and $\\mathrm{OH^{-}}$ adsorption onto surface metal and oxygen sites (step 1 in Fig. $\\mathrm{4a,c}$ ) limits the OER kinetics of $\\mathrm{LaCoO}_{3}$ and $\\mathrm{SrCoO}_{3}$ , respectively (Fig. 4d). Increasing $_{\\mathrm{Co-O}}$ covalency from $\\mathrm{LaCoO}_{3}$ to $\\mathrm{SrCoO}_{3}$ is associated with moving the oxide Fermi level below the $\\mathrm{O}_{2}/\\mathrm{H}_{2}\\mathrm{O}$ redox energy (Fig. 1b); the oxide surface becomes more negatively charged when equilibrated with the electrolyte and preferentially acts as a Brønsted base, making deprotonation by $\\mathrm{OH^{-}}$ more difficult for $\\mathrm{sr}$ -substituted $\\mathrm{LaCoO}_{3}$ (ref. 45).  \n\n![](images/e31afe9fb0ca0dd144f1c81a5b2cf920286431b8e84b5796bf9e6e1522eccf79.jpg)  \nFigure 5 | Electrochemical oxygen intercalation into brownmillerite $\\mathsf{S r C o o}_{3-\\delta}$ followed by the OER. Galvanostatic charging (iR-corrected) of $\\mathsf{S r C o O}_{3-\\delta}$ in $\\mathsf{O}_{2}$ -saturated (solid line) and $\\mathsf{A r}$ -saturated (dotted line) 1 M KOH as a function of charge passed at $7\\mathrm{mA}\\mathrm{g}_{\\mathrm{oxide}}^{-1}$ . The oxide loading was $0.25\\mathsf{m g_{o x i d e}c m_{d i s k}^{-2}}$ mixed with a carbon loading of $0.05\\mathsf{m g_{c a r b o n}c m_{d i s k}^{-2}}$ . Inset scheme: relationship between the measured voltage of oxygen intercalation at ${\\mathsf{p H}}14$ (unit activity for $\\mathsf{O H}^{-})$ and oxide band structure.  \n\nThe formation of $^{36}\\mathrm{O}_{2}$ , most pronounced for $\\mathrm{SrCoO}_{3-\\delta}$ , requires the formation of an $^{18}\\mathrm{O}-^{18}\\mathrm{O}$ bond from two surface oxygen ions of the oxide. Although other possibilities may exist (Supplementary Fig. 12), a likely mechanism is shown in Fig. 4e. The mechanism consists of two chemical steps, which create an O–O bond, then a molecular oxygen from lattice oxygen sites via the formation of two oxygen vacancies (steps 2 and 3 in Fig. 4e). The remaining steps of the mechanism involve four concerted proton–electron transfers—two for regenerating the surface oxygen sites and two associated with deprotonating surface oxygen ions. DFT calculations show that the chemical step to form an $_{\\mathrm{O-O}}$ bond and an oxygen vacancy from surface oxygen ions of $\\mathsf{S r C o O}_{3}$ is thermodynamically favourable (Fig. 4f and Supplementary Fig. 13). On the other hand, the $_{\\mathrm{O-O}}$ bond formation from two surface oxygen ions of $\\mathrm{LaCoO}_{3}$ is energetically unfavourable (Fig. 4f ), which is consistent with previous findings that the energy penalty associated with oxygen vacancy formation increases with moving the Fermi level away from the O $2p$ -band centre20 (Supplementary Fig. 14) and the absence of lattice oxygen oxidation on $\\mathrm{LaCoO}_{3}$ .  \n\nThe $\\mathrm{\\pH}$ -dependent OER activities on the RHE scale observed for $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ require the decoupling of some proton transfers from electron transfers in the reaction schemes in Fig. 4c,e. Two likely reaction schemes are proposed in Fig. $^{4\\mathrm{g},\\mathrm{h}}$ . It is postulated here that the rate-limiting step in the concerted mechanism (Fig. 4c,e)—deprotonation of hydroxyl groups $\\mathrm{O-M}^{n+}-\\mathbf{O}\\mathbf{H}+\\mathrm{O}\\mathbf{\\bar{H}}^{-}\\to\\mathrm{O-M}^{(n-1)+}-\\mathbf{O}+\\mathrm{H}_{2}\\mathbf{O}$ (highlighted in green)—has become non-concerted and decoupled from subsequent electron transfer during the evolution of molecular oxygen, $\\bar{\\mathrm{{O-M}}}^{(n-1)+}{\\bf-O}{\\bf O}\\to\\mathrm{O-M}^{n+}{\\bf-}\\breve{\\bigsqcup}+{\\bf O}_{2}+\\mathrm{e}^{-}$ (highlighted in yellow). Nevertheless, we emphasize that decoupling of proton and electron transfers can also occur in other reaction steps in these OER mechanisms, and, ultimately, the $\\mathsf{p}K_{\\mathrm{a}}$ of such surface deprotonation determines the step that is responsible for the overall pH-dependent kinetics14,46.  \n\nThe proposed rate-limiting step for the OER kinetics of oxides with strong covalency, that is, the deprotonation of surface hydroxyl groups, is supported by the remarkably different kinetics of galvanostatic oxidation of brownmillerite $\\mathrm{SrCoO}_{3-\\delta}$ $(\\mathrm{SrCoO}_{3-\\delta}+2\\delta\\mathrm{OH}^{-}\\rightarrow$ $\\mathrm{SrCoO}_{3}+\\delta\\mathrm{H}_{2}\\mathrm{O}+2\\delta\\mathrm{e}^{-})$ in Ar-saturated and $\\mathrm{O}_{2}$ -saturated $^{1\\mathrm{{M}}}$ KOH, as shown in Fig. 5. Galvanostatic oxidation of brownmillerite $\\mathrm{SrCoO}_{3-\\delta}$ in $\\mathrm{O}_{2}$ -saturated 1 M KOH led to the well-known plateau47 of oxygen intercalation, whereas oxidation with Ar saturation exhibited no clear plateau of oxygen intercalation. Oxygen vacancy filling into $\\mathrm{SrCoO}_{3-\\delta}$ is driven by the energy difference between the $\\mathrm{H}_{2}\\mathrm{O}/$ $\\mathrm{H}_{2}$ redox couple and the $\\mathrm{Co}\\ 3d$ states at the Fermi level of oxides (Fig. 5, inset), which should not be influenced by Ar or $\\mathrm{O}_{2}$ saturation in the electrolyte. The kinetics of the oxygen filling of $\\mathrm{SrCoO}_{3-\\delta}$ oxygen vacancies in Ar-saturated electrolyte can be attributed to the slow deprotonation kinetics of surface hydroxyl groups, which is needed to intercalate oxygen ions into the oxide lattice. As our DFT calculations show that filling vacancies with ${{\\mathrm{O^{*}}}}$ is more favourable than filling with ${\\mathrm{OH}}^{*}$ on the Co-terminated (001) surface of brownmillerite $(-0.82\\mathrm{eV}$ per ${{\\cal O}^{*}}$ and $-0.40\\ \\mathrm{eV}$ per $\\mathrm{OH^{*}})$ at $1.23{\\mathrm{V}}$ versus RHE (Supplementary Fig. 15), surface oxygen vacancies in the $\\mathrm{O}_{2}$ -saturated electrolyte are filled preferentially by $\\mathrm{O}_{2}$ rather than by $\\mathrm{OH^{-}}$ . The filling of surface vacancies with ${{\\mathrm{O}}^{*}}$ rather than $\\mathrm{OH^{*}}$ allows the oxygen intercalation to proceed through an ${\\bf O-M-O O H}$ intermediate rather than O–M– OH (step 3 instead of step 1 in Fig. $^{4\\mathrm{b},\\mathrm{d})}$ ), rendering the faster kinetics observed in the $\\mathrm{O}_{2}$ -saturated electrolyte. Further support for the deprotonation being decoupled from electron transfer comes from the observation that the oxygen intercalation kinetics become more sluggish in the $\\mathrm{O}_{2}$ -saturated electrolyte when decreasing the $\\mathrm{\\pH}$ from 14 to 13 (Supplementary Fig. 16).  \n\nThe proposed OER mechanisms in Fig. 4g,h used to explain lattice oxygen oxidation and the pH-dependent OER kinetics of highly covalent oxides bridge between the conventional OER mechanism11,12,14,15 of metal oxides and those reported for electrodeposited oxide $\\mathrm{\\flms^{17,28,48}}$ . Oxygen exchange is shown to participate in the OER of ${\\mathrm{Co-Pi}}$ films electrodeposited in ${\\bf\\dot{C}}{\\bf0}^{2+}$ -phosphate-containing electrolytes under neutral conditions via $^{18}\\mathrm{O}$ -isotopic labelling28,48, while no pH-dependence of OER kinetics on the RHE scale is noted17,28, where proton-acceptor phosphate species governs the kinetics of surface deprotonation28 to enable high OER activity. Recent mechanistic studies of these catalysts move beyond charge transfer steps on metal sites and propose the involvement of oxygen redox chemistry mediated by oxygen holes (that is, ${\\mathrm{Co}}^{4+}$ described as ${\\mathrm{Co}}^{3+}$ and a hole in the O states30 promotes O–O bond formation) in OER kinetics, similar to the mechanisms proposed here28,30,48. Further support for the synergy in the OER mechanisms comes from the pH-dependent OER kinetics on the RHE scale in basic solution reported for electrodeposited (Ni,Fe) OOH film18, which has been attributed to deprotonation leading to the formation of a negatively charged oxygenated intermediate (‘active oxygen’) participating in the OER, as suggested by surface-enhanced Raman spectroscopy18. Further studies are required to understand the role of non-concerted proton–electron transfer steps on the OER kinetics of electrodeposited oxide films and highly covalent oxides and the nature of surface oxygen participating in O–O bond formation (for example, terminal oxygen, as proposed for electrodeposited films28,30,48, versus bridging oxygen, for covalent oxides in this study).  \n\nIn summary, by combining electrochemical characterization with DFT studies, we have demonstrated that the bulk electronic structure of transition-metal oxides, namely the metal–oxygen covalency, not only governs the OER activity but also the reaction mechanism. Specifically, we have shown that lattice oxygen can be activated for the OER and promote new reaction pathways, in addition to the classically studied mechanism on surface metal sites. The OER on oxygen sites can be triggered when the Fermi level becomes pinned to the top of the O $2p$ band for highly covalent oxides, resulting in electronic states near the Fermi level with substantial O $2p$ character. Moreover, OER activities on the RHE scale for perovskites with strong metal–oxygen covalency become higher with increasing $\\mathrm{\\pH_{\\mathrm{\\cdot}}}$ indicating that non-concerted proton– electron transfer steps are coupled to the activation of lattice oxygen redox reactions. These new insights open new possibilities for developing highly active catalysts using lattice oxygen redox processes and non-concerted proton–electron transfer steps, bypassing the design limitations of engineering catalysts under the conventional mechanism, which involves only concerted proton–electron transfer steps on surface metal ion sites.  \n\ndetermined using Brunauer, Emmet and Teller (BET) analysis on a Quantachrome ChemBET Pulsar from single-point BET analysis performed after $12\\mathrm{{h}}$ outgassing at $150^{\\circ}\\mathrm{C}$ (Supplementary Table 2). More details are provided in the Supplementary Information.  \n\nElectrochemical measurements of OER activities. Electrodes used for CV and galvanostatic measurements were prepared by drop-casting ink containing oxide catalyst powder on a glassy carbon electrode (GCE), as described previously49. The glassy carbon electrode surface $5\\mathrm{mm}$ diameter) was loaded with $0.25\\mathrm{~mg}_{\\mathrm{oxide}}\\mathrm{cm}_{\\mathrm{disk}}^{-2}$ and a mass ratio of 5:1:1 of oxide catalyst:acetylene black carbon: Nafion. We used an oxide:carbon mass ratio of 5:1 for the OER measurements, as the specific OER activity estimated for oxide particles from such measurements agrees well with that measured from well-defined epitaxial oxide thin-film surfaces of comparable oxide chemistry48. Electrochemical measurements were performed with a rotating disk electrode set-up using a glass electrochemical cell with $\\mathrm{Ag/AgCl}$ reference electrode and Pt counter electrode. The potential was controlled using a Biologic VSP-300 potentiostat. Ohmic losses were corrected by subtracting the ohmic voltage drop from the measured potential, using the electrolyte resistance determined by high-frequency a.c. impedance, where $i R$ -corrected potentials are denoted as $E-i R$ (i as the current and $R$ as the electrolyte resistance). Rotating ring disk electrode (RRDE) measurements were performed on a GCE disk and a Pt ring held at $0.4\\mathrm{V}$ versus RHE. RRDE measurements of $\\mathrm{SrCoO}_{3-\\delta}$ in Ar-saturated 1 M KOH (Supplementary Fig. 17) revealed that oxygen was detected only at potentials higher than ${\\sim}1.5\\mathrm{V}$ versus RHE and thus the currents measured at lower potentials are not related to OER but rather to oxygen intercalation upon oxidation of $\\mathrm{SrCoO}_{3-\\delta}$ (that is, filling of oxygen vacancies in the oxide). More details are provided in the Supplementary Information.  \n\nOnline electrochemical mass spectroscopy. OLEMS experiments50 were performed using an EvoLution mass spectrometer system (European Spectrometry Systems). Volatile reaction products were collected from the electrode interface by a small inlet tip positioned close $(\\sim10~\\upmu\\mathrm{m})$ to the electrode surface using a micrometric screw system and a camera. More details are provided in the Supplementary Information. The electrochemical cell used for these experiments is a two-compartment cell with three electrodes, with a gold wire as counter electrode and a reversible hydrogen electrode as the reference electrode. The working electrode was prepared in a comparable manner to that used for OER activity measurements but on a gold disk electrode ( $4.6\\ \\mathrm{mm}$ diameter) and with an oxide loading of $0.25\\mathrm{mg_{oxide}}\\mathrm{cm}_{\\mathrm{disk}}^{-2}$ with no carbon. We used oxide-only electrodes on Au in the OLEMS measurements to avoid any corrosion currents associated with carbon such as oxidization of carbon to form CO and $\\mathrm{CO}_{2}$ , as shown previously51,52. The OER activities measured for OLEMS measurements without iR correction were in good agreement with those from RDE measurements with $i R$ correction (Supplementary Fig. 2). Moreover, the gold electrode was shown to have no OER activity within the potential range used for the OLEMS measurements (Supplementary Fig. 18). Pristine electrodes were oxidized for $10\\mathrm{min}$ in 0.1 M KOH made with $^{\\mathrm{{\\small~\\cdot~}}_{18}}\\mathrm{{O}}$ -labelled water (GMP standard from CMR, $98\\%^{18}\\mathrm{O})$ at $1.6\\mathrm{V}$ versus gold counter electrode (no gas bubbling), to label them with $^{18}\\mathrm{O}$ ( $^{18}\\mathrm{O}$ labelling currents are provided in Supplementary Fig. 19). Electrodes were then rinsed with $^{16}\\mathrm{O}$ water to remove $\\mathrm{H}_{2}^{\\ 18}\\mathrm{O}$ and measured in $0.1\\mathrm{~M~}$ KOH solution of $\\mathrm{H}_{2}^{16}\\mathrm{O}$ at $2\\mathrm{mVs}^{-1}$ for two cycles (no gas bubbling). Because samples were rinsed with $^{16}\\mathrm{O}$ water after $^{18}\\mathrm{O}$ -labelling, it is unlikely that $^{18}\\mathrm{O}$ species (for example, ${\\mathrm{OH^{*}}}$ or $\\mathrm{H}_{2}\\mathrm{O})$ ) adsorbed on the oxide surface contributed substantially to the observed $^{16}\\mathrm{O}^{18}\\mathrm{O}$ or $^{18}\\mathrm{O}^{18}\\mathrm{O}$ signal, especially for oxides with a high Co oxidation state. $\\mathrm{Ba}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{Co}_{0.8}\\mathrm{Fe}_{0.2}\\mathrm{O}_{3-\\delta}$ , which transforms amorphous cobalt–ironcontaining oxyhydroxides in bulk during $\\mathrm{OER}^{36,37}$ , had markedly different OLEMS data (Supplementary Fig. 1) from $\\mathrm{LaCoO}_{3}$ , $\\mathrm{La}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{CoO}_{3-\\delta}$ , $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ and $\\mathrm{SrCoO}_{3-\\delta}$ particles, which remained stable upon OER19,40. $\\mathrm{Ba}_{0.5}\\mathrm{Sr}_{0.5}\\mathrm{Co}_{0.8}\\mathrm{Fe}_{0.2}\\mathrm{O}_{3-\\delta}$ had monotonically increasing lattice oxygen $\\mathrm{\\dot{(}^{18}O^{18}O)}$ signals in the positive-going and negative-going scans, confirming bulk oxide decomposition. Moreover, the tip in the open OLEMS set-up used in this study detects $^{32}\\mathrm{\\dot{O}}_{2}$ not only from OER but also from the atmosphere. This contribution from the atmosphere in the $^{32}\\mathrm{O}_{2}$ detected is evident from Supplementary Table 3, where the amount of $^{32}\\mathrm{O}_{2}$ is nearly identical for all four oxides studied. We therefore chose not to include $^{32}\\mathrm{O}_{2}$ data in Fig. 2. Each oxide was examined by OLEMS measurements of two to four different electrodes, and the results of different electrodes were comparable, generating error bars for OLEMS data analysis such as the $^{18}\\mathrm{O}/^{16}\\mathrm{O}$ ratio in Supplementary Figs 4 and 5 and Supplementary Table 3. Multiple OLEMS measurements of $\\mathrm{SrCoO}_{3-\\delta}$ (Supplementary Fig. 20) and $\\mathrm{Ba_{0.5}S r_{0.5}C o_{0.8}F e_{0.2}O_{3-\\delta}}$ (Supplementary Fig. 3) are provided in the Supplementary Information as examples.  \n\n# Methods  \n\nSynthesis and bulk characterization. Perovskite $\\mathrm{La}_{1-x}\\mathrm{Sr}_{x}\\mathrm{CoO}_{3-\\delta}$ ( $x=0$ , 0.5 and 1) and $\\mathrm{Pr}_{0.5}\\mathrm{Ba}_{0.5}\\mathrm{CoO}_{3-\\delta}$ were synthesized by a conventional solid-state route15,19. All catalysts reported in this study are single-phase, as analysed by X-ray diffraction (XRD), with lattice parameters consistent with those reported previously (Supplementary Table 1). XRD measurements were performed using a PANalytical X’Pert Pro powder diffractometer in the Bragg-Brentano geometry using copper $\\mathrm{K}_{\\upalpha}$ radiation, and data were collected using the X’Celerator detector in the $8\\mathrm{-}80^{\\circ}$ window in the 2θ range. The specific surface area of each oxide sample was  \n\nDFT studies. DFT calculations with Hubbard U $\\cdot U_{\\mathrm{eff}}=3.3\\ \\mathrm{eV},$ ) correction20,53 for the Co 3d electrons were performed with the Vienna $\\mathbf{\\nabla}_{A b}$ -initio Simulation Package $(\\mathrm{VASP})^{54,55}$ using the projector-augmented plane-wave method56. Exchangecorrelation was treated in the Perdew-Wang-91 generalized gradient approximation $(\\mathrm{GGA})^{57}$ . Fully relaxed stoichiometric bulk perovskite calculations were simulated with $2\\times2\\times2$ perovskite supercells. The double perovskites were simulated based on the reported ordered structures within the $2\\times2\\times2$ perovskite supercell. Both O $2p$ -band and metal $_{3d.}$ -band centres were determined by taking the weighted mean energy of the projected density of states of O 2p and metal $3d$ states (both occupied and unoccupied states) relative to the Fermi level. We used a symmetric slab cut along the (001) direction as a model for the (001) $\\mathrm{MO}_{2}$ terminated surface and $(2\\times2)$ surface supercells. The thermodynamic approaches for calculating the OER free energy profiles are described in ref. 44. For more details see Supplementary Information.  \n\nData availability. The original data for Figs 1– 5 in this manuscript are available from https://figshare.com/s/286742731dfc47a8f1c1. Other data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.  \n\n# Received 3 August 2016; accepted 14 November 2016; published online 9 January 2017  \n\n# References  \n\n1. Grimaud, A., Hong, W. T., Shao-Horn, Y. & Tarascon, J. M. Anionic redox processes for electrochemical devices. Nat. Mater. 15, 121–126 (2016).   \n2. Lewis, N. S. & Nocera, D. G. Powering the planet: chemical challenges in solar energy utilization. Proc. Natl Acad. Sci. 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Chem. Phys. 319, 178–184 (2005).   \n59. Goodenough, J. B., Manoharan, R. & Paranthaman, M. Surface protonation and electrochemical activity of oxides in aqueous solution. J. Am. Chem. Soc. 112, 2076–2082 (1990).  \n\n# Acknowledgements  \n\nThis work was supported in part by the Skoltech-MIT Center for Electrochemical Energy, the SMART programme, and the Department of Energy (DOE) and National Energy Technology Laboratory (NETL), Solid State Energy Conversion Alliance (SECA) Core Technology Program (Funding Opportunity Number DEFE0009435). This work is also supported in part by the Netherlands Organization for Scientific Research (NWO) within the research programme of BioSolar Cells, co-financed by the Dutch Ministry of Economic Affairs, Agriculture and Innovation. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy (contract no. DE-AC02-05CH11231).  \n\nThe authors would like to acknowledge Dane Morgan and Jean-Marie Tarascon for fruitful discussion.  \n\n# Author contributions  \n\nY.S.-H. and A.G. designed the experiments. A.G. and W.T.H. carried out the synthesis, structural and chemical analysis. A.G. and B.H. performed the electrochemical measurements. O.D.-M. and M.T.M.K. conducted the OLEMS measurements. Y.-L.L and L.G. carried out the DFT calculations. Y.S.-H. wrote the manuscript and all authors edited the manuscript.  \n\n# Additional information  \n\nSupplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to Y.S.H.  \n\n# Competing financial interests  \n\nThe authors declare no competing financial interests.  ",
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  },
  {
    "id": "10.1016_j.mattod.2016.12.001",
    "DOI": "10.1016/j.mattod.2016.12.001",
    "DOI Link": "http://dx.doi.org/10.1016/j.mattod.2016.12.001",
    "Relative Dir Path": "mds/10.1016_j.mattod.2016.12.001",
    "Article Title": "On Maxwell's displacement current for energy and sensors: the origin of nullogenerators",
    "Authors": "Wang, ZL",
    "Source Title": "MATERIALS TODAY",
    "Abstract": "Self-powered system is a system that can sustainably operate without an external power supply for sensing, detection, data processing and data transmission. nullogenerators were first developed for self-powered systems based on piezoelectric effect and triboelectrification effect for converting tiny mechanical energy into electricity, which have applications in internet of things, environmental/infrastructural monitoring, medical science and security. In this paper, we present the fundamental theory of the nullogenerators starting from the Maxwell equations. In the Maxwell's displacement current, the first term epsilon(0) partial derivative E/partial derivative t gives the birth of electromagnetic wave, which is the foundation of wireless communication, radar and later the information technology. Our study indicates that the second term partial derivative P/partial derivative t in the Maxwell's displacement current is directly related to the output electric current of the nullogenerator, meaning that our nullogenerators are the applications of Maxwell's displacement current in energy and sensors. By contrast, electromagnetic generators are built based on Lorentz force driven flow of free electrons in a conductor. This study presents the similarity and differences between pieozoelectric nullogenerator and triboelectric nullogenerator, as well as the classical electromagnetic generator, so that the impact and uniqueness of the nullogenerators can be clearly understood. We also present the three major applications of nullogenerators as micro/nullo-power source, self-powered sensors and blue energy.",
    "Times Cited, WoS Core": 1592,
    "Times Cited, All Databases": 1610,
    "Publication Year": 2017,
    "Research Areas": "Materials Science",
    "UT (Unique WOS ID)": "WOS:000398895800016",
    "Markdown": "# On Maxwell’s displacement current for energy and sensors: the origin of nanogenerators  \n\nZhong Lin Wang1,2  \n\n1 School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, United States   \n2 Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, Beijing 100083, China  \n\nSelf-powered system is a system that can sustainably operate without an external power supply for sensing, detection, data processing and data transmission. Nanogenerators were first developed for selfpowered systems based on piezoelectric effect and triboelectrification effect for converting tiny mechanical energy into electricity, which have applications in internet of things, environmental/ infrastructural monitoring, medical science and security. In this paper, we present the fundamental theory of the nanogenerators starting from the Maxwell equations. In the Maxwell’s displacement current, the first term $\\epsilon_{\\mathbf{0}}\\frac{\\partial\\pmb{E}}{\\partial t}$ gives the birth of electromagnetic wave, which is the foundation of wireless communication, radar an@td later the information technology. Our study indicates that the second term $\\textstyle{\\frac{\\partial P}{\\partial t}}$ in the Maxwell’s displacement current is directly related to the output electric current of the nanogenerator, meaning that our nanogenerators are the applications of Maxwell’s displacement current in energy and sensors. By contrast, electromagnetic generators are built based on Lorentz force driven flow of free electrons in a conductor. This study presents the similarity and differences between pieozoelectric nanogenerator and triboelectric nanogenerator, as well as the classical electromagnetic generator, so that the impact and uniqueness of the nanogenerators can be clearly understood. We also present the three major applications of nanogenerators as micro/nano-power source, self-powered sensors and blue energy.  \n\n# Self-powering  \n\nInternet of things (IoT) is a technological drive that link moving things or any things around world on internet, such as shipping objects, cargo carriers and people etc. IoT needs widely distributed sensors for health monitoring, medical care, environmental protection, infrastructure monitoring and security. The power for driving each sensor is small, but the number of such units can be huge in the order of billions to trillions. The most conventional technology is using batteries, which may not be the solution for IoT with considering the limited life time, wide distribution, high maintenance cost and environmental issues. Most of the IoT would be impossible without making the devices self-powered by harvesting energy from the working environment so that the devices can operate sustainably. This was the original motivation for developing nanogenerators based self-powering systems [1,2].  \n\nIn the last half century, the road map of electronics has been focusing on miniaturization following the Moore’s law, for example, the number of devices on a chip doubles every 18 months, which is a commercial drive rather than a nature physics law. Solid state electronics has made it possible to integrate many components on a single chip. Integrated circuits set the foundation for improving the reliability, reducing size, increasing calculation speed, reducing power consumption and more. Secondly, the next revolutionary advance is the development of wireless/mobile communication technology. By conjunction with optical fiber based information transfer and computer science, the development of internet has changed every corner of the world. Thirdly, in the last decades, adding functionality to mobile devices has closely  \n\n![](images/641247a1e897171b60732ec9efd5778cfd4bfafce9b4acf8d49300294ec83d51.jpg)  \nFIGURE 1  \n\nA summary about the major development stages of microelectronics and communication technologies as well as newly arising fields.  \n\nlinked to medical science and medical care of every one, so that one can fully utilize modern sensor technology for living in a more security and healthy world. But all of these advances can be hugely impacted if we can make the mobile electronics self-powered so that the systems can operate suitably and continuously without interruption. This is desperately needed for IoT because we mostly care about mobile objects. Regarding whatever technology, one thing is true, no electronics works without electric power! Therefore, the last huge drive is to make devices self-powered. The above discussions are thus classified and summarized in Fig. 1 as four major technological drives toward systems with: miniaturized integratebility; wireless portability, functionality, and self-powerbility. The self-powering serves as the base of the other three fields. This is what I projected future areas of exploration.  \n\n# Nanogenerators  \n\nWe first proposed the idea of self-powering in 2006 as a result of discovery of piezoelectric nanogenerators (PENGs) [3–5], which utilizes piezoelectric effect of nanowires for converting tiny mechanical energy into electricity. This study inspires the field of nanoenergy. The triboelectric nanogenerator (TENG) was first invented in 2012 [6–9]. Using the electrostatic charges created on the surfaces of two dissimilar materials when they are brought into physical contact, the contact induced triboelectric charges can generate a potential drop when the two surfaces are separated by a mechanical force, which can drive electrons to flow between the two electrodes built on the top and bottom surfaces of the two materials. Research in nanogenerators has inspired a worldwide interest because of its importance not only as a power source, but also self-powered sensors with applications ranging from IoT, environmental monitoring, health care, medical science, infrastructure monitoring and security [9].  \n\nThere are a few important forms of energy that can be harvested from our living environment for self-powered system, including solar, thermal, mechanical and biochemical. Each of these energies offers its own uniqueness, potentials and limitations, as summarized in Fig. 2. In some cases, a device that can simultaneously harvesting multiple types of energies is desirable, so called hybrid energy technology [10,11]. Our current article mainly focus on mechanical energy harvesting, which can be accomplished using effects such as electromagnetic induction, electrostatic, piezoelectric and triboelectric. Each of these effects has its own uniqueness and applications, as summarized and compared in Fig. 3. The goal of this article is about the fundamental physics for energy harvesting using piezoelectric and triboelectric effect. We will explore the relationship between nanogenerator’s output and the Maxwell’s displacement current so that a clear understanding is offered about the difference between nanogenerators from classical electromagnetic generator (EMG). Finally, some key application fields of the nanogenerators will be briefly reviewed.  \n\n# Maxwell’s displacement current for understanding nanogenerators  \n\nOur discussion starts from the fundamental Maxwell’s equations that unify electromagnetism:  \n\n$$\n\\nabla{\\cdot}\\pmb{D}=\\rho_{f}\\quad(\\mathrm{Gauss^{\\prime}s\\ L a w})\n$$  \n\n$$\n\\nabla\\times\\mathbf{E}=-{\\frac{\\partial\\mathbf{B}}{\\partial t}}\\quad{\\mathrm{(Faraday's~law)}}\n$$  \n\n$$\n\\nabla\\times\\pmb{H}=\\pmb{J}_{f}+\\frac{\\partial\\pmb{D}}{\\partial t}\n$$  \n\nAmp\\`ere’s circuital law with Maxwell’s addition  \n\nwhere the electric field $\\mathbf{E};$ the magnetic field $\\mathbf{B}$ ; magnetizing field $\\mathbf{H}_{\\cdot}$ ; the free electric charge density $\\rho\\B{\\prime}$ the free electric current density $J_{f{\\big/}}$ displacement field $\\mathbf{D}$ ,  \n\n$$\n\\pmb{{\\cal D}}=\\epsilon_{0}\\pmb{{\\cal E}}+\\pmb{{\\cal P}}\n$$  \n\nand polarization field $\\mathbf{P},$ and permittivity in vacuum $\\epsilon_{0}$ . As for an isotropic media, $\\mathbf{{\\boldsymbol{D}}}=\\epsilon\\mathbf{{\\boldsymbol{E}}} $ , where e is the permittivity of the dielectrics.  \n\nIn Eq. (1.4), the second term is the Maxwell’s displacement current defined as  \n\n$$\n\\pmb{J}_{D}=\\frac{\\partial\\pmb{D}}{\\partial t}=\\epsilon_{0}\\frac{\\partial\\pmb{E}}{\\partial t}+\\frac{\\partial\\pmb{P}}{\\partial t}\n$$  \n\nThe displacement current was first postulated by Maxwell in 1861 [12], and it was introduced on consistency consideration between Amp\\`ere’s law for the magnetic field and the continuity equation for electric charges. The displacement current is not an electric current of moving free charges, but a time-varying electric field (vacuum or media), plus a contribution from the slight motion of charges bound in atoms, dielectric polarization in materials. In Eq. (3), the firs component $\\epsilon_{0}\\frac{\\partial\\pmb{E}}{\\partial t}$ in the displacement current gives the birth of electromagnetic wave, which later being taken as the approach for developing radio, radar, TV and long distance wireless communication. We now present the relationship between the second term in the displacement current and the output signal from nanogenerators, and show the contribution of displacement current to energy and sensors in the near future.  \n\nFIGURE 2   \n\n\n<html><body><table><tr><td>Energy source</td><td>Solar</td><td>Thermal</td><td>Mechanical</td><td>Biochemical</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Harvesting Principle</td><td>Photovoltaic</td><td>Thermoelectric</td><td>Electromagnetic/Electro static/ Piezoelectric</td><td>Biochemical reactions</td></tr><tr><td>Approximate power density</td><td>5-30 mWcm-2</td><td>0.01-0.1 mWcm-2</td><td>10-100 mWcm-2</td><td>0.1-1 mWcm-2</td></tr><tr><td>Pros</td><td>Microfabrication compatible mature technology, long lifetime, DC & high power output</td><td>No moving parts required, long lifetime, high reliability, DC output</td><td>Ubiquitous and abundant in the ambient, broad frequency and power ranges, high output</td><td>Biocompatible/degrada ble, clean energy, environmentally friendly, abundant in biological entities</td></tr><tr><td>Cons</td><td>Limited by environmental conditions, not available at night</td><td>Low efficiency, large size, a large and sustained thermal gradient is required</td><td>AC output, not continuous output</td><td>Low power output, poor reliability, limited lifetime</td></tr><tr><td>Potential Applications</td><td>Remote sensing and environmental Monitoring</td><td>Structural-health monitoring for Engines and machines, wearable Biomedical devices</td><td>Remote sensing and monitoring, wearable systems, blue energy, internet of things</td><td>In vivo applications, environmental Monitoring/sensing, biocompatible application</td></tr></table></body></html>  \n\ncomparison about the harvesting of energy from solar, thermal, mechanical and biochemical for illustrating their merits and possible practical limitations.   \n\n\n<html><body><table><tr><td>Mechanical energy harvesting</td><td>Electromagnetic</td><td>Electrostatic</td><td>Piezoelectric</td><td>Triboelectric</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Harvesting principle</td><td>Electromagnetic induction</td><td>Electrostatic induction</td><td>Piezoelectric effect & Electrostatic induction</td><td>Contact electrification & Electrostatic induction</td></tr><tr><td>Impedance type</td><td>Resistive</td><td>Capacitive</td><td>Capacitive</td><td>Capacitive</td></tr><tr><td>Pros</td><td>High efficiency, easy to scale up</td><td>Light weight</td><td>Easy to scale down to nanoscale</td><td>Large output power, high efficiency, low weight, cost effective materials, simple fabrication</td></tr><tr><td>Cons</td><td>Heavy magnet required, low output for small-scale devices</td><td>Precharge required, low output, high matched impedance</td><td>Low output & low efficiency, pulsed output, high matched impedance</td><td>Pulsed output, high matched impedance</td></tr></table></body></html>\n\ncomparison about the harvesting of mechanical energy using electromagnetic, electrostatic, piezoelectric and triboelectric effect for illustrating their merit nd possible practical limitations.  \n\n# FIGURE 3  \n\n# Piezoelectric nanogenerator  \n\nThe working principle of a piezoelectric nanogenerator (PENG) is illustrated in Fig. 4a-i. An insulator piezoelectric material is covered by a top and bottom electrodes on its two surfaces. A vertical mechanical deformation results in the generation of piezoelectric polarization charges at the two ends of the material (Fig. 4a-ii). An increase of the applied force results in higher polarization charge density (Fig. 4a-iii). The electrostatic potential created by the polarization charges is balanced by the flow of electrons from one electrode to the other through an external load. This is the process of converting mechanical energy into electric power. If the density of the piezoelectric polarization charges on the surface is $\\sigma_{p}(z)$ , and the corresponding charge density of free electrons in the electrode is $\\sigma(x)$ , which is a function of the thickness of the piezoelectric material z with considering the strain introduced by applied force.  \n\nAs for a piezoelectric material that is usually anisotropic, the piezoelectric equation and constituter equations under a small uniform mechanical strain are given by [13–15]:  \n\n$$\n\\begin{array}{l}{P_{i}=(\\pmb{e})_{i j k}(\\pmb{s})_{j k}}\\\\ {\\left\\{\\begin{array}{l l}{\\pmb{T}=\\pmb{c}_{E}\\pmb{s}-\\pmb{e}^{T}\\pmb{E}}\\\\ {\\pmb{D}=\\pmb{e}\\pmb{s}+\\pmb{k}\\pmb{E}}\\end{array}\\right.}\\end{array}\n$$  \n\nwhere $\\textbf{s}$ is the mechanical strain; the third order tensor $({\\pmb e})_{i j k}$ is the piezoelectric tensor; ${\\bf T}$ and $\\scriptstyle\\mathbf{c}_{E}$ are the stress tensor and the elasticity tensor, respectively; $\\pmb{k}$ is the dielectric tensor. The displacement current from the media polarization is:  \n\n$$\nJ_{D i}=\\frac{\\partial P_{i}}{\\partial t}=(\\pmb{e})_{i j k}\\left(\\frac{\\partial\\pmb{s}}{\\partial t}\\right)_{j k}\n$$  \n\nEq. (5) means that the changing rate of the applied strain is proportional to the output current density of the PENG.  \n\n# (a) Piezoelectric nanogenerator  \n\n![](images/dc7fa67b9f8a9ef73888577d6b881e5b4df7aaff63bd1ab292d49257f34b58d2.jpg)  \n\n# FIGURE 4  \n\nIllustrations about the working mechanisms of (a) piezoelectric nanogenerator with the increase of the applied stress, and (b) triboelectric nanogenerator with the increase of contact cycles.  \n\nFor a case there is no external electric field applied as show in Fig. 4a and the polarization is along the $z$ -axis, the displacement field is the polarization vector, $D_{z}=P_{z}=\\sigma_{p}(z)$ in the media, so that the displacement current is  \n\n$$\nJ_{D z}=\\frac{\\partial P_{z}}{\\partial t}=\\frac{\\partial\\sigma_{p}(z)}{\\partial t}\n$$  \n\nEq. (6) means that the changing rate of the surface polarization charges is the observed output current for a PENG. The magnitude of the open circuit voltage for PENG is  \n\n$$\nV_{o c}=z\\sigma_{p}(z)/\\epsilon\n$$  \n\nWith considering the presence of an external load $R$ as shown in Fig. 4a-ii, the current transport equation for PENG is  \n\n$$\nR A\\frac{d\\sigma}{d t}=z[\\sigma_{p}(z)-\\sigma(z)]/\\epsilon\n$$  \n\nwhere $A$ is the area of the electrode. In a case that the applied strain is a relatively slow process, so that $z$ is a function of time $t.$ The output characteristics of a PENG can be derived from Eq. (8). In analogy, the output of a pyroelectric nanogenerator can also be described accordingly.  \n\n# Triboelectric nanogenerator  \n\nWe start from the very basic model of the TENG for illustrating its theory. Starting from a four layer TENG in contact-separation mode, with two dielectrics with permittivity of $\\epsilon_{1}$ and $\\epsilon_{2}$ and thicknesses $d_{1}$ and $d_{2},$ respectively (Fig. 4b-i). Once the two dielectrics are driven to be in physical contact, electrostatic charges are transferred to the surfaces of the two owing to the contact electrification effect (triboelectricity). The surface is partially charged and the charges are non-mobile static charges (Fig. 4b-ii), and surface charge density $\\sigma_{c},$ builds up as a number of contacts between the two dielectric media and finally reaches a saturation, and it is independent of the gap distance $z$ . The electrostatic field built by the triboelectric charges drives electrons to flow through the external load, resulting in an accumulation of free electrons in the electrode, $\\sigma_{I}(z,t)$ , which is a function of the gap distance $z(t)$ between the two dielectrics. This is the process of converting mechanical energy into electricity.  \n\nAs shown in Fig. 4b-iii, the electric field in dielectric 1 and 2 are $E_{z}=\\sigma_{I}(z,t)/\\epsilon_{1}$ and $E_{z}=\\sigma_{I}(z,t)/\\epsilon_{2},$ respectively. In the gap, $E_{z}=(\\sigma_{I}(z,t)$ $-\\sigma_{c})/\\epsilon_{0}$ . The relative voltage drop between the two electrodes is  \n\n$$\nV=\\sigma_{I}(z,t)[d_{1}/\\epsilon_{1}+d_{2}/\\epsilon_{2}]+z[\\sigma_{I}(z,t)-\\sigma_{c}]/\\epsilon_{0}\n$$  \n\nUnder short-circuit condition, $V=0$ ,  \n\n$$\n\\sigma_{I}(z,t)=\\frac{z\\sigma_{c}}{d_{1}\\epsilon_{0}/\\epsilon_{1}+d_{2}\\epsilon_{0}/\\epsilon_{2}+z}\n$$  \n\nFrom Eq. (4), the corresponding displacement current density is  \n\n$$\nJ_{D}=\\frac{\\partial D_{z}}{\\partial t}=\\frac{\\partial\\sigma_{I}(z,t)}{\\partial t}=\\sigma_{c}\\frac{d z}{d t}\\frac{d_{1}\\epsilon_{0}/\\epsilon_{1}+d_{2}\\epsilon_{0}/\\epsilon_{2}}{\\left[d_{1}\\epsilon_{0}/\\epsilon_{1}+d_{2}\\epsilon_{0}/\\epsilon_{2}+z\\right]^{2}}\n$$  \n\nThis equation means that the displacement current density is proportional to the charge density on the dielectric surface and the speed at which the two dielectrics are being separated or contacted. This is the output characteristics of the TENG.  \n\nWith considering the presence of an external load $R$ as shown in Fig. 4a-ii, the current transport equation for TENG is  \n\n$$\nR A\\frac{d\\sigma_{I}(z,t)}{d t}=z\\sigma_{c}/\\epsilon_{0}-\\sigma_{I}(z,t)[d_{1}/\\epsilon_{1}+d_{2}/\\epsilon_{2}+z/\\epsilon_{0}]\n$$  \n\nwhere z is a function of time $t$ depending on the dynamic process that the force is applied. Starting from Eq. (12), we have systematically established the theories for all of the four modes for TENG once it is connected with a load, regarding to the power output, the optimization of the experimental parameters [16–20].  \n\n# Capacitive model  \n\nBoth piezoelectric and triboelectric nanogenerators are referred to as the capacitive conduction, in which the displacement current is the only conduction mechanism for electricity transport. The power is transmitted not via flow of free charges across the electrodes of the capacitor, but via electromagnetic wave and induction. Based on a capacitor model, the output current of a nanogenerator can be represented by  \n\n$$\nI={\\frac{d Q}{d t}}=C{\\frac{d V}{d t}}+V{\\frac{d C}{d t}}\n$$  \n\nwhere $Q$ is the stored charges in the capacitor, the first term is the current introduced by a change in the applied voltage; the second term is the current introduced by the variation in capacitance.  \n\nAs for a PENG, the change in capacitance is rather small because the strain induced change in crystal size/thickness is extremely small, so that the current is mainly due the change in strain induced voltage  \n\n$$\nI\\approx C\\frac{d V}{d t}=\\left(\\epsilon\\frac{A}{z}\\right)\\frac{d}{d t}\\left(\\frac{\\sigma z}{\\epsilon}\\right)\\approx A\\frac{d\\sigma}{d t},\n$$  \n\nwhere $A$ is the area of the electrode. Under short circuit condition, $\\sigma=\\sigma_{p}(z)$ , the result from Eq. (14) is just the result derived from the displacement current in Eq. (6).  \n\n$$\nI=A{\\frac{d\\sigma_{p}}{d t}}=A{\\frac{d\\sigma_{p}}{d z}}{\\frac{d z}{d t}},\n$$  \n\nAs for TENG, since the change in gap distance is rather large, so that both terms in Eq. (13) contribute to the observed output current,  \n\n$$\nI={\\frac{d Q}{d t}}=A{\\frac{d\\sigma_{I}}{d t}}\n$$  \n\nwhich will lead to the same result as for displacement current in Eq. (11). Therefore, the foundation of the capacitive model is the Maxwell’s displacement current. Our study proves the equivalence of the different models. Using Eq. (16), in conjunction with $\\mathrm{{Ohm^{\\prime}s}}$ law, we have systematically established the theories for all of the four modes for TENG once it is connected with a load, regarding to the power output, the optimization of the experimental parameters [16–20].  \n\n# Electromagnetic generators  \n\nIn Maxwell Eq. (1.4), the first term ${\\bf J}_{f}$ is the current density as a result of free electron flow. As for EMG, the Lorentz force induced electron flow in a conductor is the power generation process  \n\n$$\n\\pmb{F}=-e\\pmb{\\nu}\\pmb{\\times}\\pmb{B},\n$$  \n\nwhere $\\nu$ is the moving speed of the conductor across the electric field. The motion of the electrons is accelerated by the magnetic field, but the related resistance due to inelastic collisions with atoms and electrons limits their flow, reaching a steady state current. This means that the EMG is a resistive conduction.  \n\n# Four fundamental working modes of TENG  \n\nEver since the first report of the TENG in 2012 by Wang et al., TENG’s output area power density reaches $500\\mathrm{W}/\\mathrm{m}^{2}$ [21], an instantaneous conversion efficiency of ${\\sim}50\\%$ have been demonstrated [22]. TENG is effective for harvesting energy from human motion, walking, vibration, mechanical triggering, rotating tire, wind, flowing water and more. A TENG can also be used as a selfpowered sensor for actively detecting the static and dynamic processes arising from mechanical agitation using the voltage and current output signals of the TENG, respectively, with potential applications as mechanical sensors and for touch pad and smart skin technologies. We now present the four basic working modes of a TENG (Fig. 5) [7,8]. The contact-separation mode uses the polarization in vertical direction. The lateral sliding mode uses the polarization in lateral direction as a result of relative sliding between two dielectrics [23,24]. The single electrode mode was introduced for harvesting energy from a freely moving object without attaching a conduction line [25]. The Freestanding triboelectric-layer mode is designed for power generation using electrostatic induction between a pair of electrode [26]. In many cases, two or more modes can work in conjunction.  \n\n# Vertical contact-separation mode  \n\nA physical contact between the two dielectric films with distinct electron affinity (at least one is insulator) creates oppositely charged surfaces. Once the two surfaces are separated by a gap, a potential drop is created between electrodes deposited on the top and the bottom surfaces of two dielectric films, as demonstrated in Fig. 5a. If the two electrodes are electrically connected by a load, free electrons in one electrode would flow to the other electrode in order to balance the electrostatic field. Once the gap is closed, the potential drop created by the triboelectric charges disappears, the induced electrons will flow back. A periodic contact and separation between the two materials drives the induced electrons to flow back and forth between the two electrodes, resulting in an AC output in the external circuit [6,26]. This mode is the basic mode of TENG and it can be easily achieved in practice.  \n\n# In-plane sliding mode  \n\nWhen two materials with opposite triboelectric polarities are brought into contact, surface charge transfer takes place due to  \n\n![](images/8eaa8cd4c4536ea6111969686c40dbd847d276797afc245d1d90bfe5d8aa5cbb.jpg)  \n\n# FIGURE 5  \n\nThe four fundamental working modes of the triboelectric nanogenerators. (a) The vertical contact-separation mode. (b) The lateral sliding mode. (c) The single-electrode mode. (d) The free-standing mode.  \n\nthe triboelectrification effect (Fig. 5b) [27]. When the two surfaces are fully matched there is no current flow, because the positive charges at one side are fully compensated by the negative ones. Once a relative displacement is introduced by an externally applied force in the direction parallel to the interface, triboelectric charges are not fully compensated at the displaced/mismatched areas, resulting in the creation of an effective dipole polarization in parallel to the direction of the displacement. Therefore, a potential difference across the two electrodes is generated. The sliding mode can be made into fully packaged and even in rotation mode so that it can operate in vacuum.  \n\n# Single-electrode mode  \n\nFor a dielectric and metal plate, as shown in Fig. 5c, induction current is created in the metal plate if the charged dielectric approaches it to balance the field. Once the dielectric moves away from the metal plate, the current flows back to the ground. This mode works in a way that relies on the charge exchange between ground and metal plate [25]. This mode is most useful for utilizing the energy from a moving object without attaching an electric connection, such as human walking, moving car, finger typing and more.  \n\n# Free-standing triboelectric-layer mode  \n\nIf we make a pair of symmetric electrodes underneath a dielectric layer and the size of the electrodes are of the same order as the size of the moving object, and there is a small gap between the object and the electrode, the object’s approaching to and/or departing from the electrodes create an asymmetric charge distribution via induction in the media, provided the object was prior-charged by a triboelectric process, which causes the electrons to flow between the two electrodes to balance the local potential distribution (Fig. 5d) [28]. The oscillation of the electrons between the paired electrodes in responding to the back and forth motion of the object produces an AC current output. This mode carries the advantages of harvesting the energy from a moving object but with the entire system mobile without grounding.  \n\n# Major applications of TENG  \n\nTo provide a comparison, Fig. 6 gives a comparison between EMG and TENG, through which one can see the distinction differences between the TENG and EMG. The major applications of TENG are in three directions (Fig. 7): as sustainable nano/micro-power source for small devices to achieve self-powering; as active sensors for medical, infrastructure, human–machine, environmental monitoring and security; and as basic networks units for harvesting water motion energy at low frequency toward the dream of blue energy [9].  \n\n# TENGs as sustainable nano/micro-power source for self-powered systems  \n\nThe ultimate goal of nanogenerators is to build up self-powered systems (Fig. 8), in which multifunctional electronic devices can be powered up by the nanogenerators through collecting ambient mechanical energies [29]. TENGs can convert irregular and mostly low-frequency energy from almost any mechanical motion from human, machine to nature into electricity. Such pulsed energy cannot be directly used to drive conventional electronics that  \n\nFIGURE 6   \n\n\n<html><body><table><tr><td colspan=\"2\">Electromagnetic generator</td><td>Triboelectric nanogenerator</td></tr><tr><td>Mechanism</td><td>Electromagnetic induction; Resistive free electron conduction driven by Lorentz force</td><td>Contact electrification and electrostatic induction; Capacitive displacement current arising from time-dependent electrostatic induction and slight motion of bonded electrostatic charges</td></tr><tr><td>Pros</td><td>High current, low voltage; High efficiency at high frequency; High durability, long life</td><td>High voltage,low current; High efficiency atlow frequency; Low cost, low density,low weight; Multiple working modes; Diverse choice of materials; Diverse use of fields; Broad use as sensors</td></tr><tr><td>Cons</td><td>●Lowimpedance; Heavy, high density; ）Highcost</td><td>●High impedance; Low durability</td></tr></table></body></html>  \n\nA comparison about the electromagnetic generator and triboelectric nanogenerator in mechanisms, advantages and disadvantages.  \n\nrequire a continuous and constant input. A new power management system is required for lowering the output voltage but without scarifying energy, so that the generated electricity can be directly stored as electrochemical energy by a battery or capacitor [30]. Such a power management cannot be accomplished by a classical transformer, which usually has a very low efficiency at low-frequency. An integration of a TENG, power management circuit and storage unit form a self-charging power unit. This unit can be used as a sustainable power source for powering any electronics as long as the power output is sufficient. This is the first major applications of TENG.  \n\n# TENGs as self-powered active sensors  \n\nTENG is a technology for converting mechanical energy into electricity. Reversely, the electric output signals of the TENG directly reflect the impact of the mechanical triggering, so that TENG can be used as an active sensor in responding to external excitation [31]. From Eq. (7), the output voltage is a direct measurement of the gap distance z, while the output current represents the impact speed, dz/dt. The TENG based sensor is different from the conventional sensor that has to be driven by a power, otherwise there is no output signal. In contrast, the TENG sensor gives an output electric signal itself without applying a power to the sensor tip. TENG based sensor can be used for sensing of motion, vibration, human triggering and object contacts.  \n\nWe use keyboard based TENG as an example [32]. Keyboard is an indispensable input component for many personal electronics like computers and cell phones. We recently invented an intelligent and self-powered keyboard as an advanced security safeguard against unauthorized access to computers. Based on the triboelectric effect between human fingers and keys, the intelligent keyboard (IKB) could convert typing motions into localized electric signals that could be identified as personalized physiological information. The core part of the IKB was composed of multilayered transparent thin film materials to form a typical singleelectrode TENG. Fig. 9a shows the schematic structure and a photograph of a fully-assembled IKB with the same size as a commercial keyboard. The working principle of the IKB as an  \n\n![](images/81b58fe77ced8fb3f77739269e2ca70afd37a18730cf4cddbc9acbe529e77198.jpg)  \n\n# FIGURE 7  \n\nA summary about the three major application fields of nanogenerators as micro-/nano-energy source, for blue energy and self-powered sensors. The photos around the three directions are the ones we have demonstrated in our experiments in the last few years.  \n\n![](images/2c5bb14d03153da24dc76009aefef5487541b90eb61d6563ebc27992345f49bb.jpg)  \nself-powered system by integrating a nanogenerator, power management circuit and energy storage unit as a self-charging power cell.  \n\n# FIGURE 8  \n\n![](images/fe11d17c9d108a49bafd0409d6f83fbb47e29514c2df2f90e696a7d9813c36c3.jpg)  \nRESEARCH  \n\n# FIGURE 9  \n\nenergy harvester was similar with the single-electrode TENG. A finger movement during typing would then induce change of the potential difference between the pair of ITO electrodes, driving electrons flow through the external load or data collection system. The electric signals generated by three people by typing the same phrase or words are completely different, which can be used to identify the user of the computer for system protection (Fig. 9b).  \n\n# TENGs as basic units for large-scale blue energy  \n\nOur earth is largely covered by water. Energy offered by ocean can be huge, but harvesting such energy is extremely challenging because the low efficiency of electromagnetic generators, especially at low frequency [33]. TENG is much more effective than EMG for harvesting energy in the frequency range of $<5\\mathrm{Hz},$ which is ideally suited for our daily life and in nature (not EMG works only effective at relatively high frequency such a $s{>}s\\operatorname{Hz}$ ). More importantly, the EMGs are heavy and high cost, and they are not easily to be installed in sea floor or at water surface for collecting the water wave energy. I proposed the idea of using TENG networks for harvesting water motion energy at a large scale in 2014 [34]. The idea is that the TENG is made of mostly organic materials and it is partially filled up with air, so that the network made of millions of TENGs as fishing net would flow at the vicinity of the water surface (Fig. 10). Any wave motion would drive the TENG to perform contact-separation and sliding motions, so that the mechanical energy can be collected. Our initial estimation indicates that the power can be generated on average is $1\\mathrm{MW/km}^{2}$ , which could be improved for at least 10 times based on near future progress in materials and structure design [35] The advantage offered by TENG networks is low-cost, occupy no land, no natural disaster, independent of day, night or even weather, and there is no big security concern. I believe that this blue energy dream will offer a new energy path for human kind.  \n\n![](images/365fe486244e07a3d9c5f030ebc764a921e9bb308057b27d9ade2480f262fe0a.jpg)  \nPersonalized keystroke dynamics for self-powered human-machine interfacing. (a) Schematic illustrations of the keyboard. (b) Evaluation of the performance of the biometric authentication system using triboelectrification enabled keystroke dynamics by repeatedly typing the same phase of words by three different people. The output electric signals are completely different from one person to the other, but each person’s pattern is self-reproducible.   \nFIGURE 10  \n\nA blue energy dream by networking millions of spherical balls based triboelectric nanogenerators for harvesting low-frequency water wave energy. The inset is the designed spherical TENG. The lower-right corner is an imaginary structure of the networks.  \n\n# Summary  \n\nThe goal of this review article is to present the linkage between the Maxwell’s displacement current and the output of nanogenerators, so that the differences between EMG and PENG/TENG are clearly elaborated. The displacement current has two components (Fig. 11). The first component $\\epsilon_{0}\\frac{\\partial\\pmb{E}}{\\partial t}$ represents the electricity to magnetism induction effect, so that it represents the existence of electromagnetic waves and the theory of light. As a result, it is the foundation of antenna, telegram, radio, TV, and most recently wireless communication technology. It means that the first component of displacement current is the foundation of today’s wireless information technology, which has driven the development of the world for the last 50 years.  \n\nIn parallel, the second term $\\textstyle{\\frac{\\partial{\\pmb P}}{\\partial t}}$ in the displacement current is related to the polarization of media, from which the fundamental characteristics of piezoelectric nanogenerator and triboelectric nanogenerator can all be derived. Besides the applications in capacitors, the second term gives the birth of new energy technology and self-powered sensors, for example, our nanogenerators, which could have extensive applications in IoT, sensor networks, blue energy and even big data. The industry as generated by the second component of the displacement current could possibly drive the development of the world in energy and sensors in the next 50 years at least!  \n\nOur study indicates that the second term $\\textstyle{\\frac{\\partial P}{\\partial t}}$ in the displacement current is directly related to the output electric current of the nanogenerator. In other words, the applications of displacement current in energy and sensors are our nanogenerators. Based on this future prediction, we like to emphasize here is a lacking of  \n\n![](images/80b6bb9cc83efc2d59d3ada18d7b4fe78c1f0605c60063e995c41d047708c2b2.jpg)  \n\n# FIGURE 11  \n\nMajor fundamental science, technologies and practical impacts that have been derived from the two components of the Maxwell’s displacement current. The left hand-side column is the electromagnetic wave that has impacted the development of the world in the last century in communication; the right-hand side is the new technologies derived from displacement current for energy and sensors that are likely to impact the world for the future.  \n\nfundamental understanding about the phenomenon of triboelectrification. Although this phenomenon is known for over thousands years, its basic physics interpretation is unclear. Why? I believe that there was not enough study devoted to it because triboelectricity has been attributed as a negative effect, so people would think that it is not important or rather difficult to understand. However, as now, triboelectric nanogenerator finds the true applications of contact electrification and it is time to study this phenomenon and explore the core physics, for example, the physics each and every one of us experiences every day! We now have the urgency and practical implication for studying charging effect at dielectric surfaces. Once a clear physical picture is presented, developing effective triboelectrification process and structure would hugely impact the technological development of nanogenerators and their commercial products.  \n\n# Acknowledgements  \n\nResearch was supported by the Hightower Chair foundation, and the ‘thousands talents’ program for pioneer researcher and his innovation team, China, the National Key R&D Project from Minister of Science and Technology (2016YFA0202704).  \n\n# References  \n\n[1] Z.L. Wang, Sci. Am. 298 (2008) 82–87. [2] S. Xu, et al. Nat. Nanotechnol. 5 (2010) 366–373. [3] Z.L. Wang, J. Song, Science 312 (2006) 242–246. [4] R. Yang, et al. Nat. Nanotechnol. 4 (2009) 34–39.   \n[5] Z.L. Wang, Nanogenerators for Self-Powered Devices and Systems, Georgia Institute of Technology, 2011 http://smartech.gatech.edu/handle/1853/39262. [6] F.-R. Fan, Z.-Q. Tian, Z.L. Wang, Nano Energy 1 (2012) 328–334. [7] Z.L. Wang, ACS Nano 7 (2013) 9533–9557. [8] Z.L. Wang, J. Chen, L. Lin, Energy Environ. Sci. 8 (2015) 2250–2282. [9] Z.L. Wang, et al., Triboelectric Nanogenerators, Springer, 2016 http://www. springer.com/us/book/9783319400389.   \n[10] C. Xu, X. Wang, Z.L. Wang, J. Am. Chem. Soc. 131 (2009) 5866–5872.   \n[11] C. Xu, Z.L. Wang, Adv. Mater. 23 (2011) 873–877.   \n[12] J.C. Maxwell, Philosophical Magazine and Journal of Science, London, Edinburg and Dubline, Fourth series, p. 161.   \n[13] T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, Oxford, UK, 1996.   \n[14] G.A. Maugin, Continuum Mechanics of Electromagnetic Solids, Amsterdam, North-Holland, 1988.   \n[15] R.W. Soutas-Little, Elasticity, XVI, 431, Dover Publications, Mineola, NY, 1999.   \n[16] S. Niu, et al. Energy Environ. Sci. 6 (2013) 3576–3583.   \n[17] S. Niu, et al. Adv. Mater. 25 (2013) 6184–6193.   \n[18] S. Niu, et al. Nano Energy 12 (2015) 760–774.   \n[19] S. Niu, et al. Energy Environ. Sci. 7 (2014) 2339–2349.   \n[20] S. Niu, et al. IEEE Trans. Electron Devices 62 (2015) 641–647.   \n[21] G. Zhu, et al. Adv. Mater. 26 (2014) 3788–3796.   \n[22] Y. Xie, et al. Adv. Mater. 26 (2014) 6599–6607.   \n[23] J. Yang, et al. ACS Nano 8 (2014) 2649–2657.   \n[24] G. Zhu, et al. Nano Lett. 13 (2013) 2282–2289.   \n[25] Y. Yang, et al. ACS Nano 7 (2013) 7342–7351.   \n[26] G. Zhu, et al. Nano Lett. 12 (2012) 4960–4965.   \n[27] G. Zhu, et al. Nano Lett. 13 (2013) 847–853.   \n[28] S. Wang, et al. Adv. Mater. 26 (2014) 2818–2824.   \n[29] Z.L. Wang, W. Wu, Angew. Chem. 51 (2012) 11700–11721.   \n[30] S. Niu, et al. Nat. Commun. 6 (2015) 8975.   \n[31] S. Wang, L. Lin, Z.L. Wang, Nano Energy 11 (2015) 436–462.   \n[32] J. Chen, et al. ACS Nano 9 (2015) 105–116.   \n[33] Y. Zi, et al. ACS Nano 10 (2016) 4797–4805.   \n[34] Z.L. Wang, Faraday Discuss. 176 (2014) 447–451.   \n[35] J. Chen, et al. ACS Nano 9 (2015) 3324–3331.  ",
    "Target": null
  },
  {
    "id": "10.1016_j.jpowsour.2016.12.011",
    "DOI": "10.1016/j.jpowsour.2016.12.011",
    "DOI Link": "http://dx.doi.org/10.1016/j.jpowsour.2016.12.011",
    "Relative Dir Path": "mds/10.1016_j.jpowsour.2016.12.011",
    "Article Title": "Degradation diagnostics for lithium ion cells",
    "Authors": "Birkl, CR; Roberts, MR; McTurk, E; Bruce, PG; Howey, DA",
    "Source Title": "JOURNAL OF POWER SOURCES",
    "Abstract": "Degradation in lithium ion (Li-ion) battery cells is the result of a complex interplay of a host of different physical and chemical mechanisms. The measurable, physical effects of these degradation mechanisms on the cell can be summarised in terms of three degradation modes, namely loss of lithium inventory, loss of active positive electrode material and loss of active negative electrode material. The different degradation modes are assumed to have unique and measurable effects on the open circuit voltage (OCV) of Li-ion cells and electrodes. The presumptive nature and extent of these effects has so far been based on logical arguments rather than experimental proof. This work presents, for the first time, experimental evidence supporting the widely reported degradation modes by means of tests conducted on coin cells, engineered to include different, known amounts of lithium inventory and active electrode material. Moreover, the general theory behind the effects of degradation modes on the OCV of cells and electrodes is refined and a diagnostic algorithm is devised, which allows the identification and quantification of the nature and extent of each degradation mode in Li-ion cells at any point in their service lives, by fitting the cells' OCV. (C) 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).",
    "Times Cited, WoS Core": 1116,
    "Times Cited, All Databases": 1220,
    "Publication Year": 2017,
    "Research Areas": "Chemistry; Electrochemistry; Energy & Fuels; Materials Science",
    "UT (Unique WOS ID)": "WOS:000393003400044",
    "Markdown": "# Degradation diagnostics for lithium ion cells  \n\nChristoph R. Birkl a, Matthew R. Roberts b, Euan McTurk b, c, Peter G. Bruce b, David A. Howey a, \\*  \n\na Department of Engineering Science, University of Oxford, OX1 3PJ, Oxford, UK b Department of Materials, University of Oxford, OX1 3PH, Oxford, UK c Warwick Manufacturing Group, University of Warwick, Warwick, CV4 7AL, UK  \n\n# h i g h l i g h t s  \n\nA diagnostic algorithm to identify and quantify degradation modes in Li-ion cells.   \nExperimental proof for effects of degradation modes on the open circuit voltage.   \nState of health estimations for commercial cells by fitting open circuit voltages.  \n\n# g r a p h i c a l a b s t r a c t  \n\n![](images/e4df4857dac3336d6705ef92ff1881857a5660ee6d8bbddac6b52c6cb23da19d.jpg)  \n\n# a r t i c l e i n f o  \n\nArticle history: Received 12 July 2016 Received in revised form 15 November 2016 Accepted 4 December 2016  \n\nKeywords:   \nLithium ion   \nDegradation   \nState of health   \nDiagnostic   \nOpen circuit voltage   \nBattery management system  \n\n# a b s t r a c t  \n\nDegradation in lithium ion (Li-ion) battery cells is the result of a complex interplay of a host of different physical and chemical mechanisms. The measurable, physical effects of these degradation mechanisms on the cell can be summarised in terms of three degradation modes, namely loss of lithium inventory, loss of active positive electrode material and loss of active negative electrode material. The different degradation modes are assumed to have unique and measurable effects on the open circuit voltage (OCV) of Li-ion cells and electrodes. The presumptive nature and extent of these effects has so far been based on logical arguments rather than experimental proof. This work presents, for the first time, experimental evidence supporting the widely reported degradation modes by means of tests conducted on coin cells, engineered to include different, known amounts of lithium inventory and active electrode material. Moreover, the general theory behind the effects of degradation modes on the OCV of cells and electrodes is refined and a diagnostic algorithm is devised, which allows the identification and quantification of the nature and extent of each degradation mode in Li-ion cells at any point in their service lives, by fitting the cells' OCV.  \n\n$\\circledcirc$ 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).  \n\n# 1. Introduction  \n\nLithium ion (Li-ion) cells degrade as a result of their usage and exposure to environmental conditions [1e4]. This degradation affects the cells' ability to store energy, meet power demands and, ultimately, leads to their end of life. Any system employing Li-ion cells as its power source must be informed of the amount of energy that can be stored and the power that can be provided by the battery at any point in time. Since the rates of capacity and power fade cannot be easily inferred from operational data in a practical system, methods and models are required which utilise available parameters and measurements to generate estimates and predictions of current and future energy storage capacity and power capability.  \n\nDegradation in Li-ion cells is caused by a large number of physical and chemical mechanisms, which affect the different components of the cells: the electrodes, the electrolyte, the separator and the current collectors [5e10]. Fig. 1 illustrates some of the most commonly reported degradation mechanisms in Li-ion cells. The different causes, rates and inter-dependencies of these degradation mechanisms make them extremely challenging to model, which is why most physics-based models focus only on the most dominant mechanisms, such as the formation and growth of the solid electrolyte interphase (SEI) [11,12] or electronic contact loss through particle cracking [13,14].  \n\nPhysics-based models generally capture degradation at the micro scale, i.e. on a particle or even molecular level [13e15]. However, evidence suggests that meso and macro scale features, such as inhomogeneities in the structure of the electrodes, have a significant effect on cell degradation as a whole [16,17]. Structural nonuniformity can lead to inhomogeneous distributions of current densities and degrees of lithiation inside the electrode material, which in turn causes inhomogeneous degradation of the electrode. Evidence of such inhomogeneities has also been observed in the course of a post-mortem analysis of commercial Kokam pouch cells (described in detail in Section 2.1), which are the subject of this work. After low current rate (C/25) capacity tests, five cells were opened in an argon filled glove box and their electrode sheets visually inspected. Each cell consists of 20 sheets of positive (PE) and negative (NE) electrodes. Some of the investigated cells were fully charged, i.e. their negative graphite electrodes fully lithiated. Lithiated graphite has a golden color as opposed to the black color of delithiated graphite. Fig. 2 illustrates the difference in lithiation of graphite electrode sheets extracted from the same, fully charged cell. Whereas most NE sheets appeared to be uniformly lithiated (Fig. 2 a)), one NE sheet was clearly non-uniformly lithiated, as shown in Fig. 2 b). The cell was charged according to the test procedure specified in Table 3, with $100\\%$ SoC at the end of test. The standard deviation between the capacities of the five investigated pouch cells was less than $0.2\\%$ and the cell with one non-uniformly lithiated graphite sheet actually exhibited the highest capacity. This illustrates that meso- and macro-scale inhomogeneities can not easily be identified in commercial Li-ion cells but they may have long term effects on degradation. Bottom-up physics-based models may not be able to capture such inhomogeneities on a micro-scale.  \n\n![](images/8f07251d9f79e00da69ec35f84fa2e1e0a339ef5f77684535a251acf8163b703.jpg)  \nFig. 2. Graphite negative electrodes extracted from a fully charged Kokam pouch cell; a) uniformly lithiated, b) non-uniformly lithiated.  \n\n![](images/b22d994816fdb9e27e72faeaf2c68a6038c58b5cd1e4ab26891e71c3c14fb6f2.jpg)  \nFig. 1. Degradation mechanisms in Li-ion cells.  \n\nAs an alternative, we propose to rely more heavily on a diagnostic approach, built on frequent cell characterisations using available measurements which include information on the state of health (SoH) of the cell. One such measurement is the cell's open circuit voltage (OCV). Since the OCV is the difference between the cathode and the anode voltage, it provides a thermodynamic fingerprint of the electrodes at any point in time. Changes in this fingerprint can offer valuable information on path-dependent degradation of both the individual electrodes and the cell as a whole. Not every degradation mechanism leaves a unique fingerprint in the cell's OCV but sets of mechanisms can be clustered into so-called degradation modes, which have a measurable effect on the OCV of the cell and the electrodes [18e21]. There are three commonly reported degradation modes:  \n\n1 Loss of lithium inventory (LLI): lithium ions are consumed by parasitic reactions, such as surface film formation (e.g. SEI growth), decomposition reactions, lithium plating, etc. and are no longer available for cycling between the positive and negative electrode, leading to capacity fade. Surface films may also cause power fade. Lithium ions can also be lost if they are trapped inside electrically isolated particles of the active materials.  \n\n2 Loss of active material of the NE $(\\mathbf{LAM_{NE}})$ : active mass of the NE (or anode) is no longer available for the insertion of lithium due to particle cracking and loss of electrical contact or blocking of active sites by resistive surface layers. These processes can lead to both capacity and power fade.  \n\n3 Loss of active material of the PE $\\mathbf{(LAM_{PE})}$ : active mass of the PE (or cathode) is no longer available for the insertion of lithium due to structural disordering, particle cracking or loss of electrical contact. These processes can lead to both capacity and power fade.  \n\nA more comprehensive list of degradation mechanisms, their causes, effects and links to degradation modes is provided in Fig. 3. Note that Fig. 3 only lists the effects of degradation mechanisms and modes on the cell's thermodynamic (i.e. its OCV), not its kinetic behaviour. The primary effect of degradation on the cell's kinetics is an increase in internal resistance or cell impedance, which is easily measured by the voltage drop in response to a load. It should be noted that an increase in resistance can also lead to a decrease in useful cell capacity under load, since the lower voltage cut-off of the cell is reached sooner in a cell with a higher internal resistance. Equivalently, the higher voltage cut-off is reached sooner during charging. Methods to estimate internal cell resistance are widely reported [22e24] and not the subject of the presented work.  \n\nThe assumed links between the OCV and degradation modes have been used for SoH estimation in the literature [19,25,26]. However, to the best of our knowledge, the existence of the proposed degradation modes has never been proven experimentally and unambiguously but only in simulation. Moreover, methods for estimating degradation modes and inferring the SoH of Li-ion cells are typically based on derivatives of OCV or cell capacity, so called incremental capacity analysis (ICA) [27] or differential voltage analysis (DVA) [28]. Differentiating measurements amplifies the noise in the signal and makes it more difficult to use the resulting data for processing. This is especially problematic in practical applications where voltage measurements may be noisier than in a laboratory environment. In response to these open questions and challenges, two primary objectives were defined for this work:  \n\n1. The design and execution of experiments to verify the manifestations of LLI, $\\mathsf{L A M}_{\\mathrm{NE}}$ and $\\mathrm{\\DeltaLAM_{PE}}$ on the OCV of Li-ion cells.   \n2. The creation of a diagnostic algorithm capable of identifying and quantifying the nature and extent of degradation modes present  \n\n![](images/c89287b4f5dcc5e1f349202144e1f07d5db8da76472b31989440f275f0130d91.jpg)  \nFig. 3. Cause and effect of degradation mechanisms and associated degradation modes.  \n\nin a Li-ion cell based exclusively on the cell's OCV without performing derivative operations on the measurements.  \n\n# 2. Experimental  \n\n# 2.1. Material preparation  \n\nCoin cells were constructed with known amounts of lithium inventory and active electrode materials in order to emulate the different degradation modes. All coin cells were manufactured from electrodes harvested from commercial Kokam $740\\mathrm{mAh}$ pouch cells. The NE material of the Kokam pouch cells is graphite and the PE material is a blend of lithium cobalt oxide (LCO) and lithium nickel cobalt oxide (NCO).  \n\nThe pouch cells were opened in an argon filled glove box using a ceramic scalpel. The electrode sheets were extracted, rinsed with dimethyl carbonate and dried under vacuum for $20~\\mathrm{{min}}$ . The NE sheets consist of copper foil current collectors coated on both sides with graphite, and the PE sheets of aluminium foil current collectors coated on both sides with LCO/NCO. In order to improve electronic conductivity between the current collectors and the coin cell contacts, active material was removed from one side of the electrode sheets using $N\\cdot$ -methyl-2-pyrrolidone. The exposed $\\mathsf{C u/A l}$ surfaces were cleaned with dimethyl carbonate and the electrode disks were cut to size using hole punches. The coin cells were assembled using Celgard separators whetted with LP30 electrolyte ${\\mathrm{~}}1.0{\\mathrm{~M~LiPF}}_{6}$ solution in ethylene carbonate (EC) and dimethyl carbonate (DMC); $\\mathsf{E C/D M C}=50/50\\$ .  \n\nFive reference cells were assembled with electrodes extracted from discharged $(0\\%S_{0}C)$ and fully charged ( $100\\%$ SoC) cells. Both positive and NE disks were cut to a diameter of $15~\\mathrm{mm}$ and the separator to $20~\\mathrm{mm}$ , as illustrated in Fig. 4 a). The reference cells served as a baseline against which the degradation modes were compared, and provided a measure for the reproducibility of the coin cell manufacturing process.  \n\nTwo half-cells, one with positive and one with NE material, with $15\\mathrm{mm}$ electrode disks were made in the same way as the full cells except with lithium foil as counter electrodes (see Table 1). The half-cells served to parametrize the OCV model, as described in Section 3.2. The reference cells and half-cells used in the experiments are listed in Table 1 along with the SoC of the electrodes at the time of assembly and the electrode disk diameters. The SoC of the electrodes refers to the degree of lithiation of the respective electrode at the time of assembly. For example, in a fully charged cell, the SoC of the PE is $0\\%$ and the SoC of the NE is $100\\%$ .  \n\nLoss of lithium inventory (LLI) was created by combining the PE of a pouch cell discharged to a higher SoC with the NE of a pouch cell discharged to a lower SoC. For example, combining a PE from a cell discharged to $25\\%$ SoC with a NE from a cell discharged to $0\\%$ creates a cell with $25\\%$ LLI, since the NE is the limiting electrode during discharge and once it has reached its upper voltage limit (the lower voltage limit of the cell), no more lithium can be extracted. In coin cells with emulated LLI, both electrode disks were $15~\\mathrm{mm}$ in diameter.  \n\nLoss of active electrode material was created by reducing the diameter of the respective electrode disk. The commercial electrodes extracted from the Kokam cells were very uniformly coated with active material, so the useful capacity of the electrodes was assumed to be proportional to their surface area. Loss of NE material $(\\mathrm{LAM_{NE}})$ was created by combining a larger PE disk with a smaller NE disk. Fig. 4 b) illustrates a cell with $36\\%\\mathrm{LAM_{NE}}$ , where the NE (anode) disk is $12~\\mathrm{mm}$ in diameter and the PE (cathode) disk is $15~\\mathrm{mm}$ in diameter. Loss of PE material $\\left(\\mathrm{LAM_{PE}}\\right)$ was created by combining a larger NE disk with a smaller PE disk. Fig. 4 c) shows a cell with $36\\%$ $\\mathsf{L A M}_{\\mathrm{PE}}$ , where the PE (cathode) disk is $12\\mathrm{mm}$ in diameter and the NE (anode) disk is $15~\\mathrm{mm}$ in diameter. Theoretically, active electrode material can be lost in lithiated, delithiated and partially lithiated states. Loss of lithiated NE material $(\\mathrm{LAM_{NE,li}})$ was emulated by combining a $12\\mathrm{mm}\\mathrm{NE}$ disk with a $15\\mathrm{mm}\\mathrm{PE}$ disk, both harvested from a fully charged pouch cell $\\mathsf{T o C}=100\\%$ . Equivalently, loss of delithiated NE material $\\mathsf{L A M}_{\\mathrm{NE,de}}$ was emulated by combining a $12\\mathrm{mm}\\mathrm{NE}$ disk with a $15~\\mathrm{mm}$ PE disk, both harvested from a fully discharged pouch cell $S0C=0\\%$ . The same principle was used to create loss of lithiated PE material $(\\mathrm{LAM_{PE,li}})$ . A limitation of this experimental approach to simulate the loss of active material is the fact that lithium insertion/extraction in the overhang region of the larger electrode is limited by the lateral diffusion of lithium in the active material. The simulation of LAM using this method is therefore only valid for very slow C-rates. For this reason, a very low pseudo-OCV C-rate of $C/25$ is used in this work.  \n\n![](images/08f29b2069110a64cd191ce4561c5cc8eee89ec15cba27a0a98d3df54b539aa0.jpg)  \nFig. 4. Sizes of electrode disks and separators used in coin cells for a) reference cells and cells with LLI, b) cells with $36\\%$ $\\mathrm{LAM_{NE}}$ and c) cells with $36\\%$ $\\mathrm{\\DeltaLAM_{PE}}$ .  \n\nA combination of LLI and $\\mathsf{L A M}_{\\mathrm{PE,li}}$ was created by combining a $14~\\mathrm{mm}$ PE disk harvested from a pouch cell previously discharged to $25\\%$ SoC with a $15\\ \\mathrm{mm}\\ \\mathrm{NE}$ disk harvested from a pouch cell previously discharged to $0\\%$ SoC.  \n\nTable 1 Baseline cell and half-cell electrodes.   \n\n\n<html><body><table><tr><td>Reference cells</td><td>Coin cell ID</td><td>SoC PE</td><td>SoC NE</td><td>Diameter PE</td><td>Diameter NE</td></tr><tr><td>Full cells</td><td>Ref 1</td><td>100%</td><td>0%</td><td>15 mm</td><td>15 mm</td></tr><tr><td></td><td>Ref 2</td><td>100%</td><td>0%</td><td>15 mm</td><td>15 mm</td></tr><tr><td></td><td>Ref 3</td><td>100%</td><td>0%</td><td>15 mm</td><td>15 mm</td></tr><tr><td></td><td>Ref 4</td><td>0%</td><td>100%</td><td>15 mm</td><td>15 mm</td></tr><tr><td rowspan=\"3\">Half-cells</td><td>Ref 5</td><td>0%</td><td>100%</td><td>15 mm</td><td>15 mm</td></tr><tr><td>HCPE</td><td>100%</td><td>一</td><td>15 mm</td><td>一</td></tr><tr><td>HCNE</td><td></td><td>0%</td><td></td><td>15 mm</td></tr></table></body></html>  \n\nTable 2 Test cell electrodes.   \n\n\n<html><body><table><tr><td>Deg.mode</td><td>Coin cell ID</td><td>SoC PE</td><td>SoC NE</td><td>Diameter PE</td><td>Diameter NE</td></tr><tr><td>25% LLI</td><td>LLI25</td><td>75%</td><td>0%</td><td>20 mm</td><td>20 mm</td></tr><tr><td>50% LLI</td><td>LLI50</td><td>50%</td><td>0%</td><td>20 mm</td><td>20 mm</td></tr><tr><td>36% LAMNE.i</td><td>LAMNE,i</td><td>0%</td><td>100%</td><td>15 mm</td><td>12 mm</td></tr><tr><td>36% LAMNE,de</td><td>LAMNE,de</td><td>100%</td><td>0%</td><td>15 mm</td><td>12 mm</td></tr><tr><td>36% LAMPE.li</td><td>LAMPE.li</td><td>100%</td><td>0%</td><td>12 mm</td><td>15 mm</td></tr><tr><td>25% LLI+13% LAMPE,li</td><td>LLI + LAMPE</td><td>75%</td><td>0%</td><td>14 mm</td><td>15 mm</td></tr></table></body></html>  \n\nTable 3 Test procedure for pouch cell preparation.   \n\n\n<html><body><table><tr><td>SoC at end of test</td><td>Test step</td><td>Current</td><td>limits</td></tr><tr><td>0%</td><td>CC charge</td><td>29.6mA</td><td>Vcell > 4.2V</td></tr><tr><td rowspan=\"3\">100%</td><td>CC discharge</td><td>29.6 mA</td><td>Vcell <2.7V</td></tr><tr><td>CC charge</td><td>29.6 mA</td><td>Vcell > 4.2V</td></tr><tr><td>CC discharge</td><td>29.6 mA</td><td>Vcell <2.7V</td></tr><tr><td rowspan=\"4\">50%</td><td>CC charge</td><td>29.6mA</td><td>Vcell > 4.2V</td></tr><tr><td>CC charge</td><td>29.6 mA</td><td>Vcell > 4.2V</td></tr><tr><td>CC discharge</td><td>29.6mA</td><td>Vcell<2.7V</td></tr><tr><td>CC charge</td><td>29.6 mA</td><td>Vcell > 4.2V</td></tr><tr><td rowspan=\"6\">25%</td><td>Rest</td><td>0 mA</td><td>time>10min</td></tr><tr><td>CC discharge</td><td>29.6 mA</td><td>Qdch > 0.5 ×Qmeas</td></tr><tr><td>CC charge</td><td>29.6 mA</td><td>Vcell > 4.2V</td></tr><tr><td>CC discharge</td><td>29.6mA</td><td>Vcell <2.7V</td></tr><tr><td>CC charge</td><td>29.6 mA</td><td>Vcell > 4.2V</td></tr><tr><td>Rest CC discharge</td><td>0 mA 29.6mA</td><td>time>10min Qdch > 0.75 ×Qmeas</td></tr></table></body></html>  \n\nTable 2 provides a list of all the coin cells manufactured to emulate the different degradation modes, including the SoC of the electrodes and the diameter of the electrode disks.  \n\n# 2.2. Cell testing  \n\nAll cell tests were conducted in thermal chambers at $30^{\\circ}\\mathsf C$ using BioLogic potentiostats of type MPG-205 and SP-150. Before the start of tests, pouch cells and coin cells were stored in the thermal chambers for $^{3\\mathrm{~h~}}$ for thermal equilibration. The pouch cells from which the electrodes were extracted for the coin cell manufacturing were prepared according to the test schedule listed in Table 3. Firstly, the capacities of the pouch cells $\\left(\\mathrm{Q}_{\\mathrm{meas}}\\right)$ were measured during a C/25 ( $29.6\\ \\mathrm{\\mA})$ discharge following a $C/25$ charge. Secondly, following another C/25 charge, the SoC of the pouch cells was adjusted by a $C/25$ discharge to the levels required for the respective coin cells, based on the initial capacity measurements.  \n\nThe reference coin cells (full cells and PE/NE half-cells) and the coin cells with induced degradation modes were tested according to the test schedule in Table 4, which consisted primarily of one cycle at C/2 and one cycle at $C/25$ . Partial charges/discharges served to adjust the cells' SoC in preparation for the full cycles, as listed in Table 4. The $C/2$ cycle served to assess the general performance of the cells, based on which under-performing cells were discarded, and the $C/25$ cycle served as pseudo-OCV measurement.  \n\nAt a current rate of C/25, the voltage drop in the coin cells was measured to be on the order of $9\\times10^{-4}\\mathrm{mV}$ , which was considered negligible and any voltage measurements recorded at a current rate of $C/25$ were treated as pseudo-OCV. These pseudo-OCV measurements were used in the degradation model to estimate the degradation modes. The expected capacities of the coin cells were calculated based on the active surface areas of the electrodes. The total electrode surface area of the pouch cells was $600~\\mathrm{cm}^{2}$ and their average capacity measured at $C/25$ was $759~\\mathrm{\\mAh}$ with a standard deviation of $<0.2\\%$ Given a surface area of $1.767\\mathrm{cm}^{2}$ in all coin cells with electrode diameters of $1.5\\mathrm{cm}$ , the expected nominal coin cell capacity is $2.236\\ \\mathrm{\\mAh}$ , which gives a $C/25$ current of $0.089\\mathrm{mA}$ and a $C/2$ current of $1.118~\\mathrm{{mA}}$ The standard deviation of the capacities of all five reference coin cells measured at a $c/2$ current rate was $5.4\\%$ , which served as the measure of uncertainty for the estimation of degradation modes.  \n\nTable 4 Test procedure for coin cells.   \n\n\n<html><body><table><tr><td></td><td>Test ID</td><td>Test step</td><td>Current</td><td>Limits</td></tr><tr><td>Full cells</td><td>Part. charge</td><td>CC charge</td><td>1.118 mA</td><td>Vcell > 4.2V</td></tr><tr><td></td><td>C/2 cycle</td><td>CC discharge</td><td>1.118 mA</td><td>Vcell<2.7V</td></tr><tr><td></td><td></td><td>CC charge</td><td>1.118mA</td><td>Vcell > 4.2V</td></tr><tr><td></td><td>Part. discharge</td><td>CC discharge</td><td>1.118 mA</td><td>Vcell<3.75V</td></tr><tr><td></td><td>Part. charge</td><td>CC charge</td><td>0.089mA</td><td>Vcell> 4.2V</td></tr><tr><td></td><td>C/25 cycle</td><td>CC discharge</td><td>0.089mA</td><td>Vcell<2.7V</td></tr><tr><td>PE half-cell</td><td></td><td>CC charge</td><td>0.089mA</td><td>Vcell > 4.2V</td></tr><tr><td></td><td>Part. charge</td><td>CC charge</td><td>1.118 mA</td><td>Vcell > 4.5V</td></tr><tr><td></td><td>C/2 cycle</td><td>CC discharge</td><td>1.118 mA</td><td>Vcell<3.5V</td></tr><tr><td></td><td>Part. discharge</td><td>CC charge</td><td>1.118 mA</td><td>Vcell >4.5V</td></tr><tr><td></td><td></td><td>CC discharge</td><td>1.118 mA</td><td>Vcell<4.0V</td></tr><tr><td></td><td>Part. charge</td><td>CC charge</td><td>0.089mA</td><td>Vcell >4.5V</td></tr><tr><td></td><td>C/25 cycle</td><td>CC discharge</td><td>0.089mA</td><td>Vcell<3.5V</td></tr><tr><td>NE half-cell</td><td></td><td>CC charge</td><td>0.089mA</td><td>Vcell > 4.5V</td></tr><tr><td></td><td>Part. discharge</td><td>CC discharge</td><td>1.118 mA</td><td>Vcell <0.001V</td></tr><tr><td></td><td>C/2 cycle</td><td>CC charge</td><td>1.118 mA</td><td>Vcell>1.3V</td></tr><tr><td></td><td></td><td>CC discharge</td><td>1.118 mA</td><td>Vcell<0.001V</td></tr><tr><td></td><td>Part. charge</td><td>CC charge</td><td>1.118 mA</td><td>Vcell >0.1V</td></tr><tr><td></td><td>Part. charge</td><td>CC charge</td><td>0.089mA</td><td>Vcell >1.3V</td></tr><tr><td></td><td>C/25 cycle</td><td>CC discharge</td><td>0.089mA</td><td>Vcell<0.001V</td></tr><tr><td></td><td></td><td>CC charge</td><td>0.089mA</td><td>Vcell > 1.3V</td></tr></table></body></html>  \n\n# 3. Model development  \n\n# 3.1. Theory  \n\nThe theory underlying the proposed degradation modes and their effects on the OCV of cells and electrodes is well documented in the literature [19,26,29]. This section explains the approach and the extensions of the presented work.  \n\nIn Li-ion cells, the end of charge (EoC; $100\\%$ SoC) and the end of discharge (EoD; $0\\%$ SoC) are defined by a corresponding maximum and minimum cell voltage, in order to ensure safe operation. The lithium cycled within these limits constitutes the cell's useful capacity. During charge, the PE is limiting, since its rising voltage, resulting from delithiation, triggers the cell's EoC voltage limit (in this case $4.2\\mathrm{~V~}$ ). Analogously, the NE is limiting during discharge, triggering the EoD voltage limit (in this case $2.7\\:\\mathrm{V}$ .  \n\nFig. 5 a) shows the base case of a pristine cell. The bars on the left symbolise the anode (NE, in red) and the cathode (PE, blue). The areas of the bars represent the electrode capacities, not to scale. The golden area represents the cyclable lithium, which corresponds to the cell capacity, in this case intercalated in the NE in a fully charged cell. In commercial Li-ion cells, there is generally an excess of NE material, which is illustrated by the larger NE bar. The misalignment of the two bars indicates which electrode is limiting at $100\\%$ SoC and $0\\%$ SoC, respectively. The plot on the right of Fig. 5 a) depicts the OCV curves of the PE in blue, the NE in red and the cell in grey, as functions of the cell's normalised capacity, denoted as $S o C_{\\mathrm{Cell}}$ in $\\%$ . The horizontal dash/dot lines highlight the upper and lower voltage limits of the cell at $4.2\\:\\mathrm{V}$ and $2.7\\mathrm{V}.$ , respectively, and the vertical dash/dot lines highlight the corresponding maximum and minimum SoC of the whole cell. The points on the OCV of the PE and NE that correspond to the cell's upper and lower voltage limits are indicated by circular markers on the respective OCV curves. As LLI, $\\mathsf{L A M}_{\\mathrm{NE}}$ and $\\mathrm{\\DeltaLAM_{PE}}$ come into effect, the utilised portions of the electrodes change, which is reflected in their OCVs. This means that the OCV of the electrodes at EoC and EoD may also change accordingly. The EoC and EoD OCVs of the electrodes are further affected by maintaining constant upper and lower cell voltage limits of $4.2~\\mathrm{V}$ and $2.7~\\mathrm{V}$ , respectively. Imposing these voltage limits can lead to a stoichiometric offset between the electrodes, which has not been addressed in the literature but is an important addition of this work. The effects of these offsets are discussed individually for each degradation mode.  \n\n![](images/7f1ab5981f57a39a9dea48468975f22d50d59eb9b4086bf88cd33100b5d6149f.jpg)  \nFig. 5. Examples of the different degradation modes. The bars in the left column illustrate the utilisation of the electrodes as a result of the degradation modes, compared to the base case (not to scale). The plots on the right show the corresponding OCV of the electrodes and the cell.  \n\nIf cyclable lithium is lost a smaller fraction of the electrodes' capacities is used, due to the increased offset between the positive and negative electrode. The example of $30\\%$ LLI is illustrated by both the shift of the bars on the left of Fig. 5 b) and the shift of the negative electrode's OCV curve in the plot on the right. The OCV curves of the base case are depicted for comparison as broken lines. The yellow areas in the plots on the right of Fig. 5 indicate lost cell capacity. The OCVs of the electrodes at the EoC and EoD are also affected by the stoichiometric offset due to the imposed cell voltage limits. When a cell which has lost $30\\%$ of its lithium inventory approaches its EoD $(0\\%S_{0}C)$ , the PE voltage is significantly higher than it would be in a pristine cell at EoD (compare the two circular blue markers on the PE OCV curve at the EoD in Fig. 5 b)). If the cut-off voltage of the NE remained unchanged, the minimum cell voltage of $2.7\\:\\mathrm{V}$ could not be reached. In reality, as the cell approaches its EoD and the cell voltage is driven towards $2.7\\mathrm{V}.$ , more lithium is extracted from the NE and inserted into the positive electrode, leading to a steep rise in the NE voltage, ultimately reaching the minimum cell voltage of $2.7~\\mathrm{V}$ (compare the two circular red markers on the NE OCV curves at the EoD in Fig. 5 b)). Equivalently, as a cell with $30\\%$ LLI approaches its EoC, the NE voltage is higher than in a pristine cell. Therefore, as the cell is driven toward its upper voltage limit of $4.2\\:\\mathrm{V}$ , more lithium is extracted from the PE and inserted into the negative electrode, until the rise in PE voltage triggers the upper cut-off condition of the cell (compare the two circular blue markers on the PE voltage curve at the EoC in Fig. 5 b). In the case of $30\\%$ LLI, this stoichiometric offset causes a noticeable increase in cell capacity, on the order of $2\\%$ , as indicated by the green area in the OCV plot of Fig. 5 b). Driving the PE to ever higher voltages can destabilize the structure and, in the worst case, the delithiated cathode material reacts exothermically with the electrolyte, triggering thermal runaway [30].  \n\nThe loss of lithiated active material in the NE $(\\mathrm{LAM_{NE,li}})$ can occur as a result of particle cracking or electronic contact loss between particles of the active electrode material or between the active material and the current collector. The lithium trapped inside the isolated graphite particles and can no longer be cycled, leading to a decrease in the cell's capacity. This capacity loss is illustrated by a shortened red/golden NE bar on the left of Fig. 5 c) and the yellow area in the OCV plot on the right of the figure. Less NE material remains to receive and release lithium, which is manifested in a shrinkage of the OCV curve of the NE, as shown in Fig. 5 c). Less NE material means that the current density on the remaining material is increased during cycling, which could in turn lead to accelerated aging of the NE and, in the worst case, to lithium plating on the surface if the charging rates exceed the rates of lithium diffusion into graphite. In the event that only fully lithiated NE material is lost, the EoC voltage of the NE remains the same (as indicated by the padlock symbol in the plot of Fig. 5 c). The EoD voltage of the positive and negative electrodes change in the same manner as they would in the case of LLI (indicated by the circular markers).  \n\nThe loss of delithiated NE material $\\left(\\mathrm{LAM}_{\\mathrm{NE,de}}\\right)$ initially only has a small effect on the capacity of the cell, since there is an excess of NE material and the OCV of the NE at the EoD remains constant (see Fig. 5 d)). However, the OCV of the NE at the EoC gradually decreases as a result of the loss of active material, which limits the PE to a lower OCV at the EoC thus extracting less lithium. Once the remaining capacity of the NE is smaller than the original cell capacity, the cell loses capacity at the same rate as it loses active NE material. The capacity loss is equivalent to the portion of lithium inventory that remains trapped in the PE at higher voltages, which is indicated by the blue/gold striped area in the PE bar of Fig. 5 d). The example of $30\\%$ LAMNE,de illustrated in Fig. 5 d), effectively leads to a capacity loss of $12\\%$ . Moreover, since the cell is still driven to its upper voltage limit of $4.2\\:\\mathrm{V}$ the NE can be forced to negative voltages, which initiates lithium plating. This is a mechanism that must be prevented in practical applications, since it can lead to dendrite formation and internal short-circuits, which in turn cause catastrophic cell failure [31,32].  \n\nThe loss of lithiated PE material $(\\mathrm{LAM_{PE,li}})$ is analogous to $\\mathsf{L A M}_{\\mathrm{NE,li}}$ ; it is a result of electronic contact loss to lithiated PE particles. An example of $30\\%\\mathrm{LAM_{PE,li}}$ is given in Fig. 5 e). The OCV curve of the PE shrinks compared to its original extent, since a smaller amount of active material contains less lithium and is discharged faster. At the cell's EoC, this means that less lithium is inserted into the NE, leaving it at a higher OCV, which must be matched by a higher OCV of the PE in order to reach the cell's upper voltage limit of $4.2\\mathrm{V}.$ A similar scenario arises as discussed for the case of LLI - increasing $\\mathsf{L A M}_{\\mathrm{PE,li}}$ leads to ever higher PE voltages, potentially destabilizing the PE material.  \n\nA loss of delithiated PE material $\\left(\\mathrm{LAM}_{\\mathrm{PE,de}}\\right)$ can potentially affect the cell capacity at early stages, since there is only a smaller buffer of PE material compared to that in the negative electrode. A scenario of $30\\%\\ \\mathrm{LAM_{PE,de}}$ is illustrated in Fig. 5 f). The OCV of both electrodes at the cell's EoC remains constant but the OCV of the PE at the cell's EoD decreases as the OCV curve of the PE shrinks compared to its original extent. In the case of commercial LCO/NCO material, there can be a steep drop in OCV below $\\sim3.4\\mathsf{V}.$ Such a drop leads to an equivalently lower OCV of the NE at the cell's EoD (indicated by the circular markers in the OCV plot of Fig. 5 f)). The lithium inserted in the NE at higher NE voltages can no longer be accessed for cycling (illustrated by the red/gold striped pattern in the NE bar of Fig. 5 f)), which causes the observed capacity loss of $24\\%$ .  \n\n# 3.2. The OCV model  \n\nIn previous work, we developed a parametric OCV model which can be used as a functional expression of the OCV of both the electrodes and the cell [33]. This OCV model is used as the basis for modelling and estimating the degradation modes discussed in this work. A brief account of the application of the OCV model is provided as follows. For a more detailed description, the reader is referred to [33].  \n\nThe normalised capacity of an electrode can be expressed as the ratio $x$ of occupied to available lattice sites in a host structure, ranging from 0 to 1. In solid multi-phase intercalation materials, $x$ can be calculated as a function of the open circuit voltage $E^{O C}$ by  \n\n$$\nx\\Big(E^{O C}\\Big)=\\sum_{i=1}^{N}\\frac{\\Delta x_{i}}{1+\\exp^{\\left(E^{O C}-E_{0,i}\\right)a_{i}e/k T}}\n$$  \n\nwhere $N$ is the number of phases in the material, $i=\\{1,2,...N\\},\\Delta x_{i}$ is the fraction of material attributed to phase i, $E_{0,i}$ is the energy of lattice sites in phase i, $a_{i}$ is an approximation of the interaction energy between intercalated ions, $e$ is the elementary charge, $k$ is the Boltzmann constant and $T$ the temperature in Kelvin.  \n\nIn the first step of the OCV model parametrization, Equation (1) was fitted to the pseudo-OCV measurements of the PE and NE halfcells. In previous work, a minimum of four phases were identified in both the PE and NE material for this particular cell chemistry [33]. In this work, high qualities of fit of electrode OCVs are paramount in order to achieve accurate estimates of degradation modes. For this reason, an additional phase was added to the OCV model in order to improve the fit qualities from a root mean squared error (RMSE) of $7\\mathrm{mV}$ for the PE and $12{\\mathrm{~mV}}$ for the NE [33] to $<3\\mathrm{mV}$ for both electrodes in this work, as demonstrated in Fig. 6. Equation (1) can not be expressed explicitly as $E^{O C}(x)$ for multiple phases $N$ and is therefore solved iteratively during the optimisation. The objective function used for the optimisation is  \n\n$$\n\\mathrm{arg}_{\\theta}\\mathrm{min}\\mathrm{RMSE}=\\sqrt{\\frac{\\sum_{i}^{n}\\left(\\widehat{E}_{i}^{0C}(\\theta)-E_{i}^{0C}\\right)^{2}}{n}}\n$$  \n\nwhere $n$ is the number of OCV measurements and $\\widehat{E}_{i}^{O C}$ is the fitted electrode OCV. $\\theta$ are the model parameters obtbained for each electrode, summarised by  \n\n$$\n\\theta=\\left[{\\begin{array}{c c c}{E_{0,1}}&{\\Delta x_{1}}&{a_{1}}\\\\ {E_{0,2}}&{\\Delta x_{2}}&{a_{2}}\\\\ {\\vdots}&{\\vdots}&{\\vdots}\\\\ {E_{0,5}}&{\\Delta x_{5}}&{a_{5}}\\end{array}}\\right].\n$$  \n\nIn the second step of the OCV model parametrization, the OCVs of the electrodes were fitted simultaneously with the OCV of the cell - using pseudo-OCV measurements recorded on a full cell - using parameters $\\theta$ of each electrode as an initial guess. The OCV of the cell was calculated by  \n\n![](images/1bf5280d09eddf4fcd9ffab1da573a02cffa7f7a9e11be89c54bbc0c8f785281.jpg)  \nFig. 6. OCV model fitting results.  \n\nThe objective function used for the second optimisation was  \n\n$$\n\\arg_{\\theta_{\\mathrm{cell}}}\\operatorname*{min}{\\mathrm{RMSE}}=\\sqrt{\\frac{\\sum_{i}^{n}\\left(\\widehat{E}_{C e l l,i}^{O C}(\\theta_{C e l l})-E_{C e l l,i}^{O C}\\right)^{2}}{n}}\n$$  \n\n$$\n+\\sqrt{\\frac{\\displaystyle{\\sum_{i}^{n}\\left(\\widehat{E}_{P E,i}^{O C}\\left(\\theta_{C e l l,P E}\\right)-E_{P E,i}^{O C}\\right)^{2}}}{n}}\n$$  \n\n$$\n+\\sqrt{\\frac{\\displaystyle\\sum_{i}^{n}\\left(\\widehat{E}_{N E,i}^{O C}\\big(\\theta_{C e l l,N E}\\big)-E_{N E,i}^{O C}\\right)^{2}}{n}}.\n$$  \n\nParameters $\\theta_{C e l l}$ can be summarised by  \n\n$$\n\\theta_{C e l l}=\\left[\\begin{array}{c c c c c c c}{E_{0,P E,1}}&{\\Delta x_{P E,1}}&{a_{P E,1}}&{E_{0,N E,1}}&{\\Delta x_{N E,1}}&{a_{N E,1}}\\\\ {E_{0,P E,2}}&{\\Delta x_{P E,2}}&{a_{P E,2}}&{E_{0,N E,2}}&{\\Delta x_{N E,2}}&{a_{N E,2}}\\\\ {\\vdots}&{\\vdots}&{\\vdots}&{\\vdots}&{\\vdots}&{\\vdots}&{\\vdots}\\\\ {E_{0,P E,5}}&{\\Delta x_{P E,5}}&{a_{P E,5}}&{E_{0,N E,5}}&{\\Delta x_{N E,5}}&{a_{N E,5}}\\end{array}\\right].\n$$  \n\n$\\theta_{C e l l,P E}$ and $\\theta_{C e l l,N E}$ in Equation (5) include the electrode parameters as detailed in Equation (3). Using the parameters $\\theta_{C e l l}$ , Equations (1) and (4) are applied in the degradation model to compute the OCVs and normalised capacities of electrodes and cells for the base case, i.e. a pristine cell without any signs of degradation. It is important to emphasize that the OCV model is only parameterized in this fashion once for the base case. Fitting the OCV of degraded cells, thus identifying the degradation modes, is achieved using the degradation model described below. This is based on the assumption that the degradation does not impact the individual phases of the electrode materials in different ways.  \n\n# 3.3. The degradation model  \n\nThe degradation model is designed to estimate three parameters only; the degradation modes LLI, $\\mathsf{L A M}_{\\mathrm{NE}}$ and $\\mathrm{\\DeltaLAM_{PE}}$ . The objective of the model is to estimate the extent of the different degradation modes at any point in a cell's life by fitting the cell's OCV. Only the full cell's OCV measurement is required for this. The parameters of the OCV model described in Section 3.2 remain unaltered.  \n\nThe degradation modes affect the electrodes' capacity ranges in terms of (i) their offset, increased by LLI, (ii) their scaling, affected by $\\mathsf{L A M}_{\\mathrm{NE}}$ and $\\mathrm{\\DeltaLAM_{PE}}$ and (iii) the stoichiometric offset, at EoC $(\\Delta x_{E O C})$ and EoD $(\\Delta x_{E O D})$ due to the constant upper and lower cell voltage limits.  \n\nEquations (7)e(10) describe how LLI, $\\mathrm{LAM_{PE}}$ , $\\mathsf{L A M}_{\\mathrm{NE}}$ , $\\Delta x_{E O C}$ and $\\Delta x_{E O D}$ affect the normalised capacity of the PE at the cell's EoC $(x_{P E,E O C})$ and EoD $\\left({{x}_{P E,E O D}}\\right)$ and the normalised capacity of the NE at the cell's EoC $(x_{N E,E o C})$ and EoD $(x_{N E,E o D})$ .  \n\n$$\nx_{P E,E o C}=\\frac{\\Delta x_{E o C}}{1-\\mathrm{LAM_{PE}}}\n$$  \n\n$$\nx_{P E,E o D}=\\frac{\\Delta x_{E o D}+1-\\mathrm{LLI}+\\mathrm{LAM_{PE}}}{1-\\mathrm{LAM_{PE}}}\n$$  \n\n$$\nx_{N E,E o C}=\\frac{\\Delta x_{E o C}+\\mathrm{LLI}-\\mathrm{LAM_{NE}}}{1-\\mathrm{LAM_{NE}}}\n$$  \n\n$$\nE_{C e l l}^{O C}=E_{P E}^{O C}-E_{N E}^{O C}.\n$$  \n\n$$\nx_{N E,E o D}=\\frac{\\Delta x_{E o D}}{1-\\mathrm{LAM}_{\\mathrm{NE}}}\n$$  \n\nLLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ in Equations (7)e(10) range from 0 to 1, where 1 is equivalent to the cell's original capacity; e.g. $\\mathrm{LLI}=0.1$ means that the loss of lithium inventory is equivalent to $10\\%$ of the cell's original capacity. LAMPE and $\\mathsf{L A M}_{\\mathrm{NE}}$ refer to the loss of active material as a fraction of the active material originally utilised within the capacity range of the full cell. The normalised capacities of the positive and negative electrodes at EoC and EoD are linked through the cell's upper and lower voltage limits, denoted as ECOeCll;high and ECOeCll;low, respectively, according to  \n\n$$\nE_{C e l l,h i g h}^{O C}-\\widehat{E}_{P E,E o C}^{O C}\\left(x_{P E,E o C}\\right)+\\widehat{E}_{N E,E o C}^{O C}\\left(x_{N E,E o C}\\right)=0\n$$  \n\nand  \n\n$$\nE_{C e l l,l o w}^{O C}-\\widehat{E}_{P E,E o D}^{O C}\\left(x_{P E,E o D}\\right)+\\widehat{E}_{N E,E o D}^{O C}\\left(x_{N E,E o D}\\right)=0.\n$$  \n\nEquations (11) and (12) define the OCV of electrodes and cell at $100\\%$ SoC and $0\\%$ SoC, respectively, when upper and lower cell voltage limits are imposed. The respective points on the OCV curves of cell and electrodes are marked by the vertical, dash/dotted lines in Fig. 5. ECell;high and ECOeCll;low have predefined values, in this case $4.2\\mathrm{~V~}$ and $2.7~\\mathrm{V}$ , respectively. $\\widehat{\\boldsymbol{E}}_{P E/N E,E o C}^{O C}$ and $\\widehat{E}_{P E/N E,E o D}^{O C}$ are the modelled OCV of the PE and N bat the EoC andbEoD, respectively. The OCVs of the electrodes at the cell's EoC and EoD in Equations (11) and (12) $(\\widehat{\\boldsymbol{E}}_{P E,E o C}^{O C},\\widehat{\\boldsymbol{E}}_{N E,E o C}^{O C},\\widehat{\\boldsymbol{E}}_{P E,E o D}^{O C},E_{N E,E o D}^{O C})$ are calculated using Equation (1). $\\Delta x_{E O C}$ and $\\Delta x_{E o D}$ can be calculated by substituting Equations (7)e(10) into Equations (11) and (12) and solving the linear system of equations. The estimated LLI, $\\mathrm{\\DeltaLAM_{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ are inserted into Equations (7)e(10) during the optimisation described below.  \n\nThe normalised capacity ranges of the electrodes in a degraded cell are defined by vectors of discrete values limited by the normalised capacities at the EoC and EoD according to Equations (13) and (14). The number of elements in the vectors depends on the number of sampling points obtained for the pseudo-OCV measurements.  \n\n$$\n\\begin{array}{r l}&{\\widehat{\\mathbf{x}}_{\\mathbf{PE}}=\\big\\{x_{P E,E o C},...x_{P E,E o D}\\big\\}}\\\\ &{}\\\\ &{\\widehat{\\mathbf{x}}_{\\mathbf{NE}}=\\big\\{x_{N E,E o C},...x_{N E,E o D}\\big\\}}\\end{array}\n$$  \n\nThe OCVs of the electrodes in a degraded cell $(\\widehat{E}_{P E}^{O C},\\widehat{E}_{N E}^{O C})$ can thus be calculated by numerically solving Equation (b1) fobr $E^{O C}$ of the respective electrode, using the normalised electrode capacity ranges $\\widehat{\\mathbf{x}}_{\\mathbf{PE}}$ and $\\widehat{\\mathbf{x}}_{\\mathbf{NE}}$ .  \n\nAt thbe cell levbel, the normalised cell capacity at EoC and EoD is calculated using Equations (15) and (16).  \n\n$$\n\\begin{array}{l}{x_{C e l l,E o C}=\\Delta x_{E o C}}\\\\ {\\qquad\\ }\\\\ {x_{C e l l,E o D}=1-\\mathrm{LLI}+\\Delta x_{E o D}}\\end{array}\n$$  \n\nIn a pristine cell $100\\%$ SoC is equivalent to $x_{C e l l,E o C}=0$ and $0\\%$ SoC to $x_{C e l l,E o D}=1$ . The cell's original capacity is normalised, so that $x_{C e l l,E o D}-x_{C e l l,E o C}=1$ . The difference between $x_{C e l l,E o D}$ and xCell;EoC in a degraded cell corresponds to the cell's normalised capacity as a fraction of its original capacity, e.g. $x_{C e l l,E o D}-x_{C e l l,E o C}=0.9$ means that the cell has lost $10\\%$ of its original capacity. The normalised capacity range of the cell is defined by a vector of discrete values limited by $x_{C e l l,E o C}$ and $x_{C e l l,E o D}$ according to  \n\n$$\n\\widehat{\\mathbf{x}}_{\\mathbf{Cell}}=\\big\\{x_{C e l l,E o C},...x_{C e l l,E o D}\\big\\}.\n$$  \n\nFinally, the OCV of the degraded cell, $\\widehat{\\boldsymbol{E}}_{C e l l,d e g}^{O C}$ , is caOlCculated fOoCr capacity range $\\widehat{\\mathbf{x}}_{\\mathbf{cell}}$ , by solving Equationb(2) using $\\widehat{E}_{P E}^{0\\mathfrak{C}}$ and $\\widehat{E}_{N E}^{\\cup\\mathfrak{c}}$ . Parameters LLI, $\\mathsf{L A M}_{\\mathrm{NE}}$ and $\\mathrm{\\DeltaLAM_{PE}}$ are estimated by mibnimisingbthe objective function  \n\n$$\n{\\arg}{\\theta_{\\mathrm{deg}}\\operatorname*{min}\\mathrm{RMSE}}=\\sqrt{\\frac{\\displaystyle{\\sum_{i}^{n}\\left(\\widehat{E}_{C e l l,d e g}^{O C}\\left(\\theta_{d e g}\\right)-E_{C e l l,d e g}^{O C}\\right)^{2}}}{n}}\n$$  \n\nwhere $\\widehat{\\boldsymbol{E}}_{C e l l,d e g}^{O C}$ is the calculated OCV of the degraded cell, ECell;deg is the meabsured pseudo-OCV of the degraded cell, $n$ is the number of measurements and $\\theta_{d e g}$ are the parameters  \n\n$$\n\\begin{array}{r}{\\theta_{d e g}=[\\mathrm{LLI},\\mathrm{LAM}_{\\mathrm{PE}},\\mathrm{LAM}_{\\mathrm{NE}}].}\\end{array}\n$$  \n\nThe fitting procedure is carried out in Matlab, using the activeset algorithm in Matlab's fmincon optimisation function. In order to ensure convergence to the global minimum, the optimisation is run repeatedly (100 times) from different starting points using Matlab's global optimisation function multistart.  \n\nSince the cell's OCV drops off rapidly near the EoD, errors calculated at low OCV are generally greater than errors at higher OCV where the OCV curve is flat. In order to avoid a bias of the fit toward the lower end of the OCV curve, the calculation of the RMSE as described in Equation (18) was confined to the part of the OCV curve with a gradient of $\\frac{\\Delta E_{C e l l,d e g}^{O C}}{\\Delta{\\sf S o C}}<0.1$  \n\n# 4. Results and discussion  \n\n# 4.1. OCV model fitting  \n\nThe OCV model described in Section 3.2 and [33] was fitted to the pseudo-OCV measurements recorded on the electrode halfcells and a reference coin cell in order to obtain the OCV model parameters. High qualities of fit were achieved for both the OCV of the electrodes and the cell with root mean squared errors (RMSE) $<3\\mathrm{mV}$ (see Fig. 6). The solid lines in Fig. 6 show the fitted OCV results of the cell and the electrodes over the SoC range of the pristine cell. The high fitting accuracy is essential for the degradation model since any deviation from the actual OCV makes it more difficult to identify degradation modes, which may only have very slight effects on the OCV of a degraded cell. The estimated OCV model parameters - $\\boldsymbol{E}_{0,i},\\boldsymbol{a}_{i}$ and $\\Delta x_{i}$ - for each of the phases $i=\\{1...5\\}$ of the PE and the NE are listed in Table 5.  \n\n# 4.2. Test of the diagnostic algorithm using synthesized data  \n\nWith the OCV model parameterized, the degradation model described in Section 3.3 was used in ‘forward mode’ to create a number of scenarios of cells with known amounts of LLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ . The OCVs of the hypothetical degraded cells were subsequently used to test the ability of the diagnostic algorithm to identify the different degradation modes. It is important to point out that the model estimates the total amounts of lost active materials $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ , both lithitated and delithiated. Any lithium contained in lost active electrode material is included in the estimate of the total LLI; i.e. the total estimated LLI includes both the lithium lost through pure LLI (e.g. by SEI build up) and the lithium lost in lithiated active material $(\\mathrm{LAM_{PE,li}}$ and $\\mathsf{L A M}_{\\mathrm{NE,li}}.$ . For example, $10\\%$ of pure LLI and $5\\%$ of $\\mathsf{L A M}_{\\mathrm{NE,li}}$ gives a total of $15\\%$ LLI. The reason for the diagnostic algorithm to be designed in this manner is that a combination of e.g. LLI and $\\mathsf{L A M}_{\\mathrm{NE,de}}$ creates the same OCV signature as an equal amount of $\\mathsf{L A M}_{\\mathrm{NE,li}}$ . The same holds true for combinations of LLI and $\\mathsf{L A M}_{\\mathrm{PE}}$ . The fractions of lithiated and delithiated LAM can therefore not be uniquely identified if the assumption is that LLI can occur simultaneously, resulting from a different mechanism. An exceptional case would be one where LAM is detected but no LLI. In such a case, the respective LAM could be uniquely identified as loss of delithiated active material. In realworld scenarios of Li-ion cell degradation, there is no reason to assume that the loss of active electrode material occurs exclusively in lithiated or delithiated states - it is likely to occur over a range of different stages of lithiation. The approach to separate the loss of lithium contained in lost active electrode material from the loss of the active electrode material itself allows to account for these more realistic scenarios.  \n\nTable 5 Estimated OCV model parameters.   \n\n\n<html><body><table><tr><td rowspan=\"3\"></td><td colspan=\"3\"></td><td colspan=\"3\">NE</td></tr><tr><td>Eo,PE,i [V]</td><td>apE,i [1]</td><td>△XpE,i [1]</td><td>Eo,NE,i [V]</td><td>aNE,i [1]</td><td>△XNE,i [1]</td></tr><tr><td>P1</td><td>5.038</td><td>1.753</td><td>0.021</td><td>0.226</td><td>-18.072</td><td>0.025</td></tr><tr><td>P2</td><td>4.079</td><td>0.178</td><td>0.523</td><td>0.219</td><td>-0.165</td><td>0.112</td></tr><tr><td>P3</td><td>3.936</td><td>0.681</td><td>0.124</td><td>0.173</td><td>-1.188</td><td>0.243</td></tr><tr><td>P4</td><td>3.900</td><td>3.074</td><td>0.136</td><td>0.132</td><td>-14.773</td><td>0.254</td></tr><tr><td>P5</td><td>3.688</td><td>0.470</td><td>0.178</td><td>0.094</td><td>-6.690</td><td>0.365</td></tr></table></body></html>  \n\nThree artificial scenarios were created to test the diagnostic algorithm by running the degradation model in ‘forward mode’. The scenarios are listed in Table 6. The values in Table 6 are given as percentage of the cell's original capacity. Values of LAM refer to the loss of active material as a fraction of the active material originally utilised within the capacity range of the full cell.  \n\nThe diagnostic algorithm was used to fit the degradation model to the synthetically generated cell OCVs, thereby identifying the amounts of total LLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ . The fitting results are depicted in Fig. 7, which shows the fitted cell and electrode OCVs on the left (Fig. 7 a), c) and d)) and the amounts of real and estimated LLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ on the right (Fig. 7 b), d) and f)). The dash/ dotted horizontal lines in the OCV plots of Fig. 7 indicate the cell's upper and lower voltage limits and the vertical lines indicate the corresponding relative capacity. The broken lines show the OCVs of the electrodes and cell of the reference (pristine) cell. Areas filled with yellow indicate capacity loss. For all three scenarios, perfect fits were obtained and all degradation modes accurately identified, which proves the ability of the diagnostic algorithm to uniquely identify the three different degradation modes by fitting the OCV of a degraded cell.  \n\n# 4.3. Validation of the diagnostic algorithm using coin cell data  \n\nThe diagnostic algorithm was used to estimate the degradation modes engineered by means of the customised coin cells (described in Section 2.1). The degradation model was fitted to the pseudoOCV measurements recorded on the coin cells and the results are displayed in Fig. 8; OCV fitting results on the left and the extent of real and estimated degradation modes on the right. The broken lines in the OCV plots on the left of the figure show the cell and electrode OCV of the reference (pristine) cell. The filled areas at the EoC and EoD indicate capacity loss (in yellow) or capacity gain (in green) of the respective cell as a result of degradation. Capacity gain refers to the extraction of additional lithium from the PE as a result of the imposed upper cell voltage limit, leading to a stoichiometric offset. This can be detrimental to the cell, since the PE is driven to higher voltages, which may accelerate its degradation. The capacity gain at the EoC is never greater than the accompanying capacity loss at the EoD but it can slightly reduce the overall capacity loss. The RMSE values displayed in the OCV plots were calculated from the measured and the fitted cell voltages for the entire cell voltage window of $2.7\\:\\mathrm{V}{-4.2\\:\\mathrm{V}}.$ The error bars on the bar charts on the right of Fig. 8 are based on the standard deviation of the capacities of the reference coin cells $(5.4\\%)$ , as described in Section 2.2. It should be emphasized that the uncertainty of $5.4\\%$ reflects the reproducibility of the coin cell manufacturing. For applications on commercial cells, high accuracies can be expected for estimations of degradation modes, as demonstrated in Section 4.2. The results are discussed individually for each degradation scenario.  \n\nTable 6 Scenarios for synthetic OCV data.   \n\n\n<html><body><table><tr><td>Scenario</td><td>LLI (pure)</td><td>LAMNE,li</td><td>LAMNE,de</td><td>LAMPE,li</td><td>LAMPE,de</td><td>LLI (total)</td></tr><tr><td></td><td>12%</td><td>0%</td><td>23%</td><td>6%</td><td>0%</td><td>18%</td></tr><tr><td>II</td><td>21%</td><td>4%</td><td>0%</td><td>0%</td><td>7%</td><td>25%</td></tr><tr><td>III</td><td>9%</td><td>0%</td><td>14%</td><td>0%</td><td>11%</td><td>9%</td></tr></table></body></html>  \n\n# 4.3.1. $25\\%$ LLI  \n\nThe fitted OCV and estimated degradation mode of cell LLI25 ( $25\\%$ LLI) are shown in Fig. 8 a) and b), respectively. The diagnostic algorithm accurately estimated the extent of LLI within the margin of error (see Fig. 8 b)). The coin cell's OCV was fitted with a RMSE of $6.7~\\mathrm{mV}.$ The $25\\%$ LLI led to the expected increased offset between the positive and negative electrode's OCV, illustrated by a left-shift of the negative electrode's OCV in Fig. 8 a), which directly translates into capacity loss of the cell. The offset and imposed upper cell voltage limit forced the PE to a slightly higher OCV, extracting a small amount of additional lithium at the EoC, offsetting $\\sim1\\%$ of the overall capacity loss. This is indicated by the dash/dotted vertical lines at the cell's EoC and the circular markers on the positive and negative electrode's OCV in Fig. 8 a).  \n\n# 4.3.2. $50\\%$ LLI  \n\nFig. 8 c) and d) show the fitted OCV and estimated degradation mode of cell LLI50 ( $50\\%$ LLI). $50\\%$ LLI was accurately estimated and other degradation modes were found to be negligible within the margin of error. The RMSE of the OCV fit was $11.9~\\mathrm{mV}$ . The same trends were observed as for cell LLI25, albeit to a greater extent; a large offset between the positive and NE and a noticeably higher PE OCV at EoC, extracting an additional $\\sim2\\%$ of lithium from the positive electrode. The results obtained for cells LLI25 and LLI50 confirm the theory of LLI discussed in Section 3.1.  \n\n# 4.3.3. 36% LAMNE,li  \n\nFig. 8 e) and f) show the fitted OCV and estimated degradation mode of cell $\\mathsf{L A M}_{\\mathrm{NE,li}}$ $(36\\%\\ \\mathrm{LAM_{NE,li}})$ . $\\mathsf{L A M}_{\\mathrm{NE}}$ was successfully identified as a major degradation mode, although to a slightly smaller extent than expected, exceeding the margin of error by $\\sim4\\%$ . Against expectations, a small amount of $\\mathrm{\\DeltaLAM_{PE}}$ was detected by the diagnostic algorithm, exceeding the margin of error by $\\sim5\\%$ This discrepancy could be explained as an artifact of the coin cell manufacturing technique. The hole punch used to cut the $12~\\mathrm{mm}$ electrode disks slightly crimped the disks around the edge, causing a small rim which was bent away from the interface of electrode and separator. A rim of merely $0.3~\\mathrm{mm}$ around a $12~\\mathrm{mm}$ disk constains an additional $5\\%$ of the electrode's capacity. Due to the lack of contact between the rim around the NE disk (which was cut in a fully lithiated state) and the separator, the lithium contained in the NE could not be fully extracted during the pseudo-OCV discharge of the cell, which would appear as a loss of PE material. The fact that the extent of $\\mathsf{L A M}_{\\mathrm{NE}}$ was underestimated by roughly the same amount as the $\\mathrm{\\DeltaLAM_{PE}}$ was overestimated, namely between $4\\%$ and  \n\n![](images/65c9c8876f78fe3960c936b088471af30943955ef43401fdbd2c6faa8ea37f44.jpg)  \nFig. 7. Estimation of known amounts of LLI, $\\mathrm{\\DeltaLAM_{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ ; a) and b) Scenario I: $12\\%$ (pure) LLI, $23\\%$ $\\mathrm{LAM}_{\\mathrm{NE,de}},$ $6\\%$ $\\mathsf{L A M}_{\\mathrm{PE,li}},$ c) and d) Scenario II: $21\\%$ (pure) LLI, $4\\%$ $\\mathsf{L A M}_{\\mathrm{NE,li}}$ $7\\%$ $\\mathrm{LAM_{PE,de}}$ , e) and f) Scenario III: $9\\%$ (pure) LLI, $14\\%$ $\\mathsf{L A M}_{\\mathrm{NE,de}}$ , $11\\%$ $\\mathrm{LAM}_{\\mathrm{PE,de}}$ .  \n\n$5\\%$ , supports this theory. The lithium contained within the lost active NE material (LLI) was also slightly underestimated - to a similar degree as the $\\mathsf{L A M}_{\\mathrm{NE}}$ , within the margin of error. Overall, the predominant degradation modes were correctly identified - LLI and $\\mathsf{L A M}_{\\mathrm{NE}}$ . In this scenario, all of the lost litihium was contained in the lithiated negative electrode. The diagnostic algorithm correctly identified equal amounts of LLI and $\\mathsf{L A M}_{\\mathrm{NE}}$ , suggesting that the prevalent mechanism was $\\mathsf{L A M}_{\\mathrm{NE,li}}$ .  \n\n# 4.3.4. 36% LAMNE,de  \n\nThe diagnostic results for cell $\\mathsf{L A M}_{\\mathrm{NE,de}}$ are displayed in Fig. $8\\:\\mathrm{g}^{\\cdot}$ ) and h) - fitted OCV and estimated degradation modes, respectively. Since a $36\\%$ loss of delithiated NE material exceeds the additional NE capacity buffer, lithium plating on the NE was expected in this scenario during charging. The onset of lithium plating occurs once the NE voltage drops below $0{\\:}\\mathsf{V}.$ . The cells used in this work have a NE capacity buffer of $\\sim25\\%$ , which can be inferred from the OCV of the reference cell's negative electrode, shown as broken red line in the plots on the left of Fig. 8. The amount of $\\mathsf{L A M}_{\\mathrm{NE,de}}$ required to consume all of the NE buffer and reach the onset of lithium plating $\\left(\\mathrm{LAM}_{\\mathrm{NE,pl}}\\right)$ can be calculated as follows. Firstly, the normalised capacity of the NE at $E_{N E}^{O C}=0\\mathsf{V}$ , scaled with respect to the normalised cell capacity $(x_{N E,C e l l,p l})$ , is calculated. $x_{N E,C e l l,p l}$ is equivalent to the normalised and scaled capacity of the positive electrode, where $E_{P E}^{O C}=4.2\\mathrm{V}.$ The latter can be calculated using Equation (1) for the PE with $4.2\\mathrm{V}$ plugged in as $E_{P E}^{O C}$ , multiplying the result by the ratio of cell capacity to PE capacity and adding the offset between the normalised cell capacity and the normalised PE capacity, which is equivalent to $x_{P E}$ at $E_{C e l l}^{O C}=4.2\\mathrm{V}$ (as obtained by the OCV model). Now $\\mathsf{L A M}_{\\mathrm{NE,pl}}$ can be calculated using Equation (20).  \n\n$$\n\\mathsf{L A M}_{\\mathrm{NE,pl}}=1-\\frac{x_{N E,c e l l,p l}}{x_{N E,C e l l,m a x}}\n$$  \n\nwhere $x_{N E,m a x}=1.25$ is the maximum of the normalised capacity of the negative electrode, scaled with respect to the cell capacity, in the reference cell. Equation (20) yields $\\mathrm{LAM_{NE,pl}}=26.4\\%$ , which means that any loss of delithiated NE material exceeding $26.4\\%$ causes the onset of lithium plating on the NE and leads to further capacity loss due to irreversible deposition of metallic lithium. Although some of the lithium plated during charging may be recovered by stripping during discharging [34], the assumption in this work is that the amount of stripped lithium is negligible and any plated lithium leads to LLI. The amount of LLI resulting from plating can be approximated by calculating the difference between $\\mathsf{L A M}_{\\mathrm{NE,de}}$ and $\\mathsf{L A M}_{\\mathrm{NE,pl}}$ , in this case amounting to $9.6\\%$ LLI. The diagnostic algorithm accurately identified both the amount of $\\mathsf{L A M}_{\\mathrm{NE}}$ and LLI within the margin of error, as shown in Fig. 8 h). The capacity loss attributed to $\\mathsf{L A M}_{\\mathrm{NE,de}}$ is illustrated by the yellow area at the cell's EoC in Fig. 8 g). Note that capacity lost due to lithium trapped inside the PE (yellow area in Fig. $\\mathbf{8\\cdotg}^{\\cdot}$ ) at the cell's EoC), as a result of $\\mathsf{L A M}_{\\mathrm{NE,de}}$ , is not included in the total amount of LLI. The capacity lost as a result of lithium plating is marked by the yellow area at the cell's EoD in Fig. $\\mathbf{8\\cdotg}^{\\cdot}$ ). The small amount of $\\mathrm{LAM_{PE}}$ which was also identified exceeds the margin of error by only $\\sim1\\%$ and is therefore considered negligible.  \n\n![](images/fdd79bcf18e1d4ac8c356854843aa75c757248d93941f9f5a4a079f96c77cb09.jpg)  \nFig. 8. Results of degradation diagnostics; fitted cell OCVs (left column) and estimated degradation mode (right column). a) and b): cell LLI25, c) and d): cell LLI50, e) and f): cell $\\mathsf{L A M}_{\\mathrm{NE,li}},\\mathsf{g})$ and $\\mathbf{h}$ ): cell $\\mathrm{LAM_{NE,de},i)}$ and j): cell $\\mathsf{L A M}_{\\mathrm{PE,li}},\\mathsf{k}$ ) and l): cell $\\mathrm{LLI}+\\mathrm{LAM}_{\\mathrm{PE,li}}$ .  \n\n# 4.3.5. 36% LAMPE,li  \n\nFig. 8 i) and j) display the fitted OCV and estimated degradation mode of cell $\\mathsf{L A M}_{\\mathrm{PE,li}}$ $36\\%$ of $\\mathsf{L A M}_{\\mathrm{PE},\\mathrm{li}}\\cdot$ ). Fig. 8 i) illustrates the effect of lost lithiated PE material on the cell's capacity, which is significantly reduced at the EoC. The correct amounts of $\\mathsf{L A M}_{\\mathrm{PE}}$ and LLI contained in the lost electrode material were estimated using the diagnostic algorithm.  \n\n# 4.3.6. $25\\%L L I+13\\%L A M_{P E,l i}$  \n\nFig. $\\boldsymbol{8\\mathrm{~k~}}$ ) and l) show the fitted OCV and estimated degradation mode of cell $\\mathrm{LLI}+\\mathrm{LAM}_{\\mathrm{PE,li}}$ ( $25\\%$ of LLI and $13\\%$ of $\\mathsf{L A M}_{\\mathrm{PE,li}},$ . In this case, two degradation modes were combined. The total amount of LLI $(38\\%)$ includes the pure LLI $(25\\%)$ and the lithium lost inside the active PE material $(13\\%)$ . The yellow area at the EoC in Fig. $8\\mathrm{~k~}$ ) represents the capacity loss due to $\\mathsf{L A M}_{\\mathrm{PE,li}}$ and the yellow area at the EoD the capacity loss due to pure LLI. As shown in Fig. 8 l), the total amount of LLI was accurately estimated and the $\\mathbf{LAM_{PE}}$ was slightly overestimated, exceeding the margin of error by $1.6\\%.$ A small amount of $\\mathsf{L A M}_{\\mathrm{NE}}$ was detected, exceeding the margin of error by $\\sim1\\%.$ . This could be due to the same effect as described for cell $\\mathsf{L A M}_{\\mathrm{NE,li}}$ . Overall, the predominant degradation modes were successfully identified, even in the presence of two independent degradation modes.  \n\n# 5. Conclusions  \n\nThis work has built on the theory of degradation modes in Li-ion cells as a manifestation of a host of different physical and chemical mechanisms. The general theory behind the degradation modes LLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ has been expanded to account for the effects of imposed upper and lower cell voltage limits on the different degradation modes. This expanded theory was used to create a diagnostic algorithm to identify and quantify LLI, $\\mathrm{\\DeltaLAM_{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ using only pseudo-OCV measurements of full cells. The diagnostic algorithm was validated using test cells with known amounts of LLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ . The results led to three key findings:  \n\nExperimental proof of the effects of LLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ on the cell's OCV.   \nA diagnostic algorithm has been demonstrated to successfully identify and quantify LLI, LAMPE and $\\mathsf{L A M}_{\\mathrm{NE}}$ .   \n\u0001 The diagnostic algorithm can identify the onset of potentially dangerous processes such as excessively high voltages on the PE and lithium plating on the NE.  \n\nExperimental evidence has been presented to prove, for the first time, that the OCV of Li-ion cells can be used to provide accurate estimates of LLI, $\\mathsf{L A M}_{\\mathrm{PE}}$ and $\\mathsf{L A M}_{\\mathrm{NE}}$ . The diagnostic algorithm was evaluated for six different scenarios of degradation modes. Once the OCV model has been parameterized, the diagnostic algorithm requires only pseudo-OCV measurements. Since the algorithm uses the pseudo-OCV measurements as a direct input, rather than the derivative of voltage or capacity, it is less sensitive to noise compared to other techniques proposed in the literature. These attributes, combined with the low computational complexity of the diagnostic algorithm, make it ideal for BMS applications in order to keep track of the cells' SoH and to maintain safe operation.  \n\nTheoretically, the presented diagnostic technique can be applied to any Li-ion cell chemistry. This possibility will be investigated in future work. Further work also includes the identification and quantification of degradation modes in commercial Li-ion cells aged in a variety of use cases and throughout their service life. This should enable projections of the end-of-life for commercial Li-ion cells.  \n\nFinancial support of EPSRC UK (EP/K504518/1) and Jaguar Land Rover is gratefully acknowledged.  \n\n# Acknowledgements  \n\n# References  \n\n[1] S.S. Choi, H.S. 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    "Target": null
  },
  {
    "id": "10.1126_sciadv.1603015",
    "DOI": "10.1126/sciadv.1603015",
    "DOI Link": "http://dx.doi.org/10.1126/sciadv.1603015",
    "Relative Dir Path": "mds/10.1126_sciadv.1603015",
    "Article Title": "Machine learning of accurate energy-conserving molecular force fields",
    "Authors": "Chmiela, S; Tkatchenko, A; Sauceda, HE; Poltavsky, I; Schütt, KT; Müller, KR",
    "Source Title": "SCIENCE ADVANCES",
    "Abstract": "Using conservation of energy-a fundamental property of closed classical and quantum mechanical systems-we develop an efficient gradient-domain machine learning (GDML) approach to construct accurate molecular force fields using a restricted number of samples from ab initio molecular dynamics (AIMD) trajectories. The GDML implementation is able to reproduce global potential energy surfaces of intermediate-sized molecules with an accuracy of 0.3 kcal mol(-1) for energies and 1 kcal mol(-1) angstrom(-1) for atomic forces using only 1000 conformational geometries for training. We demonstrate this accuracy for AIMD trajectories of molecules, including benzene, toluene, naphthalene, ethanol, uracil, and aspirin. The challenge of constructing conservative force fields is accomplished in our work by learning in a Hilbert space of vector-valued functions that obey the law of energy conservation. The GDML approach enables quantitative molecular dynamics simulations for molecules at a fraction of cost of explicit AIMD calculations, thereby allowing the construction of efficient force fields with the accuracy and transferability of high-level ab initio methods.",
    "Times Cited, WoS Core": 852,
    "Times Cited, All Databases": 949,
    "Publication Year": 2017,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000401955300043",
    "Markdown": "# A P P L I E D M A T H E M A T I C S  \n\n# Machine learning of accurate energy-conserving molecular force fields  \n\n2017 $\\circledcirc$ The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).  \n\nStefan Chmiela,1 Alexandre Tkatchenko, $^{2,3_{*}}$ Huziel E. Sauceda,3 Igor Poltavsky,2 Kristof T. Schütt,1 Klaus-Robert Müller1,4,5\\*  \n\nUsing conservation of energy—a fundamental property of closed classical and quantum mechanical systems— we develop an efficient gradient-domain machine learning (GDML) approach to construct accurate molecular force fields using a restricted number of samples from ab initio molecular dynamics (AIMD) trajectories. The GDML implementation is able to reproduce global potential energy surfaces of intermediate-sized molecules with an accuracy of $0.3\\ \\mathsf{k c a l\\ m o l}^{-1}$ for energies and $1\\ k\\mathbf{c}\\mathbf{a}|\\ \\mathbf{mol}^{-1}\\ \\mathring{\\mathsf{A}}^{-1}$ for atomic forces using only 1000 conformational geometries for training. We demonstrate this accuracy for AIMD trajectories of molecules, including benzene, toluene, naphthalene, ethanol, uracil, and aspirin. The challenge of constructing conservative force fields is accomplished in our work by learning in a Hilbert space of vector-valued functions that obey the law of energy conservation. The GDML approach enables quantitative molecular dynamics simulations for molecules at a fraction of cost of explicit AIMD calculations, thereby allowing the construction of efficient force fields with the accuracy and transferability of high-level ab initio methods.  \n\n# INTRODUCTION  \n\nWithin the Born-Oppenheimer (BO) approximation, predictive simulations of properties and functions of molecular systems require an accurate description of the global potential energy hypersurface $V_{\\mathrm{BO}}(\\overrightarrow{r_{1}},\\overrightarrow{r_{2}},...,\\overrightarrow{r_{N}})$ , where $\\overrightarrow{r_{i}}$ indicates the nuclear Cartesian coordinates. Although $V_{\\mathrm{BO}}$ could, in principle, be obtained on the fly using explicit ab initio calculations, more efficient approaches that can access the long time scales are required to understand relevant phenomena in large molecular systems. A plethora of classical mechanistic approximations to $V_{\\mathrm{BO}}$ have been constructed, in which the parameters are typically fitted to a small set of ab initio calculations or experimental data. Unfortunately, these classical approximations may suffer from the lack of transferability and can yield accurate results only close to the conditions (geometries) they have been fitted to. Alternatively, sophisticated machine learning (ML) approaches that can accurately reproduce the global potential energy surface (PES) for elemental materials (1–9) and small molecules (10–16) have been recently developed (see Fig. 1, A and B) (17). Although potentially very promising, one particular challenge for direct ML fitting of molecular PES is the large amountofdatanecessarytoobtainanaccuratemodel.Often,manythousands or even millions of atomic configurations are used as training data for ML models. This results in nontransparent models, which are difficult to analyze and may break consistency (18) between energies and forces.  \n\nA fundamental property that any force field $\\mathbf{F}_{i}(\\overrightarrow{r_{1}},\\overrightarrow{r_{2}},...,\\overrightarrow{r_{N}})$ must satisfy is the conservation of total energy, which implies that $\\mathbf{F}_{i}(\\overrightarrow{r_{1}},\\overrightarrow{r_{2}},...,\\overrightarrow{r_{N}})=-\\nabla_{\\overrightarrow{r_{i}}}V(\\overrightarrow{r_{1}},\\overrightarrow{r_{2}},...,\\overrightarrow{r_{N}})$ . Any classical mechanistic expressions for the potential energy (also denoted as classical force field) or analytically derivable ML approaches trained on energies satisfy energy conservation by construction. However, even if conservation of energy is satisfied implicitly within an approximation, this does not imply that the model will be able to accurately follow the trajectory of the true ab initio potential, which was used to fit the force field. In particular, small energy/force inconsistencies between the force field model and ab initio calculations can lead to unforeseen artifacts in the PES topology, such as spurious critical points that can give rise to incorrect molecular dynamics (MD) trajectories. Another fundamental problem is that classical and ML force fields focusing on energy as the main observable have to assume atomic energy additivity—an approximation that is hard to justify from quantum mechanics.  \n\nHere, we present a robust solution to these challenges by constructing an explicitly conservative ML force field, which uses exclusively atomic gradient information in lieu of atomic (or total) energies. In this manner, with any number of data samples, the proposed model fulfills energy conservation by construction. Obviously, the developed ML force field can be coupled to a heat bath, making the full system (molecule and bath) non–energy-conserving.  \n\nWe remark that atomic forces are true quantum-mechanical observables within the BO approximation by virtue of the Hellmann-Feynman theorem. The energy of a molecular system is recovered by analytic integration of the force-field kernel (see Fig. 1C). We demonstrate that our gradient-domain machine learning (GDML) approach is able to accurately reproduce global PESs of intermediate-sized molecules within $0.{\\dot{3}}\\mathrm{kcalmol}^{-1}$ for energies and $1\\mathrm{\\kcal\\mol^{-1}\\mathring{A}^{-1}}$ for atomic forces relative to the reference data. This accuracy is achieved when using less than 1000 training geometries to construct the GDML model and using energy conservation to avoid overfitting and artifacts. Hence, the GDML approach paves the way for efficient and precise MD simulations with PESs that are obtained with arbitrary high-level quantumchemical approaches. We demonstrate the accuracy of GDML by computing AIMD-quality thermodynamic observables using pathintegral MD (PIMD) for eight organic molecules with up to 21 atoms and four chemical elements. Although we use density functional theory (DFT) calculations as reference in this development work, it is possible to use any higher-level quantum-chemical reference data. With state-of-the-art quantum chemistry codes running on current highperformance computers, it is possible to generate accurate reference data for molecules with a few dozen atoms. Here, we focus on intramolecular  \n\n# Descriptor encodes molecular structure.  \n\n$$\nD_{i j}={\\left\\{\\begin{array}{l l}{\\|{\\vec{r_{i}}}-{\\vec{r_{j}}}\\|^{-1}}&{{\\mathrm{for~}}i>j}\\\\ {0}&{{\\mathrm{for~}}i\\leq j}\\end{array}\\right.}\n$$  \n\nKernel function measures the similarity between pairs of inputs.  \n\n$$\n\\kappa:\\langle\\phi(\\mathbf{D}),\\phi(\\mathbf{D}^{\\prime})\\rangle_{\\mathcal{H}}\n$$  \n\n# Problem:  \n\nEnergy-based model lacks detail in undersampled regions.  \n\n![](images/67158f81682638f196dd1dbd0eb83acd035a0f66ea5fdae8e5e9a7877b3cf466.jpg)  \n\n![](images/b066b2797980961c27e462458e9885ab238aaa22625137d5f53ace1d29410b1d.jpg)  \nFig. 1. The construction of ML models: First, reference data from an MD trajectory are sampled. (A) The geometry of each molecule is encoded in a descriptor. This representation introduces elementary transformational invariances of energy and constitutes the first part of the prior. A kernel function then relates all descriptors to form the kernel matrix—the second part of the prior. The kernel function encodes similarity between data points. Our particular choice makes only weak assumptions: It limits the frequency spectrum of the resulting model and adds the energy conservation constraint. Hess, Hessian. (C) These general priors are sufficient to reproduce good estimates from a restricted number of force samples. (B) A comparable energy model is not able to reproduce the PES to the same level of detail.  \n\n# Solution:  \n\nTraining in the force domain accurately reproduces PES topology.  \n\n$$\n\\hat{f}_{F}(\\vec{x})=\\sum_{i=1}^{M}\\sum_{j=1}^{3N}(\\vec{\\alpha}_{i})_{j}\\frac{\\partial}{\\partial x_{j}}\\nabla\\kappa(\\vec{x},\\vec{x}_{i})\n$$  \n\n$$\n\\hat{f}_{E}(\\vec{x})=\\sum_{i=1}^{M}\\sum_{j=1}^{3N}(\\vec{\\alpha}_{i})_{j}\\frac{\\partial}{\\partial x_{j}}\\kappa(\\vec{x},\\vec{x}_{i})\n$$  \n\n![](images/fa483a98b1fb60ca462674242e90ce79d60fe2d0ae907c3c8bbbb022a3c3a524.jpg)  \nFig. 2. Modeling the true vector field (leftmost subfigure) based on a small number of vector samples With GDML, a conservative vector field estimate $\\hat{\\pmb f}_{\\sf F}$ is obtained directly. A naïve estimator $\\hat{\\pmb f}_{\\sf F}^{-}$ with independent predictions for each element of the output vector is not capable of imposing energy conservation constraints. We perform a Helmholtz decomposition of this nonconservative vector field to show the error component that violates the law of energy conservation. This is the portion of the overall prediction error that was avoided with GDML because of the addition of the energy conservation constraint.  \n\nforces in small- and medium-sized molecules. However, in the future, the GDML model should be combined with an accurate model for intermolecular forces to enable predictive simulations of condensed molecular systems. Widely used classical mechanistic force fields are based on simple harmonic terms for intramolecular degrees of freedom. Our GDML model correctly treats anharmonicities by using no assumptions whatsoever on the analytic form on the interatomic potential energy functions within molecules.  \n\n![](images/6b88459475cbbd19499a5dbd47b257890ccdd80a9c28f0c8ba37a1f0ea797f8b.jpg)  \nFig. 3. Efficiency of GDML predictor versus a model that has been trained on energies. (A) Required number of samples for a force prediction performance of MAE $(1\\ \\mathsf{k c a l\\ m o l}^{-1}\\ \\mathring{\\mathsf{A}}^{-1};$ with the energy-based model (gray) and GDML (blue). The energy-based model was not able to achieve the targeted performance with the maximum number of 63,000 samples for aspirin. (B) Force prediction errors for the converged models (same number of partial derivative samples and energy samples). (C) Energy prediction errors for the converged models. All reported prediction errors have been estimated via cross-validation.  \n\n# METHODS  \n\nThe GDML approach explicitly constructs an energy-conserving force field, avoiding the application of the noise-amplifying derivative operator to a parameterized potential energy model (see the Supplementary Materials for details). This can be achieved by directly learning the functional relationship  \n\n$$\n\\hat{f}_{\\mathrm{F}}:(\\overrightarrow{r_{1}},\\overrightarrow{r_{2}},...,\\overrightarrow{r_{N}})_{i}\\stackrel{\\mathrm{ML}}{\\rightarrow}\\mathbf{F}_{i}\n$$  \n\nbetween atomic coordinates and interatomic forces, instead of computing the gradient of the PES (see Fig. 1, C and B). This requires constraining the solution space of all arbitrary vector fields to the subset of energy-conserving gradient fields. The PES can be obtained through direct integration of $\\dot{\\boldsymbol{f}}_{\\mathrm{F}}$ up to an additive constant.  \n\nTo construct $\\hat{\\boldsymbol f}_{\\mathrm{F}}^{\\phantom{\\dagger}}$ we used a generalization of the commonly used kernel ridge regression technique for structured vector fields (see the Supplementary Materials for details) (19–21). GDML solves the normal equation of the ridge estimator in the gradient domain using the Hessian matrix of a kernel as the covariance structure. It maps to all partial forces of a molecule simultaneously (see Fig. 1A)  \n\n$$\n\\bigl(\\mathbf{K}_{\\mathrm{Hess}(\\kappa)}+\\lambda\\mathbb{I}\\bigr)\\overrightarrow{\\mathrm{d}}=\\nabla V_{B O}=-\\mathbf{F}\n$$  \n\nWe resorted to the extensive body of research on suitable kernels and descriptors for the energy prediction task (10, 13, 17).  \n\nFor our application, we considered a subclass from the parametric Matérn family (22–24) of (isotropic) kernel functions  \n\n$$\n\\upkappa:C_{\\nu=n+\\frac{1}{2}}(d)=\\exp\\left(-\\frac{\\sqrt{2\\nu}d}{\\upsigma}\\right)P_{n}(d),\n$$  \n\n$$\nP_{n}(d)=\\sum_{k=0}^{n}{\\frac{(n+k)!}{(2n)!}}{\\binom{n}{k}}\\left({\\frac{2{\\sqrt{2\\nu}}d}{\\sigma}}\\right)^{n-k}\n$$  \n\nwhere $d=\\|{\\vec{x}}-{\\vec{x}}^{\\prime}\\|$ is the Euclidean distance between two molecule descriptors. It can be regarded as a generalization of the universal Gaussian kernel with an additional smoothness parameter $n$ . Our parameterization $n=2$ resembles the Laplacian kernel, as suggested by Hansen et al. (13), while being sufficiently differentiable.  \n\nTo disambiguate Cartesian geometries that are physically equivalent, we use an input descriptor derived from the Coulomb matrix (see the Supplementary Materials for details) (10).  \n\nThe trained force field estimator collects the contributions of the partial derivatives 3N of all training points $M$ to compile the prediction. It takes the form  \n\n$$\n\\hat{f}_{\\mathrm{F}}\\big(\\overrightarrow{\\mathbfit{x}}\\big)=\\sum_{i=1}^{M}\\sum_{j=1}^{3N}\\big(\\overrightarrow{\\mathbfit{a}}_{i}\\big)_{j}\\frac{\\partial}{\\partial x_{j}}\\nabla\\kappa\\big(\\overrightarrow{\\mathbfit{x}},\\overrightarrow{\\mathbfit{x}}_{i}\\big)\n$$  \n\nand a corresponding energy predictor is obtained by integrating $\\hat{f}_{\\mathrm{F}}(\\vec{x})$ with respect to the Cartesian geometry. Because the trained model is a (fixed) linear combination of kernel functions, integration only affects the kernel function itself. The expression  \n\n$$\n\\hat{f}_{\\mathrm{E}}\\big(\\overrightarrow{\\textbf{\\textit{x}}}\\big)=\\sum_{i=1}^{M}\\sum_{j=1}^{3N}\\big(\\overrightarrow{\\textbf{\\textit{a}}}_{i}\\big)_{j}\\frac{\\partial}{\\partial x_{j}}\\upkappa\\big(\\overrightarrow{\\textbf{\\textit{x}}},\\overrightarrow{\\textbf{\\textit{x}}}_{i}\\big)\n$$  \n\nfor the energy predictor is therefore neither problem-specific nor does it require retraining.  \n\nWe remark that our PES model is global in the sense that each molecular descriptor is considered as a whole entity, bypassing the need for arbitrary partitioning of energy into atomic contributions. This allows the GDML framework to capture chemical and longrange interactions. Obviously, long-range electrostatic and van der Waals interactions that fall within the error of the GDML model will have to be incorporated with explicit physical models. Other approaches that use ML to fit PESs such as Gaussian approximation potentials (3, 8) have been proposed. However, these approaches consider an explicit localization of the contribution of individual atoms to the total energy. The total energy is expressed as a linear combination of local environments characterized by a descriptor that acts as a nonunique partitioning function to the total energy. Training on force samples similarly requires the evaluation of kernel derivatives, but w. r.t. those local environments. Although any partitioning of the total energy is arbitrary, our molecular total energy is physically meaningful in that it is related to the atomic force, thus being a measure for the deflection of every atom from its ground state.  \n\n![](images/0ff20e3569ddceef875592c5f684e30a4d1e5a98e958e203dd1bf5ed05ca4aa4.jpg)  \nFig. 4. Results of classical and PIMD simulations. The recently developed estimators based on perturbation theory were used to evaluate structural and electronic observables (30). (A) Comparison of the interatomic distance distributions, $\\begin{array}{r}{h(r)=\\langle\\frac{2}{N(N-1)}{\\sum}_{i<j}^{N}\\delta(r-||\\overrightarrow{r_{i}}-\\overrightarrow{r_{j}}^{*}||)\\rangle_{P,t},}\\end{array}$ obtained from GDML (blue line) and DFT (dashed red line) with classical MD (main frame), and PIMD (inset). a.u., arbitrary units. (B) Probability distribution of the dihedral angles (corresponding to carboxylic acid and ester functional groups) using a 20 ps time interval from a total PIMD trajectory of $200~{\\mathsf p}{\\mathsf s}$ .  \n\nWe first demonstrate the impact of the energy conservation constraint on a toy model that can be easily visualized. A nonconservative force model $\\hat{f}_{\\mathrm{F}}^{-}$ was trained alongside our GDML model $\\hat{\\b{f}}_{\\mathrm{F}}$ on a synthetic potential defined by a two-dimensional harmonic oscillator using the same samples, descriptor, and kernel.  \n\nWe were interested in a qualitative assessment of the prediction error that is introduced as a direct result of violating the law of energy conservation.  \n\nFor this, we uniquely decomposed our naïve estimate  \n\n$$\n\\hat{\\boldsymbol f}_{\\mathrm{F}}^{-}=-\\boldsymbol\\nabla E+\\boldsymbol\\nabla\\times\\boldsymbol A\n$$  \n\ninto a sum of a curl-free (conservative) and a divergence-free (solenoidal) vector field, according to the Helmholtz theorem (see Fig. 2) (25). This was achieved by subsampling $\\hat{f}_{\\mathrm{F}}^{-}$ on a regular grid and numerically projecting it onto the closest conservative vector field by solving Poisson’s equation (26)  \n\n$$\n-\\nabla^{2}E\\overset{!}{=}\\nabla\\hat{f}_{\\mathrm{F}}^{-}\n$$  \n\nwith Neumann boundary conditions. The remaining solenoidal field represents the systematic error made by the naïve estimator. Other than in this example, our GDML approach directly estimates the conservative vector field and does not require a costly numerical projection on a dense grid of regularly spaced samples.  \n\n# RESULTS  \n\nWe now proceed to evaluate the performance of the GDML approach by learning and then predicting AIMD trajectories for molecules, including benzene, uracil, naphthalene, aspirin, salicylic acid, malonaldehyde, ethanol, and toluene (see table S1 for details of these molecular data sets). These data sets range in size from $150\\mathrm{~k~}$ to nearly $^\\mathrm{~1~M~}$ conformational geometries with a resolution of 0.5 fs, although only a drastically reduced subset is necessary to train our energy and GDML predictors. The molecules have different sizes, and the molecular PESs exhibit different levels of complexity. The energy range across all data points within a set spans from 20 to $48\\ \\mathrm{kcal\\mol^{-1}}$ . Force components range from 266 to $570\\ensuremath{~\\mathrm{kcal~mol}^{-1}\\mathrm{\\AA}^{-1}}$ . The total energy and force labels for each data set were computed using the PBE $^+$ vdW-TS electronic structure method (27, 28).  \n\nThe GDML prediction results are contrasted with the output of a model that has been trained on energies. Both models use the same kernel and descriptor, but the hyperparameter search was performed individually to ensure optimal model selection. The GDML model for each data set was trained on ${\\sim}1000$ geometries, sampled uniformly according to the $\\mathrm{MD}@\\mathrm{DFT}$ trajectory energy distribution. For the energy model, we multiplied this amount by the number of atoms in one molecule times its three spatial degrees of freedom. This configuration yields equal kernel sizes for both models and therefore equal levels of complexity in terms of the optimization problem. We compare the models on the basis of the required number of samples (Fig. 3A) to achieve a force prediction accuracy of $1\\ \\mathrm{kcal\\mol^{-1}\\ \\AA^{-1}}$ . Furthermore, the prediction accuracy of the force and energy estimates for fully converged models (w.r.t. number of samples) (Fig. 3, B and C) are judged on the basis of the mean absolute error (MAE) and root mean square error performance measures.  \n\nIt can be seen in Fig. 3A that the GDML model achieves a force accuracy of $1\\mathrm{kcalmol}^{-1}\\mathring{\\mathrm{A}}^{-1}$ using only ${\\sim}1000$ samples from different data sets. Conversely, a pure energy-based model would require up to two orders of magnitude more samples to achieve a similar accuracy. The superior performance of the GDML model cannot be simply attributed to the greater information content of force samples. We compare our results to those of a naïve force model along the lines of the toy example shown in Fig. 2 (see tables S1 and S3 for details on the prediction accuracy of both models). The naïve force model is nonconservative but identical to the GDML model in all other aspects. Note that its performance deteriorates significantly on all data sets compared to the full GDML model (see the Supplementary Materials for details). We note here that we used DFT calculations, but any other high-level quantum chemistry approach could have been used to calculate forces for 1000 conformational geometries. This allows AIMD simulations to be carried out at the speed of ML models with the accuracy of correlated quantum chemistry calculations.  \n\nIt is noticeable that the GDML model at convergence (w.r.t. number of samples) yields higher accuracy for forces than an equivalent energybased model (see Fig. 3B). Here, we should remark that the energybased model trained on a very large data set can reduce the energy error to below $0.1\\mathrm{kcalmol}^{-1}$ , whereas the GDML energy error remains at $0.2\\mathrm{kcalmol}^{-1}$ for ${\\sim}1000$ training samples (see Fig. 3C). However, these errors are already significantly below thermal fluctuations $(k_{B}T)$ at room temperature $({\\sim}0.6\\mathrm{kcalmol^{-1}}.$ ), indicating that the GDML model provides an excellent description of both energies and forces, fully preserves their consistency, and reduces the complexity of the ML model. These are all desirable features of models that combine rigorous physical laws with the power of data-driven machines.  \n\nThe ultimate test of any force field model is to establish its aptitude to predict statistical averages and fluctuations using MD simulations. The quantitative performance of the GDML model is demonstrated in  \n\nFig. 4 for classical and quantum MD simulations of aspirin at $T=300\\mathrm{K}.$ . Figure 4A shows a comparison of interatomic distance distributions, $h(r)$ , from $\\mathrm{MD}@\\mathrm{DFT}$ and $\\mathbf{MD}@\\mathbf{GDML}$ . Overall, we observe a quantitative agreement in $h(r)$ between DFT and GDML simulations. The small differences in the distance range between 4.3 and $4.7\\textup{\\AA}$ result from slightly higher energy barriers of the GDML model in the pathway from A to B corresponding to the collective motions of the carboxylic acid and ester groups in aspirin. These differences vanish once the quantum nature of the nuclei is introduced in the PIMD simulations (29). In addition, long–time scale simulations are required to completely understand the dynamics of molecular systems. Figure 4B shows the probability distribution of the fluctuations of dihedral angles of carboxylic acid and ester groups in aspirin. This plot shows the existence of two main metastable configurations A and B and a shortlived configuration C, illustrating the nontrivial dynamics captured by the GDML model. Finally, we remark that a similarly good performance as for aspirin is also observed for the other seven molecules shown in Fig. 3. The efficiency of the GDML model (which is three orders of magnitude faster than DFT) should enable long–time scale PIMD simulations to obtain converged thermodynamic properties of intermediate-sized molecules with the accuracy and transferability of high-level ab initio methods.  \n\nIn summary, the developed GDML model allows the construction of complex multidimensional PES by combining rigorous physical laws with data-driven ML techniques. In addition to the presented successful applications to the model systems and intermediate-sized molecules, our work can be further developed in several directions, including scaling with system size and complexity, incorporating additional physical priors, describing reaction pathways, and enabling seamless coupling between GDML and ab initio calculations.  \n\n# SUPPLEMENTARY MATERIALS  \n\nSupplementary material for this article is available at http://advances.sciencemag.org/cgi/ content/full/3/5/e1603015/DC1   \nsection S1. Noise amplification by differentiation   \nsection S2. Vector-valued kernel learning   \nsection S3. Descriptors   \nsection S4. Model analysis   \nsection S5. Details of the PIMD simulation   \nfig. S1. The accuracy of the GDML model (in terms of the MAE) as a function of training set size: Chemical accuracy of less than 1 kcal/mol is already achieved for small training sets.   \nfig. S2. Predicting energies and forces for consecutive time steps of an MD simulation of uracil at 500 K. table S1. Properties of MD data sets that were used for numerical testing.   \ntable S2. GDML prediction accuracy for interatomic forces and total energies for all data sets. table S3. Accuracy of the naïve force predictor.   \ntable S4. Accuracy of the converged energy-based predictor.   \nReferences (31–36)  \n\n# REFERENCES AND NOTES  \n\n1. J. Behler, M. Parrinello, Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 98, 146401 (2007).   \n2. J. 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Csányi, Gaussian approximation potentials: A brief tutorial introduction. Int. J. Quantum Chem. 115, 1051–1057 (2015).   \n9. S. De, A. P. Bartók, G. Csányi, M. Ceriotti, Comparing molecules and solids across structural and alchemical space. Phys. Chem. Chem. Phys. 18, 13754–13769 (2016).   \n10. M. Rupp, A. Tkatchenko, K.-R. Müller, O. A. von Lilienfeld, Fast and accurate modeling of molecular atomization energies with machine learning. Phys. Rev. Lett. 108, 058301 (2012).   \n11. G. Montavon, M. Rupp, V. Gobre, A. Vazquez-Mayagoitia, K. Hansen, A. Tkatchenko, K.-R. Müller, O. A. von Lilienfeld, Machine learning of molecular electronic properties in chemical compound space. New J. Phys. 15, 095003 (2013).   \n12. K. Hansen, G. Montavon, F. Biegler, S. Fazli, M. Rupp, M. Scheffler, O. A. von Lilienfeld, A. Tkatchenko, K.-R. Müller, Assessment and validation of machine learning methods for predicting molecular atomization energies. J. Chem. Theory Comput. 9,   \n3404–3419 (2013).   \n13. K. Hansen, F. Biegler, R. Ramakrishnan, W. Pronobis, O. A. von Lilienfeld, K.-R. Müller, A. Tkatchenko, Machine learning predictions of molecular properties: Accurate many-body potentials and nonlocality in chemical space. J. Phys. Chem. Lett. 6, 2326–2331 (2015).   \n14. M. Rupp, R. Ramakrishnan, O. A. von Lilienfeld, Machine learning for quantum mechanical properties of atoms in molecules. J. Phys. Chem. Lett. 6, 3309–3313 (2015).   \n15. V. Botu, R. Ramprasad, Learning scheme to predict atomic forces and accelerate materials simulations. Phys. Rev. B 92, 094306 (2015).   \n16. M. Hirn, N. Poilvert, S. Mallat, Quantum energy regression using scattering transforms. CoRR arXiv:1502.02077 (2015).   \n17. J. Behler, Perspective: Machine learning potentials for atomistic simulations. J. Chem. Phys. 145, 170901 (2016).   \n18. Z. Li, J. R. Kermode, A. De Vita, Molecular dynamics with on-the-fly machine learning of quantum-mechanical forces. 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Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. Angew. Math. 1858, 25–55 (2009).   \n26. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, ed. 3, 2007).   \n27. J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).   \n28. A. Tkatchenko, M. Scheffler, Accurate molecular Van Der Waals interactions from ground-state electron density and free-atom reference data. Phys. Rev. Lett. 102, 073005 (2009).   \n29. M. Ceriotti, J. More, D. E. Manolopoulos, i-PI: A Python interface for ab initio path integral molecular dynamics simulations. Comput. Phys. Commun. 185, 1019–1026 (2014).   \n30. I. Poltavsky, A. Tkatchenko, Modeling quantum nuclei with perturbed path integral molecular dynamics. Chem. Sci. 7, 1368–1372 (2016).   \n31. A. J. Smola, B. Schölkopf, Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (MIT Press, 2001).   \n32. J. C. Snyder, M. Rupp, K. Hansen, K.-R. Müller, K. Burke, Finding density functionals with machine learning. Phys. Rev. Lett. 108, 253002 (2012).   \n33. J. C. Snyder, M. Rupp, K.-R. Müller, K. Burke, Nonlinear gradient denoising: Finding accurate extrema from inaccurate functional derivatives. Int. J. Quantum Chem. 115,   \n1102–1114 (2015).   \n34. B. Schölkopf, A. Smola, K.-R. Müller, Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299 (1998).   \n35. B. Schölkopf, S. Mika, C. J. C. Burges, P. Knirsch, K.-R. Müller, G. Ratsch, A. J. Smola, Input space versus feature space in kernel-based methods. IEEE Trans. Neural Netw. Learn. Syst. 10,   \n1000–1017 (1999).   \n36. K.-R. Müller, S. Mika, G. Rätsch, K. Tsuda, B. Schölkopf, An introduction to kernel-based learning algorithms. IEEE Trans. Neural Netw. Learn. Syst. 12, 181–201 (2001).  \n\n# Acknowledgments  \n\nFunding: S.C., A.T., and K.-R.M. thank the Deutsche Forschungsgemeinschaft (project MU 987/ 20-1) for funding this work. A.T. is funded by the European Research Council with ERC-CoG grant BeStMo. K.-R.M. gratefully acknowledges the BK21 program funded by the Korean National Research Foundation grant (no. 2012-005741). Additional support was provided by the Federal Ministry of Education and Research (BMBF) for the Berlin Big Data Center BBDC (01IS14013A). Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics, which is supported by the NSF. Author contributions: S.C. conceived, constructed, and analyzed the GDML models. S.C., A.T., and K.-R.M. developed the theory and designed the analyses. H.E.S. and I.P. performed the DFT calculations and MD simulations. H.E.S. helped with the analyses. K.T.S. and A.T. helped with the figures. A.T., S.C., and K.-R.M. wrote the paper with contributions from other authors. All authors discussed the results and commented on the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data sets used in this work are available at http://quantum-machine.org/datasets/. Additional data related to this paper may be requested from the authors.  \n\nSubmitted 1 December 2016   \nAccepted 7 March 2017   \nPublished 5 May 2017   \n10.1126/sciadv.1603015  \n\nCitation: S. Chmiela, A. Tkatchenko, H. E. Sauceda, I. Poltavsky, K. T. Schütt, K.-R. Müller, Machine learning of accurate energy-conserving molecular force fields. Sci. Adv. 3, e1603015 (2017).  \n\n# ScienceAdvances  \n\n# Machine learning of accurate energy-conserving molecular force fields  \n\nStefan Chmiela, Alexandre Tkatchenko, Huziel E. Sauceda, Igor Poltavsky, Kristof T. Schütt and Klaus-Robert Müller  \n\nSci Adv 3 (5), e1603015. DOI: 10.1126/sciadv.1603015  \n\nARTICLE TOOLS  \n\nSUPPLEMENTARY MATERIALS  \n\nREFERENCES  \n\nThis article cites 30 articles, 0 of which you can access for free http://advances.sciencemag.org/content/3/5/e1603015#BIBL  \n\nPERMISSIONS  \n\nUse of this article is subject to the Terms of Service  ",
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  {
    "id": "10.1038_ncomms14101",
    "DOI": "10.1038/ncomms14101",
    "DOI Link": "http://dx.doi.org/10.1038/ncomms14101",
    "Relative Dir Path": "mds/10.1038_ncomms14101",
    "Article Title": "Intragranular cracking as a critical barrier for high-voltage usage of layer-structured cathode for lithium-ion batteries",
    "Authors": "Yan, PF; Zheng, JM; Gu, M; Xiao, J; Zhang, JG; Wang, CM",
    "Source Title": "NATURE COMMUNICATIONS",
    "Abstract": "LiNi1/3Mn1/3Co1/3O2-layered cathode is often fabricated in the form of secondary particles, consisting of densely packed primary particles. This offers advantages for high energy density and alleviation of cathode side reactions/corrosions, but introduces drawbacks such as intergranular cracking. Here, we report unexpected observations on the nucleation and growth of intragranular cracks in a commercial LiNi1/3Mn1/3Co1/3O2 cathode by using advanced scanning transmission electron microscopy. We find the formation of the intragranular cracks is directly associated with high-voltage cycling, an electrochemically driven and diffusion-controlled process. The intragranular cracks are noticed to be characteristically initiated from the grain interior, a consequence of a dislocation-based crack incubation mechanism. This observation is in sharp contrast with general theoretical models, predicting the initiation of intragranular cracks from grain boundaries or particle surfaces. Our study emphasizes that maintaining structural stability is the key step towards high-voltage operation of layered-cathode materials.",
    "Times Cited, WoS Core": 804,
    "Times Cited, All Databases": 864,
    "Publication Year": 2017,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000392095500001",
    "Markdown": "# Intragranular cracking as a critical barrier for high-voltage usage of layer-structured cathode for lithium-ion batteries  \n\nPengfei $\\mathsf{Y a n}^{1,\\star}$ , Jianming Zheng2,\\*, Meng $\\mathsf{G u}^{1}$ , Jie Xiao2, Ji-Guang Zhang2 & Chong-Min Wang1 $\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/3}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ -layered cathode is often fabricated in the form of secondary particles, consisting of densely packed primary particles. This offers advantages for high energy density and alleviation of cathode side reactions/corrosions, but introduces drawbacks such as intergranular cracking. Here, we report unexpected observations on the nucleation and growth of intragranular cracks in a commercial $\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/3}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ cathode by using advanced scanning transmission electron microscopy. We find the formation of the intragranular cracks is directly associated with high-voltage cycling, an electrochemically driven and diffusion-controlled process. The intragranular cracks are noticed to be characteristically initiated from the grain interior, a consequence of a dislocation-based crack incubation mechanism. This observation is in sharp contrast with general theoretical models, predicting the initiation of intragranular cracks from grain boundaries or particle surfaces. Our study emphasizes that maintaining structural stability is the key step towards high-voltage operation of layered-cathode materials.  \n\nExmpelcohriangi lmitshihuams -lion1 1b6eatetneray  aLcItBiv) erlescetarrocdhe odepigcrafdoar itohne degradation allows us to design better electrode materials. In the case of layered transition metal (TM) oxide cathode degradation, three mechanisms have been identified2–5,12,17,18: (1) Layer to spinel/rock salt phase transformation, which is characteristically initiated from the individual particle surface and gradually propagated inwards with battery cycling. (2) Side reactions between the cathode and electrolyte, leading to electrolyte decomposition and passivation of the solid electrode. (3) Corrosion and dissolution of the cathode materials in the electrolyte. These findings lead to the application of coating techniques and other surface treatments to stabilize vulnerable surfaces on the cathode materials. Such coating and surface treatments have been frequently verified as effective methods for improving cathode cycling stability19–28. Besides chemical instability, another degradation mechanism is associated with the volume change of the material upon lithium (Li) ion extraction and reinsertion. Non-uniform accommodation of such a volume change will generate stress, which can lead to mechanical failure29. In fact, intergranular crack formation is one of the most well-known material degradation mechanisms29–35.  \n\nDuring the charge process of layered TM oxides, Li ions are extracted from the lattice, which usually causes lattice expansion along the $\\mathbf{\\Psi}_{c}$ direction and shrinkage along the $^a$ and $b$ directions29,36–38, which is reversed upon discharging. This type of lattice expansion and shrinkage is generally termed as lattice breathing, which has been theoretically and experimentally verified. For example, when $\\mathrm{LiNi_{1/3}M n_{1/3}C o_{1/3}O_{2}}$ (NMC333) is delithiated to $\\mathrm{Li}_{0.5}\\mathrm{Ni}_{1/3}\\mathrm{Mn}_{1/3}\\mathrm{Co}_{1/3}\\mathrm{O}_{2}.$ , Yoon et al.37 found that the lattice could expand $2.0\\%$ along the $\\mathbf{\\Psi}_{c}$ direction and shrink $1.4\\%$ along the $^a$ direction, inducing significant strain for oxides with ionic bonds.  \n\n![](images/2774bf4e5192074bbe16dd5dee1cda648e0706d8a46770d19c401e16379b6077.jpg)  \nFigure 1 | Electrochemical performance and observations of fracture. (a) Specific capacity as a function of cycle number, revealing the $\\mathsf{L i}/\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/}$ ${\\phantom{}_{3}}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ half cell’s capacity fading has strong dependence on the high cutoff voltages, (b–d) charge/discharge profiles of $\\mathsf{L i}/\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/3}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ half cells at different high cutoff voltages, and (e–g) low magnification HAADF images of $\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/3}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ after 100 cycles at different high cutoff voltages. The red arrows indicate voids and the yellow arrows in $\\pmb{\\mathscr{g}}$ indicate intragranular cracks. Scale bars, $500\\mathsf{n m}$ $(\\pmb{\\theta}\\mathbf{-}\\pmb{\\theta})$ .  \n\nTo increase the volumetric energy density of the electrode, the packing density of the active electrode component should be increased. One way to accomplish this in commercial LIB cathodes is to use primary particles to form densely packed secondary particles. However, such secondary particles always generate intergranular cracks during battery charge/discharge cycling, due to the anisotropic expansion and shrinkage of each primary particle31,32,39. Such strain-induced cracking has been considered to be one of the major degradation mechanisms for the cathode for the following reasons: (1) Cracks can result in poor grain-to-grain connections, leading to poor electrical conductivity and even loss of active materials due to fragmentation; (2) Cracks create fresh surfaces that will be exposed to electrolytes and generate new sites for surface phase transformation, corrosion and side reactions, consequently accelerating cell degradation.  \n\nBesides intergranular cracks, intragranular cracks were also observed in several cathode materials after prolonged cycling39–41. For example, Chen et $a l.^{40}$ found cracks in the bc planes of $\\mathrm{LiFePO_{4}}.$ , Wang et al.41 noted cracks in $\\mathrm{LiCoO}_{2}$ particles, and Kim et al.39 observed cracks in $\\mathrm{LiNi_{0.6}M n_{0.2}C o_{0.2}O_{2}}$ after 150 cycles at $60^{\\circ}\\mathrm{C}.$ . Compared with intergranular cracks, intragranular cracks are smaller in size but much higher in density, and thus, they can generate many more fresh surfaces that will be exposed to electrolytes. Moreover, intragranular cracking is not only a mechanical failure but also more likely to be a structural degradation under severe electrochemical conditions. Therefore, the previously proposed effective design concepts (such as surface coating) for preventing intergranular cracking31,39,42,43 may not solve the intragranular cracking problem. Mitigation of intragranular cracking requires a stable structural framework of the cathode material and careful controls of cycle conditions. A systematic investigation on intragranular cracking in cathode materials is still lacking.  \n\nIn this work, we report detailed observations on the cracking phenomenon in NMC333 layered-cathode materials by using advanced scanning transmission electron microscopy (STEM). In particular, the intragranular cracking process is comprehensively investigated. We find that the density of intragranular cracks in NMC333 cathodes abruptly increases when cycled at a high cutoff voltage of $4.7\\mathrm{V}$ . In contrast expectations, we also observe the intragranular cracks to actually initiate from the grain interior, which is in sharp contrast with general theoretical models predicting the surface or grain boundary to be the preferred sites for intragranular crack initiation $^{42,44-46}$ . We also verify that the edge dislocation core can assist the incubation of intragranular cracks, and that intragranular cracking is an electrochemically driven and diffusion-controlled process, mimicking the classic model of slow crack growth during fatigue process of materials.  \n\n# Results  \n\nHigh cutoff voltage cycling induced intragranular cracking. The performance of NMC333 cathode electrodes in Li/NMC333 half-cells cycled at different voltage ranges, that is, $2.7\\sim4.2\\:\\mathrm{V}$ , $2.7\\sim4.5\\:\\mathrm{V}$ , and $2.7\\sim4.7\\mathrm{V}$ , are shown in Fig. 1a, and the corresponding charge/discharge voltage profile evolutions are shown in Fig. 1b–d, respectively. Electrochemical data indicate that the cycling stability of the NMC333 cathode shows strong dependence on the charge cutoff voltages that are applied for battery cycling. Generally, the higher the charge cutoff voltage, the faster degradation of the battery performance. When the battery was cycled at a low charge cutoff voltage of $4.2\\mathrm{V}$ , the NMC333 shows excellent cycling stability along with very limited voltage decay. However, with the increase of charge cutoff voltage, the NMC333 shows obvious voltage fading and capacity decay. Particularly, serious voltage decay and capacity fading occur when cycling at $4.7\\mathrm{V}$ .  \n\nBecause excessive Li metal was used as anode for these three cells, it is believed that their performance difference should be mainly associated with instability of the cathode and electrolyte. It is known that a higher charge voltage can result in aggravated degradation of the cathode and electrolyte due to the side reactions between the cathode and electrolyte, formation of a thicker phase transformation layer on the surface of the cathode, and severe surface corrosion of the cathode18,47. For the densely packed secondary particles, the formation of intergranular cracks also contributes to the degradation of the cell. The general features of these intergranular cracks are representatively identified in the cross-sectional scanning electron microscopy (SEM) images in Fig. $\\operatorname{le-g},$ Fig. 2a,b and Supplementary Fig. 1. Comparing the pristine and cycled samples, it is obvious that after 100 cycles, the samples cycled at different high cutoff voltages of 4.2, 4.5, and $4.7\\mathrm{V}$ exhibit no significant differences in terms of the intergranular cracking features.  \n\nIntragranular cracks are one of the significant differences for the samples cycled at different charge cutoff voltages. In the sample cycled at $4.7\\mathrm{V}$ , the number of intragranular cracks were significantly higher than the samples cycled at 4.2 and $4.5\\mathrm{V}$ , as shown representatively in Fig. 1e–g. Intragranular cracks are hardly seen in the samples cycled at 4.2 and $4.5\\mathrm{V}$ , but are universally observed in the sample cycled at $4.7\\mathrm{V}$ . In the sample cycled at $4.7\\mathrm{V}$ , it would be expected that the intragranular cracking characteristics would substantially contribute to the faster capacity degradation as compared with those cycled at 4.2 and $4.5\\mathrm{V}$ . The abrupt increase of the density of intragranular cracks also indicates the high cutoff cycle voltage is the direct driving force for intragranular crack generation, suggesting a critical cycle voltage between 4.5 and $\\mathrm{\\bar{4.7V}}$ for initializing the intragranular cracks in NMC333.  \n\n![](images/aea9d7a0de532cc42adc8d4203904cc73e2a5a85efc12de8c779de99f5872c77.jpg)  \nFigure 2 | Intergranular and intragranular cracks. Cross-sectional SEM images of secondary particles from (a) the pristine material and $(\\pmb{6})$ the cycled one (100 cycles at the high cutoff voltage of $4.7\\vee$ . (c) and (d) are HAADF images from cycled $\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/3}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ cathode particles, showing intragranular cracks along (001) plane. The yellow arrows indicate real cracks and the pink arrows indicate incubation cracks. Scale bars, $5\\upmu\\mathrm{m}$ $(\\mathbf{a},\\mathbf{b})$ ; $50\\mathsf{n m}$ (c); and $10\\mathsf{n m}$ (d).  \n\n![](images/c261fe572360fd58c46529b4724f5c38e9a0aba4c75c398631ba46dfc04abcfb.jpg)  \nFigure 3 | Lattice images of premature cracks. Each pair of HAADF and ABF images are taken simultaneously. (a,b) [010] axis. (c) The corresponding lattice model. (d,e) A crack tip; (f) The corresponding model. (g,h) [1–10] axis. (i) Strain map at Mode I crack tip, which matches the strain contrast in h. Scale bars, $2{\\mathsf{n m}}$ .  \n\nStructural features of the intragranular cracks. SEM and STEM observations reveal the details of these intergranular cracks as marked by the yellow arrows in Figs 1g and 2c, d. As the intragranular cracks are mostly observed in the sample cycled at $4.7\\mathrm{V}$ , we focused our effort on the NMC333 sample that was subjected to 100 cycles at $4.7\\mathrm{V}$ in order to reveal the detailed structure of the intragranular cracks and understand their formation mechanism.  \n\nTwo types of intragranular cracks can be uniquely identified. One type possesses the classical term of crack, which is featured by two free surfaces as indicated by the yellow arrows in the STEM high-angle annular dark-field (HAADF) images of Fig. 2c, d. The two free surfaces appear to be parallel along the whole crack except at the very tip, which is markedly different from a wedge-shaped crack formed by fast extension of crack under stress. Furthermore, cracks are predominantly parallel to (003) planes in the layer structure. These morphological features of the cracks are associated with their formation process, which will be discussed in detail in the subsequent sections.  \n\nThe other type of crack appears as narrow, dark strips when observed under STEM-HAADF imaging as indicated by the pink arrows in Fig. 2c, d. The dark strips are all parallel to the (003) planes (the layers) and distribute randomly with various spacing among the strips. The closest distance between two strips is a single layer of TM. As shown in the inset of Fig. 2c, the (003) plane spacing is $0.48\\mathrm{nm}$ , while the dark contrasted strip corresponds to a widened TM plane spacing, ranging from 0.6 to $0.8\\mathrm{nm}$ . Therefore, these dark contrasted strips appear to be formed by a parallel splitting of two adjacent TM layers, leading to a wider $\\boldsymbol{c}$ plane spacing.  \n\nIt should be noted that, not always, but for some cases, the dark contrasted strip is spatially connected to the real crack as representatively shown in Fig. 2d. This observation likely indicates that the dark contrasted strip is a premature crack. The reason we term the dark contrasted strip as a premature crack is because it does not have two free surfaces. With the continued cycling of the battery, the premature crack will further develop into a real crack.  \n\nIt is interesting to note that the dark strip contrasted region still possesses internal structural features. As shown in Fig. 3a,b and Supplementary Fig. 2, high-resolution STEM-HAADF and annular bright-field (ABF) images were simultaneously collected from one premature crack. As verified by the ABF image, the dark contrasted strip that appeared in the STEM-HAADF image is actually not empty. As denoted by the pink arrow in Fig. 3b, there are some black dots that appeared with a rock-salt-like structure. Simulated HAADF/ABF images are shown in Supplementary Fig. 3 to support our interpretation. Thus, the dark contrasted strip, in fact, still contains an internal structure with material of low density. Its crystal model is illustrated in Fig. 3c. The very tip of the dark contrasted strip shows bending of the TM atomic row, forming a ‘V’ shape that looks just as the configuration of a crack tip. Fig. 3d, e are imaged from [010] axis and Fig. 3g, h are from [1–10] axis. The crystal model of the dark contrasted strip is illustrated in Fig. 3f. The ABF image shown in Fig. 3h even shows a strain contour at the tip, which is very similar to the strain contour of a real crack generated by tensile stress (Mode I crack, Fig. 3i). These structural features clearly demonstrate that the dark contrasted strips are premature cracks, which were formed by splitting the two neighbouring TM slabs and propagated along (003) planes.  \n\nAnother significant feature of the intragranular cracking is that a large fraction of the intragranular cracks terminate within the grain interior, as representatively shown in Fig. 4, for which the red arrows highlight the intragranular end-to-end cracks that were fully terminated within the grain interior. These observations indicate that the intragranular cracks are initiated from the grain interior, which is in contrast with cracking models that predict the surface or grain boundary should be the preferred crack initiation site $^{42,44-46}$ . However, based on thermal analogy analysis, Kalnaus et al.48 predicted cracking may initiate from the centre of the particle. Operando X-ray diffraction measurement has indicated the inhomogeneous lithiation/delithiation within a single cathode crystal49. Therefore, the variation of the $\\mathbf{\\Psi}_{c}$ plane spacing at different delithiation states within a single grain can lead to a complex strain pattern within the grain interior. Figure 4e gives a general illustration on the intragranular crack formation process under tensile stress. Inhomogenous Li distribution is believed to be the direct cause of such tensile stress. This internal cracking model also matches our proposed Mode I crack mechanism based on the observations on the crack tips in Fig. 3.  \n\nGeneration of dislocations in the primary particles. Within the densely packed secondary particles, dislocation activity in the primary particle is another unique feature. High density of dislocations in both pristine and cycled samples were observed based on STEM-BF imaging, as representatively shown in Fig. $^{5\\mathrm{a},\\mathrm{b},}$ for which the blue arrows highlight the dislocations. The dislocation density is in the range of $\\mathrm{\\check{10}}^{11}\\mathrm{\\check{c}m}^{-2}$ . The observation of high-density dislocation in the primary particles is in marked contrast to the case of using nano-sized particles to assemble the battery electrode, where dislocation activity is hardly visible within the layer-structured particles. The high-density dislocations in the primary particle, as indicated in Fig. $^{5\\mathrm{a},\\mathrm{b}}$ , is the consequence of the formation of secondary particles by packing the smaller primary particles. Within the densely packed secondary particles, the primary particles are randomly oriented and in direct contact with their adjacent primary particles. Therefore, thermally (material synthesis process) and electrochemically (battery cycling) induced strain between neighbouring primary particles cannot be concordantly accommodated, which can inherently initiate dislocation in the primary particles. Figure 5c is a high-resolution STEM-HAADF image showing an end-on edge dislocation with Burger’s vector $\\stackrel{\\rightharpoonup}{b}=1\\big/3[1\\bar{1}01]\\approx0.5\\mathrm{nm}$ , which is a whole dislocation as shown in Fig. 5e. At the dislocation core area, due to the lattice mismatch and distortion, strain can build up accordingly. Our geometric phase analysis $\\left(\\mathrm{GPA}\\right)^{50}$ on Fig. 5c is shown in Fig. 5d. It can be seen in this out-of-plane strain map $(\\varepsilon_{y y})$ that the left side with the extra plane is under compressive lattice strain, while the right side is under tensile strain.  \n\n![](images/d1d4b08f02eb9fe41263db6fc1d810593e471fcd42245c4a759bca44c9fc72a0.jpg)  \nFigure 4 | Intragranular cracks in $\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/3}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ (NMC333). The NMC333 particles are cycled 100 times with the high cutoff voltage of $4.7\\mathrm{V}.$ In a–d, the arrows highlight the crack tips that are terminated in grain interior, and (e) a schematic diagram showing crack formation in the grain interior due to tensile stress. Scale bars, $100\\mathsf{n m}$ (a,b); $200\\mathsf{n m}(\\pmb{\\mathsf{c}});$ and $50\\mathsf{n m}$ (d).  \n\nThe effects of dislocation on the battery properties can be evaluated from the following two aspects: firstly, the role of the dislocation itself on the ionic transport characteristics; and secondly, the evolution of dislocations and their effect on the structural stability of the material. In terms of dislocation itself, it is known that the dislocation core can act as a fast channel for ionic transport49. At the same time, the strain field associated with the dislocation can affect the active ion distribution and transport characteristics in the lattice. As shown in the GPA $\\varepsilon_{y y}$ strain map in Fig. 5d, the lattice strain field introduced by the edge dislocation goes well beyond the dislocation core (indicated by the yellow arrow in Fig. 5c); in fact, previous Bragg coherent diffraction imaging49 shows the strain field of an edge dislocation can reach more than $100\\mathrm{nm}$ . Therefore, the high-density dislocations in the primary particles will definitely affect, either detrimentally or beneficially, the properties of cathode materials41,49,51. To our best knowledge, this is the first report on discovering the high-density dislocations in layered cathodes when the primary particles are packed as dense secondary particles.  \n\n# Discussion  \n\nThere have been many studies on the cracking mechanisms of cathode materials for LIBs32,33,44–46,51,52. However, these research efforts are mostly based on theoretical modelling of the stress-strain evolution. Fundamental understanding on the crack incubation is still far from clear. On the basis of what we have observed, the dark contrast strip (the premature crack) is the predecessor of the intragranular crack. The transition from the dark contrasted strip to the crack is a diffusion-controlled process, which is in essence an electrochemical driving process. Now, the key question is the origin of the dark contrasted strip or the premature cracks. On the basis of intensive observation using high-resolution STEM-HAADF imaging, we found that there exists a close correlation between edge dislocations and premature crack, as typically shown in Fig. 6. Fig. 6a and b show the nucleation of a premature crack at the dislocation core as indicated by the red arrows. Fig. 6d–f shows the association of an edge dislocation with premature cracks. These observations indicate that edge dislocation core can act as the nucleation site for crack incubation. From an energy point of view, Li and O ions will be preferentially removed from the tensile part of the dislocation core region to release strain. Kinetically, the dislocation core is a fast diffusion pathway. Therefore, the nucleation of the premature crack from the dislocation core is an electrochemically driven, but diffusion limited, process as schematically shown in Fig. 6c, which is similar to the fatigueinduced cracks in the slip band53. Our proposed mechanism is also supported by previous theoretical calculation work done by Huang et al.51 who proposed dislocation-based cracking models.  \n\n![](images/e447e48dede564bb0489c3b9b45657be8e29ffa40c21f52e5654e3fcbe4c8002.jpg)  \nFigure 5 | Dislocations in both pristine and cycled $\\mathsf{L i N i}_{1/3}\\mathsf{M n}_{1/3}\\mathsf{C o}_{1/3}\\mathsf{O}_{2}$ (NMC333). High density of dislocations are shown in the bright-field images of (a) pristine and (b) cycled NMC333 (after 100 cycles at the high cutoff voltage of 4.7 V). (c) A HAADF image showing an end-on edge dislocation in pristine NMC333, (d) is the corresponding strain map by GPA and (e) shows the dislocation model of (c). Scale bars, $200\\mathsf{n m}$ (a,b); 5 nm (c,d).  \n\nWe have drawn the conclusion that high-voltage cycling is the direct driving force for intragranular crack generation as evidenced by the drastic increase of intragranular crack density in the $4.7\\mathrm{V}$ sample (Fig. 1g). Higher cycle voltage will result in deeper Li-ion extraction, which, on one hand, can aggravate structure instability, and on the other hand, can amplify the internal strain within a grain. Therefore, when the cycle voltage exceeds a critical value, in this case, some point between 4.5 and  \n\n$4.7\\mathrm{V}_{:}$ intragranular cracks can be massively initiated as shown in Fig. 7a. As dislocation can act as a nucleation site for incubation of intragranular crack, we schematically illustrate in Fig. 7b the overall formation process of intragranular cracking based on a dislocation-nucleation mechanism. One of the fundamental questions is how the battery cycling rate contributes to the generation of intragranular cracks. To test the rate effect, the battery was cycled at 1C (as compared with 0.1C) at a cutoff voltage of 4.2 and $4.5\\mathrm{V}$ . No premature cracks were identified at these samples. The results clearly point out the effect of cycle voltage on the intragranular crack formation.  \n\nThe present observation has relevant implications for electrode design and battery operation. Although many surface coating methods have been used to minimize the surface-initiated structure degradation and cation dissolution in layer-structured cathode materials, these methods cannot be used to prevent the intrinsic, intragranular cracks that were initiated inside the primary particles and aggravated by the high-voltage charge process. During the Li-ion extraction and insertion processes, the lattice of crystal will be subject to change (either expansion or contraction). Although these changes are reversible within the certain limit, too large a lattice change, such as those induced by a high-voltage charge process, will lead to the irreversible formation of the dislocations and cracks, which will in turn be detrimental for the performance of the battery, as what has been observed in the present case. Therefore, on one hand, for the currently available NMC materials, the charge voltage has to be well controlled to minimize the electrochemically induced intragranular cracks. On the other hand, in order to push the NMC-layerbased materials for high-voltage applications, efforts have to be made to adjust the chemistry and structure of the material such that it can alleviate the internal grain strain, cause minimal Li distribution inhomogeneity, and retain a stable lattice during charge and discharge cycling.  \n\n![](images/8f32e32a9e46ec6c306a95fab02a5dbc9891a9daeed7da25c102065d398c87ab.jpg)  \nFigure 6 | Dislocation associated with cracks in cycled $\\mathbf{LiNi_{1/3}M n_{1/3}C o_{1/}}$ $30_{2}.$ (a,b) are the early incubation stages, showing vacancy condensation at dislocation core and (c) is the corresponding model. (d–f) show dislocations associated with cracks. Red arrows indicate crack tips. Scale bars, $2{\\mathsf{n m}}$ ; except f $(5\\mathsf{n m})$ .  \n\n# Methods  \n\nCathode material and cell test. NMC333 pristine electrode laminates were provided by the Cell Analysis, Modelling, and Prototyping Facility at Argonne National Laboratory (pristine powders are commercially available and are manufactured by TODA KOGYO Company, Japan). The electrode laminates were punched into electrode disks that were $\\%$ inches in diameter and dried at $75^{\\circ}\\mathrm{C}$ overnight under a vacuum. Coin cells were assembled with the dried cathode electrode, metallic lithium foil as an anode electrode, Celgard2500 polyethylene (PE) membrane as separator, and 1 M lithium hexafluorophosphate $\\left(\\mathrm{LiPF}_{6}\\right)$ 1 dissolved in ethylene carbonate and dimethyl carbonate (1:2 in volume) as an electrolyte in an argon-filled MBraun glovebox. All the cathode electrodes were cycled at $\\mathrm{C}/10$ rate $\\mathsf{\\check{(}1C=180\\ m A\\ g^{-1}},$ ) in the voltage range of 2.7–4.2 V, 2.7–4.5 V, $2.7{-}4.7\\:\\mathrm{V}$ and $2.7–4.8\\mathrm{V}$ .  \n\nMicrostructure characterization and simulation. FIB/SEM imaging and TEM specimen preparation by FIB lift out were conducted on a FEI Helios DualBeam Focused Ion Beam operating at $2{-}30\\mathrm{kV}$ . Firstly, $1.2\\upmu\\mathrm{m}$ thick Pt layer $(200\\mathrm{nm}$ e-beam deposition followed by $1\\upmu\\mathrm{m}$ ion beam deposition) was deposited on the particles to be lifted out to avoid Ga ion beam damage. After lift out, the specimen was thinned to $200\\mathrm{nm}$ using $30\\mathrm{kV}$ Ga ion beam. A final polishing was performed using $2\\mathrm{kV}$ Ga ion to remove the surface damage layer and further thinning to electron transparency. After a $2\\mathrm{kV}$ Ga ion polish, the surface damage layer was believed to be $<1\\mathrm{nm}$ (ref. 54). The FIB-prepared NMC333 samples were investigated by using a JEOL JEM-ARM200CF microscope at $200\\mathrm{kV}$ . This microscope is equipped with a probe spherical aberration corrector, enabling sub-angstrom imaging using STEM-HAADF/ABF detectors. For STEM-HAADF imaging, the inner and outer collection angles of an annular dark-field detector were set at 68 and 280 mrad, respectively. For STEM-ABF imaging, the inner and outer collection angles are 10 and 23 mrad, respectively. [010] and [1–10] zone axis STEM-HAADF/ABF images are simulated by using the $\\mathrm{\\dot{Q}S T E M}^{55}$ , which is a suite of software for quantitative image simulation of electron microscopy images, including model building and TEM/STEM/CBED image simulation. The collection angles for HAADF and ABF are 68–280 mrads and 10–20 mrads, respectively. A probe size of $0.8\\mathring\\mathrm{A}$ is used with 27.5 mrad as convergence angle at $200\\mathrm{kV}$ .  \n\n![](images/88cb67fbdf439eb476c0916163c7a5a7f90622cfe5b4443bf3c9ad5f5d846240.jpg)  \nFigure 7 | Cycle voltage governed intragranular cracking and underlying dislocation-based mechanism. (a) HAADF images overlaid diagram shows the apparent dependence of intragranular cracking on the cycle voltage; when cycled below $4.5\\mathsf{V},$ intragranular crack can be hardly generated, while above $4.7\\mathsf V,$ intragranular density shows a drastic increase; and (b) schematic diagrams to illustrate the dislocation-assisted crack incubation, propagation and multiplication process.  \n\nDifferent sample thicknesses (5, 10, 20, 30, 40 and $50\\mathrm{nm}$ ) with different focus values $(-5,~-4,~-3,~-2,~-1,0,1,2,3,4,5\\mathrm{nm})$ are simulated.  \n\nData availability. 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L., Zhang, Z. J., Zhang, P., Wang, Z. G. & Zhang, Z. F. Controllable fatigue cracking mechanisms of copper bicrystals with a coherent twin boundary. Nat. 3536 (2014).  \n\n54. Mayer, J., Giannuzzi, L. A., Kamino, T. & Michael, J. TEM sample preparation and FIB-induced damage. MRS Bull. 32, 400–407 (2011). 55. Woo, J., Borisevich, A., Koch, C. & Guliants, V. V. Quantitative analysis of HAADF–STEM images of MoVTeTaO M1 phase catalyst for propane ammoxidation to acrylonitrile. ChemCatChem 7, 3731–3737 (2015).  \n\n# Acknowledgements  \n\nWe thank Dr Yuanyuan Zhu for help on the GPA analysis. This work was supported by the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, Subcontract No. 6951379 under the Advanced Battery Materials Research (BMR) program. The microscopic analysis in this work was conducted in the William R. Wiley Environmental Molecular Sciences Laboratory (EMSL), a national scientific user facility sponsored by DOE’s Office of Biological and Environmental Research and located at PNNL. PNNL is operated by Battelle for the Department of Energy under Contract DE-AC05-76RLO1830.  \n\n# Author contributions  \n\nC.-M.W., J.X., J.Z. and J.-G.Z. initiated this research project. P.Y., J.Z., J.-G.Z. and C.-M.W. designed the experiment. J.Z. carried out material preparation and battery test. P.Y. conducted TEM experimental work and drafted the manuscript. All authors were involved in revising the manuscript.  \n\n# Additional information  \n\nSupplementary Information accompanies this paper at http://www.nature.com/ naturecommunications  \n\nCompeting financial interests: The authors declare no competing financial interests.  \n\nReprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/  \n\nHow to cite this article: Yan, P. et al. Intragranular cracking as a critical barrier for high-voltage usage of layer-structured cathode for lithium-ion batteries. Nat. Commun. 8, 14101 doi: 10.1038/ncomms14101 (2017).  \n\nPublisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\nThis work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/  ",
    "Target": null
  },
  {
    "id": "10.1038_ncomms15893",
    "DOI": "10.1038/ncomms15893",
    "DOI Link": "http://dx.doi.org/10.1038/ncomms15893",
    "Relative Dir Path": "mds/10.1038_ncomms15893",
    "Article Title": "Origin of fast ion diffusion in super-ionic conductors",
    "Authors": "He, XF; Zhu, YZ; Mo, YF",
    "Source Title": "NATURE COMMUNICATIONS",
    "Abstract": "Super-ionic conductor materials have great potential to enable novel technologies in energy storage and conversion. However, it is not yet understood why only a few materials can deliver exceptionally higher ionic conductivity than typical solids or how one can design fast ion conductors following simple principles. Using ab initio modelling, here we show that fast diffusion in super-ionic conductors does not occur through isolated ion hopping as is typical in solids, but instead proceeds through concerted migrations of multiple ions with low energy barriers. Furthermore, we elucidate that the low energy barriers of the concerted ionic diffusion are a result of unique mobile ion configurations and strong mobile ion interactions in super-ionic conductors. Our results provide a general framework and universal strategy to design solid materials with fast ionic diffusion.",
    "Times Cited, WoS Core": 727,
    "Times Cited, All Databases": 794,
    "Publication Year": 2017,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000403770700001",
    "Markdown": "# Origin of fast ion diffusion in super-ionic conductors  \n\nXingfeng ${\\mathsf{H}}{\\mathsf{e}}^{1},$ Yizhou Zhu1 & Yifei Mo1,2  \n\nSuper-ionic conductor materials have great potential to enable novel technologies in energy storage and conversion. However, it is not yet understood why only a few materials can deliver exceptionally higher ionic conductivity than typical solids or how one can design fast ion conductors following simple principles. Using ab initio modelling, here we show that fast diffusion in super-ionic conductors does not occur through isolated ion hopping as is typical in solids, but instead proceeds through concerted migrations of multiple ions with low energy barriers. Furthermore, we elucidate that the low energy barriers of the concerted ionic diffusion are a result of unique mobile ion configurations and strong mobile ion interactions in super-ionic conductors. Our results provide a general framework and universal strategy to design solid materials with fast ionic diffusion.  \n\nScoloidmpmonatenrtisals nwith feacstt oicohneicmitcralnspeonret ayre isntodrisapgensabnlde electrochemical membranes1–6, which are critical in the societal shift to renewable energy. These electrochemical devices can further improve through the use of super-ionic conductor (SIC) materials, which have several orders of magnitude higher ionic conductivity than typical solids. For example, lithium SICs, including $\\mathrm{Li}_{10}\\mathrm{GeP}_{2}\\mathrm{S}_{12}$ (LGPS)7, $\\mathrm{Li}_{7}\\mathrm{P}_{3}\\mathrm{S}_{11}$ (ref. 8), lithium garnet (for example, $\\mathrm{Li}_{7}\\mathrm{La}_{3}\\mathrm{Zr}_{2}\\mathrm{O}_{12}$ (refs 9,10)), and $\\mathrm{Li^{+}}$ -conducting NASICON (for example, $\\mathrm{Li_{1.3}A l_{0.3}T i_{1.7}(P O_{4})_{3}}$ (ref. 11)), achieve high Li ionic conductivity, $\\sim1-10\\mathrm{mS}\\mathrm{cm}^{-1}$ at room temperature (RT), and low activation energy, ${\\sim}0.2\\mathrm{-}0.3\\mathrm{eV}$ . These SIC materials are promising solid electrolytes for the development of next-generation all-solid-state Li-ion batteries, which provide improved safety, higher energy density, and better thermal stability than current organic electrolyte-based Li-ion batteries6,7,12,13. Despite significant research efforts, only a few materials out of tens of thousands of known inorganic materials have been identified as SICs. It is of great scientific interests to understand why these SICs can achieve several orders of magnitude faster ionic diffusion than other solid materials, and to enable a rationally guided materials design strategy for fast ion conductors.  \n\nCurrent understanding of ionic diffusion in solids is based on the classical diffusion model, which describes ionic transport as the hopping of individual ions from one lattice site to another through inter-connected diffusion channels in the crystal structural framework14 (Fig. 1). The crystal structural framework determines the energy landscape of the ion migration. During ion diffusion, a mobile ion migrates through the energy landscape, and the highest energy of the energy landscape along the diffusion path determines the energy barrier $E_{\\mathrm{a}}$ of ionic diffusion. A low activation energy $E_{\\mathrm{a}}$ and a high concentration $n_{c}$ of mobile ion carriers (such as vacancies or interstitials) are required to achieve high ionic conductivity $\\sigma$ which is proportional to $n_{c}\\cdot\\exp(-E_{\\mathrm{a}}/k_{\\mathrm{B}}\\mathrm{\\bar{}}T)$ at temperature $T$ On the basis of this classical diffusion model, current research efforts in the design and discovery of fast ion conductors target materials with crystal structural frameworks that yield an energy landscape of low barriers. For example, the structural framework with body-centred cubic (bcc) anion packing yields the flattest energy landscape with the lowest $\\mathrm{Li^{+}}$ migration barrier, for example, $\\mathrm{\\sim}0.2\\mathrm{e}\\bar{\\mathrm{V}}$ in lithium-containing sulfides, whereas non-bcc structural frameworks such as in face-centred cubic or hexagonal close-packed exhibit significantly higher energy barriers13. Unfortunately, bcc anion packing is a rare structural feature in Li-containing oxides and sulfides, and among known Li SICs is only found in LGPS and $\\mathrm{Li}_{7}\\mathrm{P}_{3}\\mathrm{S}_{11}$ . Other well-known SICs, such as lithium garnet and NASICON, do not exhibit bcc anion packing, but still achieve high $\\mathrm{Li^{+}}$ ionic conductivity of $\\stackrel{\\bullet}{\\sim}1\\mathrm{m}\\check{\\mathrm{S}}\\mathrm{cm}^{-1}$ at RT.  \n\nSuper-ionic conduction is known to be activated at high mobile-ion concentration $n_{\\mathrm{c}}$ and in specific mobile ion sublattice configuration achieved through materials doping. For example, Li garnet achieves the highest RT Li conductivity, $\\sigma_{\\mathrm{RT}}=\\sim0.1$ to $\\mathrm{\\check{1}m S c m^{-1}}$ $(E_{\\mathrm{a}}=\\sim0.3\\mathrm{eV})$ ), at $6.4\\substack{-7.0\\mathrm{Li}}$ per formula unit in the doped, cubic-phase $\\mathrm{Li}_{7}\\mathrm{La}_{3}\\mathrm{Zr}_{2}\\mathrm{O}_{12}$ compositions10,15,16, whereas $\\mathrm{Li}_{5}\\mathrm{La}_{3}\\mathrm{Ta}_{2}\\mathrm{O}_{12}$ composition of the same crystal structural framework only exhibits $\\sigma_{\\mathrm{RT}}=\\sim10^{-3}\\mathrm{mScm}^{-1}$ $(E_{\\mathrm{a}}=\\sim0.5\\mathrm{eV})^{17}$ . $\\mathrm{Li^{+}}$ -conducting NASICON $\\mathrm{Li}_{1+x}\\mathrm{Al}_{x}\\mathrm{Ti}_{2-x}(\\mathrm{PO}_{4})_{3}$ achieves high ionic conductivity $\\sigma_{\\mathrm{RT}}=\\sim1\\mathrm{mScm}^{-1}$ $(E_{\\mathrm{a}}=\\sim0.3\\mathrm{eV})$ at $\\scriptstyle x=0.2-0.3$ (ref. 11), whereas $\\operatorname{LiTi}_{2}(\\operatorname{PO}_{4})_{3}$ composition has only $\\sigma_{\\mathrm{RT}}=\\sim10^{-3}\\mathrm{mScm}^{-1}$ $\\left(E_{\\mathrm{a}}=\\sim0.45\\mathrm{eV}\\right)^{11,}$ 18. Therefore, the super-ionic conduction in these materials is only activated at certain doped compositions with particular $\\dot{\\mathrm{Li}}^{+}$ sublattice ordering. However, the classical diffusion model, which predicts similar migration barriers for the same crystal framework, fails to capture such super-ionic conduction in these materials. For example, the classical model cannot explain why $\\mathrm{Li^{+}}$ migrations suddenly exhibit significantly lower activation energy barriers in the same crystal structural framework with similar energy landscape as seen in doped Li garnet and NASICON. The answer to this question may help guide the design of SIC materials, especially those with distinctive crystal structural frameworks that deviate from the optimal bcc anion packing.  \n\n![](images/1bf781272e19c0855f2298481ddc286b9f3c90a3044f349553912c2b853c13b3.jpg)  \nFigure 1 | Schematic illustration of single-ion migration versus multi-ion concerted migration. For single-ion migration (upper insets), the migration energy barrier is the same as the barrier of the energy landscape. In contrast, the concerted migration of multiple ions (lower insets) has a lower energy barrier as a result of strong ion-ion interactions and unique mobile ion configuration in super-ionic conductors.  \n\nIn this study, we reveal the origin of fast ionic diffusion in SIC materials with distinctive structural frameworks. We demonstrate a general understanding of fast ionic diffusion across a range of materials using a diffusion model with explicit consideration of the unique mobile-ion sublattice at super-ionic states. Furthermore, we establish a simple conceptual framework for activating fast ion conduction with low migration barriers through materials design, which is generally applicable to any ion-conducting materials.  \n\n# Results  \n\nConcerted ion migration in super-ionic conductors. We performed ab initio molecular dynamics (AIMD) simulations to study diffusion mechanism in the model SIC materials,  \n\n![](images/036e866b5a20af404b999b407b21dc8b8e7914b2097cd081d4d0e9862dec45e7.jpg)  \nFigure 2 | Li ion diffusion in super-ionic conductors. (a–c) Crystal structures of (a) LGPS, (b) LLZO and (c) LATP marked with Li sites (partially filled green spheres), ${\\mathsf{L i}}^{+}$ diffusion channels (green bars), and polyanion groups (purple and blue polyhedra). (d–f) The probability density of ${\\mathsf{L i}}^{+}$ spatial occupancy during AIMD simulations. The zoom-in subsets show the elongation feature of probability density along the migration channel (Li: green; ${\\mathsf{O}}/{\\mathsf{S}}$ : yellow). The isosurfaces are $6\\rho_{0},$ $,6\\rho_{0},2\\rho_{0}$ for LGPS, LLZO, LATP, respectively, where $\\rho_{0}$ is the mean probability density in each structure and the inner isosurfaces have twice the density of the outer isosurfaces. $({\\pmb g}-{\\bf\\dot{i}})$ Van Hove correlation functions of ${\\mathsf{L i}}^{+}$ dynamics on distinctive Li ions during AIMD simulations.  \n\nLGPS, cubic-phase $\\mathrm{Li}_{7}\\mathrm{La}_{3}\\mathrm{Zr}_{2}\\mathrm{O}_{12}$ (LLZO) and $\\mathrm{Li_{1.3}A l_{0.3}T i_{1.7}(P O_{4})_{3}}$ (LATP) (Fig. 2 and Supplementary Fig. 1), which have different anion packing (that is, bcc in LGPS versus non-bcc in LLZO and LATP). The high Li ionic conductivities and low activation energies calculated from AIMD simulations are in good agreement with experimental values7,9,11 (Supplementary Table 1 and Supplementary Fig. 1). By analysing $\\bar{\\mathrm{Li^{+}}}$ dynamics from AIMD simulations, we found that most Li ions migrate in a highly concerted fashion, that is, multiple ions hop simultaneously into their nearest sites within a few picoseconds (Supplementary Note 1 and Supplementary Fig. 2). The strong time correlation in $\\mathrm{Li^{+}}$ hopping during the concerted migration is confirmed by the van Hove correlation function (Fig. 2g–i) of $\\mathrm{Li^{+}}$ dynamics. In addition, to characterize the extent of concerted migrations, we calculated the correlation factor related to the Haven ratio. Whereas a correlation factor of 1.0 corresponds to isolated single-ion diffusion, the correlation factor is calculated as 3.0, 3.0 and 2.1 for LGPS, LLZO and LATP, respectively, in the AIMD simulations at $900~\\mathrm{K},$ corresponding to correlated hopping of approximately two to three ions on average in these SICs. Therefore, the concerted migration is the dominant mechanism for fast diffusion in SICs, as it is in liquids19,20.  \n\nThe concerted migration extracted from AIMD simulations (illustrated as insets Fig. 3a–c) is simultaneous hopping of multiple adjacent ions into their nearest sites. In LGPS, a typical concerted migration involves four Li ions occupying Li1 and Li3 sites hopping simultaneously along the c channel into their nearest-neighbour Li3 and Li1 sites, respectively (Fig. 3a), as observed in a previous study21. In LLZO, Li ions partially co-occupy tetrahedral (T) sites and octahedral (O) sites. During concerted migration in LLZO, T-site Li ions hop to the nearest-neighbour O sites and the Li ions occupying these O sites hop into their nearest neighbour T sites, resulting in concerted hopping of multiple Li ions along the garnet diffusion channel (Fig. 3b) similar to previous modelling studies22,23. In LATP, the typical concerted migration mode is that two Li ions at adjacent M1 and M2 sites migrate in pair. The $\\mathrm{Li^{+}}$ on the M1 site hops into the unoccupied M2 site, and at the same time the $\\mathrm{Li^{+}}$ on the M2 site hops into the next M1 site (Fig. 3c). The migration barriers of these concerted migrations were calculated using nudged-elastic-band (NEB) methods based on ab initio computation (Fig. 3a–c), and were found to be 0.20, 0.26 and $0.27\\mathrm{eV}$ in LGPS, LLZO and LATP, respectively. Given the highly disordered nature of the Li sublattice, various modes of concerted migration mechanisms involving different number of Li and different Li configurations were observed during AIMD simulations and are illustrated in Supplementary Note 2 and Supplementary Fig. 3. The other modes of concerted migration also show migration barriers similar to the typical modes in Fig. 3. The calculated energy barriers of concerted migration are in good agreement with the activation energies obtained from the AIMD simulations and from experiments7,9,11 (Supplementary Table 1). Therefore, these typical concerted migrations observed in AIMD simulations represent the key diffusion mechanisms in SICs.  \n\n![](images/6941cee2430d4d7b80551fa14734c97a536e9890564f0bf5996cd9f98327ba1a.jpg)  \nFigure 3 | Concerted migration and energy landscape in super-ionic conductors. (a–c) Migration energy barrier in $\\mathbf{\\eta}(\\mathbf{a})$ LGPS, $(\\pmb{6})$ LLZO, (c) LATP for concerted migration of multiple Li ions hopping into the next sites along the diffusion channel. Insets show the ${\\mathsf{L i}}^{+}$ path (green spheres) and $0/5$ ions (yellow spheres). (d–f) The energy landscape of single ${\\mathsf{L i}}^{+}$ along the migration channel (shown in insets) across multiple Li sites (partially filled green sphere) and ${\\mathsf{L i}}^{+}$ pathway (red spheres).  \n\nOrigin of concerted migration with low barriers. Given such low energy barriers for multi-ion concerted migration, a relatively flat energy landscape along $\\mathrm{Li^{+}}$ diffusion channels is expected. Surprisingly, the energy landscapes have barriers of 0.47, 0.58 and $0.4\\bar{9}\\mathrm{eV}$ for LGPS, LLZO and LATP, respectively (Fig. 3d–f), which are significantly higher than the energy barrier of concerted migration. On the basis of the classical diffusion model, these high barriers of the energy landscape would lead to even higher activation energy $E_{\\mathrm{a}},$ as each migrating ion feels the high barriers of the energy landscape along the diffusion channel. Thus, the low-barrier concerted migration of multiple ions cannot be explained by the classical diffusion model. As super-ionic conduction is only activated at specific composition with high Li concentration, the mobile-ion configuration and the interactions among these ions, which are neglected in the classical diffusion model, must be considered in order to properly describe the concerted migration in SICs.  \n\nTo reveal the mechanism of multi-ion concerted migration, here we performed a simple diffusion model on the basis of the classical diffusion model by taking into account the configuration of mobile ions and Coulomb interactions among them. In this model, we chose an energy landscape (Fig. 4a) with a $0.6\\mathrm{eV}$ barrier, similar to that in LLZO (Fig. 3e), and included Coulomb interaction among mobile ions with a strength $K$ of ${\\sim}2{\\mathrm{-}}4\\mathrm{eV}\\mathring\\mathrm{A}$ fitted to ab initio calculations (Supplementary Note 3). In addition, the unique $\\mathrm{Li^{+}}$ configuration in SIC materials (Fig. 3d–f) was also considered in this model. In SICs, the mobile ions occupy the high-energy sites (Fig. 4a,b), such as the octahedral O sites in LLZO (Fig. 3e) and the M2 sites in LATP (Fig. 3f), which are near the highest energy point along the diffusion path. At high Li concentration of these SIC materials, the high-energy sites in SICs are occupied because the low-energy sites (for example, tetrahedral $\\mathrm{\\DeltaT}$ sites in LLZO and M1 sites in LATP) are preferably occupied and cannot accommodate all Li ions inserted. The extra Li ions occupying high-energy sites are stabilized by Coulomb interactions from nearby mobile ions (within $\\sim2\\dot{-}3\\mathring\\mathrm{A};$ ) during the minimization of the overall lattice energy.  \n\nOur model shows that such a unique mobile-ion configuration under strong mobile ion-ion interactions is the key for achieving low-barrier concerted migration in these SICs. At typical $K$ values of $2{\\-}4\\mathrm{eV}\\mathring\\mathrm{A}$ in these SICs, the concerted migration of multiple ions shows a significantly lower migration barrier of ${\\sim}0.2{\\mathrm{-}}0.4\\ \\mathrm{\\bar{e}V}$ (Fig. 4d and Supplementary Fig. 4), which is in good agreement with those from NEB calculations (Fig. 3a–c) and AIMD simulations (Supplementary Table 1). Therefore, this simple diffusion model captures the key physics of concerted migration in the SICs. This model demonstrates that low energy barrier of multi-ion concerted migration is a result of the unique mobileion configuration with high-energy site occupancy. During the concerted migration of multiple ions, the ions located at the highenergy sites migrate downhill, which cancels out a part of the energy barrier felt by other uphill-climbing ions. As a result, concerted migration of multiple ions has a significantly lower energy barrier than the energy landscape of the crystal structural framework.  \n\nIn addition, high-energy sites should have locally low barriers and flat energy landscapes, in order to activate low-barrier concerted migration. As observed in the AIMD simulations (Fig. 2d,e), the high-energy sites are associated with elongated spatial occupancy density of mobile Li ions. For example, the Li probability density is elongated at the octahedral (O) sites in LLZO (Fig. 2e). That elongated density indicates a locally flat energy landscape for $\\mathrm{Li^{+}}$ to hop out. The easy migration of ions occupying high-energy sites may facilitate the onset of multi-ion concerted migration24. Otherwise, for the energy landscape in  \n\n![](images/87ad001e12842b7c9d11654844c1aa781cb445d51a788339836aed5c09c94c7f.jpg)  \nFigure 4 | Diffusion model for concerted migration. (a,b) The potential energy of the structural framework with low $\\mathbf{\\eta}(\\mathbf{a})$ or high $(\\pmb{\\ b})$ barriers at the high-energy sites. The mobile ion (grey sphere) configurations and the migration paths (arrows) are illustrated. (c) The energy profile for the concerted migration in the energy landscape (a) and $\\mathbf{(6)}$ at ${\\cal K}=3\\mathsf{e V}\\mathsf{A}$ (d) The energy barrier of concerted migration at different Coulomb interaction strength $K$ .  \n\nFig. 4b as in non-SIC materials (Supplementary Note 4 and Supplementary Fig. 5), multiple ions would simultaneously climb uphill, leading to higher energy barrier for concerted migration (Fig. 4c).  \n\n# Discussion  \n\nIn summary, our theory demonstrates a simple conceptual framework for understanding fast ion diffusion in SICs. Specifically, mobile ions occupying high-energy sites can activate concerted migration with a reduced migration energy barrier. In addition to lithium garnet and NASICON SICs, this mechanism is observed in other SIC materials, such as $\\mathrm{Li}_{7}\\mathrm{P}_{3}\\mathrm{S}_{11},$ $\\mathrm{\\sf{\\beta-Li}}_{3}\\mathrm{\\sf{PS}}_{4}$ , LISICON, $\\mathrm{Li}_{x}\\mathrm{La}_{2/3-x/3}\\mathrm{TiO}_{3}$ (LLTO) perovskite25, and Na þ - conducting NASICON, where high-energy sites are occupied along the diffusion path and the concerted migration with low energy barrier is confirmed in ab initio modelling (Supplementary Note 5 and Supplementary Figs 6–11). The concerted migration of multiple ions is also reported for low-barrier diffusion in other Li ionic conductors, for example, $\\mathrm{Li}_{3}\\mathrm{OX}$ $(\\mathrm{X}=\\mathrm{Cl},\\ \\mathrm{Br})^{26}$ and doped $\\mathrm{Li}_{3}\\mathrm{PO}_{4}$ (refs 27,28). In addition to $\\mathrm{Li^{+}}$ and $\\mathrm{Na^{+}}$ conductors, our proposed model is generally applicable to conductors of other ions. For example, $\\mathbf{Ag}$ super-ionic conductor $\\mathrm{AgI}^{29}$ is known for highly concerted migration, and oxygen ionic conductors of fluorite structure (for example, $\\mathrm{Bi}_{2}\\mathrm{O}_{3}.$ ) (Supplementary Fig. 12) and $\\mathrm{La}_{1-x}\\mathrm{Ba}_{1+x}\\mathrm{GaO}_{4-x/2}$ (ref. 30) also show concerted migration behaviour. Therefore, our proposed theory and identified mechanism are universally applicable to fast diffusion in a broad range of ion-conducting materials.  \n\nMoreover, our theory provides a simple strategy for designing super-ionic conductor materials, that is, inserting mobile ions into high-energy sites to activate concerted ion migration with lower barriers. This explains how super-ionic conduction in lithium garnet and NASICON SICs is activated at certain compositions with increased Li concentration. Here, we demonstrate this strategy by designing a number of novel fast ion conducting materials to activate concerted migration with reduced diffusion barrier. We select ${\\mathrm{LiTaSiO}}_{5}$ and $\\mathrm{LiAlSiO_{4}}$ (details of structures in Supplementary Note 6 and Supplementary Figs 13 and 14), which have structures with a decent bottleneck size of diffusion channels and well-connected $\\mathrm{Li^{+}}$ percolation network, but have not been studied for $\\mathrm{Li^{+}}$ transport. The original structures show low $\\mathrm{Li^{+}}$ conductivities and high activation energies similar to their high-barrier energy landscapes (Supplementary Figs 13 and 14). Extra Li ions are inserted into the high-energy sites of ${\\mathrm{LiTaSiO}}_{5}$ and $\\mathrm{LiAlSiO_{4}}$ by aliovalent substitution of non-Li cations with lower valences. For the doped materials, AIMD simulations show $\\mathrm{Li^{+}}$ concerted migrations with significantly reduced migration barriers of $0.2\\bar{3}{-0.28}\\mathsf{e V}$ and $\\bar{\\mathrm{Li}^{+}}$ conductivities of $1{-}4\\mathrm{mS}\\mathrm{cm}^{-1}$ at RT (Supplementary Figs 13 and 14), which are comparable to many known Li SICs. These results demonstrate that the design strategy based on our simple conceptual framework can be successfully utilized to design novel fast ion conducting materials. In addition, this strategy for facilitating diffusion is generally applicable to any ion-conducting materials.  \n\n# Methods  \n\nDensity functional theory computation. All density functional theory (DFT) calculations in this study were performed using Vienna $\\mathbf{\\nabla}_{A b}$ initio Simulation package $(\\mathrm{VASP})^{31}$ within the projector augmented-wave approach. Perdew–Burke–Ernzerhof $(\\mathrm{PBE})^{3\\bar{2}}$ generalized-gradient approximation (GGA) functionals were adopted in all calculations. The parameters in static DFT calculations were consistent with the Materials Project33–35. The nudged elastic band (NEB) calculations were performed in supercell models using a $T$ -centred $2\\times2\\times2$ k-point grid.  \n\nAb initio molecular dynamics simulation. $\\vert A b$ initio molecular dynamics (AIMD) simulations were performed in supercell models using non-spin-polarized DFT calculations with a $\\Gamma$ -centred $k$ -point. The time step was set to $2\\mathrm{fs}$ . The initial structures were statically relaxed and were set to an initial temperature of $100\\mathrm{K}$ . The structures were then heated to targeted temperatures (300–1500 K) at a constant rate by velocity scaling over a time period of $2\\mathrm{ps}$ . The NVT ensemble using a Nose´–Hoover thermostat36 was adopted. The total time of AIMD simulations were in the range of 100 to $600\\mathrm{ps}$ until the total mean square displacement of Li ions was ${\\phantom{-}}>250\\mathrm{\\AA}^{2}$ in each AIMD simulation and until the diffusivity was converged.  \n\nAs in previous studies37–39, the diffusivity $D$ was calculated as the mean square displacement over time interval $\\Delta t$ :  \n\n$$\nD{=}\\frac{1}{2N d\\Delta t}{\\sum_{i=1}^{N}{\\left\\langle|{{\\bf{r}}_{i}}(t+\\Delta t)-{{\\bf{r}}_{i}}(t)|^{2}}\\right\\rangle_{t}},\n$$  \n\nwhere $d=3$ is the dimension of the diffusion system, $N$ is the total number of diffusion ions, ${\\bf r}_{i}(t)$ is the displacement of the $i$ -th ion at time $t,$ and the bracket represents averaging over $t.$ The ionic conductivity was calculated based on the Nernst-Einstein relationship using  \n\n$$\n\\sigma{=}\\frac{n q^{2}}{k_{\\mathrm{B}}T}D,\n$$  \n\nwhere $n$ is the number of mobile ions per unit volume and $q$ is the ionic charge. The probability density of mobile ions was calculated as the fraction of time that each spatial location was occupied.  \n\nTime correlation of ${\\pmb{\\lfloor\\dot{\\imath}\\dagger}}$ dynamics. The van Hove correlation function $^{40}$ was calculated from the AIMD simulations. The distinctive part $G_{\\mathrm{d}}$ describes the radial distribution of different ions after time interval $\\Delta t$ with respect to the initial ion,  \n\n$$\nG_{\\mathrm{d}}(r,\\Delta t)=\\frac{1}{4\\pi r^{2}N n}\\Bigg\\langle\\sum_{i=1}^{N}\\sum_{j=1,j\\neq i}^{N}\\delta\\big(r-\\big|\\mathbf{r}_{i}(t+\\Delta t)-\\mathbf{r}_{j}(t)\\big|\\big)\\Bigg\\rangle_{t},\n$$  \n\nwhere $\\delta$ is the Dirac delta function. The correlation function is averaged over the time t.  \n\nThe Haven ratio is often used to measure the correlation effect in ionic diffusion41. In this study, we defined a similar correlation factor $f$ to quantify the correlation of ion migration:  \n\n$$\nf=\\frac{N D_{\\sigma}}{D}.\n$$  \n\nWhile $D$ is the self-diffusion diffusivity of individual ions, $D_{\\sigma}$ is the diffusivity of the centre of all diffusion ions and is calculated as:  \n\n$$\nD_{\\sigma}=\\frac{1}{2d\\Delta t}\\left\\langle\\left|\\frac{1}{N}\\sum_{i=1}^{N}\\mathbf{r}_{i}(t+\\Delta t)-\\frac{1}{N}\\sum_{i=1}^{N}\\mathbf{r}_{i}(t)\\right|^{2}\\right\\rangle_{t}.\n$$  \n\nEnergy landscape of single-ion migration. The energy landscape of a single $\\mathrm{Li^{+}}$ along the migration channel (Fig. 3d–f) was calculated using the NEB methods. In LGPS, three $\\mathrm{Li^{+}}$ ions in the $\\scriptstyle{c}$ channel were removed and the energy landscape was obtained by migrating a $\\mathrm{Li^{+}}$ across the $\\mathbf{\\Psi}_{c}$ channel. The energy landscape of cubic-phase LLZO corresponds to single $\\mathrm{Li^{+}}$ migration between two neighbouring tetrahedral sites after removing a Li ion. The energy landscape of LATP corresponds to single $\\mathrm{Li^{+}}$ migration between two neighbouring M1 sites after removing a Li ion. In the NEB calculations, the charge states of all ions were maintained by inserting extra electrons into the system as in the previous study13. To avoid excessive relaxation of the $\\mathrm{Li^{+}}$ sublattice, only non-Li cations and anions were relaxed during the NEB calculations.  \n\nDiffusion model for concerted migration. In the diffusion model illustrated in Fig. 4, four mobile ions were arranged in a 1D lattice consisting of two unit cells. The 1D unit cell has a period of $L=6\\mathring\\mathrm{A}$ , which is similar between two nearest-neighbour M1 sites in LATP (Figs 2f and 3f). The total energy $E$ of the entire mobile lattice is given by the sum of the potential energy $\\phi$ from the crystal framework (that is, the energy landscape) and the Coulomb interaction among mobile ions:  \n\n$$\nE=\\sum_{i}\\phi(x_{i})+\\sum_{i,j,i\\neq j}{\\frac{K}{\\left|x_{i}-x_{j}\\right|}},\n$$  \n\nwhere $x_{i}$ is the position of the ion $i$ and $K$ is the Coulomb interaction strength between two mobile ions. The lattice energy landscape $\\phi$ considered in the main text (Fig. 4a,b) is defined as follows. The energy landscape in Fig. 4a is given by  \n\n$$\n\\phi_{\\mathrm{a}}(x)=E_{\\mathrm{a}}\\cdot\\frac{\\cos\\theta-0.25\\cos2\\theta-C_{1}}{C_{2}},\n$$  \n\nwhere $\\theta=2\\pi x/L-\\pi$ and the normalization factors $C_{1}$ and $C_{2}$ are $-1.25$ and 2.00, respectively. The highest point of the energy landscape is $E_{\\mathrm{a}}=0.6\\mathrm{eV}$ , which is set similar to the single Li-ion energy landscape of LLZO (Fig. 3e). The energy landscape in Fig. 4b is given by  \n\n$$\n\\phi_{b}(x)=E_{\\mathrm{a}}\\cdot\\frac{\\cos\\theta-1.5\\cos2\\theta-C_{1}}{C_{2}},\n$$  \n\nwhere $C_{1}=-2.50$ and $C_{2}=4.08$ . This energy landscape has the same highest energy point of $0.6\\mathrm{eV}$ , but has a higher local barrier of $0.3\\mathrm{eV}$ at the high-energy sites (Fig. 4b).  \n\nMaterials. The crystal structures investigated were obtained from the Inorganic Crystal Structure Database42 and Materials Project35. The structures with disordered site occupancies were ordered using the same method used in previous studies38,39. The structure of LATP was derived from the $\\operatorname{LiTi}_{2}(\\operatorname{PO}_{4})_{3}$ structure by partially substituting Ti with Al and by inserting extra Li atoms into M2 sites (Fig. 2f). The occupancy of Al/Ti and Li were ordered to obtain the structure.  \n\nData availability. 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First principles study of the $\\mathrm{Li}_{10}\\mathrm{GeP}_{2}\\mathrm{S}_{12}$ lithium super ionic conductor material. Chem. Mater. 24, 15–17 (2012).   \n39. He, X. & Mo, Y. Accelerated materials design of $\\mathrm{Na}_{0.5}\\mathrm{Bi}_{0.5}\\mathrm{TiO}_{3}$ oxygen ionic conductors based on first principles calculations. Phys. Chem. Chem. Phys. 17, 18035–18044 (2015).   \n40. Van Hove, L. Correlations in space and time and born approximation scattering in systems of interacting particles. Phys. Rev. 95, 249 (1954).   \n41. Compaan, K. & Haven, Y. Correlation factors for diffusion in solids. Trans. Faraday Soc. 52, 786–801 (1956).   \n42. Belsky, A., Hellenbrandt, M., Karen, V. L. & Luksch, P. New developments in the Inorganic Crystal Structure Database (ICSD): accessibility in support of materials research and design. Acta Crystallogr. Sect. B: Struct. Sci. 58, 364–369 (2002).  \n\n# Acknowledgements  \n\nWe acknowledge the support by A. James Clark School of Engineering, University of Maryland. This research used computational facilities from the University of Maryland supercomputing resources, the Maryland Advanced Research Computing Center (MARCC), and the Extreme Science and Engineering Discovery Environment (XSEDE) supported by National Science Foundation Award No. DMR150038. We thank Alexander Epstein for reviewing the manuscript.  \n\n# Author contributions  \n\nY.M. conceived and oversaw the project. Y.M. and X.H. designed the computation and analyses, and X.H. performed them. Y.M. and X.H. wrote the manuscript. All authors contributed to discussions and revisions of the manuscript.  \n\n# Additional information  \n\nSupplementary Information accompanies this paper at http://www.nature.com/ naturecommunications  \n\nCompeting interests: The authors declare no competing financial interests.  \n\nReprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/  \n\nHow to cite this article: He, X. et al. Origin of fast ion diffusion in super-ionic conductors. Nat. Commun. 8, 15893 doi: 10.1038/ncomms15893 (2017).  \n\nPublisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\nOpen Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/  \n\n$\\copyright$ The Author(s) 2017  ",
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  {
    "id": "10.1016_j.cpc.2017.09.033",
    "DOI": "10.1016/j.cpc.2017.09.033",
    "DOI Link": "http://dx.doi.org/10.1016/j.cpc.2017.09.033",
    "Relative Dir Path": "mds/10.1016_j.cpc.2017.09.033",
    "Article Title": "WannierTools: An open-source software package for novel topological materials",
    "Authors": "Wu, QS; Zhang, SN; Song, HF; Troyer, M; Soluyanov, AA",
    "Source Title": "COMPUTER PHYSICS COMMUNICATIONS",
    "Abstract": "We present an open-source software package Wannier Tools, a tool for investigation of novel topological materials. This code works in the tight-binding framework, which can be generated by another software package Wannier90 (Mostofi et al., 2008). It can help to classify the topological phase of a given material by calculating the Wilson loop, and can get the surface state spectrum, which is detected by angle resolved photoemission (ARPES) and in scanning tunneling microscopy (STM) experiments. It also identifies positions of Weyl/Dirac points and nodal line structures, calculates the Berry phase around a closed momentum loop and Berry curvature in a part of the Brillouin zone (BZ). Program summary Program title: WannierTools Program Files doi: http://dx.doi.org/10.17632/ygsmh4hyh6.1 Licensing provisions: GNU General Public Licence 3.0 Programming language: Fortran 90 External routines/libraries used: BIAS (http://www/netlib.org/blas) LAPACK (http://www.netlib.org/lapack) Nature of problem: Identifying topological classificatiOns of crystalline systems including insulators, semimetals, metals, and studying the electronic properties of the related slab and ribbon systems. Solution method: Tight-binding method is a good approximation for solid systems. Based on that, Wilson loop is used for topological phase classification. The iterative Green's function is used for obtaining the surface state spectrum. (C) 2017 Elsevier BV. All rights reserved.",
    "Times Cited, WoS Core": 1950,
    "Times Cited, All Databases": 2057,
    "Publication Year": 2018,
    "Research Areas": "Computer Science; Physics",
    "UT (Unique WOS ID)": "WOS:000424726700035",
    "Markdown": "# WannierTools: An open-source software package for novel topological materials  \n\nQuanSheng Wu a,\\*, ShengNan Zhang b, Hai-Feng Song c, Matthias Troyer a, Alexey A. Soluyanov a,d  \n\na Theoretische Physik and Station Q Zurich, ETH Zurich, 8093 Zurich, Switzerland b Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China c Institute of Applied Physics and Computational Mathematics, Beijing 100094, China d Department of Physics, St. Petersburg State University, St. Petersburg, 199034, Russia  \n\n# a r t i c l e i n f o  \n\n# a b s t r a c t  \n\nArticle history: Received 1 March 2017 Accepted 29 September 2017 Available online 18 October 2017  \n\nKeywords:   \nNovel topological materials   \nTopological number   \nSurface state   \nTight-binding model  \n\nWe present an open-source software package WannierTools, a tool for investigation of novel topological materials. This code works in the tight-binding framework, which can be generated by another software package Wannier90 (Mostofi et al., 2008). It can help to classify the topological phase of a given material by calculating the Wilson loop, and can get the surface state spectrum, which is detected by angle resolved photoemission (ARPES) and in scanning tunneling microscopy (STM) experiments. It also identifies positions of Weyl/Dirac points and nodal line structures, calculates the Berry phase around a closed momentum loop and Berry curvature in a part of the Brillouin zone (BZ).  \n\n# Program summary  \n\nProgram title: WannierTools   \nProgram Files doi: http://dx.doi.org/10.17632/ygsmh4hyh6.1   \nLicensing provisions: GNU General Public Licence 3.0   \nProgramming language: Fortran 90   \nExternal routines/libraries used: BLAS (http://www/netlib.org/blas) • LAPACK (http://www.netlib.org/lapack)  \n\nNature of problem: Identifying topological classifications of crystalline systems including insulators, semimetals, metals, and studying the electronic properties of the related slab and ribbon systems. Solution method: Tight-binding method is a good approximation for solid systems. Based on that, Wilson loop is used for topological phase classification. The iterative Green’s function is used for obtaining the surface state spectrum.  \n\n$\\mathfrak{C}$ 2017 Elsevier B.V. All rights reserved.  \n\n# 1. Introduction  \n\nNovel topological states have attracted much attention during the past decades. They give a lot of opportunities to explore new physics and to realize new quantum devices [1,2]. Since quantum spin hall effect (QSHE) was predicted in 2005 [3], and realized in HgTe/CdTe quantum wells in 2007 [4,5], more and more topological novel states were discovered, such as 3D topological insulators [6–8], Dirac [9,10], Weyl [11–13] semimetals, Hourglass fermions [14], Nodal line semimetals [15–17] and nodal chain metals [18] et al. Nowadays, a lot of new topological phases are emerging, and more and more materials are identified to be topologically non-trivial.  \n\nOne important feature of topological materials is topologically protected surface [19] states, which are robust against the white noise disorder and have novel transport properties [5] due to spin-momentum locked features. In experiments, the surface states and transport properties are detectable, therefore, taken as an evidence for non-trivial topology in bulk band structures [5,6,20]. The logic here is that the bulk-edge correspondence principle, which tells that there are topologically protected $d-1$ -dimension edge states if the topological property is non-trivial in $d$ -dimension bulk. In the theoretical part, besides calculation of surface-state spectrum and transport properties, we could also calculate topological numbers such as the $\\mathbb{Z}_{2}$ number [21], Chern number [22] and Wilson loops [23] (identical to Wannier charge centers [24]) to study the topology of energy band structures of a material directly.  \n\nAt present, there are more and more groups joining the investigation of novel topological materials. However, there are only a few software packages that can be used so far. Z2pack [25] is a package that uses the Wannier charge centers (WCCs) to classify the topological properties of real compounds. By using Z2pack, the WCCs can be obtained either from tight-binding (TB) model or from first-principle packages, such as VASP, ABINIT and Quantum-espresso. PythTB [26] is another software package with a Python implementation of TB models. There are many functions to build and solve TB models of the electronic structure of systems for arbitrary dimensional systems (crystals, slabs, ribbons, clusters, etc.) in PythTB, and it can compute Berry phases and related properties.  \n\nHere we introduce a TB dependent open-source software package called WannierTools, which can be used for novel topological materials investigation. Unlike Z2pack, it can be used to calculate the surface states of materials, and, being parallelized with MPI, it is faster than PythTB. It is a user friendly and efficient single program implemented in Fortran90. The only thing needed to run it is an input file, which contains some parameters describing your systems, and a TB model written in Wannier90_hr.dat format [27]. With WannierTools, topological numbers like the $\\mathbb{Z}_{2}$ numbers or Wilson loops for the bulk system can be calculated in order to explore the topological properties of a material. It can also help to search for Weyl/Dirac points or nodal loop structures in the BZ of metallic systems. There are plenty of other functions, e.g. studying the electron structure properties for slab and ribbon systems, studying the Berry curvature for bulk systems, studying the Berry phase around one momentum loop in the BZ for nodal-line systems and so on.  \n\nThis paper is organized as follows. In Section 2, we review briefly some basic theories related to this package. In Section 3, we introduce the capabilities of this package. In Section 4, we introduce the installation and basic usages. In Section 5, we introduce a new topological material HfPtGe in order to show you how to use WannierTools to explore a new topological phase.  \n\n# 2. Methods  \n\n# 2.1. TB method  \n\nTB method is a semi-empirical approach to study electronic structures of solid-state systems by projecting the Hamiltonian of the system onto a series of local orbitals. There are several ways to construct TB models, such as Slater–Koster method [28], maximum localized Wannier functions (MLWF) [29], and discretization of $k$ ·p model [30] onto a lattice. Among these methods, the MLWF method [29] is widely used by the people who are interested in real materials simulations. MLWF is implemented in Wannier90, which has many interfaces with different first-principle software packages like VASP, WIEN2k, et al. Therefore MLWF TB models can be automatically obtained from first-principle calculations together with Wannier90 [27].  \n\nIn different TB methods, the basis functions could be mutually orthogonal or non-orthogonal. However, WannierTools is only capable of dealing with the TB models with orthogonal basis functions. Fortunately, the Wannier functions (WFs) for MLWF TB method fulfill this limitation. In this section, we give some brief introductions to the general orthogonal TB methods. The details of how to construct MLWF TB models can be found in Refs. [31,29].  \n\nLet i label the atoms, $\\mu$ label the orbitals, $m$ label the combination of $\\{i\\mu\\}$ , R label the lattice vectors in 3D crystal, and ${\\pmb{\\tau}}_{i}$ label the position of atoms in a home unit cell. The local orbital for the i’th atom centered at ${\\pmb R}+\\pmb{\\tau}_{i}$ can be written as  \n\n$$\n\\phi_{R m}(\\pmb{r})\\equiv\\phi_{m}(\\pmb{r}-\\pmb{R})\\equiv\\varphi_{i\\mu}(\\pmb{r}-\\pmb{R}-\\pmb{\\tau}_{i}).\n$$  \n\nThe orthogonality of orbitals requires $\\langle\\phi_{R m}\\vert\\phi_{R^{\\prime}n}\\rangle=\\delta_{R R^{\\prime}}\\delta_{m n}.$ . TB parameters of the Hamiltonian that have the translational symmetry due to Bloch theorem can be calculated via  \n\n$$\nH_{m n}({\\pmb R})=\\langle\\phi_{{\\pmb0}m}|\\hat{H}|\\phi_{{\\pmb R}n}\\rangle\n$$  \n\nOnce we have the TB Hamiltonian $H_{m n}(\\pmb{R})$ , the Hamiltonian in k space can be obtained by a Fourier transformation (FT) [29]. There are two conventions [26] for FTs. One is  \n\n$$\nH_{m n}({\\pmb k})=\\sum_{\\pmb{R}}e^{i{\\pmb k}\\cdot{\\pmb R}}H_{m n}({\\pmb R})\n$$  \n\nThe other one is  \n\n$$\nH_{m n}({\\pmb k})=\\sum_{{\\pmb R}}e^{i{\\pmb k}\\cdot({\\pmb R}+{\\pmb\\tau}_{m}-{\\pmb\\tau}_{n})}H_{m n}({\\pmb R})\n$$  \n\nIt can be demonstrated that eigenvalues for these two conventions are the same, but the eigenvectors are different. The eigenvectors of the first convention Eq. (3) are analogous to the Bloch wave functions $\\psi_{n\\pmb{k}}(\\pmb{r})$ . The eigenvectors of the second convention Eq. (4) are analogous to the periodic part of the Bloch wave functions $u_{n k}(\\pmb{r})=\\psi_{n k}(\\pmb{r})e^{-i\\pmb{k}\\pmb{r}}$ , which is of great importance in Berry phase and Berry curvature or the Wannier centers calculations. Therefore, the second convention is used in WannierTools.  \n\nAccording to the bulk-edge correspondence, there are topologically protected surface states if the topology of bulk energy bands is non-trivial. In order to study such surface states, we have to construct a slab system which is periodic along two directions at the surface. In practice, a new unit cell is defined with lattice vectors R′1,2,3,  \n\n$$\n\\begin{array}{r}{\\mathbf{R}_{1}^{\\prime}=U_{11}\\mathbf{R}_{1}+U_{12}\\mathbf{R}_{2}+U_{13}\\mathbf{R}_{3}}\\\\ {\\mathbf{R}_{2}^{\\prime}=U_{21}\\mathbf{R}_{1}+U_{22}\\mathbf{R}_{2}+U_{23}\\mathbf{R}_{3}}\\\\ {\\mathbf{R}_{3}^{\\prime}=U_{31}\\mathbf{R}_{1}+U_{32}\\mathbf{R}_{2}+U_{33}\\mathbf{R}_{3}}\\end{array}\n$$  \n\nwhere $\\mathbf{R}_{1,2,3}$ are lattice vectors of the original unit cell of the bulk system, ${\\pmb R}_{1}^{\\prime}$ and ${\\bf R}_{2}^{\\prime}$ are two lattice vectors in the target slab surface, ${\\bf R}_{3}^{\\prime}$ is the other lattice vector which is out of the surface and fulfills the volume fixed condition,  \n\n$$\n\\mathbf{R}_{1}^{\\prime}\\cdot(\\mathbf{R}_{2}^{\\prime}\\times\\mathbf{R}_{3}^{\\prime})=\\mathbf{R}_{1}\\cdot(\\mathbf{R}_{2}\\times\\mathbf{R}_{3})\n$$  \n\nSince the slab system is non-periodic along the ${\\pmb R}_{3}^{\\prime}$ direction, the Hamiltonian of a slab system with a 2D momentum $k_{\\parallel}$ can be calculated by the following FT  \n\n$$\nH_{m n}^{s l a b}({\\bf k}_{\\parallel})=\\sum_{\\parallel{\\bf R}\\parallel}e^{i{\\bf k}_{\\parallel}\\cdot{\\bf R}}H_{m n}^{s l a b}({\\bf R})\n$$  \n\nwhere $\\begin{array}{r}{\\mathbf{R}=a^{\\prime}\\mathbf{R}_{3}^{\\prime}+b^{\\prime}\\mathbf{R}_{2}^{\\prime}+c^{\\prime}\\mathbf{R}_{3}^{\\prime}}\\end{array}$ and $\\|\\mathbf R\\|$ is a restriction that the summation is only carried on $a^{\\prime}$ and $b^{\\prime}$ with different $c^{\\prime}$ . We label the layer index along ${\\pmb R}_{3}^{\\prime}$ as i, j. As a consequence, the Hamiltonian of a slab system with $n_{s}$ layers can be written in the layer index matrix form,  \n\n$$\nH_{m n}^{s l a b}({\\bf k}_{\\parallel})=\\left(\\begin{array}{c c c c}{H_{m n,11}({\\bf k}_{\\parallel})}&{H_{m n,12}({\\bf k}_{\\parallel})}&{\\cdot\\cdot}&{H_{m n,1n_{s}}({\\bf k}_{\\parallel})}\\\\ {H_{m n,21}({\\bf k}_{\\parallel})}&{H_{m n,22}({\\bf k}_{\\parallel})}&{\\cdot\\cdot}&{H_{m n,2n_{s}}({\\bf k}_{\\parallel})}\\\\ {\\vdots}&{\\vdots}&{\\cdot}&{\\vdots}\\\\ {H_{m n,n_{s}1}({\\bf k}_{\\parallel})}&{H_{m n,n_{s}2}({\\bf k}_{\\parallel})}&{\\cdot\\cdot}&{H_{m n,n_{s}n_{s}}({\\bf k}_{\\parallel})}\\end{array}\\right)\n$$  \n\nwhere the diagonal elements of the Hamiltonian are the intra-plane ones, and the off diagonal elements are the inter-plane ones. The element in Eq. (10) can be read explicitly as,  \n\n$$\nH_{m n,i j}(\\mathbf{k}_{\\parallel})=\\sum_{\\mathbf{R}=\\{\\mathbf{R}_{1}^{\\prime},\\mathbf{R}_{2}^{\\prime},(i-j)\\mathbf{R}_{3}^{\\prime}\\}}e^{i\\mathbf{k}_{\\parallel}\\cdot\\mathbf{R}}H_{m n}(\\mathbf{R})\n$$  \n\nFinally the energy band of the slab system can be obtained straightforwardly to diagonalize Eq. (10).  \n\nBy the way, the Hamiltonian for the ribbon system can be obtained in the same way as the slab system does. The difference is that there are two confined directions $\\mathbf{R}_{1}^{\\prime},\\mathbf{R}_{2}^{\\prime}$ in ribbon systems, which enlarges the size of Hamiltonian.  \n\n# 2.2. Wannier charge center calculation  \n\n$\\mathbb{Z}_{2}$ topological number [21] and Chern number [22] are applied to classify topological properties for time-reversal invariant and time-reversal symmetry breaking systems respectively. In inversion symmetric invariant system, the $\\mathbb{Z}_{2}$ topological number can be calculated by multiplying the parities of the occupied bands at time reversal invariant momenta (TRIMs) in the Brillouin zone [32]. There are several methods [33,23,24] to calculate the $\\mathbb{Z}_{2}$ number in inversion symmetry breaking systems. Among them it was demonstrated that the Wilson loop [23] and Wannier charge centers (WCCs) [24] method are equivalent to each other, and are also valid for time-reversal symmetry breaking systems. In WannierTools, we take the algorithm presented in Refs. [24,25]. The hybrid WFs [34] are defined as  \n\n$$\n|n k_{x}l_{y}\\rangle=\\frac{1}{2\\pi}\\int_{0}^{2\\pi}d k_{y}e^{-i k_{y}l_{y}}|\\psi_{n\\mathbf{k}}\\rangle\n$$  \n\nwhere $|\\psi_{n\\mathbf{k}}\\rangle$ is the Bloch wave function. The hybrid Wannier centers are defined as  \n\n$$\n\\begin{array}{l}{\\displaystyle\\bar{y}_{n}(k_{x})=\\langle n k_{x}0|y|n k_{x}0\\rangle}\\\\ {\\displaystyle=\\frac{i}{2\\pi}\\int_{-\\pi}^{\\pi}d k_{y}\\langle u_{n,k_{x},k_{y}}|\\partial_{k_{y}}|u_{n,k_{x},k_{y}}\\rangle}\\end{array}\n$$  \n\nwhere $\\vert u_{n,k_{x},k_{y}}\\rangle$ is the periodic part of Bloch function $|\\psi_{n\\mathbf{k}}\\rangle$ . In practice , the integration over $k_{y}$ is transformed by a summation over the discretized $k_{y}^{-}$ . Eq. (14) can be reformulated using the discretized Berry phase formula [35],  \n\n$$\n\\bar{y}_{n}(k_{x})=-\\frac{1}{2\\pi}\\mathrm{Im}\\ln\\prod_{j}M_{n n}^{(j)}\n$$  \n\nwhere the gauge-dependent overlap matrix $M_{m n}^{(j)}=\\langle u_{m,k_{x},k_{y_{j}}}|u_{n,k_{x},k_{y_{j+1}}}\\rangle$ is introduced. However, the summation of the hybrid Wannier centers $\\bar{y}_{n}(k_{x})$ over $k_{x}$ is gauge invariant [36], which is the total electronic polarization. As shown in Ref. [35,37], there is another way to obtain $\\bar{y}_{n}(k_{x})$ . Firstly, we get the ‘‘unitary part’’ $\\tilde{M}^{(j)}$ of each overlap matrix $M_{m n}^{\\{j\\}}$ by carrying out the single value decomposition $M=V\\Sigma W^{\\dagger}$ , where $\\mathsf{V}$ and W are unitary and $\\Sigma$ is real-positive and diagonal. Then we set $\\tilde{M}^{(j)}=\\bar{V}W^{\\bar{\\dagger}}$ . The eigenvalues $\\lambda_{n}$ of matrix $\\begin{array}{r}{\\varLambda=\\prod_{j}\\tilde{M}^{(j)}}\\end{array}$ are all of unit modulus. The hybrid Wannier centers are defined with the phases of $\\lambda_{n}$ eventually,  \n\n$$\n\\bar{y}_{n}(k_{x})=-\\frac{1}{2\\pi}\\mathrm{Im}\\ln{\\lambda_{n}}\n$$  \n\nWe can get the topological properties of $k_{x}-k_{y}$ plane from the evolution of $\\bar{y}_{n}(k_{x})$ along a $k_{x}$ string. The details of such classification of WCCs or Wilson loop are discussed in Refs. [23,37,24]. More information can be found in Ref. [25].  \n\n# 2.3. Berry phase and Berry curvature  \n\nIn this section, we give the basic formalism for computing Berry phase [38,39] and Berry curvature [40,41] of Bloch states. Firstly, we introduce the single band case, where the energy bands are isolated to each other. A Berry phase $\\phi_{n}$ is a geometric phase associated with the phase evolution of the $n$ ’th state over a closed curve $c$ in external parameter space $k$ , defined as  \n\n$$\n\\phi_{n}=\\oint_{\\mathcal{C}}\\pmb{A}_{n}\\cdot d\\pmb{k}\n$$  \n\nwhere the Berry connection is $A_{n,\\alpha}=i\\langle u_{n,\\pm}|\\partial_{\\alpha}u_{n,\\pm}\\rangle$ , $\\alpha=k_{x},k_{y},k_{z}$ , and Berry curvature is introduced  \n\n$$\n\\begin{array}{r l}&{\\Omega_{n,\\alpha\\beta}=\\partial_{\\alpha}A_{n,\\beta}-\\partial_{\\beta}A_{n,\\alpha}}\\\\ &{\\qquad=i\\langle\\partial_{\\alpha}u_{n,\\boldsymbol{k}}|\\partial_{\\beta}u_{n,\\boldsymbol{k}}\\rangle-i\\langle\\partial_{\\beta}u_{n,\\boldsymbol{k}}|\\partial_{\\alpha}u_{n,\\boldsymbol{k}}\\rangle}\\\\ &{\\qquad=-2\\operatorname{Im}\\langle\\partial_{\\alpha}u_{n,\\boldsymbol{k}}|\\partial_{\\beta}u_{n,\\boldsymbol{k}}\\rangle}\\end{array}\n$$  \n\nSecondly, for multi-band case, it is often to treat the occupied $N_{o c c}$ bands as a joint band manifold, which is referred to as the ‘‘non-Abelian’’ case. Generalizations for the formalism of Berry phase and Berry curvature from single band to multi-band case are as follows,  \n\n$$\n\\phi=\\oint_{\\cal C}\\mathrm{Tr}[A]\\cdot d{\\bf k}\n$$  \n\nwhere the Berry connection [39] is $\\mathcal{A}_{m n,\\alpha}=i\\langle u_{m,k}|\\partial_{\\alpha}u_{n,k}\\rangle,\\alpha=k_{x},k_{y},k_{z}$ , and Berry curvature is  \n\n$$\n\\begin{array}{r l}&{\\Omega_{m n,\\alpha\\beta}=\\partial_{\\alpha}A_{m n,\\beta}-\\partial_{\\beta}A_{m n,\\alpha}-i[\\mathcal{A}_{\\alpha},\\mathcal{A}_{\\beta}]_{m n}}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\\\ &{\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad}\\quad}\\end{array}\n$$  \n\nand define  \n\n$$\n\\mathcal{\\Omega}_{\\alpha\\beta}=\\mathrm{Tr}\\mathcal{\\Omega}_{m n,\\alpha\\beta}\n$$  \n\nwhere Tr denotes a trace over the occupied bands.  \n\nIn practice, the integration of Eq. (21) is implemented on a discrete $k$ mesh. The loop $c$ is discretized into a series of closely space points $k_{j}$ . Accordingly, the Berry phase becomes  \n\n$$\n\\phi=-\\sum_{j}\\mathrm{Im}\\ln\\mathrm{det}M^{(j)}=-\\mathrm{Im}\\ln\\prod_{j}\\mathrm{det}M^{(j)}\n$$  \n\nwhere the overlap matrix $M_{m n}^{(j)}$ is the same as in Eq. (15), i.e., $M_{m n}^{(j)}=\\langle u_{m,\\pmb{k}_{j}}|u_{n,\\pmb{k}_{j+1}}\\rangle$ .  \n\n# 2.4. Calculation of surface states  \n\nTheoretically, we have two methods to get surface spectrum corresponding to the bulk topology. One is that we calculate the band structure of a slab system, which was introduced in Section 2.1. The other one is to calculate the surface Green’s function (SGF) for a semiinfinite system that will be introduced in this section. In the 1970s, one of the most popular GF approaches was based on the ‘‘effective field’’ and transfer matrix [42–44], which are relatively of slow convergence especially near singularities. Now, the extensively used schemes to obtain the SGFs is the iterative Green’s function method developed in the 1980s [45,46]. With an effective concept of principle layers (The layer that is large enough to guarantee that hoppings between the next nearest layers are negligible.), the iterative procedure can save quite an amount of computational time. The method [45,46] involves replacing the principle layer by an effective two principle layers, and these effective layers interact through energy-dependent residual interactions which are weaker than those of the original ones. This replacement can be repeated iteratively until the residual interactions between the effective layers become as small as desired. Each new iteration doubles the number of the original layers included in the new effective layer. That is, after $n$ iterations, one has a chain of lattice constant $2^{n}$ times the original one, and each effective layer replacing $2^{n}$ original layers. The details of the algorithm are presented in Ref. [47]. For the integrity of the paper, we list the main iterations that we reused in WannierTools. The most important parameters for iteration i are the following  \n\n$$\n\\begin{array}{r l}&{\\alpha_{i}=\\alpha_{i-1}(\\omega-\\varepsilon_{i-1})^{-1}\\alpha_{i-1}}\\\\ &{\\beta_{i}=\\beta_{i-1}(\\omega-\\varepsilon_{i-1})^{-1}\\beta_{i-1}}\\\\ &{\\varepsilon_{i}=\\varepsilon_{i-1}+\\alpha_{i-1}(\\omega-\\varepsilon_{i-1})^{-1}\\beta_{i-1}+\\beta_{i-1}(\\omega-\\varepsilon_{i-1})^{-1}\\alpha_{i-1}}\\\\ &{\\varepsilon_{i}^{s}=\\varepsilon_{i-1}^{s}+\\alpha_{i-1}(\\omega-\\varepsilon_{i-1})^{-1}\\beta_{i-1}}\\\\ &{\\tilde{\\varepsilon}_{i}^{s}=\\tilde{\\varepsilon}_{i-1}^{s}+\\beta_{i-1}(\\omega-\\varepsilon_{i-1})^{-1}\\alpha_{i-1}}\\end{array}\n$$  \n\nwith the initialization $\\varepsilon_{0}=\\tilde{\\varepsilon}_{0}=\\tilde{\\varepsilon}_{0}^{s}=H_{00}(\\mathbf{k}_{\\parallel}),\\alpha_{0}=H_{01}(\\mathbf{k}_{\\parallel}),\\beta_{0}=H_{01}^{\\dagger}(\\mathbf{k}_{\\parallel})$ , where $H_{00}(\\mathbf{k}_{\\parallel})$ is the intra-hopping parameters in the principle layers, $H_{01}(\\mathbf{k}_{\\parallel})$ is the inter-hopping parameters between the nearest neighbor of principle layers.  \n\nIteration of Eq. (27) should be converged until $\\varepsilon_{n}\\simeq\\varepsilon_{n-1}$ and $\\tilde{\\varepsilon}_{n}^{s}\\simeq\\tilde{\\varepsilon}_{n-1}^{s}$ . The SGFs $G_{s}(\\mathbf{k}_{\\parallel},\\omega)$ and the bulk GF $G_{b}(\\mathbf{k}_{\\|},\\omega)$ can be obtained as  \n\n$$\n\\begin{array}{r}{G_{s}(\\mathbf{k}_{\\parallel},\\omega)\\simeq(\\omega-\\varepsilon_{n}^{s})^{-1}}\\\\ {\\tilde{G}_{s}(\\mathbf{k}_{\\parallel},\\omega)\\simeq(\\omega-\\tilde{\\varepsilon}_{n}^{s})^{-1}}\\\\ {G_{b}(\\mathbf{k}_{\\parallel},\\omega)\\simeq(\\omega-\\varepsilon_{n})^{-1}}\\end{array}\n$$  \n\nwhere $\\tilde{G}_{s}$ is the SGF of the dual surface. The surface spectrum function $A(\\mathbf{k}_{\\parallel},\\omega)$ can be obtained from the imaginary part of SGF  \n\n$$\nA(\\mathbf{k}_{\\parallel},\\omega)=-\\frac{1}{\\pi}\\operatorname*{lim}_{\\eta\\rightarrow0^{+}}\\mathrm{Im}\\mathrm{Tr}G_{s}(\\mathbf{k}_{\\parallel},\\omega+i\\eta)\n$$  \n\nTable 1 Main capabilities of WannierTools: Bulk topology studies.   \n\n\n<html><body><table><tr><td>Control flag in wt .in</td><td>Description</td></tr><tr><td>BulkBand_calc</td><td>Energy bands for a 3D bulk system.</td></tr><tr><td>BulkFS_calc</td><td>3D Fermi surface in 1st BZ.</td></tr><tr><td>FindNodes_calc</td><td>Locate Weyl, Dirac point positions and nodal line structures in 1st BZ.</td></tr><tr><td>BulkGap_plane_calc</td><td>Gap function in a 3D k plane.</td></tr><tr><td>Wanniercenter_calc</td><td>WCCs [24] for a 3D k plane.</td></tr><tr><td>BerryPhase_calc</td><td>Berry phase for a closed path in 3D k space.</td></tr><tr><td>BerryCurvature_calc</td><td>Berry curvature in a 3D k plane.</td></tr></table></body></html>  \n\nTable 2 Main capabilities of WannierTools: related responses from the bulk topology.   \n\n\n<html><body><table><tr><td>Control flag in wt .in</td><td>Description</td></tr><tr><td>Dos_calc</td><td>Density of state of a 3D bulk system</td></tr><tr><td>JDos_calc</td><td>Joint density of state [54] of a 3D bulk system</td></tr><tr><td>SlabBand_calc</td><td>Energy bands of a 2D slab system.</td></tr><tr><td>WireBand_calc</td><td>Energy bands of a 1D ribbon system.</td></tr><tr><td>SlabSS_calc</td><td>Surface state spectrum along some k lines at different energies.</td></tr><tr><td>SlabArc_calc</td><td>Surface state spectrum in the 2D BZ at a fixed energy.</td></tr><tr><td>SlabQPI_calc</td><td>Quasi-particle interference (QPl) [55] pattern of surface state.</td></tr><tr><td>SlabSpintexture_calc</td><td>Spin texture [56] of surface state.</td></tr></table></body></html>  \n\nThe spin texture of surface states can also be obtained with [48]  \n\n$$\n\\mathbf{S}(\\mathbf{k}_{\\parallel},\\omega)=-\\frac{1}{\\pi}\\operatorname*{lim}_{\\eta\\rightarrow0^{+}}\\mathrm{Im}\\mathrm{Tr}\\left[\\sigma G_{s}(\\mathbf{k}_{\\parallel},\\omega+i\\eta)\\right]/A(\\mathbf{k}_{\\parallel},\\omega)\n$$  \n\nwhere $\\pmb{\\sigma}$ are the Pauli matrices.  \n\n# 2.5. Algorithm for searching nodal points/lines  \n\nNodal point is a gapless point between the highest valence band and the lowest conduction band. In terms of degeneracy character, nodal point could be classified into Weyl, Triple, Dirac, hyper-Dirac point with 2-fold degeneracy, 3-fold degeneracy, 4-fold degeneracy or higher degeneracy respectively. They also can be sorted into nodal point and nodal line by the connectivity between them. Searching the Weyl/Dirac points and the nodal-line structures is very important for such nodal systems. Some symmetry protected nodal points or nodal lines which are located in high symmetry lines or mirror planes are easy to find. While the other nodes which are located anywhere in the BZ need more efforts to be found. Here we introduce an algorithm trying to find all the nodal points.  \n\nBasically, node points are local minima of the energy gap function in 3D BZ. Local minima can be obtained by using some well known multidimensional minimization methods, e.g., Nelder and Mead’s Downhill Simplex Method [49], Conjugate Gradient Methods [50], QuasiNewton Methods [51] et al. However, the local minimum obtained from those methods depends on a initial point. One initial point gives only one local minimum. So, in order to find all the nodes, we have to choose different initial points in the whole 3D BZ. WannierTools takes a uniform mesh of the 1st BZ as a set of initial points for the Nelder and Mead’s Downhill Simplex Method [49]. Eventually, the nodes will be selected out from a set of local minima. It is easy to check the convergence of the number of nodes by increasing the initial point mesh. This algorithm is very suitable for high throughput search of new Weyl, Dirac semimetals and nodal-line metals. It has been checked to be very efficient to find Weyl points in $\\mathsf{W T e}_{2}$ [13,52], MoTe2 [53].  \n\n# 3. Capabilities of WannierTools  \n\nThere are two kinds of tasks that WannierTools can do to study novel topological materials. a. One is to get the topology of materials’ band structure. b. The other one is to explore the properties of surface states corresponding to the bulk topology. For part a, we need to study the bulk band structure, 3D Fermi surface, density of state (DOS) to check whether the bulk material is a band insulator or a metal. Further, WCCs calculations are applied to get the $\\mathbb{Z}_{2}$ topological index or Chern number for band insulators, the nodes searching algorithm and the energy gap function calculation are applied to search for Weyl/Dirac point positions or nodal-line structures; The Berry phase and Berry curvature calculations are also aided to the classification of topology. After the topological classification is done, one can turn to part b, which means to study the bulk topology related properties, such as joint density of state (JDOS), which is related to the optical conductivity [54], electronic structure of the slab and wire systems, spin-texture of the surface states, Quasi-particle interference (QPI) pattern of surface states et al. These two main capabilities of WannierTools are listed in Tables 1 and 2. The meaning of control flags in Tables 1 and 2 is illustrated in the documentation, which is distributed with WannierTools.  \n\n# 4. Installation and usage  \n\nIn this section, we will show how to install and use the WannierTools software package.  \n\n# 4.1. Get WannierTools  \n\nWannierTools is an open source free software package. It is released on Github under the GNU General Public Licence 3.0 (GPL), and it can be downloaded directly from the public code repository: https://github.com/quanshengwu/wannier_tools.  \n\n# 4.2. Build WannierTools  \n\nTo build and install WannierTools, a Fortran 90 compiler, BLAS, and LAPACK linear algebra libraries are needed. An MPI-enabled Fortran 90 compiler is needed if you want to compile a parallel version. WannierTools can be successfully compiled using the state-of-art Intel Fortran compiler. Most of the MPI implementations, such as MPICH, MVAPICH and Intel MPI are compatible with WannierTools. The downloaded WannierTools software package is likely a compressed file with a zip or tar.gz suffix. One should uncompress it firstly, then move into the wannier_tools/soc folder and edit the Makefile file to configure the compiling environment. It is noteworthy that one should set up the Fortran compiler, BLAS and LAPACK libraries manually by modifying the following lines in Makefile file according to the user’s particular system.  \n\n$$\n\\begin{array}{l}{\\mathbf{f}90}\\\\ {\\mathbf{libs}}\\end{array}=\n$$  \n\nOnce the compiling environment is configured, the executable binary wt.x will be compiled by typing the following command in the current directory (wannier_tools/soc)  \n\n$\\$8$ make  \n\n# 4.3. Running WannierTools  \n\nBefore running WannierTools, the user must provide two files wannier90_hr.dat1 and wt.in. The file called wannier90_hr.dat containing the TB parameters has fixed format which is defined in software Wannier90 [27]. It can be generated by the software Wannier90 [27], or generated by users with a toy TB model, or generated from a discretization of $k\\cdot p$ model onto a cubic lattice. The other file wt.in is the master input file for WannierTools. It is designed to be simple and user friendly. The details of wt.in are described comprehensively in the documentation that contained within the WannierTools distribution. An example file is provided in Appendix.  \n\nAfter putting wt.in and wannier90_hr.dat in the same folder, one can run it in single processor in the same folder like this  \n\n\\$ wt.x & or in multiprocessor $\\$1$ mpirun -np 4 wt.x &  \n\nSome important information during the running process are written in WT.out, from which, you can check the running status. After the whole program is done, you would obtain two kinds of files other than WT.out. One is the data file suffixed with dat. and the other one is a plotting script for software gnuplot suffixed with gnu. You can get nice plots with gnuplot [57]. Taking a bulk band structure calculation from the examples, two files bulkek.dat and bulkek.gnu are accomplished after a successful running of WannierTools. A band structure plot bulkek.png will be generated with the following command  \n\ngnuplot bulkek.gnu  \n\n# 5. Examples  \n\nIn the past few years, WannierTools has been successfully applied in many projects, such as finding type-II Weyl semimetals $\\mathsf{W I e}_{2}$ [13,52], MoTe2 [53], triple point metals [58] ZrTe, TaN, nodal chain metals IrF4 [18], topological phase in $\\mathrm{InAs}_{1-x}{\\sf S}{\\sf b}_{x}$ [59] et al. Besides, more and more groups notice this package, and becoming users. There are several examples in the wannier_tools/examples directory like ${\\tt B i}_{2}\\mathsf{S e}_{3}$ , $\\mathsf{W T e}_{2}$ , $\\mathrm{IrF}_{4}$ . MLWF TB Hamiltonians for those materials and the necessary input files for generating those Hamiltonians can be downloaded from the Github repository [60]. The detailed hands-on tutorials for those examples are listed in the wiki of Github [61]. In this paper, a new series of topological materials called ternary silicides and ternary germanides TiPtSi, ZrPtSi, ZrPtGe, HfPtSi and HfPtGe [62,63] are exhibited as an example to show how to study topological properties of new materials with WannierTools.  \n\n# 5.1. Crystal structure and band structure  \n\nTernary silicides and ternary germanides are crystallized with the orthorhombic TiNiSi type structure in a nonsymmorphic orthorhombic space group No.62 (Pnma) [64,62], containing three glide reflections $G_{x}=\\{m_{x}|{\\textstyle\\frac{1}{2}},{\\textstyle\\frac{1}{2}},{\\textstyle\\frac{1}{2}}\\},G_{z}=\\{m_{z}|{\\textstyle\\frac{1}{2}},0,{\\textstyle\\frac{1}{2}}\\},\\tilde{G}_{y}=\\{m_{y}|0,{\\textstyle\\frac{1}{2}},0\\}$ [65], three screw rotations $S_{x}=\\{R_{2x}|\\textstyle{\\frac{1}{2}},\\ {\\frac{1}{2}},\\ {\\frac{1}{2}}\\},S_{z}=\\{R_{2z}|\\textstyle{\\frac{1}{2}},0,\\textstyle{\\frac{1}{2}}\\},S_{y}=\\{R_{2y}|0,\\textstyle{\\frac{1}{2}},0\\}$ and an inversion symmetry I. These materials seem to be interesting systems in the search for new superconducting intermetallic compounds [63]. Due to the same crystal structure and similar chemical properties, these materials show very similar band structures. In this paper, we are only focusing on HfPtGe compound.  \n\nAs mentioned in Section 4.3, we need a TB model of HfPtGe for WannierTools. Firstly, a first-principle calculation within Vienna Ab initio Simulation Package (VASP) [66,67] using Gamma centered $K$ -points mesh $8\\times13\\times7$ and energy cut $360\\mathrm{eV}$ for plane wave expansions was performed. Then the band structure and the partial density of states (PDOS) shown in Fig. 1c and d were analyzed, where the PDOS indicate that the relevant orbitals close to the Fermi level are dominated by Hf 5d orbitals and Pt 5d orbitals, besides, they are also hybridized with Ge 4p and Pt s orbitals. In the end, a 112-band MLWF TB Hamiltonian with Hf 5d, Pt 6s 5d and Ge 4p as projectors are constructed with Wannier90. Fig . 1c shows that the band structures calculated from the MLWF TB model are quite fitted to the first-principle calculated band structures. After the successful construction of MLWF TB model, the master input file wt.in is needed, which is attached in the Appendix.  \n\n![](images/45ddf1becc6bcf64850d1a023f75cac6769ce8219198e3c30b4687ee9060eac1.jpg)  \nFig. 1. a. Crystal structure of HfPtGe. The gray plane is (010) plane. b. 3D Brillouin Zone (BZ) and 2D BZ for (010) surface c. Band structure of HfPtGe with SOC, the red lines are from Wannier TB model, the black lines are from first-principle calculation. d. Partial density of states (PDOS). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)  \n\n![](images/836966da5b161fbebcd1abab722a4661d46112b5916f90f26a959c8e1625a61f.jpg)  \nFig. 2. Energy gap $\\varDelta E(k_{x},k_{z})$ between the lowest conduction band and the topest valence band at $k_{y}=0$ plane. a. Without SOC, it is a nodal loop. b. With SOC, The nodal loop will be gapped out. c. The nodal loop distribution in the momental space $k_{x}–k_{z}$ and energy space E. The blue plane is a energy fixed plane $E=E_{F}$ . d. Fermi surface plot, purple and cyan pockets represent hole and electron pockets respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)  \n\n# 5.2. Energy gap shape  \n\nThere are two ‘‘Dirac like’’ cones along Y –Γ –Z direction which are shown in the band structure plot Fig. 1c. From the literature, we know that such cones could be originated from a nodal line structure without SOC [16]. Calculations of the energy gap at $k_{y}=0$ plane both for without SOC and with SOC cases were performed to study the details of positions of these ’’Dirac like’’ cones. The results are shown in Fig. 2a and b, where there is a gapless nodal line protected by the $\\tilde{G}_{y}$ mirror symmetry [16] in the ${\\sf k}_{\\mathrm{y}}=0$ plane without SOC , while, the gapless line will be gapped when SOC turns on. One thing should mentioned is that the SOC strength of this material is very weak, and the opened gap is not very big. The smallest gap is about $0.1\\mathrm{meV}$ where the $k$ points are located at $(\\pm0.4,0.0,\\pm0.229)$ .  \n\nWe can further study the nodal line distribution in momentum and energy space, which shown in Fig. 2c. It is clearly shown that the nodal line is not in the same energy plane. There are six nodes crossing the Fermi level. In such case, there are electron pockets and hole pockets which link together on the Fermi surface at the same time, which are shown in Fig. 2d. Those compensated hole and electron pockets will cause extremely large positive magnetoresistance [68].  \n\nThe related settings in the master input file wt.in for this section are as follows  \n\n&CONTROL   \nBulkGap_plane_calc = T   \nBulkFS_calc = T   \n/   \n&PARAMETERS   \n$\\tt N K1=\\ 101$   \nNK2= 101   \nNK3= 101   \n/   \nKPLANE_BULK   \n-0.50 0.00 -0.50 ! Original point for 3D k plane 1.00 0.00 0.00 ! The first vector to define 3d k space plane 0.00 0.00 1.00 ! The second vector to define 3d k space plane  \n\n# 5.3. Wannier charge center  \n\nFrom the gap shape calculation for the whole BZ in the SOC case, we can conclude that HfPtGe is a semimetal with a continuous finite energy gap between electron-like and hole-like bands. Similar to classification of band insulators [69], $\\mathbb{Z}_{2}$ topological indices $\\left(\\nu_{0},\\nu_{1}\\nu_{2}\\nu_{3}\\right)$ are still appropriate for such a semimetal. $\\mathbb{Z}_{2}$ number of a bulk material can be obtained through calculations of WCCs in six time reversal invariant planes $\\mathtt{k_{x}}=0$ , $\\pi$ , ${\\bf k}_{\\mathrm{y}}=0$ , $\\pi$ and $\\boldsymbol{\\mathrm{k}}_{\\mathrm{z}}=\\boldsymbol{0}$ , $\\pi$ plane. The results calculated by WannierTools are shown in Fig. 3. It shows that the $Z_{2}$ invariant numbers are 1 for $k_{x}=0,k_{y}=0,k_{z}=0$ plane, while zeros for other planes. Eventually, the topological index is (1, 000), which indicates that HfPtGe is a ‘‘strong’’ topological material in all three reciprocal lattice directions. The related settings for this section in wt.in are as follows.  \n\n![](images/77b9b1f5a70ac67983b4160412f6d4c14bf44bf8b063b1decfa944e0f83f0692.jpg)  \nFig. 3. a. Wannier charge center evolution for the time-reversal invariant planes of HfPtGe. (a) $\\mathbb{Z}_{2}=1$ for $k_{x}=0\\left(\\mathbf{b}\\right)\\mathbb{Z}_{2}=1$ for $k_{y}=0\\left({\\mathrm{c}}\\right)\\mathbb{Z}_{2}=1$ for $k_{z}=0\\left({\\bf d}\\right)\\mathbb{Z}_{2}=0$ for $k_{x}$ $\\scriptstyle={\\frac{\\pi}{a}}$ (e) $\\mathbb{Z}_{2}=0$ for $\\begin{array}{r}{k_{y}=\\frac{\\pi}{b}}\\end{array}$ (f) $\\mathbb{Z}_{2}=0$ for $\\begin{array}{r}{k_{z}=\\frac{\\pi}{c}}\\end{array}$ .  \n\n&CONTROL   \nZ2_3D_calc = T   \n/  \n\n# 5.4. Surface state spectrums  \n\nDue to the bulk-edge correspondence, there should be topologically protected surface states of any cuts of surface for a strong topological material [69]. Here we study the (010) surface which is shown as a gray plane in Fig. 1a. The surface state spectrums calculated by WannierTools are shown in Fig. 4a, b. For a 3D strong topological insulator, there is surface Dirac cone at the $\\boldsymbol{{\\Gamma}}$ point [7]. Indeed, there is a Dirac like cone of HfPtGe at the $\\boldsymbol{{\\Gamma}}$ point. However, the dispersion of this cone is highly anisotropic and even tiled in the momentum space. Such tiled cone is result from that the nodal line is located at different energies. Fig. 4b shows a $E=E_{F}$ iso-energy plot of the surface state spectrum. From this plot, we can learn that the surface states originate from the $k$ points that have the smallest gap. The related settings for this section in wt.in are as follows.  \n\n&CONTROL   \nSlabSS_calc = T   \nSlabArc_calc $\\mathbf{\\Psi}=\\mathbf{\\Psi}_{\\mathrm{T}}$   \n/   \n&PARAMETERS   \nE_arc = 0.0 ! Fixed energy Fermi arc calculation   \n/   \nKPATH_SLAB   \n4 ! numker of k lines for a slab system   \nX -0.50 0.00 G 0.00 0.00 ! k path for a slab system   \nG 0.00 0.00 Z 0.00 0.50   \nZ 0.00 0.50 R -0.50 0.50   \nR -0.50 0.50 X -0.50 0.00   \nKPLANE_SLAB   \n-0.5 -0.5 1.0 0.0 0.0 1.0  \n\n# 5.5. Surface state spin texture  \n\nThe key ingredient to generate topological non-trivial properties is the SOC interaction, which have a similar behavior as a Lorentz force in quantum Hall states. The SOC interaction will make a spin and a momentum locked to each other, forming a spin texture in momentum space. For different topological phase, the spin texture will be different [70]. In topological insulator ${\\tt B i}_{2}{\\tt S e}_{3}$ , the spin texture is Dirac type. The spin texture calculated by WannierTools for HfPtGe is shown in Fig. 4c. The related settings for spin texture calculations in wt.in are as follows.  \n\n![](images/4aaea6a175086f00ee259e476252eb4df353fee6c666cabd9d59234cbd24b4b5.jpg)  \nFig. 4. Surface state spectrum (SSS) of HfPtGe in the presence of SOC. a. SSS along high symmetry $k$ -line at different energies. b. SSS in 2D BZ at a fixed energy $E-E_{F}=0.{\\mathrm{c}}$ Spin texture for the ${\\mathsf{S S S i n b}}$ .  \n\n&CONTROL   \nSlabSpintexture_calc $\\mathbf{\\sigma}=\\mathbf{\\sigma}$ T   \n/  \n\nKPLANE_SLAB -0.50 -0.50 1.00 0.00 0.00 1.00  \n\n# 6. Conclusions  \n\nIn conclusion, we presented an open-source software package called WannierTools. It is very user-friendly and is written in Fortran90, using MPI techniques to get excellent performance in computer cluster. We showed how to use this software package to identify the topological properties for a new material and to get the surface state spectrum which can be compared with experimental data. As an example, we explored a new topological material HfPtGe, which was identified as a Dirac nodal line semimetal.  \n\n# Acknowledgments  \n\nQSW would like to thank Xi Dai, Zhong Fang, Lei Wang, Li Huang, Dominik Gresch, Zhida Song for helpful discussions. Especially, QSW appreciates Rui Yu and Haijun Zhang for their kindly helps at the beginning of this project. QSW, AAS, MT were supported by Microsoft Research, and the Swiss National Science Foundation through the National Competence Centers in Research MARVEL and QSIT, ERC Advanced Grant SIMCOFE. QSW was also supported by the National Natural Science Foundation of China (11404024). SNZ was supported by NSF-China under Grants No. 11074174. HFS was supported by National High Technology Research and Development Program of China under Grant 2015AA01A304, and Science Challenge Project No. JCKY2016212A502. This job was started in IOP CAS, finished in ETH Zurich.  \n\n# Appendix. wt.in for HfPtGe  \n\n&TB_FILE   \nHrfile $\\mathbf{\\sigma}=\\mathbf{\\sigma}$ \"wannier90_hr.dat\"   \n/   \n&CONTROL   \nBulkBand_calc ${\\begin{array}{l}{=\\mathrm{~T~}}\\\\ {=\\mathrm{~T~}}\\\\ {=\\mathrm{~T~}}\\\\ {=\\mathrm{~T~}}\\\\ {=\\mathrm{~T~}}\\\\ {=\\mathrm{~T~}}\\\\ {=\\mathrm{~T~}}\\end{array}}$   \nBulkFS_calc   \nBulkGap_plane_calc   \nZ2_3D_calc   \nSlabSS_calc   \nSlabArc_calc   \nSlabSpintexture_calc   \n/   \n&SYSTEM   \nNumOccupied $\\mathbf{\\sigma}=\\mathbf{\\sigma}$ 64   \n$\\mathsf{S}0\\mathsf{C}=\\mathsf{\\Omega}1$   \nE_FERMI $\\mathbf{\\tau}=\\mathbf{\\tau}$ 8.4551   \n/  \n\n&PARAMETERS E_arc $\\mathit{\\Theta}=\\ 0.0$  \n\nOmegaNum $\\mathit{\\Theta}=\\ 100\\$   \nOmegaMin $\\mathbf{\\epsilon}=\\mathbf{\\epsilon}-\\mathbf{0}.6$   \nOmegaMax $\\mathbf{\\tau}=\\mathbf{\\tau}$ 0.6   \n$\\mathtt{N k1}\\ =\\ 101$   \n$\\mathtt{N k}2\\ =\\ 201$   \n$\\mathtt{N k3}\\ =\\ 101$   \n${\\tt N P}=2$   \nGap_threshold = 0.05   \n/ LATTICE   \nAngstrom   \n6.6030000 0.0000000 0.0000000 0.0000000 3.9500000 0.0000000 0.0000000 0.0000000 7.6170000  \n\nATOM_POSITIONS   \n12   \nDirect   \nHf 0.029900 0.250000 0.186300   \nHf 0.470100 -0.250000 0.686300   \nHf -0.029900 0.750000 -0.186300   \nHf 0.529900 0.250000 0.313700   \nGe 0.750600 0.250000 0.621500   \nGe -0.250600 -0.250000 1.121500   \nGe -0.750600 0.750000 -0.621500   \nGe 1.250600 0.250000 -0.121500   \nPt 0.142500 0.250000 0.561700   \nPt 0.357500 -0.250000 1.061700   \nPt -0.142500 0.750000 -0.561700   \nPt 0.642500 0.250000 -0.061700  \n\n# PROJECTORS  \n\n5 5 5 5 3 3 3 3 6 6 6 6   \nHf dxy dyz dxz dx2-y2 dz2 Hf dxy dyz dxz dx2-y2 dz2 Hf dxy dyz dxz dx2-y2 dz2 Hf dxy dyz dxz dx2-y2 dz2 Ge px py pz   \nGe px py pz   \nGe px py pz   \nGe px py pz   \nPt s dxy dyz dxz dx2-y2 dz2 Pt s dxy dyz dxz dx2-y2 dz2 Pt s dxy dyz dxz dx2-y2 dz2 Pt s dxy dyz dxz dx2-y2 dz2  \n\nSURFACE $\\begin{array}{c c c}{{1}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{1}}\\\\ {{0}}&{{-1}}&{{0}}\\end{array}$  \n\n<html><body><table><tr><td colspan=\"8\">KPATH_BULK</td></tr><tr><td>9</td><td>Y -0.50000</td><td>0.00000</td><td>0.00000</td><td>G</td><td>0.00000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>G</td><td>0.00000</td><td>0.00000</td><td>0.00000</td><td>Z</td><td>0.00000</td><td>0.00000</td><td>0.50000</td></tr><tr><td>Z</td><td>0.00000</td><td>0.00000</td><td>0.50000</td><td></td><td>T -0.50000</td><td>0.00000</td><td>0.50000</td></tr><tr><td>T</td><td>-0.50000</td><td>0.00000</td><td>0.50000</td><td>Y</td><td>-0.50000</td><td>0.00000</td><td>0.00000</td></tr><tr><td>Y</td><td>-0.50000</td><td>0.00000</td><td>0.00000</td><td>S</td><td>-0.50000</td><td>0.50000</td><td>0.00000</td></tr><tr><td>S</td><td>-0.50000</td><td>0.50000</td><td>0.00000</td><td>X</td><td>0.00000</td><td>0.50000</td><td>0.00000</td></tr><tr><td>X</td><td>0.00000</td><td>0.50000</td><td>0.00000</td><td>U</td><td>0.00000</td><td>0.50000</td><td>0.50000</td></tr><tr><td>U</td><td>0.00000</td><td>0.50000</td><td>0.50000</td><td></td><td>R -0.50000</td><td>0.50000</td><td>0.50000</td></tr><tr><td></td><td>R -0.50000</td><td>0.50000</td><td>0.50000</td><td></td><td>S -0.50000</td><td>0.50000</td><td>0.00000</td></tr></table></body></html>  \n\nX -0.50 0.00 G 0.00 0.00   \nG 0.00 0.00 Z 0.00 0.50   \nZ 0.00 0.50 R -0.50 0.50   \nR -0.50 0.50 X -0.50 0.00  \n\nKPLANE_SLAB -0.5 -0.5 1.0 0.0 0.0 1.0  \n\nKPLANE_BULK -0.50 0.00 -0.50 1.00 0.00 0.00 0.00 0.00 1.00  \n\nKCUBE_BULK   \n-0.50 -0.50 -0.50   \n1.00 0.00 0.00   \n0.00 1.00 0.00   \n0.00 0.00 1.00  \n\n# References  \n\n[1] X.-L. 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    "Target": null
  },
  {
    "id": "10.1103_PhysRevLett.120.145301",
    "DOI": "10.1103/PhysRevLett.120.145301",
    "DOI Link": "http://dx.doi.org/10.1103/PhysRevLett.120.145301",
    "Relative Dir Path": "mds/10.1103_PhysRevLett.120.145301",
    "Article Title": "Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties",
    "Authors": "Xie, T; Grossman, JC",
    "Source Title": "PHYSICAL REVIEW LETTERS",
    "Abstract": "The use of machine learning methods for accelerating the design of crystalline materials usually requires manually constructed feature vectors or complex transformation of atom coordinates to input the crystal structure, which either constrains the model to certain crystal types or makes it difficult to provide chemical insights. Here, we develop a crystal graph convolutional neural networks framework to directly learn material properties from the connection of atoms in the crystal, providing a universal and interpretable representation of crystalline materials. Our method provides a highly accurate prediction of density functional theory calculated properties for eight different properties of crystals with various structure types and compositions after being trained with 104 data points. Further, our framework is interpretable because one can extract the contributions from local chemical environments to global properties. Using an example of perovskites, we show how this information can be utilized to discover empirical rules for materials design.",
    "Times Cited, WoS Core": 1518,
    "Times Cited, All Databases": 1718,
    "Publication Year": 2018,
    "Research Areas": "Physics",
    "UT (Unique WOS ID)": "WOS:000429451000012",
    "Markdown": "# Crystal Graph Convolutional Neural Networks for an Accurate and Interpretable Prediction of Material Properties  \n\nTian Xie and Jeffrey C. Grossman Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA  \n\n![](images/223cc87ba20d1fa320f9de6875fa13da8d8be5a0dd0cb9df369c331c81bef3df.jpg)  \n\n(Received 18 October 2017; revised manuscript received 15 December 2017; published 6 April 2018)  \n\nThe use of machine learning methods for accelerating the design of crystalline materials usually requires manually constructed feature vectors or complex transformation of atom coordinates to input the crystal structure, which either constrains the model to certain crystal types or makes it difficult to provide chemical insights. Here, we develop a crystal graph convolutional neural networks framework to directly learn material properties from the connection of atoms in the crystal, providing a universal and interpretable representation of crystalline materials. Our method provides a highly accurate prediction of density functional theory calculated properties for eight different properties of crystals with various structure types and compositions after being trained with $10^{4}$ data points. Further, our framework is interpretable because one can extract the contributions from local chemical environments to global properties. Using an example of perovskites, we show how this information can be utilized to discover empirical rules for materials design.  \n\nDOI: 10.1103/PhysRevLett.120.145301  \n\nMachine learning (ML) methods are becoming increasingly popular in accelerating the design of new materials by predicting material properties with accuracy close to ab initio calculations, but with computational speeds orders of magnitude faster [1–3]. The arbitrary size of crystal systems poses a challenge as they need to be represented as a fixed length vector in order to be compatible with most ML algorithms. This problem is usually resolved by manually constructing fixed length feature vectors using simple material properties [1,3–6] or designing symmetry-invariant transformations of atom coordinates [7–9]. However, the former requires a case-by-case design for predicting different properties, and the latter makes it hard to interpret the models as a result of the complex transformations.  \n\nIn this Letter, we present a generalized crystal graph convolutional neural networks (CGCNN) framework for representing periodic crystal systems that provides both material property prediction with density functional theory (DFT) accuracy and atomic level chemical insights. Recent advances in “deep learning” have enabled learning from a very raw representation of data, e.g., pixels of an image, making it possible to build general models that outperform traditionally expert designed representations [10]. By looking into the simplest form of crystal representation, i.e., the connection of atoms in the crystal, we directly build convolutional neural networks on top of crystal graphs generated from crystal structures. The CGCNN achieves similar accuracy with respect to DFT calculations as DFT compared with experimental data for eight different properties after being trained with data from the Materials Project [11], indicating the generality of this method. We also demonstrate the interpretability of the CGCNN by extracting the energy of each site in the perovskite structure from the total energy, an example of learning the contribution of local chemical environments to the global property. The empirical rules generalized from the results are consistent with the common knowledge for discovering more stable perovskites and can significantly reduce the search space for high throughput screening.  \n\nThe main idea in our approach is to represent the crystal structure by a crystal graph that encodes both atomic information and bonding interactions between atoms, and then build a convolutional neural network on top of the graph to automatically extract representations that are optimum for predicting target properties by training with DFT calculated data. As illustrated in Fig. 1(a), a crystal graph $\\mathcal{G}$ is an undirected multigraph which is defined by nodes representing atoms and edges representing connections between atoms in a crystal (the method for determining atom connectivity is explained in the Supplemental Material [12]). The crystal graph is unlike normal graphs since it allows multiple edges between the same pair of end nodes, a characteristic for crystal graphs due to their periodicity, in contrast to molecular graphs. Each node $i$ is represented by a feature vector $\\boldsymbol{\\nu}_{i:}$ , encoding the property of the atom corresponding to node $i$ . Similarly, each edge $(i,j)_{k}$ is represented by a feature vector $\\pmb{u}_{(i,j)_{k}}$ corresponding to the kth bond connecting atom $i$ and atom $j$ .  \n\nThe convolutional neural networks built on top of the crystal graph consist of two major components: convolutional layers and pooling layers. Similar architectures have been used for computer vision [22], natural language processing [23], molecular fingerprinting [24] and general graph-structured data [25,26], but not for crystal property prediction to the best of our knowledge. The convolutional layers iteratively update the atom feature vector $\\mathbf{\\nabla}\\nu_{i}$ by “convolution” with surrounding atoms and bonds with a nonlinear graph convolution function,  \n\n![](images/6766dc8c1eef962e14d0687e24a428cc2ee1b79f27eaef388d4fd042d80e1580.jpg)  \nFIG. 1. Illustration of the crystal graph convolutional neural networks. (a) Construction of the crystal graph. Crystals are converted to graphs with nodes representing atoms in the unit cell and edges representing atom connections. Nodes and edges are characterized by vectors corresponding to the atoms and bonds in the crystal, respectively. (b) Structure of the convolutional neural network on top of the crystal graph. $R$ convolutional layers and $L_{1}$ hidden layers are built on top of each node, resulting in a new graph with each node representing the local environment of each atom. After pooling, a vector representing the entire crystal is connected to $L_{2}$ hidden layers, followed by the output layer to provide the prediction.  \n\n$$\n\\pmb{\\nu}_{i}^{(t+1)}=\\mathbf{C}\\mathrm{onv}\\left(\\pmb{\\nu}_{i}^{(t)},\\pmb{\\nu}_{j}^{(t)},\\pmb{u}_{(i,j)_{k}}\\right),\\quad(i,j)_{k}\\in\\mathcal{G}.\n$$  \n\nAfter $R$ convolutions, the network automatically learns the feature vector viðRÞ for each atom by iteratively including its surrounding environment. The pooling layer is then used for producing an overall feature vector $\\pmb{\\nu}_{c}$ for the crystal, which can be represented by a pooling function,  \n\n$$\n\\pmb{\\nu}_{c}=\\mathrm{Pool}(\\pmb{\\nu}_{0}^{(0)},\\pmb{\\nu}_{1}^{(0)},...,\\pmb{\\nu}_{N}^{(0)},...,\\pmb{\\nu}_{N}^{(R)})\n$$  \n\nthat satisfies permutational invariance with respect to atom indexing and size invariance with respect to unit cell choice. In this work, a normalized summation is used as the pooling function for simplicity, but other functions can also be used. In addition to the convolutional and pooling layers, two fully connected hidden layers with the depths of $L_{1}$ and $L_{2}$ are added to capture the complex mapping between crystal structure and property. Finally, an output layer is used to connect the $L_{2}$ hidden layer to predict the target property $\\hat{y}.$ .  \n\nThe training is performed by minimizing the difference between the predicted property $\\hat{y}$ and the DFT calculated property $y_{:}$ defined by a cost function $J(y,\\hat{y})$ . The whole CGCNN can be considered as a function $f$ parametrized by weights W that maps a crystal $\\mathcal{C}$ to the target property $\\hat{y}$ . Using backpropagation and stochastic gradient descent (SGD), we can solve the following optimization problem by iteratively updating the weights with DFT calculated data:  \n\n$$\n\\operatorname*{min}_{W}J(y,f({\\mathcal{C}};W))\n$$  \n\nthe learned weights can then be used to predict material properties and provide chemical insights for future materials design.  \n\nIn the Supplemental Material (SM) [12], we use a simple example to illustrate how a CGCNN composed of one linear convolution layer and one pooling layer can differentiate two crystal structures. With multiple convolution layers, pooling layers, and hidden layers, the CGCNN can extract any structure differences based on the atom connections and discover the underlaying relations between structure and property.  \n\nTo demonstrate the generality of the CGCNN, we train the model using calculated properties from the Materials Project [11]. We focus on two types of generality in this work: (1) the structure types and chemical compositions for which our model can be applied and (2) the number of properties that our model can accurately predict.  \n\nThe database we used includes a diverse set of inorganic crystals ranging from simple metals to complex minerals. After removing ill-converged crystals, the full database has 46 744 materials covering 87 elements, 7 lattice systems, and 216 space groups. As shown in Fig. 2(a), the materials consist of as many as seven different elements, with $90\\%$ of them binary, ternary, and quaternary compounds. The number of atoms in the primitive cell ranges from 1 to 200, and $90\\%$ of crystals have less than 60 atoms (Fig. S2). Considering most of the crystals originate from the Inorganic Crystal Structure Database [27], this database is a good representation of known stoichiometric inorganic crystals.  \n\nThe CGCNN is a flexible framework that allows variance in the crystal graph representation, neural network architecture, and training process, resulting in different $f$ in Eq. (3) and prediction performance. To choose the best model, we apply a train-validation scheme to optimize the prediction of formation energies of crystals. Each model is trained with $60\\%$ of the data and then validated with $20\\%$ of the data, and the best-performing model in the validation set is selected. In our study, we find that the neural network architecture, especially the form of convolution function in Eq. (1), has the largest impact on prediction performance. We start with a simple convolution function,  \n\n$$\n\\boldsymbol{\\nu}_{i}^{(t+1)}=g\\Biggl[\\biggl(\\sum_{j,k}\\boldsymbol{\\nu}_{j}^{(t)}\\oplus\\boldsymbol{u}_{(i,j)_{k}}\\biggr)\\boldsymbol{W}_{c}^{(t)}+\\boldsymbol{\\nu}_{i}^{(t)}\\boldsymbol{W}_{s}^{(t)}+\\boldsymbol{b}^{(t)}\\Biggr],\n$$  \n\n![](images/df8964f1403efe102e368038833d91a07e93e3193a380b9c273975032572f590.jpg)  \nFIG. 2. Performance of CGCNN on the Materials Project database [11]. (a) Histogram representing the distribution of the number of elements in each crystal. (b) Mean absolute error as a function of training crystals for predicting formation energy per atom using different convolution functions. The shaded area denotes the MAEs of DFT calculations compared with experiments [28]. (c) 2D histogram representing the predicted formation per atom against DFT calculated value. (d) Receiver operating characteristic curve visualizing the result of metalsemiconductor classification. It plots the proportion of correctly identified metals (true positive rate) against the proportion of wrongly identified semiconductors (false positive rate) under different thresholds.  \n\nwhere $\\oplus$ denotes concatenation of atom and bond feature vectors, $\\boldsymbol{W}_{c}^{(t)}$ , $\\boldsymbol{W}_{s}^{(t)}$ , and $\\mathbf{\\boldsymbol{b}}^{(t)}$ are the convolution weight matrix, self-weight matrix, and bias of the tth layer, respectively, and $g$ is the activation function for introducing nonlinear coupling between layers. By optimizing hyperparameters in Table S1, the lowest mean absolute error (MAE) for the validation set is $0.108\\ \\mathrm{eV},$ =atom. One limitation of Eq. (4) is that it uses a shared convolution weight matrix $\\mathbf{\\Delta}W_{c}^{(t)}$ for all neighbors of $i$ , which neglects the differences of interaction strength between neighbors. To overcome this problem, we design a new convolution function that first concatenates neighbor vectors $\\boldsymbol{z}_{(i,j)_{k}}^{(t)}=\\boldsymbol{\\nu}_{i}^{(t)}\\oplus\\boldsymbol{\\nu}_{j}^{(t)}\\oplus\\boldsymbol{u}_{(i,j)_{k}}$ , then perform convolution by  \n\n$$\n\\begin{array}{c}{{\\pmb{\\nu}_{i}^{(t+1)}=\\pmb{\\nu}_{i}^{(t)}+\\displaystyle\\sum_{j,k}\\sigma\\Bigl(z_{(i,j)_{k}}^{(t)}{\\pmb{W}}_{f}^{(t)}+{\\pmb{b}}_{f}^{(t)}\\Bigr)}}\\\\ {{\\odot g\\Bigl(z_{(i,j)_{k}}^{(t)}{\\pmb{W}}_{s}^{(t)}+{\\pmb{b}}_{s}^{(t)}\\Bigr),}}\\end{array}\n$$  \n\nwhere $\\odot$ denotes element-wise multiplication and $\\sigma$ denotes a sigmoid function. In Eq. (5), the $\\sigma(\\cdot)$ functions as a learned weight matrix to differentiate interactions between neighbors and adding $\\mathbf{\\boldsymbol{\\nu}}_{i}^{(t)}$ makes learning deeper networks easier [29]. We achieve MAE on the validation set of $0.039\\ \\mathrm{eV}.$ =atom using the modified convolution function, a significant improvement compared to Eq. (4). In Fig. S3, we compare the effects of several other hyperparameters on the MAE which are much smaller than the effect of the convolution function.  \n\nFigures 2(b) and 2(c) show the performance of the two models on 9350 test crystals for predicting the formation energy per atom. We find a systematic decrease of the MAE of the predicted values compared with DFT calculated values for both convolution functions as the number of training data is increased. The best MAEs we achieved with Eqs. (4) and (5) are 0.136 and $0.039\\ \\mathrm{eV},$ =atom, respectively, and $90\\%$ of the crystals are predicted within 0.3 and $0.08\\ \\mathrm{eV}.$ =atom errors. In comparison, Kirklin et al. reports that the MAE of the DFT calculation with respect to experimental measurements in the Open Quantum Materials Database is $0.081{-}0.136~\\mathrm{eV/atom}$ depending on whether the energies of the elemental reference states are fitted, although they also find a large MAE of $0.082\\ \\mathrm{eV}_{I}$ atom between different sources of experimental data. Given the comparison, our CGCNN approach provides a reliable estimation of DFT calculations and can potentially be applied to predict properties calculated by more accurate methods like GW [30] and quantum Monte Carlo calculations [31].  \n\nAfter establishing the generality of the CGCNN with respect to the diversity of crystals, we next explore its prediction performance for different material properties. We apply the same framework to predict the absolute energy, band gap, Fermi energy, bulk moduli, shear moduli, and Poisson ratio of crystals using DFT calculated data from the Materials Project [11]. The prediction performance of Eq. (5) is improved compared to Eq. (4) for all six properties (Table S4). We summarize the performance in Table I and the corresponding 2D histograms in Fig. S4. As we can see, the MAEs of our model are close to or higher than DFT accuracy relative to experiments for most properties when ${\\sim}10^{4}$ training data are used. For elastic properties, the errors are higher since less data are available, and the accuracy of DFT relative to experiments can be expected if ${\\sim}10^{4}$ training data are available (Fig. S5).  \n\nTABLE I. Summary of the prediction performance of seven different properties on test sets.   \n\n\n<html><body><table><tr><td>Property</td><td># of train data</td><td>Unit MAEmodel</td><td>MAEDFT</td></tr><tr><td>Formation</td><td>28046</td><td>eV/atom 0.039</td><td>0.081-0.136 [28]</td></tr><tr><td>energy Absolute</td><td>28046</td><td>eV/atom</td><td>0.072</td></tr><tr><td>energy Band gap</td><td>16458</td><td>eV</td><td>0.388 0.6 [32]</td></tr><tr><td>Fermi energy</td><td>28046</td><td>eV</td><td>0.363</td></tr><tr><td>Bulk moduli</td><td>2041</td><td>log(GPa)</td><td>0.054 0.050 [13]</td></tr><tr><td>Shear moduli</td><td>2041</td><td>log(GPa)</td><td>0.087 0.069 [13]</td></tr><tr><td>Poisson ratio</td><td>2041</td><td></td><td>0.030</td></tr></table></body></html>  \n\nRecently, Jong et al. [33] developed a statistical learning (SL) framework using multivariate local regression on crystal descriptors to predict elastic properties using the same data from the Materials Project. By using the same number of training data, our model achieves root mean squared error (RMSE) on test sets of $0.105\\log(\\mathrm{GPa})$ and $0.127\\ \\log(\\mathrm{GPa})$ for the bulk and shear moduli, which is similar to the RMSE of SL on the entire data set of 0.0750 $\\mathrm{log(GPa)}$ and $0.1378~\\mathrm{log(GPa)}$ . Comparing the two methods, the CGCNN predicts properties by extracting features only from the crystal structure, while SL depends on crystal descriptors like cohesive energy and volume per atom. Recently, 1585 new crystals with elastic properties have been uploaded to the Materials Project database. Our model in Table I achieves MAE of $0.077\\ \\log(\\mathrm{GPa})$ for bulk moduli and $0.114\\ \\log(\\mathrm{GPa})$ for shear moduli on these crystals, showing good generalization to materials from potentially different crystal groups.  \n\nIn addition to predicting continuous properties, the CGCNN can also predict discrete properties by changing the output layer. By using a softmax activation function for the output layer and a cross entropy cost function, we can predict the classifications of metal and semiconductor with the same framework. In Fig. 2(d), we show the receiver operating characteristic curve of the prediction on 9350 test crystals. Excellent prediction performance is achieved with the area under the curve at 0.95. By choosing a threshold of 0.5, we get metal prediction accuracy at 0.80, semiconductor prediction accuracy at 0.95, and overall prediction accuracy at 0.90.  \n\nModel interpretability is a desired property for any ML algorithm applied in materials science, because it can provide additional information for material design which may be more valuable than simply screening a large number of materials. However, nonlinear functions are needed to learn the complex structure-property relations, resulting in ML models that are difficult to interpret. The CGCNN resolves this dilemma by separating the convolution and pooling layers. After the $R$ convolutional and $L_{1}$ hidden layers, we map the last atom feature vector viðRÞ to a scalar ˜vi and perform a linear pooling to predict the target property directly without the $L_{2}$ hidden layers (details discussed in SM [12]). Therefore, we can learn the contribution of different local chemical environments, represented by $\\tilde{v}_{i}$ for each atom, to the target property while maintaining a model with high capacity to ensure the prediction performance.  \n\nWe demonstrate how this local chemical environment related information can be used to provide chemical insights and guide the material design by a specific example: learning the energy of each site in perovskites from the total energy above hull data. Perovskite is a crystal structure type with the form of $A B X_{3}$ , where the site $A$ atom sits at a corner position, the site $B$ atom sits at a body centered position, and site $X$ atoms sit at face centered positions [Fig. 3(a)]. The database [34] we use includes the energy above hull of 18 928 perovskite crystals, in which $A$ and $B$ sites can be any nonradioactive metals and $X$ sites can be one or several elements from O, N, S, and F. We use the CGCNN with a linear pooling to predict the total energy above hull of perovskites in the database, using Eq. (4) as the convolution function. The resulting MAE on 3787 test perovskites is $0.130\\ \\mathrm{eV},$ =atom as shown in Fig. 3(b), which is slightly higher than using a complete pooling layer and $L_{2}$ hidden layers $\\operatorname{\\langle0.099\\eV/}$ atom as shown in Fig. S6) due to the additional constraints introduced by the simplified pooling layer. However, this CGCNN allows us to learn the energy of each site in the crystal while training with the total energy above hull, providing additional insights for material design. Figures 3(c) and 3(d) visualize the mean of the predicted site energies when each element occupies the $A$ and $B$ site, respectively. The most stable elements that occupy the $A$ site are those with large radii due to the space needed for 12 coordinations. In contrast, elements with small radii like Be, B, and Si are the most unstable for occupying the $A$ site. For the $B$ site, elements in groups 4, 5, and 6 are the most stable throughout the periodic table. This can be explained by crystal field theory, since the configuration of $d$ electrons of these elements favors the octahedral coordination in the $B$ site. Interestingly, the visualization shows that large atoms from groups 13–15 are stable in the $A$ site, in addition to the well-known region of groups 1–3 elements. Inspired by this result, we applied a combinatorial search for stable perovskites using elements from groups 13–15 as the $A$ site and groups 4–6 as the $B$ site. Because of the theoretical inaccuracies of DFT calculations and the possibility of metastable phases that can be stabilized by temperature, defects, and substrates, many synthesizable inorganic crystals have positive calculated energies above hull at $0\\mathrm{~K~}$ . Some metastable nitrides can even have energies up to $0.2{\\mathrm{~eV}},$ =atom above hull as a result of the strong bonding interactions [35]. In this work, since some of the perovskites are also nitrides, we choose to set the cutoff energy for potential synthesizability at $0.2\\ \\mathrm{eV}$ =atom. We discovered 33 perovskites that fall within this threshold out of 378 in the entire data set, among which 8 are within the cutoff out of 58 in the test set (Table S5). Many of these compounds like ${\\mathrm{PbTiO}}_{3}$ [36], $\\mathrm{Pb}{\\mathrm{Zr}}\\mathrm{O}_{3}$ [36], $\\mathrm{SnTaO}_{3}$ [37], and $\\mathrm{PbMoO}_{3}$ [38] have been experimentally synthesized. Note that $\\mathrm{PbMoO}_{3}$ has calculated energy of $0.18\\ \\mathrm{eV}_{/}$ =atom above hull, indicating that our choice of cutoff energy is reasonable. In general, chemical insights gained from the CGCNN can significantly reduce the search space for high throughput screening. In comparison, there are only 228 potentially synthesizable perovskites out of 18 928 in our database: the chemical insight increased the search efficiency by a factor of 7.  \n\n![](images/6a2af6ea80c6176da76eb63a78fd7ada69d9e4c2b96e63feaaf9dfcbadaeda12.jpg)  \nFIG. 3. Extraction of site energy of perovskites from total formation energy. (a) Structure of perovskites. (b) 2D histogram representing the predicted total energy above hull against DFT calculated value. (c),(d) Periodic table with the color of each element representing the mean of the site energy when the element occupies $A$ site (c) or $B$ site (d).  \n\nIn summary, the crystal graph convolutional neural networks present a flexible machine learning framework for material property prediction and design knowledge extraction. The framework provides a reliable estimation of DFT calculations using around $10^{4}$ training data for eight properties of inorganic crystals with diverse structure types and compositions. As an example of knowledge extraction, we apply this approach to the design of new perovskite materials and show that information extracted from the model is consistent with common chemical insights and significantly reduces the search space for high throughput screening.  \n\nThe code for the CGCNN is available from Ref. [39].  \n\nThis work was supported by Toyota Research Institute. Computational support was provided through the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231, and the Extreme Science and Engineering Discovery Environment, supported by National Science Foundation Grant No. ACI-1053575.  \n\n[1] A. Seko, A. Togo, H. Hayashi, K. Tsuda, L. Chaput, and I. Tanaka, Phys. Rev. Lett. 115, 205901 (2015).   \n[2] F. A. Faber, A. Lindmaa, O. A. von Lilienfeld, and R. Armiento, Phys. Rev. Lett. 117, 135502 (2016).   \n[3] D. Xue, P. V. Balachandran, J. Hogden, J. Theiler, D. Xue, and T. Lookman, Nat. Commun. 7, 11241 (2016).   \n[4] O. Isayev, C. Oses, C. Toher, E. Gossett, S. Curtarolo, and A. Tropsha, Nat. Commun. 8, 15679 (2017). [5] L. M. Ghiringhelli, J. Vybiral, S. V. Levchenko, C. Draxl, and M. Scheffler, Phys. Rev. Lett. 114, 105503 (2015). [6] O. Isayev, D. Fourches, E. N. Muratov, C. Oses, K. Rasch, A. 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Phys. 73, 33 (2001).   \n[32] A. Jain, G. Hautier, C. J. Moore, S. P. Ong, C. C. Fischer, T. Mueller, K. A. Persson, and G. Ceder, Comput. Mater. Sci. 50, 2295 (2011).   \n[33] M. De Jong, W. Chen, R. Notestine, K. Persson, G. Ceder, A. Jain, M. Asta, and A. Gamst, Sci. Rep. 6, 34256 (2016).   \n[34] I. E. Castelli, T. Olsen, S. Datta, D. D. Landis, S. Dahl, K. S. Thygesen, and K. W. Jacobsen, Energy Environ. Sci. 5, 5814 (2012).   \n[35] W. Sun, S. T. Dacek, S. P. Ong, G. Hautier, A. Jain, W. D. Richards, A. C. Gamst, K. A. Persson, and G. Ceder, Sci. Adv. 2, e1600225 (2016).   \n[36] G. Shirane, K. Suzuki, and A. Takeda, J. Phys. Soc. Jpn. 7, 12 (1952).   \n[37] J. Lang, C. Li, X. Wang et al., Mater. Today: Proc. 3, 424 (2016).   \n[38] H. Takatsu, O. Hernandez, W. Yoshimune, C. Prestipino, T. Yamamoto, C. Tassel, Y. Kobayashi, D. Batuk, Y. Shibata, A. M. Abakumov et al., Phys. Rev. B 95, 155105 (2017).   \n[39] CGCNN website, https://github.com/txie-93/cgcnn.  ",
    "Target": null
  },
  {
    "id": "10.1103_PhysRevLett.120.143001",
    "DOI": "10.1103/PhysRevLett.120.143001",
    "DOI Link": "http://dx.doi.org/10.1103/PhysRevLett.120.143001",
    "Relative Dir Path": "mds/10.1103_PhysRevLett.120.143001",
    "Article Title": "Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics",
    "Authors": "Zhang, LF; Han, JQ; Wang, H; Car, R; Weinull, E",
    "Source Title": "PHYSICAL REVIEW LETTERS",
    "Abstract": "We introduce a scheme for molecular simulations, the deep potential molecular dynamics (DPMD) method, based on a many-body potential and interatomic forces generated by a carefully crafted deep neural network trained with ab initio data. The neural network model preserves all the natural symmetries in the problem. It is first-principles based in the sense that there are no ad hoc components aside from the network model. We show that the proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and molecules. In all these cases, DPMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size.",
    "Times Cited, WoS Core": 1319,
    "Times Cited, All Databases": 1452,
    "Publication Year": 2018,
    "Research Areas": "Physics",
    "UT (Unique WOS ID)": "WOS:000429119100003",
    "Markdown": "# Deep Potential Molecular Dynamics: A Scalable Model with the Accuracy of Quantum Mechanics  \n\nLinfeng Zhang and Jiequn Han Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA  \n\nHan Wang\\* Institute of Applied Physics and Computational Mathematics, Fenghao East Road 2, Beijing 100094, People’s Republic of China and CAEP Software Center for High Performance Numerical Simulation, Huayuan Road 6, Beijing 100088, People’s Republic of China  \n\nRoberto Car Department of Chemistry, Department of Physics, Program in Applied and Computational Mathematics, Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA  \n\nWeinan E†   \nDepartment of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA   \nand Center for Data Science, Beijing International Center for Mathematical Research, Peking University, Beijing Institute of Big Data Research, Beijing 100871, People’s Republic of China  \n\n(Received 3 August 2017; published 4 April 2018)  \n\nWe introduce a scheme for molecular simulations, the deep potential molecular dynamics (DPMD) method, based on a many-body potential and interatomic forces generated by a carefully crafted deep neural network trained with ab initio data. The neural network model preserves all the natural symmetries in the problem. It is first-principles based in the sense that there are no ad hoc components aside from the network model. We show that the proposed scheme provides an efficient and accurate protocol in a variety of systems, including bulk materials and molecules. In all these cases, DPMD gives results that are essentially indistinguishable from the original data, at a cost that scales linearly with system size.  \n\nDOI: 10.1103/PhysRevLett.120.143001  \n\nMolecular dynamics (MD) is used in many disciplines, including physics, chemistry, biology, and materials science, but its accuracy depends on the model for the atomic interactions. $A b$ initio molecular dynamics (AIMD) [1,2] has the accuracy of density functional theory (DFT) [3], but its computational cost limits typical applications to hundreds of atoms and time scales of ${\\sim}100$ ps. Applications requiring larger cells and longer simulations are currently accessible only with empirical force fields (FFs) [4–6], but the accuracy and transferability of these models is often in question.  \n\nDeveloping FFs is challenging due to the many-body character of the potential energy. Expansions in two- and three-body interactions may capture the physics [7], but are strictly valid only for weakly interacting systems. A large class of potentials, including the embedded atom method (EAM) [8], the bond order potentials [9], and the reactive FFs [10], share the physically motivated idea that the strength of a bond depends on the local environment, but the functional form of this dependence can only be given with crude approximations.  \n\nMachine learning (ML) methodologies are changing this state of affairs [11–20]. When trained on large data sets of atomic configurations and corresponding potential energies and forces, ML models can reproduce the original data accurately. In training these models, the atomic coordinates cannot be used as they appear in MD trajectories because their format does not preserve the translational, rotational, and permutational symmetry of the system. Different ML models address this issue in different ways. Two successful schemes are the Behler-Parrinello neural network (BPNN) [13] and the gradient-domain machine learning (GDML) method [19]. In the BPNN, symmetry is preserved by mapping the coordinates onto a large set of two- and threebody symmetry functions, which are, however, largely ad hoc. Fixing the symmetry functions may become painstaking in systems with many atomic species. In the GDML, the same goal is achieved by mapping the coordinates onto the eigenvalues of the Coulomb matrix, whose elements are the inverse distances between all distinct pairs of atoms. It is not straightforward how to use the Coulomb matrix in extended periodic systems.  \n\nSo far, GDML has only been used for relatively small molecules.  \n\nIn this Letter, we introduce a neural network (NN) based scheme for MD simulations, called deep potential molecular dynamics (DPMD), which overcomes the limitations associated with auxiliary quantities like the symmetry functions or the Coulomb matrix (All the examples presented in this work are tested using the DeePMD-kit package [21], which is available at [22]). In our scheme, a local reference frame and a local environment is assigned to each atom. Each environment contains a finite number of atoms, whose local coordinates are arranged in a symmetry preserving way following the prescription of the deep potential method [23], an approach that was devised to train a NN with the potential energy only. With typical AIMD data sets, this is insufficient to reproduce the trajectories. DPMD overcomes this limitation. In addition, the learning process in DPMD improves significantly over the deep potential method thanks to the introduction of a flexible family of loss functions. The NN potential constructed in this way reproduces accurately the AIMD trajectories, both classical and quantum (path integral), in extended and finite systems, at a cost that scales linearly with system size and is always several orders of magnitude lower than that of equivalent AIMD simulations.  \n\nIn DPMD, the potential energy of each atomic configuration is a sum of “atomic energies” $E=\\textstyle\\sum_{i}E_{i}$ , where $E_{i}$ is determined by the local environment of atom $i$ within a cutoff radius $R_{c}$ and can be seen as a realization of the embedded atom concept. The environmental dependence of $E_{i}$ , which embodies the many-body character of the interactions, is complex and nonlinear. The NN is able to capture the analytical dependence of $E_{i}$ on the coordinates of the atoms in the environment in terms of the composition of the sequence of mappings associated with the individual hidden layers. The additive form of $E$ naturally preserves the extensive character of the potential energy. Because of the analyticity of the atomic energies, DPMD is, in principle, a conservative model.  \n\n$E_{i}$ is constructed in two steps. First, a local coordinate frame is set up for every atom and its neighbors inside $R_{c}$ [24]. This allows us to preserve the translational, rotational, and permutational symmetries of the environment, as shown in Fig. 1, which illustrates the format adopted for the local coordinate information $\\{D_{i j}\\}$ . The $1/R_{i j}$ factor present in $D_{i j}$ reduces the weight of the particles that are more distant from atom $i$ .  \n\nNext, $\\{D_{i j}\\}$ serves as input of a deep neural network (DNN) [25], which returns $E_{i}$ in output (Fig. 2). The DNN is a feed forward network, in which data flow from the input layer to the output layer $(E_{i})$ , through multiple hidden layers consisting of several nodes that input the data $d_{l}^{\\mathrm{in}}$ from the previous layer and output the data $d_{k}^{\\mathrm{{out}}}$ to the next layer. A linear transformation is applied to the input data, i.e., $\\tilde{\\boldsymbol{d}}_{k}=$ $\\sum_{l}w_{k l}d_{l}^{\\mathrm{in}}+b_{k}$ , followed by action of a nonlinear function $\\varphi$ on $\\tilde{\\boldsymbol{d}}_{k}$ , i.e., $d_{k}^{\\mathrm{out}}=\\varphi(\\tilde{d}_{k})$ . In the final step from the last hidden layer to $E_{i}$ , only the linear transformation is applied. The composition of the linear and nonlinear transformations introduced above provides the analytical representation of $E_{i}$ in terms of the local coordinates. The technical details of this construction are discussed in the Supplemental Material [26]. In our applications, we adopt the hyperbolic tangent for $\\varphi$ and use five hidden layers with decreasing number of nodes per layer, i.e., 240, 120, 60, 30, and 10 nodes, respectively, from the innermost to the outermost layer. It is known empirically that the hidden layers greatly enhance the capability of neural networks to fit complex and highly nonlinear functional dependences [27,28]. In our case, only by including a few hidden layers could DPMD reproduce the trajectories with sufficient accuracy.  \n\n![](images/4eec028f19f8266035bc1fb27a0dd2bd39e579ea0be08abc94d7daee995ba4c4.jpg)  \nFIG. 1. Schematic plot of the neural network input for the environment of atom $i$ , taking water as an example. Atom $j$ is a generic neighbor of atom $i$ , $(e_{x},e_{y},e_{z})$ is the local frame of atom i, $e_{x}$ is along the $_\\mathrm{O-H}$ bond, $\\boldsymbol{e}_{z}$ is perpendicular to the plane of the water molecule, $\\boldsymbol{e}_{y}$ is the cross product of $\\pmb{e}_{z}$ and $e_{x}$ , and $(x_{i j},y_{i j},z_{i j})$ are the Cartesian components of the vector $\\pmb{R}_{i j}$ in this local frame. $R_{i j}$ is the length of $\\pmb{R}_{i j}$ . The neural network input $D_{i j}$ may either contain the full radial and angular information of atom $j$ , i.e., $D_{i j}=\\{1/R_{i j},x_{i j}/R_{i j}^{2},y_{i j}/R_{i j}^{2},z_{i j}/R_{i j}^{2}\\}$ or only the radial information, i.e., $\\dot{D_{i j}}=\\{1/R_{i j}\\}$ . We first sort the neighbors of atom $i$ according to their chemical species, e.g., oxygens first then hydrogens. Within each species, we sort the atoms according to their inverse distances to atom $i$ , i.e., $1/R_{i j}$ . We use $\\{D_{i j}\\}$ to denote the sorted input data for atom $i$ .  \n\n![](images/af817281967fc2948c7458859df215c40568a2296108185e339cb8e726d03848.jpg)  \nFIG. 2. Schematic plot of the DPMD model. The frame in the box is an enlargement of a DNN. The relative positions of all neighbors with respect to atom $i$ , i.e., $\\{R_{i j}\\}$ , is first converted to $\\{D_{i j}\\}$ , then passed to the hidden layers to compute $E_{i}$ .  \n\nWe use the Adam method [29] to optimize the parameters $w_{k l}$ and $b_{k}$ of each layer with the family of loss functions  \n\n$$\nL(p_{\\epsilon},p_{f},p_{\\xi})=p_{\\epsilon}\\Delta\\epsilon^{2}+\\frac{p_{f}}{3N}\\sum_{i}\\lvert\\Delta F_{i}\\rvert^{2}+\\frac{p_{\\xi}}{9}\\lvert\\lvert\\Delta\\xi\\rvert\\rvert^{2}.\n$$  \n\nHere $\\Delta$ denotes the difference between the DPMD prediction and the training data, $N$ is the number of atoms, $\\epsilon$ is the energy per atom, $\\boldsymbol{F}_{i}$ is the force on atom $i$ , and $\\xi$ is the virial tensor $\\begin{array}{r}{\\Xi=-\\frac{1}{2}\\sum_{i}{\\pmb{R}}_{i}\\otimes{\\pmb{F}}_{i}}\\end{array}$ divided by $N.$ In Eq. (1), $p_{\\epsilon},~p_{f}$ , and $p_{\\xi}$ are tunable prefactors. When virial information is missing from the data, we set $p_{\\xi}=0$ . In order to minimize the loss function in Eq. (1) in a well balanced way, we vary the magnitude of the prefactors during training. We progressively increase $p_{\\epsilon}$ and $p_{\\xi}$ and decrease $p_{f}$ , so that the force term dominates at the beginning, while energy and virial terms become important at the end. We find that this strategy is very effective and reduces the total training time to a few core hours in all the test cases.  \n\nTo test the method, we have applied DPMD to extended and finite systems. As representative extended systems, we consider (a) liquid water at $P=1$ bar and $T=300~\\mathrm{K}$ , at the path-integral AIMD (PI-AIMD) level, (b) ice Ih at $P=1$ bar and $T=273\\mathrm{~K~}$ , at the PI-AIMD level, (c) ice Ih at $P=1$ bar and $T=330~\\mathrm{K}$ , at the classical AIMD level, and (d) ice Ih at $P=2.13$ kbar and $T=238\\mathrm{~K~}$ , which is the experimental triple point for ice I, II, and III, at the classical AIMD level. The variable periodic simulation cell contains $64~\\mathrm{H}_{2}\\mathrm{O}$ molecules in the case of liquid water and 96 $\\mathrm{H}_{2}\\mathrm{O}$ molecules in the case of ices. We adopt $R_{c}=6.0\\mathrm{~\\AA~}$ and use the full radial and angular information for the 16 oxygens and the 32 hydrogens closest to the atom at the origin, while retaining only radial information for all the other atoms within $R_{c}$ . All the ice simulations include proton disorder. Deuterons replace protons in simulations (c) and (d). The hybrid version of Perdew-Burke-Ernzerhof PBE0 $+$ Tkatchenko-Scheffler TS [30,31] functional is adopted in all cases. As representative finite systems, we consider benzene, uracil, naphthalene, aspirin, salicylic acid, malonaldehyde, ethanol, and toluene, for which classical AIMD trajectories with the Perdew-Burke-Ernzerhof PBE $+$ TS functional [31,32] are available [33]. In these systems, we set $R_{c}$ large enough to include all the atoms and use the full radial and angular information in each local frame.  \n\nWe discuss the performance of DPMD according to four criteria: (i) generality of the model; (ii) accuracy of the energy, forces, and virial tensor; (iii) faithfulness of the trajectories; and (iv) scalability and computational cost [34].  \n\nGenerality.—Bulk and molecular systems exhibit different levels of complexity. The liquid water samples include quantum fluctuations. The organic molecules differ in composition and size, and the corresponding data sets include large numbers of conformations. Yet DPMD produces satisfactory results in all cases, using the same methodology, network structure, and optimization scheme. The excellent performance of DPMD in systems so diverse suggests that the method should be applicable to harder systems such as biological molecules, alloys, and liquid mixtures.  \n\nTABLE I. The RMSE of the DPMD prediction for water and ices in terms of the energy, the forces, and/or the virial. The RMSEs of the energy and the virial are normalized by the number of molecules in the system.   \n\n\n<html><body><table><tr><td>System</td><td>Energy (meV)</td><td>Force (meV/A)</td><td>Virial (meV)</td></tr><tr><td>Liquid water</td><td>1.0</td><td>40.4</td><td>2.0</td></tr><tr><td>Ice Ih (b)</td><td>0.7</td><td>43.3</td><td>1.5</td></tr><tr><td>Ice Ih (c)</td><td>0.7</td><td>26.8</td><td>…</td></tr><tr><td>Ice Ih (d)</td><td>0.8</td><td>25.4</td><td>...</td></tr></table></body></html>  \n\nAccuracy.—We quantify the accuracy of energy, forces, and virial predictions in terms of the root-mean-square error (RMSE) in the case of water and ices (Table I) and in terms of the mean absolute error (MAE) in the case of the organic molecules (Table II). No virial information was used for the latter. In the water case, the RMSE of the forces is comparable to the accuracy of the minimization procedure in the original AIMD simulations, in which the allowed error in the forces was less than $10^{-3}$ Hartree/Bohr. In the case of the molecules, the predicted energy and forces are generally slightly better than the GDML benchmark.  \n\nMD trajectories.—In the case of water and ices, we perform path-integral or classical DPMD simulations at the thermodynamic conditions of the original models, using the I-PI software [35], but with much longer simulation time (300 ps). The average energy $\\bar{E}_{:}$ density $\\bar{\\rho}.$ , radial distribution functions (RDFs), and a representative angular distribution function (ADF), i.e., a three-body correlation function, are reproduced with high accuracy. The results are summarized in Table III. The RDFs and ADF of the quantum trajectories of water are shown in Fig. 3. The  \n\nTABLE II. The MAE of the DPMD prediction for organic molecules in terms of the energy and the forces. The numbers in parentheses are the GDML results [19].   \n\n\n<html><body><table><tr><td>Molecule</td><td>Energy (meV)</td><td>Force (meV/A)</td></tr><tr><td>Benzene</td><td>(3.0)</td><td>7.6 (10.0)</td></tr><tr><td>Uracil</td><td>(4.0)</td><td>9.8 (10.4)</td></tr><tr><td>Naphthalene</td><td>(5.2)</td><td>7.1 (10.0)</td></tr><tr><td>Aspirin</td><td>(11.7)</td><td>19.1 (42.9)</td></tr><tr><td>Salicylic acid</td><td>(5.2)</td><td>10.9 (12.1)</td></tr><tr><td>Malonaldehyde</td><td>(6.9)</td><td>12.7 (34.7)</td></tr><tr><td>Ethanol</td><td>(6.5)</td><td>8.3 (34.3)</td></tr><tr><td>Toluene</td><td>(5.2)</td><td>8.5 (18.6)</td></tr></table></body></html>  \n\nTABLE III. The equilibrium energy and density, $\\bar{\\boldsymbol{E}}$ and $\\bar{\\rho}$ , of water and ices, with DPMD and AIMD. The numbers in square brackets are the AIMD results. The numbers in parentheses are statistical uncertainties in the last one or two digits. The training AIMD trajectories for the ices are shorter and more correlated than in the water case.   \n\n\n<html><body><table><tr><td>System</td><td>E (eV/HO)</td></tr><tr><td>Liquid water</td><td>-467.678(2) [-467.679(6)]1.013(5)[1.013(20)]</td></tr><tr><td>Ice Ih (b）-467.750(1)</td><td>[-467.747(4)] 0.967(1）[0.966(6)]</td></tr><tr><td></td><td>Ice Ih (c)-468.0478(3)[-468.0557(16)] 0.950(1) [0.949(2)]</td></tr><tr><td></td><td>Ice Ih (d)-468.0942(2) [-468.1026(9)] 0.986(1) [0.985(2)]</td></tr></table></body></html>  \n\nRDFs of ice are reported in the Supplemental Material. A higher-order correlation function, the probability distribution function of the $_{0-0}$ bond orientation order parameter $Q_{6}$ [36], is additionally reported in the Supplemental Material and shows excellent agreement between DPMD and AIMD trajectories. In the case of the molecules, we perform DPMD at the same temperature of the original data, using a Langevin thermostat with a damping time $\\tau=0.1$ ps. The corresponding distributions of interatomic distances are very close to the original data (Fig. 4).  \n\nScalability and computational cost.—All the physical quantities in DPMD are sums of local contributions. Thus, after training on a relatively small system, DPMD can be directly applied to much larger systems. The computational cost of DPMD scales linearly with the number of atoms. Moreover, DPMD can be easily parallelized due to its local decomposition and the near-neighbor dependence of its atomic energies. In Fig. 5, we compare the cost of DPMD fixed-cell simulations (NVT) of liquid water with that of equivalent simulations with AIMD and the empirical FF TIP3P (transferable intermolecular potential with 3 points) [41] in units of CPU core seconds/step/molecule.  \n\nWhile in principle the environmental dependence of $E_{i}$ is analytical, in our implementation, discontinuities are present in the forces, due to adoption of a sharp cutoff radius, limitation of angular information to a fixed number of atoms, and abrupt changes in the atomic lists due to sorting. These discontinuities are similar in magnitude to those present in the AIMD forces due to finite numerical accuracy in the enforcement of the Born-Oppenheimer condition. In both cases, the discontinuities are much smaller than thermal fluctuations and perfect canonical evolution is achieved by coupling the systems to a thermostat. We further note that long-range Coulomb interactions are not treated explicitly in the current implementation, although implicitly present in the training data. Explicit treatment of Coulombic effects may be necessary in some applications and deserves further study.  \n\n![](images/c5c6d1590661c9caab6bc0aaf2cf3e3d21939ff9539c93df5c5ef8760a7695ae.jpg)  \nFIG. 3. Correlation functions of liquid water from DPMD and PI-AIMD. (Left) RDFs. (Right) The O-O-O ADF within a cutoff radius of $3.7\\mathrm{~\\AA~}$ .  \n\n![](images/4d0b597e39d0ab200bff642f5cb54b9cb0d452be9c9de6ed3d83ddffc2a4210a.jpg)  \nFIG. 4. Interatomic distance distributions of the organic molecules. The solid lines denote the DPMD results. The dashed lines denote the AIMD results.  \n\n![](images/418f1a91ae075ce8a9611a78fae6b93fd59f55b9b64aa0d966b8abc806a7f573.jpg)  \nFIG. 5. Computational cost of MD step versus system size, with DPMD, TIP3P, $\\mathrm{PBE+TS}$ , and $\\mathrm{PBE0+TS}$ . All simulations are performed on a Nersc Cori supercomputer with the Intel Xeon CPU E5-2698 v3. The TIP3P simulations use the Gromacs codes (version 4.6.7) [42]. The $\\mathrm{PBE}+\\mathrm{TS}$ and $\\mathrm{PBE0+TS}$ simulations use the Quantum Espresso codes [43].  \n\nIn conclusion, DPMD realizes a paradigm for molecular simulation, wherein accurate quantum mechanical data are faithfully parametrized by machine learning algorithms, which make possible simulations of DFT-based AIMD quality on much larger systems and for much longer time than with direct AIMD. While substantially more predictive than empirical FFs, DFT is not chemically accurate [44]. In principle, DPMD could be trained with chemically accurate data from high-level quantum chemistry [45] and/or quantum Monte Carlo calculations [46], but so far this has been prevented by the large computational cost of these calculations.  \n\nDPMD should also be very useful to coarse grain the atomic degrees of freedom, for example, by generating a NN model for a reduced set of degrees of freedom while using the full set of degrees of freedom for training. The above considerations suggest that DPMD should enhance considerably the realm of AIMD applications by successfully addressing the dilemma of accuracy versus efficiency that has confronted the molecular simulation community for a long time.  \n\nThe authors acknowledge H.-Y. Ko and B. Santra for sharing the AIMD data on water and ice. The work of J. H. and W. E. is supported in part by Major Program of NNSFC under Grant No. 91130005, ONR Grant No. N00014-13-1- 0338, DOE Awards No. DE-SC0008626 and No. DESC0009248, and NSFC Grant No. U1430237. The work of R. C. is supported in part by DOE-SciDAC Grant No. DESC0008626. The work of H. W. is supported by the National Science Foundation of China under Grants No. 11501039 and No. 91530322, the National Key Research and Development Program of China under Grants No. 2016YFB0201200 and No. 2016YFB0201203, and the Science Challenge Project No. JCKY2016212A502.  \n\n[10] A. C. T. van Duin, S. Dasgupta, F. Lorant, and W. A. Goddard, J. Phys. Chem. A 105, 9396 (2001).   \n[11] A. P. Thompson, L. P. Swiler, C. R. Trott, S. M. Foiles, and G. J. Tucker, J. Comput. Phys. 285, 316 (2015).   \n[12] T. D. Huan, R. Batra, J. Chapman, S. Krishnan, L. Chen, and R. Ramprasad, npj Comput. Mater. 3, 37 (2017).   \n[13] J. Behler and M. Parrinello, Phys. Rev. Lett. 98, 146401 (2007).   \n[14] J. Behler, J. Chem. 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Phys. 23, 629 (2018).   \n[24] Some flexibility can be used in the definition of the local frame of atom i. Usually we define it in terms of the two atoms closest to $i$ , independent of their species. Exceptions to this rule are discussed in the Supplemental Material.   \n[25] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, Cambridge, MA, 2016).   \n[26] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.120.143001 for the technical details of the construction of the DNN model.   \n[27] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle, in Advances in Neural Information Processing Systems (MIT Press, Cambridge, 2007), p. 153.   \n[28] A. Krizhevsky, I. Sutskever, and G. E. Hinton, Advances in Neural Information Processing Systems (Curran, Red Hook, 2012), p. 1097.   \n[29] D. Kingma and J. Ba, arXiv:1412.6980.   \n[30] C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 (1999).   \n[31] A. Tkatchenko and M. Scheffler, Phys. Rev. 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Pronk, S. Páll, R. Schulz, P. Larsson, P. Bjelkmar, R. Apostolov, M. Shirts, J. Smith, P. Kasson, D. van der  \n\nSpoel, B. Hess, and E. Lindahl, Bioinformatics 29, 845 (2013). [43] P. Giannozzi et al., J. Phys. Condens. Matter 29, 465901 (2017). [44] Conventionally, chemical accuracy corresponds to an error of $1~\\mathrm{kcal/mol}$ in the energy. [45] J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 (1993). [46] D. Ceperley and B. Alder, Science 231, 555 (1986).  ",
    "Target": null
  },
  {
    "id": "10.1016_j.envpol.2018.02.050",
    "DOI": "10.1016/j.envpol.2018.02.050",
    "DOI Link": "http://dx.doi.org/10.1016/j.envpol.2018.02.050",
    "Relative Dir Path": "mds/10.1016_j.envpol.2018.02.050",
    "Article Title": "Adsorption of antibiotics on microplastics",
    "Authors": "Li, J; Zhang, KN; Zhang, H",
    "Source Title": "ENVIRONMENTAL POLLUTION",
    "Abstract": "Microplastics and antibiotics are two classes of emerging contaminullts with proposed negative impacts to aqueous ecosystems. Adsorption of antibiotics on microplastics may result in their long-range transport and may cause compound combination effects. In this study, we investigated the adsorption of 5 antibiotics [sulfadiazine (SDZ), amoxicillin (AMX), tetracycline (TC), ciprofloxacin (CIP), and trimethoprim (TMP)] on 5 types of microplastics [polyethylene (PE), polystyrene (PS), polypropylene (PP), polyamide (PA), and polyvinyl chloride (PVC)] in the freshwater and seawater systems. Scanning Electron Microscope (SEM) and X-ray diffractometer (XRD) analysis revealed that microplastics have different surface characterizes and various degrees of crystalline. Adsorption isotherms demonstrated that PA had the strongest adsorption capacity for antibiotics with distribution coefficient (K-d) values ranged from 7.36 +/- 0.257 to 756 +/- 48.0 L kg(-1) in the freshwater system, which can be attributed to its porous structure and hydrogen bonding. Relatively low adsorption capacity was observed on other four microplastics. The adsorption amounts of 5 antibiotics on PS, PE, PP, and PVC decreased in the order of CIP > AMX > TMP > SDZ > TC with K-f correlated positively with octanol-water partition coefficients (Log K-OW). Comparing to freshwater system, adsorption capacity in seawater decreased significantly and no adsorption was observed for CIP and AMX. Our results indicated that commonly observed polyamide particles can serve as a carrier of antibiotics in the aquatic environment. (C) 2018 Elsevier Ltd. All rights reserved.",
    "Times Cited, WoS Core": 928,
    "Times Cited, All Databases": 1062,
    "Publication Year": 2018,
    "Research Areas": "Environmental Sciences & Ecology",
    "UT (Unique WOS ID)": "WOS:000431158900047",
    "Markdown": "# Adsorption of antibiotics on microplastics\\*  \n\nJia Li a, b, Kaina Zhang c, Hua Zhang a, \\*  \n\na Key Laboratory of Coastal Environmental Process and Ecology Remediation, Yantai Institute of Coastal Zone Research, Chinese Academy of Sciences, Yantai   \n264003, China   \nb University of Chinese Academy of Sciences, Beijing 100049, China   \nc School of Environment and Materials Engineering, YanTai University, Yantai 264003, China  \n\n# a r t i c l e i n f o  \n\n# a b s t r a c t  \n\nArticle history:   \nReceived 31 October 2017 Received in revised form 31 January 2018   \nAccepted 16 February 2018   \nKeywords:   \nAntibiotics   \nMicroplastics   \nAdsorption   \nDistribution coefficient  \n\nMicroplastics and antibiotics are two classes of emerging contaminants with proposed negative impacts to aqueous ecosystems. Adsorption of antibiotics on microplastics may result in their long-range transport and may cause compound combination effects. In this study, we investigated the adsorption of 5 antibiotics [sulfadiazine (SDZ), amoxicillin (AMX), tetracycline (TC), ciprofloxacin (CIP), and trimethoprim (TMP)] on 5 types of microplastics [polyethylene (PE), polystyrene (PS), polypropylene (PP), polyamide (PA), and polyvinyl chloride (PVC)] in the freshwater and seawater systems. Scanning Electron Microscope (SEM) and $\\mathsf{X}$ -ray diffractometer (XRD) analysis revealed that microplastics have different surface characterizes and various degrees of crystalline. Adsorption isotherms demonstrated that PA had the strongest adsorption capacity for antibiotics with distribution coefficient $(K_{d})$ values ranged from $7.36\\pm0.257$ to $756\\pm48.0\\mathrm{L}\\mathrm{kg}^{-1}$ in the freshwater system, which can be attributed to its porous structure and hydrogen bonding. Relatively low adsorption capacity was observed on other four microplastics. The adsorption amounts of 5 antibiotics on PS, PE, PP, and PVC decreased in the order of $\\mathrm{CIP}>\\mathrm{AMX}>\\mathrm{TMP}>\\mathrm{SDZ}>\\mathrm{TC}$ with $K_{f}$ correlated positively with octanol-water partition coefficients (Log $\\ensuremath{\\mathrm{K}}_{\\mathrm{ow}})$ . Comparing to freshwater system, adsorption capacity in seawater decreased significantly and no adsorption was observed for CIP and AMX. Our results indicated that commonly observed polyamide particles can serve as a carrier of antibiotics in the aquatic environment.  \n\n$\\circledcirc$ 2018 Elsevier Ltd. All rights reserved.  \n\n# 1. Introduction  \n\nDuring the last 60 years, plastic production increased considerably from around 0.5 million tonnes in 1950 to 311 million tonnes in 2014 (Thompson et al., 2009; Plastics, 2015), which accompanied by increasing release of plastic waste to the environment. It is estimated that 4.8e12.7 million tonnes of plastic waste washed offshore in 2010 alone (Jambeck et al., 2015). Recently, microplastics (MPs) with particle size in the micrometer range have become the focus of study due to their potential toxic impact to aquatic ecosystems. MPs have been detected in surface water (Zhao et al., 2015), water column (Nel and Froneman, 2015), and bottom sediments (Browne et al., 2011). Previous studies have showed that polyethylene (PE), polystyrene (PS), polypropylene (PP), polyamide (PA), and polyvinyl chloride (PVC) are the most frequently detected  \n\nMPs in the aquatic environment (Hidalgo-Ruz et al., 2012; Fok et al., 2017). It is believed that MPs can accumulate various toxins and chemical pollutants and serve as a carrier for long-range transport (Guo et al., 2012; Turner and Holmes, 2015; Hueffer and Hofmann, 2016).  \n\nStudies have been conducted on the adsorption behaviors of organic pollutants or heavy metals onto different types of MPs (Bakir et al., 2014a, 2014b, Velzeboer et al., 2014; Wang et al., 2015; Hueffer and Hofmann, 2016; Wu et al., 2016) as well as the effects of plastic types and environmental factors (e.g., ionic strength and pH) on pollutants adsorption processes (Wang et al., 2015). Both the sorbent and the sorbate properties can influence the adsorption extent significantly. For instance, properties of MPs such as polarity, abundance of rubbery, and degree of crystalline have great impacts on adsorption capacities of pollutants (Guo et al., 2012; Wang et al., 2015; Brennecke et al., 2016). The hydrophobicity of organic contaminants is also important in determining their adsorption on MPs (Hueffer and Hofmann, 2016). Furthermore, adsorption of organic pollutants on MPs varied in the seawater and the freshwater (Velzeboer et al., 2014), which may be due to the impacts of salinity.  \n\nAs reported by Wang et al. (2015), perfluorooctanesulfonate (PFOS) adsorption on PE and PS increased with increasing of ionic strength, while ionic strength had no effect on perfluorooctanesulfonamide (FOSA) adsorption.  \n\nAs a class of emerging contaminants, antibiotics have received increasing attention due to their impacts on the microbial community as well as the generation of resistance genes (Le et al., 2005; Yang et al., 2017). A large number of antibiotics are released into the environment every year. As evaluated by Zhang et al. (2015), only in China, 53,800 tonnes of antibiotics discharged into the receiving environment in 2013. Studies reported that tetracyclines, macrolides, fluoroquinolones, and sulfonamides are the frequently detected antibiotics in the aquatic environment worldwide (Kolpin et al., 2002; Watkinson et al., 2009; Jiang et al., 2011; Li et al., 2012). Antibiotics such as trimethoprim, fluoroquinolones, and sulfonamides were found to be stable in surface water (Lunestad et al., 1995; Lin et al., 2010). More importantly, the residual antibiotics may pose relatively high ecological risk to the relevant aquatic organisms (Xu et al., 2013). If antibiotics can be absorbed by MPs, both could have higher toxic effects on aquatic life due to the combined pollution. There is evidence that persistent organic pollutants (POPs) can transfer from MPs to Artemia nauplii and further to zebrafish via a trophic food web (Batel et al., 2016). Thus, understanding the possible reactions between different kinds of antibiotics and MPs is warranted for the evaluation of their environmental risks.  \n\nThe physicochemical properties such as specific surface area, degree of crystallinity, and pore size distribution vary substantially among different types of microplastic particles and may dominate their antibiotics adsorption capacities. To verify this hypothesis, experiments were conducted 1) to describe the structures and properties of 5 MPs using analytical techniques such as Scanning Electron Microscope (SEM) and X-ray diffractometer (XRD); 2) to evaluate the adsorption capacities of 5 types of commonly used antibiotics in the freshwater and seawater systems.  \n\n# 2. Materials and method  \n\n# 2.1. Chemicals  \n\nSulfadiazine (SDZ), amoxicillin (AMX), tetracycline (TC), ciprofloxacin (CIP), and trimethoprim (TMP) were purchased from Sigma-Aldrich (USA), with ${>}99\\%$ purity. The physicochemical properties of antibiotics was shown in Table S1. Acetonitrile were high-performance liquid chromatography (HPLC) grade and were obtained from Anaqua Chemicals Supply (ACS, USA). Ultrapure water (MQ) was obtained from a Milli- $\\cdot{\\sf Q}$ water purification system (Millipore, Billerica, MA, USA). The other reagents were analytical grade or higher. Seawater was filtered through $0.45\\upmu\\mathrm{m}$ membranes and irradiated with ultraviolet light to eliminate the influences of dissolved organic matter as much as possible. TC, AMX, and TMP were dissolved in the background solutions (i.e. ultrapure water and filtered seawater) directly to prepare the stock solutions. For SDZ and CIP, methanol was added to enhance their solubility in background solutions. All the stock solutions were kept in the dark at $4^{\\circ}\\mathsf C$ for no long than one week. Stock solutions were diluted to the desired concentrations before use.  \n\n# 2.2. Microplastic particles and analytical methods  \n\nPolyethylene (PE), polystyrene (PS), polypropylene (PP), polyamide (PA), and polyvinyl chloride (PVC) were purchased as powders from Youngling Electromechanical Technology Co., Ltd. (Shanghai, China). The physicochemical properties of 5 MPs were shown in Table S2. The particle sizes distribution of these polymers were shown in Fig. S1. For any kinds of MPs, more than $90\\%$ of polymers fell into the $75{-}180\\upmu\\mathrm{m}$ size classes. Their point of zero charge (PZC) were analyzed based on the method described by Ferro-Garcia et al. (1998). The polymers microscopic morphological characteristics were analyzed by a Scanning Electron Microscope (SEM) (S-4800, Hitachi, Japan). The crystalline compositions of MPs were measured using X-ray diffractometer (XRD) (XRD-7000, SHIMADZU, Japan) with a $c_{u-K\\mathfrak{L}}$ as the radiation source $(\\uplambda=1.5406\\tilde{\\mathrm{A}};$ . The samples were scanned over the range of $5{\\mathrm{-}}90^{\\circ}$ of 2q at a rate of $1^{\\circ}\\operatorname*{min}^{-1}$ .  \n\n# 2.3. Batch adsorption experiments  \n\nBatch adsorption experiments for 5 antibiotics were undertaken with 5 concentration gradients (i.e. 0.5, 1, 5, 10, and $15\\mathrm{mgL}^{-1}.$ at room temperature $(25^{\\circ}\\mathsf{C})$ . Specifically, $_{0.02\\mathrm{g}}$ of each plastic particle was added into the glass vials. Different volumes of background solution (ultrapure water or filtered seawater) were added according to the concentration gradient. Then, antibiotic stock solution with a concentration of $50\\mathrm{mgL}^{-1}$ was added to make up the suspension volume of $5\\mathrm{mL}$ in each glass vial capped with a Teflon gasket. To minimize the effects of cosolvent, the volume ratio of methanol in the test solution was controlled below $0.1\\%$ . The glass vials were shaken in a temperature-controlled shaking incubator (HZS-HA, Harbin, China) at a shaking speed of $180\\mathrm{rpm}$ at $25^{\\circ}\\mathsf{C}$ for 4 d. After equilibrium, the sample was filtered through a $0.22\\upmu\\mathrm{m}$ syringe filter before analysis. All the adsorption experiments were conducted in triplicate. The blank sorption experiments with the reactor system containing antibiotics without MPs were carried out. The loss of antibiotics during sorption test was calculated and subtracted from the blank loss. To improve calculation accuracy, the amounts of antibiotics adsorbed on MPs were calculated using the following equation:  \n\n$$\nq_{e}=\\frac{\\frac{m_{2}-m_{1}}{\\rho_{2}}\\left(c_{0}-c_{e}\\right)-\\frac{m_{1}-m_{0}}{\\rho_{1}}c_{e}-\\alpha}{m}\n$$  \n\nwhere $q_{\\mathrm{e}}(\\mathrm{mg}\\cdot\\mathbf{g}^{-1})$ is the equilibrium adsorption amount; $c_{0}$ and $c_{\\mathrm{e}}$ $(\\mathrm{mg}\\cdot\\mathrm{L}^{-1})$ are the initial concentration and the equilibrium concentration; $\\rho_{1}$ and $\\rho_{2}$ $(\\mathbf{g}\\ \\mathbf{m}\\mathbf{L}^{-1})$ are density of the background solution and the antibiotic stock solution, respectively; $m\\left({\\mathfrak{g}}\\right)$ is the mass of adsorbent; $m_{0}({\\bf g})$ is the mass of the adsorbent and the vial; $m_{1}$ (g) is the mass after adding the background solution; $m_{2}\\left({\\bf g}\\right)$ is the mass after adding the antibiotic stock solution; $\\alpha\\left(\\mathrm{mg}\\right)$ is blank loss.  \n\n# 2.4. Adsorption model  \n\nLinear, Freundlich, and Langmuir adsorption models were used to fit the adsorption isotherms of antibiotics. Briefly, the Linear model can be described as:  \n\n$$\n\\begin{array}{r}{q=K_{d}C_{e}}\\end{array}\n$$  \n\nwhere $q(\\mathrm{mgg}^{-1})$ is the absorbed amount of antibiotic; $C_{e}(\\mathrm{mgL}^{-1})$ is antibiotic mass in the aqueous phase at equilibrium the equilibrium; and $K_{d}(\\mathrm{L}\\ \\mathrm{g}^{-1})$ is the partition coefficient. The Freundlich model is given by:  \n\n$$\nq=K_{f}C_{e}^{n}\n$$  \n\nwhere $K_{f}(\\mathrm{L}\\ \\mathrm{g}^{-1})$ is the Freundlich adsorption coefficient which indicates adsorption capacity; $n$ is the Freundlich isotherm exponent that determines the non-linearity. The Langmuir isotherm model can be expressed as follow:  \n\n$$\nq=q_{m a x}\\frac{k_{L}C_{e}}{1+k_{L}C_{e}}\n$$  \n\nwhere $q_{\\mathrm{max}}(\\mathrm{mgg^{-1}})$ represents the maximum adsorption capacity; $K_{L}(\\mathrm{L}\\operatorname{mg}^{-1})$ is related proportionally to the affinity between plastic particles and antibiotics.  \n\n# 2.5. Detection of antibiotics  \n\nAll the selected antibiotics were detected using highperformance liquid chromatography (Exformma 1600, USA) equipped with a UV detector. Chromatographic separations were performed with a SDS HYPERSIL C18 $(250\\mathrm{mm}\\times4.6\\mathrm{mm}$ , $5\\upmu\\mathrm{m}\\dot{}$ . The column temperature was $30^{\\circ}\\mathsf C$ The flow rate was $1.0\\mathrm{mL}\\mathrm{min}^{-1}$ and injection volume was $50\\upmu\\mathrm{L}$ The mobile phase and detection wavelength varied across antibiotics and were shown in Table 1. The detection limit of 5 antibiotics ranged from 0.01 to $0.05\\mathrm{mgL}^{-1},$ and the relative standard deviation (RSD) of antibiotic measurements was less than $3.0\\%$ . For each kind of antibiotics, two standard curves (one for ultrapure water and another for seawater) with seven-point (i.e. 0.1, 0.5, 1, 5, 10, 15, $20\\mathrm{mgL}^{-1}.$ ) were used for quantitative analysis. The ${\\boldsymbol{\\mathrm{r}}}^{2}$ of standard curves were between 0.990 and 1.00.  \n\n# 3. Results and discussion  \n\n# 3.1. Characterization of MPs  \n\nFig. 1 shows the SEM micrographs of PE, PVC, PS, PP, and PA, respectively. The surface of PE particle was relatively smooth, while several micropores can be found on the surface of PVC. Reticular formation was developed on the surface of PS particles. Spherical bulges and micropores were observed on the surface and internal cross-section of PP. More importantly, the surface of PA was rough and exhibited characteristics of porous polymer.  \n\nThe XRD patterns are good indications of the degree of crystalline of plastic particles. Generally, polymer with high degree of crystalline has sharp diffraction peak. As illustrated in Fig. S2, one sharp diffraction peak appeared in the XRD pattern of PE, indicating PE had high degree of crystalline. Similarly, three obvious peaks with high intensity can be found in the XRD pattern of PP. For PS and PA, their diffraction peaks showed the similar intensity. Whereas for PVC, there was no apparent diffraction peak in the 2q range of $5{-}90^{\\circ}$ . Therefore, the degree of crystallinity followed the order as: $\\mathrm{PE}>\\mathrm{PP}>\\mathrm{PA}\\approx\\mathrm{PS}>\\mathrm{PVC}$ .  \n\n# 3.2. Adsorption isotherms  \n\nAdsorption of antibiotics on MPs in freshwater and seawater were illustrated in Fig. 2. The estimated Linear, Freundlich, and Langmuir parameters were summarized in Table 2 and Table 3. The Freundlich parameter n was related to the non-linearity of isotherms. Previous studies showed that the sorption isotherms of perfluoroalkyl acids (PFAAs), polychlorinated biphenyls (PCBs), and pharmaceuticals and personal care products (PPCPs) on various plastic particles were highly linear (Velzeboer et al., 2014; Wang et al., 2015; Wu et al., 2016). But other study suggested that the linearity of sorption isotherms varied among chemicals and plastic types (Hueffer and Hofmann, 2016). As demonstrated in Tables 2 and 3, the adsorption isotherms of CIP, TMP, and TC on MPs were obviously non-linear with Freundlich n values ranged from 0.303 to 0.842, while adsorption of SDZ and AMX were relatively linear with the larger n values $(\\mathrm{n}>0.540),$ . Correspondingly, the non-linear models (i.e. Freundlich model and Langmuir model) exhibited better goodness-of-fit for isotherms of CIP, TMP, and TC, as indicated by the high values of ${\\boldsymbol{\\mathrm{r}}}^{2}$ . On the contrary, the Linear model was more suitable to simulate the adsorption isotherms of SDZ and AMX. Hueffer and Hofmann (2016) reported that the adsorption of seven non-polar organic compounds by PE, PS, PA, and PVC yielded a better fit with Freundlich model than Langmuir model. However, Bakir et al. (2012) suggested that Langmuir model was more suitable to explain sorption isotherm of bi-sorbates by PE and PVC than Freundlich model. Based on the ${\\boldsymbol{\\mathrm{r}}}^{2}$ (Table 2), the adsorption of CIP by PP and PA yielded a better fitness for Langmuir model, while the adsorption of CIP by PS, PE, and PVC yielded a better fitness for Freundlich model. For TMP in the seawater system, Freundlich model was suitable to describe its adsorption on PP, PS, PE, and PA. For TMP in the freshwater system, Freundlich model was suitable to describe its adsorption on PA, whereas Langmuir model described its adsorption on PP and PE well. The experiment results indicate that different reaction mechanisms are involved in the adsorption of antibiotics on MPs. More in-depth understanding of the binding mechanisms is required to establish more generalized adsorption model.  \n\n# 3.3. Effects of microplastic properties  \n\nTable 1 High-performance liquid chromatography conditions for the analysis of antibiotics.   \n\n\n<html><body><table><tr><td>Antibiotic</td><td>Mobile phase A</td><td>Mobile phase B</td><td>Volume radio (A:B)</td><td>Wavelength</td></tr><tr><td>CIP</td><td>disodium hydrogen phosphate buffer (pH = 2.5)</td><td>acetonitrile</td><td>80:20</td><td>277 nm</td></tr><tr><td>SDZ</td><td>0.1% acetic acid solution</td><td> acetonitrile</td><td>78:22</td><td>268 nm</td></tr><tr><td>AMO</td><td>disodium hydrogen phosphate buffer (pH = 2.5)</td><td>acetonitrile</td><td>85:15</td><td>254 nm</td></tr><tr><td>TMP</td><td>1% acetic acid solution</td><td> acetonitrile</td><td>80:20</td><td>271 nm</td></tr><tr><td>TC</td><td>disodium hydrogen phosphate buffer (pH = 2.5)</td><td>acetonitrile</td><td>80:20</td><td>360 nm</td></tr></table></body></html>  \n\nThe adsorption capacities of antibiotics on different MPs varied significantly (Table 2). Physicochemical properties of MPs such as specific surface area, polarity, and degree of crystallinity may affect their adsorption capacities. Wang et al. (2015) suggested that polarity of MPs can influence adsorption levels of polar chemicals. The tested antibiotics are polar compounds (Kanda et al., 2015) and is expected to have strong sorption capacity to polar MPs by the polar-polar interaction. As shown in Table 2, however, only polar PA has significant higher sorption capacity for 4 antibiotics (CIP, TMP, AMX, and TC) in the freshwater system, while polar PVC shows low affinity to polar antibiotics. This indicated that polarity of MPs alone were not capable of explaining the differences in adsorption capacities. As reported by Teuten et al. (2009) and Wang et al. (2015), the rubbery plastic PE showed higher adsorption capacity to organic pollutants than the glassier plastics (i.e. PP, PS, and PVC). However, for any kinds of antibiotics, its sorption capacity on PE was not the largest, which indicating sorption extents of antibiotics had little or no correlation with rubbery state of plastic. Guo et al. (2012) suggested that the plastic with low crystallinity could accumulated more hydrophobic organic pollutants. As shown in section 3.1, the degree of crystallinity of five types of MPs followed the order as: $\\mathrm{PE}>\\mathrm{PP}>\\mathrm{PA}\\approx\\mathrm{PS}>\\mathrm{PVC}.$ . However, this order was not consistent with the order of sorption capacity of any type of antibiotic, indicating the crystallinity of MPs was not the essential factor affecting antibiotic adsorption.  \n\n![](images/5bbdc436c857a2d61b65a349d46c3a8dc343ddfd931fdea4f23691ae5595dfd0.jpg)  \nFig. 1. SEM micrographs of polyethylene (PE), polystyrene (PS), polypropylene (PP), polyamide (PA), and polyvinyl chloride (PVC).  \n\nAdsorption capacities of CIP, TMP, and SDZ on PS are higher than those on PE (Table 2). The reason may be that PS can undergo nonspecific vander Waals interactions and $\\pi-\\pi$ interactions at the aromatic surface, while PE can only undergo the vander Waals interactions (Velzeboer et al., 2014; Hueffer and Hofmann, 2016). As shown in Fig. 1, PS, PP, and especially PA have the developed pore structure, and this may be used to explain why these three MPs have higher sorption capacity than PE and PVC. Furthermore, PA has high adsorption for AMX, TC, and CIP, which can be attributed to the specific functional group (i.e. amide group). Hydrogen bonding can be formed between amide group (proton donor group) of PA and carbonyl group (proton acceptor group) of AMX, TC, and CIP (Antony et al., 2010). Other studies also demonstrated that hydrogen bonding between antibiotics (e.g. TC and CIP) and organic matter (e.g. humic substance and organic carbon) surfaces may contribute to antibiotics sorption (Tolls, 2001; Pils and Laird, 2007). Thus we suggested that the formation of hydrogen bonding as the mechanism underlying the high adsorption of AMX, TC, and CIP on PA.  \n\nThe properties of MPs exposed to the environment may change because of environmental factors and therefore influence their adsorption behaviors (Turner and Holmes, 2015). For example, the polar functional groups, carbonyl groups, have been identified in aged plastic samples collected from beaches (Zbyszewski and  \n\nCorcoran, 2011). The presence of polar functional groups may contribute to the formation of H-bonding. Furthermore, aged plastic particles tend to acquire a greater surface area through photo-oxidation, weathering, and aging (Turner and Holmes, 2011), and this will also favor the adsorption of contaminants.  \n\n# 3.4. Effects of antibiotic properties  \n\nAs shown in Table 2, the adsorption capacities of various antibiotics on a specific type of plastic differed greatly. Apart from PA, adsorption of 5 antibiotics on the other four MPs decreased in the order of $\\mathrm{CIP}>\\mathrm{AMX}>\\mathrm{TMP}>\\mathrm{SDZ}>\\mathrm{TC}$ . These results implied that antibiotics properties can influence their adsorptions on MPs. Previous studies indicated that the octanol-water partition coefficients $(\\mathrm{Log}K_{\\mathrm{ow}})$ of sorbates were essential in determining their adsorption extents on MPs (Hueffer and Hofmann, 2016; Wu et al., 2016). That is, the Log $K_{d}$ values of investigated sorbates were positively correlated with their Log $K_{\\mathrm{ow}}$ values. The Log $K_{\\mathrm{ow}}$ values of the investigated antibiotics decreased in the order of $\\mathrm{CIP}>\\mathrm{TMP}>\\mathrm{AMX}>\\mathrm{SDZ}>\\mathrm{TC}$ (Table S1). There was a significantly positive correlation between antibiotics’ Log $K_{\\mathrm{ow}}$ values and their average $K_{f}$ values on PP, PS, PE, and PVC $(\\mathtt{p}<0.05),$ . Our results suggested that hydrophobic antibiotics (with higher Log $K_{\\mathrm{ow}}$ values) had higher affinity to PP, PS, PE, and PVC. However, this correlation did not apply to PA, indicating hydrophobic interaction was insignificant for antibiotics adsorption on PA.  \n\nAntibiotics are ionisable compounds, but the ionization constant (pKa) of various antibiotics usually differed significantly because of their specific functional groups. Thus, in a specific pH condition, various antibiotics will exhibit different speciation of the cation, zwitterion, and anion. The speciation of ionic chemicals can influence their sorption extents on MPs. As reported by Wang et al. (2015), the anionic forms of PFOS showed a higher adsorption capacity on PE with positive surface at low pH than the nonionic forms of FOSA. The pKa values of 5 antibiotics and their speciation at experimental pH condition were shown in Table S1. In the freshwater system, all the tested antibiotics were zwitterions and anions; nevertheless, CIP also had a portion of cations. In the seawater system, zwitterions and anions were the main speciation for 5 antibiotics. Because the experimental pH was higher than the PZC of 5 MPs, all the 5 MPs carried a net negative charge. Thus, for  \n\n![](images/6317295d52d0344410ee94840ceb4cee4a0b037760dacc58c7920d70b36dbc8c.jpg)  \nFig. 2. Adsorption of antibiotics on MPs in the freshwater system (left column) and in the seawater system (right column). Note: the values of ${\\sf q}_{\\mathrm{e}}$ less than 0 were not shown in this figure.  \n\nCIP in the freshwater system, the cations of CIP enhanced its sorption capacity on negatively charged MPs surface due to electrostatic attraction.  \n\n# 3.5. Freshwater and seawater systems  \n\nOur results demonstrated that adsorption of antibiotics on MPs differed among freshwater and seawater systems. As shown in Fig. 2, adsorption of CIP and AMX did not occur in the seawater system. The sorption capacities of TMP, SDZ, and TC also decreased compared with the freshwater system. Differences in ionic strength and pH values may be used to explain the different sorption capacities in the freshwater and seawater systems. The pH of seawater system was higher than freshwater system (Table S2);  \n\nTable 2 Estimated Linear, Freundlich, and Langmuir parameters for antibiotics adsorption on MPs in the freshwater system.   \n\n\n<html><body><table><tr><td rowspan=\"2\">MPs</td><td colspan=\"2\">Linear</td><td colspan=\"3\">Freundlich</td><td colspan=\"3\">Langmuir</td></tr><tr><td>Ka (L kg-1)</td><td>r²</td><td>Kf(L kg)</td><td>ｎ</td><td>r²</td><td>qmax (mg g-1)</td><td>K (L mg-1)</td><td>r²</td></tr><tr><td></td><td>CIP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PP</td><td>57.1 ± 11.5</td><td>0.827</td><td>252 ± 33.3</td><td>0.345</td><td>0.938</td><td>0.615 ± 0.0299</td><td>0.844</td><td>0.990</td></tr><tr><td>PS</td><td>51.5 ± 7.76</td><td>0.846</td><td>205 ± 17.0</td><td>0.316</td><td>0.968</td><td>0.416 ± 0.0427</td><td>1.67</td><td>0.903</td></tr><tr><td>PVC</td><td>41.5 ± 7.83</td><td>0.844</td><td>184 ± 6.19</td><td>0.371</td><td>0.998</td><td>0.453 ± 0.00863</td><td>1.15</td><td>0.996</td></tr><tr><td>PE</td><td>55.1 ± 7.94</td><td>0.904</td><td>222 ± 6.59</td><td>0.393</td><td>0.994</td><td>0.200 ± 0.0143</td><td>0.443</td><td>0.990</td></tr><tr><td>PA</td><td>96.5 ± 7.81</td><td>0.968</td><td>170 ± 45.2</td><td>0.741</td><td>0.963</td><td>2.20 ± 0.657</td><td>0.0740</td><td>0.980</td></tr><tr><td></td><td>TMP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PP</td><td>9.71 ± 2.28</td><td>0.851</td><td>32.3 ± 4.01</td><td>0.450</td><td>0.964</td><td>0.102 ± 0.0142</td><td>0.498</td><td>0.934</td></tr><tr><td>PS</td><td>9.51 ± 1.07</td><td>0.963</td><td>32.1 ± 2.48</td><td>0.507</td><td>0.992</td><td>0.174 ± 0.0385</td><td>0.158</td><td>0.932</td></tr><tr><td> PVC</td><td>8.41 ± 1.20 8.38 ± 1.32</td><td>0.941</td><td>13.4 ± 6.58</td><td>0.842</td><td>0.939</td><td>0.481 ± 0.496</td><td>0.0259</td><td>0.944</td></tr><tr><td>PE</td><td>17.1 ± 1.24</td><td>0.908</td><td>22.0 ±2.59</td><td>0.560</td><td>0.986</td><td>0.154 ± 0.0413</td><td>0.116</td><td>0.939</td></tr><tr><td>PA</td><td>SDZ</td><td>0.974</td><td>36.0 ± 6.15</td><td>0.696</td><td>0.985</td><td>0.468 ± 0.128</td><td>0.0646</td><td>0.979</td></tr><tr><td> PP</td><td>7.85 ± 0.679</td><td>0.985</td><td>8.00 ±7.14</td><td>0.939</td><td>0.884</td><td>na</td><td>na</td><td></td></tr><tr><td>PS</td><td>7.39 ± 0.308</td><td>0.995</td><td>4.10 ± 2.18</td><td>1.22</td><td>0.972</td><td>na </td><td>na </td><td>na na</td></tr><tr><td> PVC</td><td>6.61 ± 0.549</td><td>0.973</td><td>3.20 ± 2.91</td><td>1.27</td><td>0.918</td><td>na </td><td>na </td><td>na </td></tr><tr><td>PE</td><td>6.19 ± 0.238</td><td>0.996</td><td>2.20 ± 3.13</td><td>1.40</td><td>0.962</td><td>na</td><td>na </td><td>na </td></tr><tr><td>PA</td><td>7.36 ± 0.257 AMX</td><td>0.996</td><td>1.10 ± 0.196</td><td>1.71</td><td>0.999</td><td>na </td><td>na</td><td>na </td></tr><tr><td>PP</td><td>17.5 ± 3.39</td><td>0.895</td><td>60.0 ± 6.55</td><td>0.540</td><td>0.720</td><td>0.294 ± 0.0702</td><td>0.376</td><td></td></tr><tr><td>PS</td><td>一</td><td>一</td><td>一</td><td>一</td><td>一</td><td></td><td>一</td><td>0.930</td></tr><tr><td>PVC</td><td>24.7 ± 1.20</td><td>0.991</td><td>20.0 ± 8.86</td><td>1.07</td><td>0.969</td><td>0.523 ± 0.368</td><td>0.0657</td><td>一 0.953</td></tr><tr><td>PE</td><td>8.40 ± 0.675</td><td>0.968</td><td>18.0 ± 2.27</td><td>0.637</td><td>0.930</td><td>0.131 ± 0.0284</td><td>0.174</td><td>0.920</td></tr><tr><td>PA</td><td>756 ± 48.0</td><td>0.980</td><td>700 ± 31.8</td><td>0.900</td><td>0.991</td><td>22.7 ± 22.6</td><td>0.0361</td><td>0.992</td></tr><tr><td>PA</td><td>TC 356 ±38.2</td><td>0.945</td><td>588 ± 128</td><td>0.699</td><td>0.943</td><td>3.84 ± 0.839</td><td>0.189</td><td>0.977</td></tr></table></body></html>\n\nNote: “-” means no antibiotic adsorption was occurred in the related system; “na” means fit did not converge, that is, Langmuir model failed to fit the adsorption isotherms.  \n\nTable 3 Estimated Linear, Freundlich, and Langmuir parameters for antibiotics adsorption on MPs in the seawater system.   \n\n\n<html><body><table><tr><td>MPs</td><td colspan=\"2\">Linear</td><td colspan=\"3\">Freundlich</td><td colspan=\"3\">Langmuir</td></tr><tr><td></td><td>Ka (L kg-1)</td><td>r²</td><td>Ky(L kg-1)</td><td>ｎ</td><td>r²</td><td>qmax (mg g-1)</td><td>K (L mg-1)</td><td>r²</td></tr><tr><td></td><td>TMP</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PP</td><td>3.93 ± 0.944</td><td>0.765</td><td>18.4 ± 2.65</td><td>0.442</td><td>0.765</td><td>0.0597 ± 0.00926</td><td>0.551</td><td>0.917</td></tr><tr><td>PS</td><td>7.30 ± 0.912</td><td>0.927</td><td>13.9 ± 2.88</td><td>0.769</td><td>0.631</td><td>0.166 ± 0.122</td><td>0.0969</td><td>0.724</td></tr><tr><td>PVC</td><td>5.45 ± 0.492</td><td>0.961</td><td>19.0 ± 2.46</td><td>0.303</td><td>0.419</td><td>0.0341 ± 0.0135</td><td>1.46</td><td>0.170</td></tr><tr><td>PE</td><td>6.47 ± 1.02</td><td>0.888</td><td>23.2 ± 0.861</td><td>0.538</td><td>0.963</td><td>0.0868 ± 0.00510</td><td>0.469</td><td>0.989</td></tr><tr><td>PA</td><td>5.89 ± 1.05</td><td>0.938</td><td>10.0 ± 1.64</td><td>0.560</td><td>0.992</td><td>0.130 ± 0.0320</td><td>0.113</td><td>0.962</td></tr><tr><td> PP</td><td>SDZ</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>PS</td><td>7.13 ± 0.952 6.80 ± 0.352</td><td>0.948 0.989</td><td>6.48 ± 8.79</td><td>1.02</td><td>0.744 0.965</td><td>na </td><td>na</td><td>na</td></tr><tr><td>PVC</td><td>5.37 ± 0.598</td><td>0.952</td><td>5.69 ± 2.91</td><td>1.07</td><td>0.994</td><td>na</td><td>na </td><td>na</td></tr><tr><td>PE</td><td>6.26 ± 0.630</td><td>0.961</td><td>0.850 ± 0.308</td><td>1.73</td><td>0.926</td><td>na</td><td>na </td><td>na</td></tr><tr><td>PA</td><td>6.56 ±0.496</td><td>0.983</td><td>3.00 ± 2.98 2.53 ± 0.226</td><td>1.29 1.38</td><td>0.999</td><td>na na </td><td>na na</td><td>na </td></tr><tr><td></td><td>TC</td><td></td><td></td><td></td><td></td><td></td><td></td><td>na</td></tr><tr><td>PA</td><td>4.44 ± 0.963</td><td>0.871</td><td>12.4 ± 5.67</td><td>0.600</td><td>0.886</td><td>0.0878 ± 0.0192</td><td>0.152</td><td>0.961</td></tr></table></body></html>\n\nNote: “na” means fit did not converge, that is, Langmuir model failed to fit the adsorption isotherms.  \n\naccordingly, the anionic speciation of antibiotics in the seawater system were more than in the freshwater system. Meanwhile, all the 5 MPs carried a net negative charge because the pH in seawater was higher than the PZC of MPs (Table S2). As a result, the enhanced electrostatic repulsions between MPs and antibiotics will reduce sorption level. Earlier study showed that sorption of PFOS on PE, PS, and PVC decreased with increasing pH (Wang et al., 2015). Tizaoui et al. (2017) also reported that increasing pH of reaction system can reduce endocrine disrupting chemicals (EDCs) adsorption on PA significantly. On the other hand, ionic strength, to a certain extent, could affect the electrostatic interactions since the electrolytes can compete with adsorbate for electrostatic sites (Shen and Gondal, 2017). When ionic strength increased, cations such as ${\\mathsf{N a}}^{+}$ and ${\\mathsf{C a}}^{2+}$ may be attracted electrostatically to the MPs surface. Further, the inorganic exchangeable cations (e.g. ${\\mathsf{N a}}^{+}$ ) can substitute the hydrogen ions of acidic groups and then inhibit the formation of Hbinding (Aristilde et al., 2010). These findings suggested that adsorption sites may decrease under high ionic strength condition. Therefore, adsorption of the tested antibiotics on MPs decreased at high ionic strength level. This result agreed with previous studies which demonstrated that the adsorption capacities of various antibiotics on different kinds of sorbents (e.g. marine sediments and soils) decreased with increasing of ionic strength (Wang et al., 2010; Xu and Li, 2010; Cao et al., 2015; Li and Zhang, 2017). Our study showed that all the tested antibiotics tend to be adsorbed by MPs in the freshwater system. The increase adsorption extents of antibiotics on MPs in the freshwater system may enhance their bioavailability and accumulation in food chain.  \n\n# 3.6. Environmental implications  \n\nAntibiotics and MPs are two classes of emerging contaminants and have attracted increasing public attention due to their potential toxicity to freshwater and marine ecosystems. Published studies have provided strong evidence that MPs can serve as a vector for the bioaccumulation of toxic chemicals (e.g. POPs and PPCPs) and the combination exhibit higher lethality than MPs themselves (Browne et al., 2013; Rochman et al., 2013; Chua et al., 2014). Although there is yet no available data on the combined effects of MPs and antibiotics, adsorption on MPs may result in the longrange transport of antibiotics and increase their exposure to aquatic ecosystem since antibiotics and MPs are ubiquitous in the aquatic environment and have the similar pollution resources (e.g. domestic wastewater and aquaculture pollution). Furthermore, MPs exposed to the environment may have higher affinity for antibiotics due to the increased specific surface area and polar groups from fragmentation and weathering. Interactions between MPs and antibiotics certainly need to be considered in evaluating the environmental risks associated with the two emerging contaminants.  \n\n# 4. Conclusion  \n\nThe adsorption behaviors of 5 antibiotics on 5 types of microplastic particles were investigated using batch type experiments. Our results showed that adsorption capacities varied among antibiotics, plastic types, and environmental conditions (e.g., ionic strength and pH). PA had high affinity for AMX, TC, and CIP in the freshwater system, potentially due to the formation of hydrogen bonding. Adsorption capacities of CIP in the freshwater system were relatively strong related to its cation speciation. All the tested antibiotics exhibited higher amount of adsorption in the freshwater than in seawater. Hydrogen bonding, hydrophobic interaction, van der Waals force, and electrostatic interaction were the main binding mechanisms between antibiotics and MPs. In the future studies, experiments should be performed 1) to investigate adsorption capacities of different speciation of antibiotics (i.e. cation, zwitterion, and anion) on MPs under different pH; 2) to reveal the effects of the major ions (e.g. $\\subset{}^{-}$ , SO3- 4, PO3- 4, ${\\mathsf{N a}}^{+}$ ) on antibiotics adsorption on MPs; 3) to evaluate the release of adsorbed antibiotics from MPs during their transport from river to ocean.  \n\n# Acknowledgements  \n\nThis study was financially supported by National Key R & D projects (2016YFC1402202) and the Key projects of international cooperation of Chinese Academy of Sciences (KYSB20160003).  \n\n# Appendix A. 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    "Target": null
  },
  {
    "id": "10.1016_j.marpolbul.2017.12.061",
    "DOI": "10.1016/j.marpolbul.2017.12.061",
    "DOI Link": "http://dx.doi.org/10.1016/j.marpolbul.2017.12.061",
    "Relative Dir Path": "mds/10.1016_j.marpolbul.2017.12.061",
    "Article Title": "Validation of ATR FT-IR to identify polymers of plastic marine debris, including those ingested by marine organisms",
    "Authors": "Jung, MR; Horgen, FD; Orski, SV; Rodriguez, CV; Beers, KL; Balazs, GH; Jones, TT; Work, TM; Brignac, KC; Royer, SJ; Hyrenbach, KD; Jensen, BA; Lynch, JM",
    "Source Title": "MARINE POLLUTION BULLETIN",
    "Abstract": "Polymer identification of plastic marine debris can help identify its sources, degradation, and fate. We optimized and validated a fast, simple, and accessible technique, attenuated total reflectance Fourier transform infrared spectroscopy (ATR FT-IR), to identify polymers contained in plastic ingested by sea turtles. Spectra of consumer good items with known resin identification codes #1-6 and several #7 plastics were compared to standard and raw manufactured polymers. High temperature size exclusion chromatography measurements confirmed ATR Fr-IR could differentiate these polymers. High-density (HDPE) and low-density polyethylene (LDPE) discrimination is challenging but a clear step-by-step guide is provided that identified 78% of ingested PE samples. The optimal cleaning methods consisted of wiping ingested pieces with water or cutting. Of 828 ingested plastics pieces from 50 Pacific sea turtles, 96% were identified by ATR FT-IR as HDPE, LDPE, unknown PE, polypropylene (PP), PE and PP mixtures, polystyrene, polyvinyl chloride, and nylon.",
    "Times Cited, WoS Core": 937,
    "Times Cited, All Databases": 996,
    "Publication Year": 2018,
    "Research Areas": "Environmental Sciences & Ecology; Marine & Freshwater Biology",
    "UT (Unique WOS ID)": "WOS:000427332900078",
    "Markdown": "# Validation of ATR FT-IR to identify polymers of plastic marine debris, including those ingested by marine organisms  \n\nMelissa R. Junga, F. David Horgena, Sara V. Orskib, Viviana Rodriguez C.b, Kathryn L. Beersb, George H. Balazsc, T. Todd Jonesc, Thierry M. Workd, Kayla C. Brignace, Sarah-Jeanne Royerf, K. David Hyrenbacha, Brenda A. Jensena, Jennifer M. Lynchg,  \n\na College of Natural and Computational Sciences, Hawai'i Pacific University, Kaneohe, HI, United States b Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, United States c Pacific Islands Fisheries Science Center, National Marine Fisheries Service, Honolulu, HI, United States d U.S. Geological Survey, National Wildlife Health Center, Honolulu Field Station, Honolulu, HI, United States e School of Ocean, Earth Science, and Technology, University of Hawai'i at Manoa, Honolulu, HI, United States f Daniel K. Inouye Center for Microbial Oceanography: Research and Education, University of Hawai'i at Manoa, Honolulu, HI, United States g Chemical Sciences Division, National Institute of Standards and Technology, Kaneohe, HI, United States  \n\n# A R T I C L E I N F O  \n\n# A B S T R A C T  \n\nKeywords:   \nSea turtles   \nPacific Ocean   \nMarine plastic debris   \nPlastic ingestion   \nFourier transform infrared spectroscopy   \nPolymer identification  \n\nPolymer identification of plastic marine debris can help identify its sources, degradation, and fate. We optimized and validated a fast, simple, and accessible technique, attenuated total reflectance Fourier transform infrared spectroscopy (ATR FT-IR), to identify polymers contained in plastic ingested by sea turtles. Spectra of consumer good items with known resin identification codes $\\#1{-}6$ and several $\\#7$ plastics were compared to standard and raw manufactured polymers. High temperature size exclusion chromatography measurements confirmed ATR FT-IR could differentiate these polymers. High-density (HDPE) and low-density polyethylene (LDPE) discrimination is challenging but a clear step-by-step guide is provided that identified $78\\%$ of ingested PE samples. The optimal cleaning methods consisted of wiping ingested pieces with water or cutting. Of 828 ingested plastics pieces from 50 Pacific sea turtles, $96\\%$ were identified by ATR FT-IR as HDPE, LDPE, unknown PE, polypropylene (PP), PE and PP mixtures, polystyrene, polyvinyl chloride, and nylon.  \n\n# 1. Introduction  \n\nPlastic is one of the most persistent and abundant types of marine debris (Rios et al., 2007). For instance, high concentrations of up to 334,271 pieces $\\ensuremath{\\mathrm{km}}^{2}$ have been estimated floating in the North Pacific central gyre, where this material is concentrated by wind-driven ocean currents (Moore et al., 2001; Howell et al., 2012). The production of plastic and associated marine plastic debris continues to rise (Geyer et al., 2017; Jambeck et al., 2015; Bakir et al., 2014; Hoarau et al., 2014), with an estimated 4.8 million metric tons to 12.7 million - metric tons of plastic debris entering the marine environment each year (Jambeck et al., 2015). As marine plastic debris continues to accumulate, long-term environmental, economic, and waste management problems grow, including significant economic costs for prevention and clean-up (Singh and Sharma, 2008; McIlgorm et al., 2011). Increasing awareness of the possible ecological impacts of marine debris has stimulated research to quantify and understand the incidence and  \n\nmagnitude of plastic ingestion by marine animals (Andrady, 2011;   \nProvencher et al., 2017).  \n\nIngestion of plastic debris has been documented in marine species across a range of sizes and biological complexity: from microscopic zooplankton to large vertebrates (Hoss and Settle, 1990; Nelms et al., 2015; Cole and Galloway, 2015; Unger et al., 2016). The size of ingested plastic debris occupies a large range, evidenced by filter feeders, like oyster larvae, which can ingest microplastics as small as $0.16\\upmu\\mathrm{m}$ diameter (Cole and Galloway, 2015), while large items such as part of a car engine cover $\\mathbf{650mm\\times235mm}$ have been found in the gastrointestinal tracts of sperm whales (Physeter macrocephalus) (Unger et al., 2016). Sea turtles are a good indicator of plastic debris occurrence in the natural environment as studies have documented ingestion around the world including coastal Florida, southern Brazil, the Central Pacific, and Mediterranean Sea (Bjorndal et al., 1994; Bugoni et al., 2001; Clukey et al., 2017; Tomás et al., 2002). Sea turtles ingest a variety of plastic items of varying types, sizes, and morphologies, including pieces of bags, rope, fishing line, foam, and fragments of less flexible plastic that range in size from microplastics $\\AA<5\\mathrm{mm}$ on largest edge) to macroplastics $\\mathrm{(>25\\:mm)}$ ) with fragments up to $10\\mathrm{cm}$ observed (Bugoni et al., 2001; Tomás et al., 2002; Clukey et al., 2017). A study by Clukey et al. (2017) showed that a total of 2880 plastic debris items were ingested by 37 olive ridley (Lepidochelys olivacea), nine green (Chelonia mydas), and four loggerhead (Caretta caretta) pelagic sea turtles that were incidentally taken by longline fisheries in the North Pacific Ocean. Plastic fragments constituted $79.5\\%$ of the total debris while $12.5\\%$ were thin plastic sheets (e.g., bags and thin packaging material) and $6.1\\%$ were line or rope (Clukey et al., 2017). While the commercial use of some ingested plastics, such as bags and fishing line, can be easily identified by visual inspection, few pieces are found completely intact and their original origin is difficult to discern (Hoss and Settle, 1990). Fortunately, plastic manufacturers and standards organizations have developed a standard identification system for general classes of plastic that can be used to help identify their likely intended commercial use.  \n\nMost plastic consumer goods are labeled with standardized resin codes marked inside a triangle (ASTM, 2013), signifying the chemical composition of the main polymer, which is used to sort and recycle compatible materials. These include polyethylene terephthalate (PETE, #1), high-density polyethylene (HDPE, #2), polyvinyl chloride (PVC, $\\#3)$ , low-density polyethylene (LDPE, #4, which also currently includes linear LDPE [LLDPE]), polypropylene (PP, $\\#5\\mathrm{\\textperthousand}$ , polystyrene (PS, $\\#6)$ , and other polymers (#7). These codes are rarely present or legible in recovered plastic debris or small plastic fragments, hence identification of the polymer must be accomplished using chemical testing. Characterizing unknown polymers helps illuminate many of the issues surrounding marine debris. Knowing the polymer structure will aid in determining the transport and fate of debris pieces in the environment, such as the effect of material density on stratification within the water column or the susceptibility of specific chemical bonds to break under environmental conditions. In addition, different polymers have different affinities for adsorbing chemical pollutants from seawater, suggesting some polymers may present a larger risk of transferring pollutants to marine organisms who ingest them (Rochman et al., 2013; Fries and Zarfl, 2012; Endo et al., 2005; Koelmans et al., 2013). Knowing the predominant polymers found in various habitats or ingested by marine organisms can help focus conservation efforts, including changes to recycling strategies, targeted waste management, or novel approaches in polymer production (Ryan et al., 2009). Furthermore, since certain polymers are more commonly recycled than others (e.g., $\\#2$ HDPE compared to $\\#4$ LDPE), it is important to be able to distinguish these to monitor the success of waste management techniques.  \n\nSeveral analytical tools have been used to identify the composition of plastic debris (Andrady, 2017). For example, environmental samples from German rivers were analyzed using thermogravimetric analysis connected to solid-phase adsorbers that were subsequently analyzed by thermal desorption gas chromatography mass spectrometry (GC/MS; Dümichen et al., 2015). Fischer and Scholz-Böttcher (2017) used pyrolysis-GC/MS to identify microplastics ingested by North Sea fish. In addition, GC/MS has been utilized to identify indicator chemicals characteristic of different polymers of plastics ingested by Laysan albatross (Phoebastria immutabilis) (Nilsen et al., 2014). These methods are limited to only volatile or ionizable compounds, such as small oligomeric fragments or additives within the bulk material. Methods that can analyze the entire sample, and often require less sample preparation, are vibrational spectroscopy measurements such as Raman microspectroscopy (Frère et al., 2016) and Fourier transform infrared (FT-IR) spectroscopy. FT-IR is becoming the most common technique for marine debris polymer identification. It has been used to identify microplastics near the surface of the Ross Sea, from the English Channel, and ingested by zooplankton (Cincinelli et al., 2017; Cole et al., 2014). Recently, Mecozzi et al. (2016) used FT-IR coupled with the Independent Component Analysis (ICA) database and Mahalanobis  \n\nDistance (MD) to identify marine plastics ingested by four loggerhead sea turtles in the Mediterranean Sea.  \n\nFT-IR spectroscopy offers a simple, efficient, and non-destructive method for identifying and distinguishing most plastic polymers, based on well-known infrared absorption bands representing distinct chemical functionalities present in the material (Verleye et al., 2001; Coates, 2000; Asensio et al., 2009; Beltran and Marcilla, 1997; Noda et al., 2007; Nishikida and Coates, 2003; Ilharco and Brito de Barros, 2000; Guidelli et al., 2011; Rotter and Ishida, 1992; Asefnejad et al., 2011). Structural isomeric polymers, such as HDPE and LDPE, are difficult, yet important, to differentiate. Asensio et al. (2009) and Nishikida and Coates (2003) reported that LDPE had a unique characteristic (yet quite small) band at $1377\\mathrm{cm}^{-1}$ , representing a $\\mathrm{CH}_{3}$ bending deformation, suggesting that even these similar polymers can be distinguished using FT-IR spectra. This band is reportedly absent in HDPE. These polymers differ by the extent of branching with HDPE being a linear PE chain with minimal branching, LLDPE having short alkyl branches off a linear backbone, and LDPE having long PE branches that represent a significant portion of the total chain length. Increased branching will reduce material density, with HDPE densities ranging from $0.94\\ \\mathrm{g/mL}$ to $0.97\\ \\mathrm{g/mL}$ and LLDPE and LDPE densities ranging from $0.90\\mathrm{g/mL}$ to $0.94\\ \\mathrm{g/mL}$ (Peacock, 2000; Verleye et al., 2001). However, chemical weathering, natural aging, and biochemical processes affecting ingested plastics can modify their spectral features, making identification difficult (Mecozzi et al., 2016), which was evident in Brandon et al. (2016) in which $30\\%$ of marine debris polyethylene (PE) samples could not be differentiated. These particularly challenging pieces produce confusing spectra due to the similar intensities of bands at $1377\\mathrm{cm}^{-1}$ and $1368~\\mathrm{{cm}^{-1}}$ . No study has yet tested or provided criteria on how to differentiate these.  \n\nThe goal of this study was to thoroughly assess the validity of attenuated total reflectance (ATR) FT-IR for identifying polymer composition of ingested plastic marine debris. This chemical technique is certainly not new and is common, but our study provides novel details that can help future studies avoid pitfalls, reduce confusion, and increase identification accuracy. We provide a clear guide with strict criteria to differentiate spectra from HDPE and LDPE. Furthermore, we identified and described the most effective cleaning method for preparing ingested plastic samples of three common polymers from pelagic, long-line caught olive ridley sea turtles to obtain high quality spectra. Sample handling was minimized to retain the original sample in a specimen bank for future additional chemical testing. To accomplish these goals, we developed an in-house spectral library from plastic consumer goods marked with resin codes. We validated our library with polymers originating from National Institute of Standards and Technology (NIST) Standard Reference Materials (SRMs) ${\\mathfrak{P}}$ , polymer standards obtained from scientific vendors, raw polymers sourced from manufacturers, and an additional set of consumer goods with polymer identity unknown to the analyst. PE materials of known density were used to confirm that ATR FT-IR is capable of discriminating between HDPE and LDPE, and to determine if a float/sink test in various dilutions of ethanol could further assist in differentiating these polymers. Using these optimized ATR FT-IR methods, we analyzed 828 ingested plastic items for polymer identity. A subset of these ingested samples was analyzed at NIST using high temperature size exclusion chromatography (HT-SEC) to confirm the accuracy of polymer identification by ATR FT-IR.  \n\n# 2. Methods  \n\n# 2.1. Plastic standards  \n\nPlastic standards were obtained from four sources with different degrees of purity or certainty (see Supplemental material Table S1 for a complete list). Four NIST SRMs and 10 polymers that were sourced from scientific/laboratory vendors (scientifically sourced) were considered the purest or best characterized. Raw materials obtained from manufacturers were considered purer than the consumer goods collected, which could contain additives. These standards represent each resin code $\\#1$ through $\\#6$ , LLDPE, and several code $\\#7$ or other polymers (Table S1). The $\\#7$ category included polymers that could be found in marine debris, including acrylonitrile butadiene styrene (ABS), cellulose acetate (CA), ethylene vinyl acetate (EVA), latex, nitrile, nylon (represented by nylon 12 and nylon 6,6), polycarbonate (PC), poly (methyl methacrylate) (PMMA or acrylic), polytetrafluoroethylene (PTFE), fluorinated ethylene propylene (FEP), and polyurethane (PU).  \n\nTwo to three consumer goods or raw materials labeled with each resin code were used to create standard spectra for each polymer. While consumer goods likely contain additives, the in-house spectral library was intentionally based on spectra from consumer goods, because they were assumed to more closely represent consumer items found in marine debris and ingested by marine organisms.  \n\nTo validate the polymer identification by the analyst from ATR FTIR spectra, eleven additional consumer goods with stamped resin codes were used in a blind test (Table S1): PETE ${\\bf(n}=2)$ ), HDPE $\\left(\\mathbf{n}=3\\right)$ , LDPE ${\\bf(n=2)}$ ), PE of unknown density $(\\mathbf{n}=1)$ ), PP ${\\bf\\Pi}({\\bf n}=2)$ , and PS ${\\mathrm{(n~}}=1{\\mathrm{)}}$ ).  \n\n# 2.2. ATR FT-IR instrument details  \n\nA Perkin Elmer FT-IR Spectrometer Spectrum Two Universal ATR was used to collect spectra from $4000~\\mathrm{{cm}^{-1}}$ to $450~\\mathrm{cm}^{-1}$ with a data interval of $1\\mathrm{cm}^{-1}$ . Resolution was set at $4\\mathrm{cm}^{-1}$ . The ATR diamond crystal was cleaned with $70\\%$ 2-propanol and a background scan was performed between each sample. Each sample was compressed against the diamond with a force of at least $80\\mathrm{N}$ to ensure good contact between sample and ATR crystal, as recommended by Perkin Elmer. Absorption bands identified using a peak height algorithm within the Perkin Elmer software were recorded and compared to absorption bands of each polymer reported in the literature and obtained from our in-house spectral library (Tables 1 and 2). A minimum of four matching absorption bands were required for accepted identification. Spectra of consumer goods of each polymer type tested are shown in Fig. 1. No pre-existing spectral library or database was used in this study. This was intentional, because comprehensive libraries can be expensive. We wanted out approach to be available to all labs regardless of their resources. Secondly, relying solely on automated library searches and statistical methods can lead to inaccurate identifications. For example, we suspect the automated approach used by Mecozzi et al. (2016) to identify plastic fragments from a sea turtle gastrointestinal tract resulted in inaccurate results. Three fragments were identified as polyethylene oxide, which is typically a liquid at environmental temperatures. Manual assessment of the spectra may have avoided this potential mistake.  \n\nWe validated the ability to differentiate HDPE and LDPE via the relative intensity of a small absorption band at $1377\\mathrm{cm}^{-1}$ , which represents the more abundant methyl group in highly branched LDPE (Asensio et al., 2009; Nishikida and Coates, 2003; Brandon et al., 2016). For samples determined to be PE, the spectral region of $1400~\\mathrm{{cm}^{-1}}$ to $1330\\mathrm{cm}^{-1}$ was examined closely by magnifying this region in Microsoft Excel scatterplots. PE spectra were binned into the following seven categories in which $1377\\mathrm{cm}^{-1}$ was 1) absent, 2) a shoulder on $1368~\\mathrm{{cm}^{-1}}$ , 3) a small bump on $1368~\\mathrm{{cm}^{-1}}$ , 4) the second largest band in this region, 5) nearly equal to $1368~\\mathrm{{cm}^{-1}}$ , 6) the strongest band in this region, 7) detected as a band by the instrument's software. The confidence of each bin to identify the PE type was assessed in three ways. Firstly, the ATR FT-IR spectral bin was recorded for each SRM, scientifically sourced or raw manufactured plastic standards of known PE. Secondly, the densities of PE standards and debris samples categorized across the bins were estimated via a float/sink test in different dilutions of ethanol (200 proof, A.C.S. reagent grade, Acros Organics, Fair Lawn, NJ) in deionized water. Dilutions were prepared volumetrically with graduated cylinders and ranged from $23\\%$ to $42\\%$ ethanol with approximately $2\\%$ increments. Density of the solutions was measured by weighing $25\\mathrm{mL}$ in a $25\\mathrm{-mL}$ graduated cylinder to the closest $_{0.0001\\ g}$ . Relative standard uncertainty in measuring the density of these solutions was $0.34\\%$ . The measured densities of all PE standards and 49 PE marine debris pieces collected from Main Hawaiian Island beaches were used to assign the piece to either HDPE or LDPE based on known densities of these polymers (Peacock, 2000; Verleye et al., 2001). The percentage of HDPE or LDPE assignments via the float test within each bin provided quantified confidence in using each bin and allowed us to set clear criteria. Thirdly, tentative ATR FT-IR assignments of ingested plastics from sea turtles (samples described below; 5 HDPE and 5 LDPE) were confirmed with HT-SEC with differential refractive index, infrared, and multi-angle light scattering detection at NIST (methods described below).  \n\nDifferentiation between LLDPE and LDPE was tested by examining the regions between $650~\\mathrm{{cm}^{-1}}$ and $1000\\mathrm{cm}^{-1}$ . According to Nishikida and Coates (2003), absorbance bands at $890~\\mathrm{{cm}^{-1}}$ (vinylidene group) and $910~\\mathrm{{cm}^{-1}}$ (terminal vinyl group) should be of similar intensities and both weak for LLDPE, whereas they state that $890~\\mathrm{{cm}^{-1}}$ should be predominant in LDPE. These spectra regions from one scientifically sourced LLDPE and three consumer goods made of LLDPE were compared to several LDPE materials.  \n\n# 2.3. Ingested plastic collection  \n\nAs described in Clukey et al. (2017), 2880 ingested plastic pieces were found in the gastrointestinal (GI) tracts of olive ridley $\\mathrm{(n}=37\\mathrm{)}$ , green $(\\mathbf{n}=9)$ , and loggerhead ${\\bf(n=4)}$ ) sea turtles caught incidentally by the Hawaiian and American Samoan longline fishery between 2012 and 2015. Pieces were removed with hexane-rinsed forceps, rinsed with nanopore deionized water, gently cleaned with cleanroom wipers, wrapped in hexane-rinsed foil, placed in a FEP bag, and archived frozen as part of the Biological and Environmental Monitoring and Archival of Sea Turtle tissues (BEMAST) project of the NIST Marine Environmental Specimen Bank (Keller et al., 2014).  \n\n# 2.4. Plastic preparation  \n\nTo minimize instrument time, a subset of pieces $\\mathbf{\\tau}(\\mathbf{n}=828)$ was selected for this study that visually represented all other pieces found in each turtle. A three-category rugosity scoring system was applied to some of the pieces and defined as (1) smooth, (2) ridged, and (3) rugose (Fig. S1). Pieces were weighed before and after FT-IR analysis, repackaged and frozen for continued archival storage by BEMAST and future chemical analysis.  \n\nEleven plastic fragments ingested by olive ridley sea turtles were chosen for testing five different cleaning methods, after being identified using absorption bands in Table 1 as HDPE ${\\mathrm{(n~}}=3{\\mathrm{)}}$ , LDPE $(\\mathbf{n}=5)$ , and PP $(\\mathbf{n}=3)\\AA$ . These fragments were analyzed by ATR FT-IR after undergoing five different treatments: (1) no additional cleaning, (2) wiping a small area with a dry cleanroom wiper, (3) wiping a new area with a cleanroom wiper that was wet with $70\\%$ 2-propanol from a LDPE squirt bottle, (4) wiping a third area with a cleanroom wiper wet with deionized water from a LDPE squirt bottle, and (5) cutting the piece with hexane-rinsed scissors or pliers to expose the inside surface of the fragment. Three spectra were generated for each cleaning method on each piece by analyzing the fragment on three non-overlapping sections of the cleaned area. The optimal cleaning method was determined as described below in statistical methods. These less destructive cleaning methods were chosen over chemical manipulation with acids and strong solvents as in Mecozzi et al. (2016) for green chemistry reasons and to minimize manipulation so that the samples could be archived by BEMAST and tested in the future for persistent organic pollutants.  \n\nTable 1 List of important vibration modes and mode assignments for the ATR FT-IR spectra of eight of 16 polymers identified. The remaining eight polymers are in Table 2. Absorption ban listed are representative of vibrations critical for polymer identification. Please consult references for full lists of absorption bands.   \n\n\n<html><body><table><tr><td>Polymer</td><td>Resin code</td><td>Chemical structure</td><td>Absorption bands (cm-1) used for identificationa</td><td>Assignment</td><td>Reference in addition to this study</td></tr><tr><td>Polyethylene terephthalate (PETE)</td><td>1</td><td></td><td>1713 (a) 1241 (b)</td><td>C=O stretch</td><td>Asensio et al., 2009; Verleye et al., 2001; Noda et al., 2007</td></tr><tr><td></td><td></td><td></td><td>1094 (c) 720 (d)</td><td>C-O stretch C-O stretch Aromatic CH out-of-</td><td></td></tr><tr><td>High-density polyethylene (HDPE)</td><td>2</td><td></td><td>2915 (a)</td><td>plane bend C-H stretch</td><td>Asensio et al., 2009; Noda</td></tr><tr><td></td><td></td><td>n</td><td>2845 (b)</td><td>C-H stretch</td><td>et al., 2007; Nishikida and</td></tr><tr><td></td><td></td><td></td><td>1472 (c)</td><td>CH bend</td><td>Coates,2003</td></tr><tr><td></td><td></td><td></td><td>1462 (d)</td><td>CH bend</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>730 (e)</td><td>CH2 rock</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Polyvinyl chloride (PVC)</td><td>3</td><td></td><td>717 (f)</td><td>CH2 rock</td><td></td></tr><tr><td></td><td></td><td>></td><td>1427 (a)</td><td>CH2 bend</td><td>Beltran and Marcilla, 1997;</td></tr><tr><td></td><td></td><td>n cl</td><td>1331 (b)</td><td>CH bend</td><td>Verleye et al., 2001; Noda</td></tr><tr><td></td><td></td><td></td><td>1255 (c)</td><td>CH bend</td><td>et al.,2007</td></tr><tr><td></td><td></td><td></td><td>1099 (d)</td><td>C-C stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>966 (e)</td><td>CH2 rock</td><td></td></tr><tr><td>Low-density polyethylene</td><td></td><td></td><td>616 (f)</td><td>C-Cl stretch</td><td></td></tr><tr><td>(LDPE) or linear LDPE</td><td>4</td><td></td><td>2915 (a)</td><td>C-H stretch</td><td>Asensio et al., 2009; Noda</td></tr><tr><td>(LLDPE)</td><td></td><td>I'n</td><td>2845 (b)</td><td>C-H stretch</td><td>et al., 2007; Nishikida and</td></tr><tr><td></td><td></td><td>IR R = H or alkyl (LLDPE), PE (LDPE)</td><td>1467 (c)</td><td>CH bend</td><td>Coates,2003</td></tr><tr><td></td><td></td><td></td><td>1462 (d)</td><td>CH bend</td><td></td></tr><tr><td></td><td></td><td></td><td>1377 (e)</td><td>CH3 bend</td><td></td></tr><tr><td></td><td></td><td></td><td>730 (f)</td><td>CH2 rock</td><td></td></tr><tr><td>Polypropylene (PP)</td><td></td><td></td><td>717 (g)</td><td>CH2 rock</td><td></td></tr><tr><td></td><td>5</td><td></td><td>2950 (a)</td><td>C-H stretch</td><td>Asensio et al., 2009; Verleye</td></tr><tr><td></td><td></td><td>n</td><td>2915 (b)</td><td>C-H stretch</td><td>et al., 2001; Noda et al., 2007</td></tr><tr><td></td><td></td><td></td><td>2838 (c)</td><td>C-H stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>1455 (d)</td><td>CH2 bend</td><td></td></tr><tr><td></td><td></td><td></td><td>1377 (e)</td><td>CH3 bend</td><td></td></tr><tr><td></td><td></td><td></td><td>1166 (f)</td><td>CH bend, CH3 rock, C-C stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>997 (g)</td><td>CH3 rock, CH bend, CH bend</td><td></td></tr><tr><td></td><td></td><td></td><td>972 (h)</td><td>CH3 rock, C-C stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>840 (i)</td><td>CH rock, C-CH3 stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>808 (j)</td><td>CH rock, C-C stretch,</td><td></td></tr><tr><td>Polystyrene (PS)</td><td>6</td><td></td><td>3024 (a)</td><td>C-CH stretch Aromatic C-H stretch</td><td>Asensio et al., 2009; Verleye</td></tr><tr><td></td><td></td><td>n</td><td>2847 (b)</td><td>C-H stretch</td><td>et al., 2001; Noda et al., 2007</td></tr><tr><td></td><td></td><td></td><td>1601 (c)</td><td>Aromatic ring stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>1492 (d)</td><td>Aromatic ring stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>1451 (e)</td><td>CH bend Aromatic CH bend</td><td></td></tr><tr><td></td><td></td><td></td><td>1027 (f) 694 (g)</td><td>Aromatic CH out-of-</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>plane bend</td><td></td></tr><tr><td></td><td></td><td></td><td>537 (h)</td><td> Aromatic ring out-of-</td><td></td></tr><tr><td>Acrylonitrile butadiene styrene</td><td>7</td><td></td><td></td><td>plane bend</td><td>Verleye et al., 2001</td></tr><tr><td>(ABS)</td><td></td><td>P (1-p-β)</td><td>2922 (a) 1602 (b)</td><td>C-H stretch Aromatic ring stretch</td><td></td></tr><tr><td></td><td></td><td>N</td><td>1494 (c)</td><td>Aromatic ring stretch</td><td></td></tr><tr><td></td><td></td><td></td><td>1452 (d)</td><td>CH2 bend</td><td></td></tr><tr><td></td><td></td><td>ABS is mixtureofcis,trans,and vinyl</td><td>966 (e)</td><td>=C-H bend</td><td></td></tr><tr><td></td><td></td><td>isomers, linear and crosslinked</td><td>759 (f)</td><td>Aromatic CH out-of- plane bend, =CH</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td>bend</td><td></td></tr><tr><td></td><td></td><td></td><td>698 (g)</td><td>Aromatic CH out-of-</td><td></td></tr><tr><td>Cellulose acetate (CA)</td><td>7</td><td></td><td>1743 (a)</td><td>plane bend C=O stretch</td><td>Ilharco and Brito de Barros,</td></tr><tr><td></td><td></td><td></td><td>1368 (b)</td><td>CH3 bend</td><td>2000; Verleye et al., 2001;</td></tr><tr><td></td><td></td><td></td><td>904 (c)</td><td>Aromatic ring stretch</td><td>Noda et al.,2007</td></tr><tr><td></td><td></td><td></td><td></td><td>or CH bend</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>600 (d)</td><td>O-H bend</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td></table></body></html>\n\na Resolution was $4\\mathrm{cm}^{-1}$ . Letters can be cross referenced to bands shown in ATR FT-IR spectra in Fig. 1.  \n\n# 2.5. Analysis of ingested plastics for polymer type  \n\nThe 828 ingested plastic pieces discovered in the turtles were analyzed by ATR FT-IR by first cleaning a small area with water and cleanroom wiper or cutting to expose a smooth clean surface. Polymers were identified based on presence of absorption bands as described in Table 1 and shown in Fig. 1. Pieces producing absorption bands consistent with both PE and PP were assigned as “mixture” (Fig. 2). Pieces that could not be identified by ATR FT-IR spectra (e.g., presence of less than four identifying absorption bands) were assigned “unknown.” A subset of pieces that were identified by ATR FT-IR was analyzed by high HT-SEC with differential refractive index, infrared, and multi-angle light scattering detection.  \n\nTable 2 List of important vibration modes and mode assignments for the ATR FT-IR spectra for the remaining eight of 16 polymers identified. Absorption bands listed are representative o   \n\n\n<html><body><table><tr><td colspan=\"6\">vibrations critical for polymer identification.Please consult references for fullists of absorption bands.</td></tr><tr><td>Polymer</td><td>Resin code Chemical structure</td><td></td><td>Absorption bands (cm -1) used for</td><td>Assignment</td><td>Reference in addition to this study</td></tr><tr><td rowspan=\"5\">Ethylene vinyl acetate (EVA)</td><td rowspan=\"5\">7</td><td></td><td>identificationa</td><td>C-H stretch</td><td rowspan=\"5\">Asensio et al., 2009; Verleye et al., 2001</td></tr><tr><td>[(1-p) 0</td><td>2917 (a) 2848 (b)</td><td>C-H stretch</td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td>1740 (c)</td><td>C=O stretch CH bend, CH3</td></tr><tr><td>1469 (d)</td><td>bend</td><td></td></tr><tr><td rowspan=\"5\"></td><td rowspan=\"5\"></td><td></td><td>1241 (e)</td><td>C(=0)0 stretch</td><td rowspan=\"5\"></td></tr><tr><td></td><td>1020 (f)</td><td>C-O stretch</td></tr><tr><td></td><td></td><td>CH rock</td></tr><tr><td></td><td>720 (g)</td><td></td></tr><tr><td></td><td>2960 (a)</td><td>C-H stretch Guidelli et al., 2011 C-H stretch</td></tr><tr><td rowspan=\"5\"></td><td rowspan=\"5\"></td><td>(1-p)</td><td>2920 (b) 2855 (c)</td><td>C-H stretch</td><td rowspan=\"5\"></td></tr><tr><td></td><td>1167 (d)</td><td>C=C stretch</td></tr><tr><td>Mixture of cis and trans; natural latex does not</td><td>1447 (e)</td><td>CH bend</td></tr><tr><td></td><td>1376 (f)</td><td>CH3 bend</td></tr><tr><td>contain styrene copolymer </\\/</td><td>2917 (a)</td><td>=C-H stretch Coates, 2000; Verleye</td></tr><tr><td rowspan=\"5\"> Nitrile</td><td rowspan=\"5\"></td><td>(1-p) Mixture of ci an trans</td><td>2849 (b)</td><td>=C-H stretch</td><td rowspan=\"5\">et al.,2001</td></tr><tr><td></td><td>2237 (</td><td>C- strtreheh</td></tr><tr><td>1440 (e)</td><td></td><td></td></tr><tr><td></td><td></td><td>CH2 bend</td></tr><tr><td>1360 (f) 1197 (g)</td><td></td><td>CH bend CH2 bend</td></tr><tr><td rowspan=\"7\">Nylon (all polyamides)</td><td rowspan=\"7\">7</td><td rowspan=\"7\"></td><td>967 (h)</td><td>=C-H bend</td><td></td><td rowspan=\"7\">Rotter and Ishida, 1992; Verleye et al., 2001;</td></tr><tr><td>3298 (a) 2932 (b)</td><td></td><td>N-H stretch CH stretch</td></tr><tr><td></td><td></td><td>CH stretch Noda et al., 2007</td></tr><tr><td>2858 (c)</td><td></td><td></td></tr><tr><td>r and b vary from 0 to 12 based on monomer 1634 ()</td><td></td><td>G= end, cb-N</td></tr><tr><td></td><td></td><td>stretch</td></tr><tr><td>1464 (f) 1372 (g)</td><td>CH bend CH bend</td><td></td></tr><tr><td rowspan=\"7\">Polycarbonate (PC)</td><td rowspan=\"7\"></td><td rowspan=\"7\"></td><td></td><td>1274 (h)</td><td>NH bend, C-N stretch</td></tr><tr><td>1199 (i)</td><td>CH bend</td><td></td></tr><tr><td>687 (i)</td><td> NH bend,C=O</td><td></td></tr><tr><td></td><td>bend</td><td>Asensio et al., 2009;</td></tr><tr><td>2966 (a) 1768 (b)</td><td>CH stretch C=O stretch</td><td>Verleye et al., 2001; Noda et al., 2007</td></tr><tr><td>1503 (c)</td><td>Aromatic ring</td><td></td></tr><tr><td>1409 (d)</td><td>stretch Aromatic ring</td><td></td></tr><tr><td rowspan=\"7\"></td><td rowspan=\"7\"></td><td rowspan=\"7\"></td><td></td><td>stretch</td><td></td></tr><tr><td>1364 (e)</td><td>CH3 bend</td><td></td></tr><tr><td>1186 (f)</td><td>C-O stretch</td><td></td></tr><tr><td>1158 (g)</td><td>C-O stretch</td><td></td></tr><tr><td>1013 (h)</td><td>Aromatic CH in-</td><td></td></tr><tr><td></td><td> plane bend</td><td></td></tr><tr><td>828 (i)</td><td>Aromatic CH out-</td><td></td></tr><tr><td rowspan=\"7\">Poly(methyl methacrylate) (PMMA or acrylic)</td><td rowspan=\"7\">7</td><td rowspan=\"7\">'n</td><td></td><td>of-plane bend</td><td></td></tr><tr><td>2992 (a)</td><td>C-H stretch</td><td>Verleye et al., 2001</td></tr><tr><td>2949 (b)</td><td>C-H stretch</td><td></td></tr><tr><td>1721 (c)</td><td>C=O stretch</td><td></td></tr><tr><td>1433 (d)</td><td>CH2 bend</td><td></td></tr><tr><td>1386 (e)</td><td>CH bend</td><td></td></tr><tr><td>1238 (f)</td><td>C-O stretch</td><td></td></tr><tr><td rowspan=\"7\"></td><td rowspan=\"7\"></td><td></td><td>1189 (g) 1141 (h)</td><td>CH3 rock C-O stretch CH rock</td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>985 (i) 964 ()</td><td></td><td>C-H bend</td><td></td></tr><tr><td>750 (k)</td><td></td><td>CH rock, C=O</td><td></td></tr><tr><td></td><td></td><td>bend</td><td>Coates, 2000; Verleye</td></tr><tr><td>1201 (a)</td><td></td><td>CF2 stretch CF2 stretch</td><td>et al.,2001</td></tr><tr><td></td><td>1147 (b)</td><td></td><td></td></tr><tr><td rowspan=\"4\">(FEP) Polyurethane (PU) 7</td><td rowspan=\"2\"></td><td></td><td>638 (d)</td><td>C-C-F bend CF2 bend</td><td></td></tr><tr><td>554 (e)</td><td></td><td></td><td></td></tr><tr><td></td><td>509 (f) 2865 (a)</td><td>CF2 bend C-H stretch</td><td>Asefnejad et al., 2011;</td></tr><tr><td colspan=\"2\"></td><td></td><td></td></tr><tr><td colspan=\"2\"></td><td></td><td>1731 (b) 1531 (c)</td><td>C=O stretch C-N stretch CH bend</td><td>Verleye et al., 2001; Noda et al., 2007</td></tr></table></body></html>\n\na Resolution was $4\\mathrm{cm}^{-1}$ . Letters can be cross referenced to bands shown in ATR FT-IR spectra in Fig. 1.  \n\n![](images/7c69f72aabeea4e17771d1122dd2fa715406680ee1a43a99fb0611fc64d6054c.jpg)  \nFig. 1. Spectra produced from plastic consumer goods labeled with resin codes of (a) polyethylene terephthalate (PETE, $\\#1\\mathrm{\\cdot}$ , (b) high-density polyethylene (HDPE, $\\#2\\rangle$ , (c) polyvinyl chloride (PVC, #3), (d) low-density polyethylene and linear low density polyethylene (LDPE and LLDPE, $\\#4)$ , (e) polypropylene (PP, $\\#5\\mathrm{\\cdot}$ , and (f) polystyrene (PS, $\\#6)$ along with ten other polymers: $\\mathbf{\\delta}(\\mathbf{g})$ acrylonitrile butadiene styrene (ABS), (h) cellulose acetate (CA), (i) ethylene vinyl acetate (EVA), (j) latex, (k) nitrile, (l) nylons, $\\mathbf{\\tau}(\\mathbf{m})$ polycarbonate (PC), (n) poly (methyl methacrylate) (PMMA), (o) polytetrafluoroethylene (PTFE) or fluorinated ethylene propylene (FEP), and (p) polyurethane (PU) using ATR FT-IR. Letters represent characteristic absorption bands $(\\mathrm{cm}^{-1})$ used to identify each polymer.  \n\nSamples for HT-SEC were sonicated in ethanol for $10\\mathrm{min}$ , followed by $10\\mathrm{min}$ sonication in nanopure deionized water $(18.2\\mathrm{M}\\Omega)$ to remove aqueous soluble contaminants and minimize the addition of biological contaminants to the instrument. Approximately $10\\mathrm{mg}$ of each sample was encased in a $5\\upmu\\mathrm{m}$ stainless steel mesh and dissolved in HPLC grade $^{1,2,4}$ -trichlorobenzene under nitrogen atmosphere for $^{\\textrm{1h}}$ prior to injection in the instrument, allowing soluble polymers to dissolve and pass through the mesh, and insoluble debris, filler, or crosslinked components to remain sequestered in the mesh. The samples were injected into a Polymer Characterization (Valencia, Spain) GPC-IR instrument with an IR 4 detector consisting of two infrared IR detection bands, $2800~\\mathrm{{cm}^{-1}}$ to $3000~\\mathrm{{cm}^{-1}}$ representing the entire $\\mathsf{C}{\\mathrm{-}}\\mathsf{H}$ stretching region (CH, $\\mathrm{CH}_{2},$ , and $\\mathrm{CH}_{3})$ ), and a narrow band at $2950\\mathrm{cm}^{-1}$ for the methyl $\\mathsf{C}{\\mathrm{-}}\\mathsf{H}$ stretch absorbances, respectively, as well as a Wyatt Technology (Santa Barbara, CA) Dawn Heleos II multi-angle light scattering (MALS) detector with 18 angles and a forward monitor (zero angle detector). Separately, the samples were also injected on a Tosoh (Tokyo, Japan) HT-Eco SEC with differential refractive index detection. Both instruments ran at $160^{\\circ}\\mathrm{C}$ with a $^{1,2,4}$ -trichlorobenzene mobile phase with 300 ppm Irganox 1010 added as an antioxidant. The stationary phase columns used in both systems are a set of three Tosoh HT2 columns (two, Tosoh TSKgel GMHhr-H (S) HT2, $13\\upmu\\mathrm{m}$ mixed bed, $7.8\\:\\mathrm{mm}\\:\\mathrm{ID}\\times30\\:\\mathrm{cm}$ columns and one, Tosoh TSKgel GMHHR-H (20) HT2, $20\\upmu\\mathrm{m}$ , $7.8\\mathrm{mm}$ $\\begin{array}{l l l}{\\mathrm{ID}}&{\\times}&{30\\mathrm{cm}}\\end{array}$ column with an exclusion limit $\\approx4\\times10^{8}\\mathrm{g/mol})$ . Sample molar masses, molar mass distribution, short chain branching content (SCB), were determined by calibration with narrow molar mass distribution PS standards, NIST SRM 1475a (linear, broad, HDPE), and NIST SRM 1478 (to determine interdetector delay and normalize photodiode response of the MALS detector), and 10 blends of linear PE and PP with systematic variation of PP content, where the total degree of short chain branching (SCB) was confirmed by nuclear magnetic resonance spectroscopy (NMR). Calibration and data analysis was performed by proprietary software from each instrument vendor. HT-SEC and NMR have many advantages, but they can only measure polymer chains that are soluble under the solvent and temperature conditions used and they are not highthroughput like ATR-FTIR, which measures the bulk sample.  \n\n![](images/f557dc94d1e0146966e3125e3bb16f538a0e8afa38d947f492133c8d93996e8c.jpg)  \nFig. 1. (continued)  \n\n![](images/849660cb500e0911401bbb4f988d690fda4a2da719767a57edb10a368e016975.jpg)  \nFig. 2. ATR FT-IR spectrum of an ingested plastic fragment assigned as a mixture of polyethylene (PE) and polypropylene (PP). Wavenumbers in boxes are characteristic of PE, underlined wavenumbers are characteristic of both PE and PP, and unmarked wavenumbers are characteristic of PP.  \n\n# 2.6. Data handling and statistical analysis  \n\nOrdination was used to synthesize the absorbance data, in order to: 1) determine if novel absorption bands at additional wavenumbers could distinguish HDPE and LDPE, and 2) investigate if clustering of “unknown” ingested pieces near known polymers could help to identify their polymer composition. MetaboAnalyst software was used and the “normalized by sum” option was chosen so that all spectral bands had equal weight and samples could be compared. Two principal component analyses (PCAs) were performed on different sample sets. PCA1 included spectra from three consumer goods of each of the following polymers: HDPE, LDPE, and PP. PCA2 included 797 ingested plastic pieces (793 identified, 4 unknown) of nine identified polymer types. Because PCA requires at least three samples of each polymer and less than three ingested pieces of PVC and nylon were discovered, it was necessary to include the spectra of consumer good items representing PVC and nylon in PCA2. PCA1 was run using bins of four wavenumbers over the entire spectral range of $4000\\mathrm{cm}^{-1}$ to $450\\mathrm{cm}^{-1}$ while PCA2 used selected absorption bands within a range o $\\mathsf{f}\\pm1\\mathrm{cm}^{-1}$ identified in Table 1 (plus additional bands from the literature) for polymers included in the analysis. All possible absorption bands were included in PCA1 to discover novel absorption bands for distinguishing HDPE from LDPE. No transformations were performed and Pareto scaling was used for both PCAs.  \n\nThe optimal cleaning method was determined in two ways: 1) determining the percent of spectra that provided visually identifiable polymer assignment (good vs. poor quality spectra), and 2) counting the number of detectable absorption bands used for identification of that particular polymer. Wavenumbers with absorption bands greater than three times the noise surrounding the absorption band were recorded as detectable wavenumbers. All variables were tested for normality using the Shapiro-Wilk tests in IBM SPSS Statistics Version 24.  \n\nBecause normality could not be accomplished even after data transformations, non-parametric Friedman's ANOVA tests followed by Wilcoxon signed-rank tests were used to compare differences in cleaning methods using two different response variables: percent of identifiable spectra and number of identifiable absorption bands greater than three times the noise. A Spearman Rank Order correlation was used to determine if rugosity had an effect on the number of detectable wavenumbers.  \n\n# 3. Results and discussion  \n\n# 3.1. ATR FT-IR polymer identification of consumer goods, raw manufactured, or scientifically sourced polymers  \n\nPlastic consumer goods from known resin codes produced spectra with expected absorption bands (Fig. 1, Table 1). When compared to the spectra of raw manufactured polymers or scientifically sourced polymers, the appearance and number of identifiable wavenumbers were nearly identical (data not shown). Absorption bands identified for these polymers were either a direct match or within four wavenumbers of the absorption bands listed in Table 1. Of the 18 polymers tested, all could be easily distinguished from each other with only three minor exceptions (Fig. 1). Spectra of FEP and PTFE showed absorbance bands at the same wavenumbers and with the same intensity for $638~\\mathrm{{cm}^{-1}}$ , $554~\\mathrm{{cm}^{-1}}$ , and $509\\mathrm{cm}^{-1}$ , but the intensity of $1201~\\mathrm{{cm}^{-1}}$ ( $\\mathrm{CF}_{2}$ stretch) and $1147~\\mathrm{{cm}}^{-1}$ ( $\\mathrm{CF}_{2}$ stretch) were $16\\%$ and $27\\%$ higher, respectively, in PTFE than FEP. All types of nylon produced the same absorbance bands, so nylon-12 cannot be distinguished from nylon 6,6 or others (Verleye et al., 2001). Differentiating among HDPE, LLDPE, and LDPE is challenging, but our goal was to develop a simple ATR FT-IR method so that discrimination by sample destructive methods, such as HT-SEC with infrared detection or differential scanning calorimetry (DSC), is not required. Our results confirm that ATR FT-IR can identify consumer goods produced from PETE, PEs, PVC, PP, PS, ABS, CA, EVA, latex, nitrile, nylons, PC, PMMA, (PTFE or FEP), and PU, but PE samples require closer inspection of the ATR FT-IR spectra to distinguish HDPE from LDPE.  \n\nThe use of ATR FT-IR for polymer identification was further confirmed via a blind test, in which 11 consumer goods consisting of diverse polymers were correctly identified by an analyst without prior knowledge of the resin code (Table S1). The five PE samples were all correctly identified as PE, but some could not be further categorized as either HDPE or LDPE. Of the three HDPE samples, one was correctly assigned and two were categorized as unknown PE. Of the two LDPE, one was correctly assigned and one was assigned unknown PE.  \n\nIn hopes of discovering additional absorbance bands to distinguish HDPE from LDPE, a PCA was performed including the spectra of consumer goods of HDPE ${\\bf(n=3)}$ ), LDPE $\\left(\\mathbf{n}=3\\right)$ ), and PP $\\left(\\mathbf{n}=3\\right)$ ). The PCA showed no separation between HDPE and LDPE (Fig. S2). The first two principal components (PC) explained $78.3\\%$ of the variance and the loadings are shown in Table S2. This biplot revealed absorbance bands that differentiate PE from PP $(700\\mathrm{cm}^{-1}$ to $730\\mathrm{cm}^{-1}$ , $\\mathrm{CH}_{2}$ rock), but no novel bands that could distinguish HDPE from LDPE (Fig. S2). While additional PCs explained more of the variation $(14.7\\%$ by PC3 and $6\\%$ by PC4), they did not provide any additional separation of HDPE from LDPE. This is because HDPE and LDPE share the same major structural unit, functional groups, chemical bonds (Asensio et al., 2009), and therefore have many identical wavenumbers (Table 1). However, the different degree of branching results in small, but important differences, in the spectral region of $1400\\mathrm{cm}^{-1}$ to $1330\\mathrm{cm}^{-1}$ with LDPE having greater intensity at $1377\\mathrm{cm}^{-1}$ due to methyl bending deformation of the branched chain ends (Asensio et al., 2009; Nishikida and Coates, 2003). This band may have been too small for the PCA to detect and must be magnified and compared to the intensity of $1368~\\mathrm{{cm}^{-1}}$ manually.  \n\nThe differentiation between LDPE and HDPE with the presence of a  \n\n![](images/f1c3b80161578ab9c7d2ceddb28b495eb187728c11d742a9f2030f2cc63e27cb.jpg)  \nFig. 3. Decision flow chart for differentiating high-density polyethylene (HDPE), linear low-density polyethylene (LLDPE), and low-density polyethylene (LDPE) using ATR FT-IR spectra and float/sink tests.  \n\n$1377\\mathrm{cm}^{-1}$ band is easy in some spectra, while others are more challenging. PE spectra fell into seven different bins based on the observation of the $1377\\mathrm{cm}^{-1}$ band being: 1.) absent, 2.) a tiny shoulder, 3.) a small bump, 4.) the second largest in the $1400~\\mathrm{{cm}^{-1}}$ to $1330\\mathrm{cm}^{-1}$ region, 5.) nearly equivalent to $1368~\\mathrm{{cm}^{-1}}$ band, 6.) the largest in this region, and 7.) detected by the instrument's software (Fig. 3). Bins at the extremes (1 and 2 are HDPE; 6 and 7 are LDPE) are clear, but those in the middle are ambiguous (bins 3, 4, and 5) and cause substantial confusion.  \n\nTable S1 describes ATR FT-IR results of each standard and consumer good tested. All but one of the SRMs, raw manufactured plastic or scientifically sourced standards of known PE $({\\mathfrak{n}}=10$ for HDPE, $\\mathtt{n}=4$ LLDPE, and ${\\mathfrak{n}}=17$ LDPE) were correctly and easily assigned because they fell in the clear bins (Table S1). One LDPE standard, SRM 1474b, fell into bin 3. Fourteen consumer good standards were stamped with HDPE $\\mathrm{(n=6)}$ , LLDPE $\\left(\\mathbf{n}=3\\right)$ , or LDPE $(\\mathbf{n}=5)$ . Of these, eight $(57\\%)$ were accurately assigned because they produced unambiguous spectra, six $(43\\%)$ produced ambiguous spectra, and one $\\left(0.1\\%\\right)$ with a clear spectrum was inaccurately assigned. Three of the six ambiguous samples were thin bags used for produce, shopping, and shipping. The incorrect standard was a grocery shopping bag stamped with $\\#2$ resin code (HDPE), but $1377\\mathrm{cm}^{-1}$ was the strongest peak. These results suggest that thin sheet bags are consistently the most ambiguous and challenging to assign to HDPE versus LDPE for reasons currently unknown.  \n\nThe other three ambiguous spectra came from all three LLDPE consumer goods tested. Because LLDPE has intermediate extents of branching, this was not surprising. Unfortunately, bins 3, 4, and 5 cannot be considered LLDPE, because materials known to be HDPE and  \n\nLDPE also produced spectra in these bins. A method to distinguish LLDPE from LDPE samples was proposed using another region of the spectra $650~\\mathrm{cm}^{-1}$ to $1000~\\mathrm{{cm}^{-1}};$ by Nishikida and Coates (2003). They report that LLDPE should have equal and weak bands at $890~\\mathrm{{cm}^{-1}}$ (vinylidene group) and $910~\\mathrm{{cm}^{-1}}$ (terminal vinyl group), whereas $890~\\mathrm{{cm}^{-1}}$ is larger in LDPE. We could not confirm this method with a close examination of this spectral region with four LLDPE and 18 LDPE standards (Table S1). The four known LLDPE standards produced variable results. The LLDPE trash bag had equally weak bands, as expected. The LLDPE biohazard bag produced a band at $890~\\mathrm{{cm}^{-1}}$ was larger but nearly equal to the $910~\\mathrm{{cm}^{-1}}$ band. However, the scientifically sourced LLDPE sample produced no band at $890~\\mathrm{{cm}^{-1}}$ and a small band at $910~\\mathrm{{cm}^{-1}}$ , and the LLDPE tubing produced a larger band at $910~\\mathrm{{cm}^{-1}}$ than $890~\\mathrm{{cm}^{-1}}$ . As expected, 14 of the 18 LDPE samples $(78\\%)$ produced a more intense $890~\\mathrm{{cm}}^{-1}$ band than $910~\\mathrm{{cm}^{-1}}$ . Three materials produced equally intense peaks (SRM 1476a, a swimmer's ear bottle, and a shipping bag), and for this reason we suspect they were produced with LLDPE. One produced a very small band at $910~\\mathrm{{cm}^{-1}}$ and no band at $890~\\mathrm{{cm}}^{-1}$ (a breastmilk storage bag). The inconsistent results within the known LLDPE standards did not give enough confidence to use this distinguishing method. Therefore, we conclude that LLDPE and LDPE cannot be distinguished from each other using ATR FT-IR.  \n\nIn the ambiguous bins 3, 4, and 5, the $1377\\mathrm{cm}^{-1}$ band appears as a small bump on the tail of the $1368~\\mathrm{{cm}^{-1}}$ band, a distinct but smaller band than $1368~\\mathrm{{cm}^{-1}}$ , or equivalent to the intensity of $1368~\\mathrm{{cm}^{-1}}$ respectively. Confidence to assign these bins to a particular PE was assessed, for the first time to our knowledge, by estimating the density of PE samples using a float/sink test in different dilutions of ethanol. Using a graduated cylinder to volumetrically prepare solutions resulted in inaccuracies of solution densities of up to $0.02\\mathrm{g/mL}$ . Because distinguishing between $0.93\\mathrm{g/mL}$ and $0.94~\\mathrm{g/mL}$ required better accuracy, the density of the solutions was determined by weighing $25\\mathrm{mL}$ in a graduated cylinder. Relative standard uncertainty in measuring the density of these solutions was $0.34\\%$ . Resulting estimated densities for each standard item are shown in Table S1. All PE standards that were not stamped or labeled with a resin code that were subsequently assigned LDPE because they fell in bins 6 and 7 $\\mathbf{\\rho}(\\mathbf{n}=4)$ ), floated in solutions $\\mathrm{>0.931g/mL}$ as expected. This added more confidence to our LDPE criteria. Furthermore, 19 of the 20 known LDPE or LLDPE standards $(95\\%)$ tested had estimated densities of $0.938\\mathrm{g/mL}$ or less; and all ten known HDPE standards had estimated densities of $0.938\\mathrm{g/mL}$ or greater. Unexpectedly, one LDPE thin shipping bag sank in solutions up to $0.950~\\mathrm{g/mL}$ . Its greater density may be attributed to a silver-colored inner layer of unknown polymer composition.  \n\nBecause our methods to differentiate HDPE from LDPE were slightly less successful in consumer goods than in raw or scientifically sourced standards, we confirmed our method using marine debris samples. Forty-nine plastic debris items collected from Main Hawaiian Island beaches that were discovered to be PE by ATR FT-IR were categorized as bin 1 $(\\boldsymbol{\\mathrm{n}}=3)$ , bin 2 ${\\bf(n=14)}$ ), bin 3 $(\\mathbf{n}=1)$ ), bin 4 $(\\mathbf{n}=5)$ , bin 5 $(\\boldsymbol{\\mathrm{n}}=13)$ , and bin 6 ${\\mathrm{(n}}=13{\\mathrm{)}}$ ). These fragments were tested for floating or sinking in solutions with targeted densities of $0.935\\mathrm{g/mL}$ and $0.941~\\mathrm{g/mL}$ . All of bin 1 samples sank, as expected for HDPE. $86\\%$ of bin 2 samples sank, providing enough confidence to conclude spectra with a shoulder at $1377\\mathrm{cm}^{-1}$ are very likely HDPE. The one sample in bin 3 sank, suggesting samples producing a very small bump at $1377\\mathrm{cm}^{-1}$ are HDPE, but our sample size was too small to have certainty. Only $40\\%$ of bin 4 and $46\\%$ of bin 5 floated in both solutions, suggesting these polymers could be either HDPE or LDPE when $1377\\mathrm{cm}^{-1}$ is the second largest band or equivalent to $1368~\\mathrm{{cm}^{-1}}$ . $85\\%$ of bin 6 samples floated in both solutions, giving us enough confidence to confirm that spectra with $1377\\mathrm{cm}^{-1}$ as the largest band are likely LDPE, even though the instrument's software does not detect it.  \n\nThe inter-laboratory comparison using HT-SEC on PE ingested plastic samples, confirmed that ATR FT-IR assignments were $100\\%$  \n\nTable 3 Comparison of identifications of ingested plastic samples analyzed by attenuated total reflectance Fourier transform infrared spectroscopy (ATR FT-IR) and high-temperature size exclusion chromatography (HT-SEC) with infrared, differential refractive index, and multi-angle light scattering detection.   \n\n\n<html><body><table><tr><td colspan=\"6\">HT-SEC results</td></tr><tr><td>Identification by ATR FT-IR</td><td>Identification by HT-SEC</td><td>RI peak magnitude</td><td>Average CH3/ 1000 total Ca</td><td>Mn (kg/ mol)c</td><td>Mw (kg/ mol)c</td></tr><tr><td>PETE</td><td>PUb</td><td></td><td></td><td></td><td></td></tr><tr><td>HDPE</td><td>HDPE</td><td>一</td><td>11.2 ± 7</td><td>1.1</td><td>36.2</td></tr><tr><td>HDPE</td><td>HDPE</td><td></td><td>10.6 ± 8</td><td>26.6</td><td>161.2</td></tr><tr><td>HDPE</td><td>HDPE</td><td></td><td>6.2±9</td><td>6.0</td><td>83.8</td></tr><tr><td>HDPE</td><td>HDPE</td><td>一</td><td>9.9 ± 15</td><td>5.0</td><td>32.8</td></tr><tr><td>HDPE</td><td>HDPE</td><td></td><td>5.7±9</td><td>15.2</td><td>80.4</td></tr><tr><td>LDPE</td><td>LDPE</td><td>一</td><td>24.0 ± 5</td><td>42.5</td><td>148.3</td></tr><tr><td>LDPE</td><td>LDPE</td><td>一</td><td>35.8 ±17</td><td>0.9</td><td>70.9</td></tr><tr><td>LDPE</td><td>LDPE</td><td>一</td><td>48.3 ± 16</td><td>2.5</td><td>65.6</td></tr><tr><td>LDPE</td><td>LDPE</td><td>一</td><td>25.7 ± 9</td><td>32.0</td><td>148.4</td></tr><tr><td>LDPE</td><td>LDPE</td><td>一</td><td>54.7 ± 12</td><td>33.2</td><td>197.1</td></tr><tr><td>PP</td><td>PP</td><td>一</td><td>338.7 ±6</td><td>42.4</td><td>196.4</td></tr><tr><td>PP</td><td>PP</td><td>一</td><td>348.4 ± 18</td><td>4.7</td><td>58.6</td></tr><tr><td>PP</td><td>PP</td><td>一</td><td>303.5 ±5</td><td>6.8</td><td>44.5</td></tr><tr><td>PS</td><td>PS</td><td>+</td><td>15.2 ± 23</td><td>21.7</td><td>52.5</td></tr><tr><td>PS</td><td>PS</td><td>+</td><td>35.2 ± 27</td><td>28.2</td><td>557.2</td></tr><tr><td>PS</td><td>PS</td><td>+</td><td>34.1 ± 51</td><td>137.6</td><td>281.7</td></tr></table></body></html>  \n\na Represents the relative methyl content of the polymer across the measured molar mass distribution. Error represents one standard deviation of the methyl content across all molar masses measured. Precision of molar mass measurements is $\\leq5\\%$ of reported value based on repeat injections of mass standards run during sample analyses.  \n\nb Measured by XPS survey scan.  \n\nc Polymer number average (Mn) and mass average (Mw) molar masses determined by MALS detection.  \n\naccurate (Table 3). These samples all produced unambiguous spectra $({\\bf n}=5$ HDPE, $\\mathtt{n}=5$ LDPE). Taken together the results suggest there is a high confidence in unambiguous ATR FT-IR spectra to distinguish HDPE from LDPE. However, ambiguous spectra (bins 3–5) cannot be assigned to a particular PE polymer without further testing, and ATR FT-IR spectra cannot be used to distinguish LDPE from LLDPE. A detailed step-by-step decision tree outlines our criteria for distinguishing HDPE from LDPE using ATR FT-IR in addition to a float/sink test (Fig. 3). Once samples are determined to be PE based on absorbance bands listed in Table 1, the spectral region between $1330\\mathrm{cm}^{-1}$ and $1400~\\mathrm{{cm}^{-1}}$ is magnified and each sample is matched to the most similar bin. Samples falling in bins 1 and 2 are assigned HDPE. Those in bins 3–5 are considered unknown PE until further testing can be done. Those in bins 6 and 7 are assigned LDPE or LLDPE. The unknown PE samples that are not air-filled can be placed into a $0.935\\mathrm{g/mL}$ solution of ethanol. If they float, they are assigned LDPE or LLDPE. If they sink, they are assigned HDPE. The approach described in this decision tree should help future studies with the often confusing, yet very important, differentiation of HDPE and LDPE.  \n\n# 3.2. Cleaning methods for polymer identification of ingested plastics  \n\nTo our knowledge, no study had addressed how digestive processes affect the FT-IR spectra of polymers, so we determined an optimal cleaning method that would also preserve the samples for future chemical testing. The spectra from the 11 ingested fragments after cleaning with water were of higher quality and easier to identify than before cleaning (Fig. S3), with the noise reduced and the absorbance bands more prominent. Thus, this test suggests that this simple treatment, involving removing surface residue with a cleanroom wiper and water, increases the ability to identify plastic polymers.  \n\nOnly half of the spectra were of good enough quality to identify the polymer when no cleaning was performed on the samples $(58\\%~\\pm~29\\%$ standard deviation), whereas $100\\%$ of spectra were identifiable with the other four cleaning methods (Fig. S4). Significant differences in the percent of spectra that could be identified were found among the five cleaning methods (Friedman's analysis of variance (ANOVA), $x^{2}(4)=26.20,\\mathrm{p~<~}0.0001,\\mathrm{n}=11\\mathrm{1}$ ). Post-hoc comparisons using Wilcoxon signed-rank tests revealed that all four sample treatments (wiping $(\\mathtt{p}=0.004$ , $\\mathbf{n}=11\\mathbf{\\dot{\\mathrm{.}}}$ , cleaning with 2-propanol $(\\mathtt{p}=0.004$ , $\\mathbf{n}=11\\mathbf{\\dot{\\mathrm{~.~}}}$ ), water $(\\mathtt{p}=0.004$ , $\\mathbf{n}=11\\mathbf{\\dot{\\Omega}}$ ), or cutting $(\\mathtt{p}=0.004$ , $\\mathbf{n}=11)$ ) resulted in a greater percentage of the spectra being identified, when compared to no cleaning (Fig. S4). No significant differences were found among the four cleaning methods. These results suggest that cleaning a polymer of ingested plastic fragments with any of the four methods should improve quality of ATR FT-IR spectra.  \n\n![](images/dca311a6925610118e576926f36811cb01d43dc4ed3238a3ef064c93f4161920.jpg)  \nFig. 4. Mean and standard deviation of the number of detected wavenumbers for five different cleaning methods on ingested high-density polyethylene (HDPE), low-density polyethylene (LDPE), and polypropylene (PP) fragments. Different letters above bars indicate significant differences among cleaning techniques within a polymer type $(\\mathtt{p}\\ <\\ 0.05$ Wilcoxon signed-rank tests).  \n\nThe number of detected peaks increased significantly for all three polymer types after performing any of the four cleaning methods (Fig. 4). The number of detected peaks and rugosity codes for each cleaning method for each fragment can be found in Table S3. Significant differences were found among the five cleaning methods for HDPE (Friedman's ANOVAs, $x^{2}(4)=27.41$ , $\\mathrm{~p~<~}0.0001\\$ , for LDPE $(x^{2}(4)=34.992,\\mathrm{p~<~}0.0001)$ and for PP $(x^{2}(4)=19.92$ , $\\begin{array}{r}{\\mathbf{p}=0.001_{-}^{}}\\end{array}$ . For HPDE, wiping (Wilcoxon $\\mathbf{p}=0.007)$ , 2-propanol $(\\mathtt{p}=0.006)$ , water $(\\mathtt{p}=0.005)$ , and cutting $(\\mathtt{p}=0.018)$ produced significantly more detectable peaks than no treatment. Cleaning HDPE fragments with water also produced significantly more detectable peaks when compared to wiping $(\\mathtt{p}=0.008)$ , 2-propanol $(\\mathtt{p}=0.008)$ , and cutting $(\\mathtt{p}=0.014)$ . Similar results were seen with PP and LDPE ingested fragments (Fig. 4). These differences suggest that cleaning the surface of ingested fragments with water will produce the spectrum with the most detectable peaks and this method might be preferred if the goal is to minimize handling so that the pieces can be used in the future for additional chemical testing, such as measuring sorbed persistent organic pollutants.  \n\nLDPE fragments with a higher rugosity code yielded fewer detectable peaks $(\\mathbf{r}_{s}=-0.803$ , $\\mathbf{n}=5$ , $\\mathtt{p}=0.102)$ , although the relationship was not significant (Fig. S5). In contrast, when fragments were cut, no significant correlation was found between the number of detectable peaks and rugosity codes $(\\mathbf{r}_{s}=0.631$ , $\\mathbf{n}=5$ , $\\mathfrak{p}=0.254\\rangle$ ). This is most likely due to rugosity being reduced when a fragment is cut. Therefore, if a rugose fragment cannot be identified after being cleaned with water, cutting may be the most effective cleaning method as it can allow for the sample to come in more direct contact with the diamond and thus evanescent wave, resulting in more detectable peaks.  \n\n![](images/051fdfe53902f72ceab4f5a9f8a1d914fd2540377be3ab4153548a202eda4919.jpg)  \nFig. 5. PCA ordination of spectra from ingested plastic samples identified as high-density polyethylene (HDPE, $\\mathrm{~n~}=58\\mathrm{{^{\\circ}}}.$ , polyvinyl chloride (PVC, $\\mathbf{n}=1\\mathbf{\\dot{2}},$ ), low-density polyethylene (LDPE, $\\mathbf{n}=310\\mathbf{\\dot{\\Omega}}.$ ), polypropylene (PP, $\\begin{array}{r}{\\mathrm{~n~}=270_{,}^{\\circ}}\\end{array}$ , polystyrene (PS, $\\mathbf{n}=7\\mathbf{\\cdot}$ ), nylon $\\mathbf{\\rho}(\\mathbf{n}=1)$ , PE/PP mixture $\\mathrm{(n=40)}$ ), unknown PE $\\left(\\mathbf{n}=106\\right)$ , and unknown ${\\mathrm{(n}}=4{\\mathrm{)}}$ . Spectra of consumer good items representing PVC $\\left(\\mathbf{n}=3\\right)$ and nylon $(\\mathbf{n}=3) $ ) were also included. The amount of variation in the data explained by each principal component is shown in parentheses.  \n\n# 3.3. Method validation with ingested polymers  \n\nOnly 30 of the 828 ingested plastic pieces analyzed $(4\\%)$ produced spectra of poor quality that could not be identified by ATR FT-IR. Criteria for differentiating HDPE and LDPE (even without the float/sink test, which was not applied to these samples) allowed assignment of $77.7\\%$ of the PE pieces, while $22.3\\%$ of PE samples fell in the unknown PE category. This proportion is similar to the $70\\%$ assignment capability of oceanic microplastics by Brandon et al. (2016).  \n\nPCA was performed on spectra from 797 representative ingested plastic pieces identified as HDPE $\\mathrm{(n}=58\\mathrm{)}$ ), PVC $(\\mathbf{n}=1)$ , LDPE or LLDPE $({\\mathfrak{n}}=310$ ; one outlier was removed), PP $(\\mathbf{n}=270$ ; one outlier was removed), PS $(\\mathbf{n}=7)$ , nylon $(\\mathbf{n}=1).$ , mixture of PP and PE $\\left({\\mathfrak{n}}=40\\right)$ , unknown PE ${\\mathrm{(n}}=106{\\mathrm{)}}$ ) and unknown ${\\bf\\Pi}({\\bf n}=4)$ ) along with plastic consumer goods of PVC and nylon. The PCA shows distinction between PEs and PP (Fig. 5; Table S4 for loadings) with $85.8\\%$ of the variance explained within the first two principal components. In an earlier version of the PCA, more of the ingested pieces, a total of 10, were originally identified as unknown (data not shown). The clustering of six of the unknown samples within the PCA, followed by further review of their ATR FT-IR spectra, allowed polymer assignment of these samples. These results suggest that PCA is a tool that can help interpret ambiguous spectra. Plastic pieces identified as a mixture were located between clusters for PE and PP as expected. In order to improve assignment of these predominantly olefinic polymers, HT-SEC with multiple detectors was used to definitively identify the samples as a mixture of PE and PP, and confirm results of the PCA.  \n\nSixteen of 17 ingested plastic samples were positively identified by HT-SEC. All HT-SEC determined identities matched those obtained by ATR FT-IR, as shown in Table 3. Example chromatograms for three samples, identified as PS, LDPE, and HDPE are shown in Fig. 6. A number of qualitative and quantitative pieces of information were used to identify the polymers analyzed by HT-SEC. First, the injected polymer samples demonstrated a positive or negative differential refractive index (RI) peak as they eluted from the columns, indicative that the polymer had a greater or lesser refractive index than the mobile phase $(\\mathtt{n}_{0}=1.56)$ . Refractive indices of commercial polymers are available from a number of sources (Brandrup et al., 1999; Mark, 2007). This qualitative identification is used to rule out general classes of polymers; for example, polyolefins (PP, HDPE, LDPE) have a refractive index $<1.56$ , so these polymers must have a negative RI elution peak (Table 3, Fig. 6b and c).  \n\n![](images/2ee659ddfb2e786eded30da6f4513b986e25c9bd1c90dd4987558767d05091a9.jpg)  \nFig. 6. Representative HT-SEC chromatograms of samples run to confirm ATR-FTIR materials for ingested samples identified as (a) polystyrene (PS) (b) low-density polyethylene (LDPE) (c) high-density polyethylene (HDPE). The $\\mathrm{CH}_{3}/1000$ total C were measured by the ratio of the two IR signals, methyl stretching bands and alkyl stretching bands at $2950\\mathrm{cm}^{-1}$ and (2800 to 3000) $\\mathrm{cm}^{-1}$ (broad detector range), being represented by the orange and red traces, respectively. The asterisk $(^{*})$ denotes an added flow rate marker, dodecane, used as an internal standard. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)  \n\nDifferentiation between PP, HDPE, and LDPE was based on the degree of short chain branching, which was measured using the HT-SEC IR detector. The flow-through IR detector measures alkyl and methyl $\\mathsf{C}{\\mathrm{-}}\\mathsf{H}$ stretching simultaneously as the separated polymer elutes. The ratio of the two absorption spectra at each elution volume (Fig. 6), when compared to a calibration curve, permit branching content to be determined across the molar mass distribution. As ATR FT-IR is a bulk measurement, the branching content measured by HT-SEC was averaged across the molar mass distribution for each sample and the average methyl content per 1000 total carbons ( $\\mathrm{{CH_{3}}/1000}$ total C) is shown in Table 3. HDPE is identified from samples that have $10~\\mathrm{CH}_{3}/\\$ 1000 total C or less, as the only $\\mathrm{CH}_{3}$ contributions in HDPE are from chain ends, which are negligible. This is also a convenient metric as the limit of detection for the IR detector is $10~\\mathrm{CH_{3}/1000}$ total C. PP was identified for polymers with a methyl content of $\\left(330\\pm33\\right)$ ) $\\mathrm{CH}_{3}/1000$ total C, which is determined from theoretical calculations based on the propylene repeat unit. The $10\\%$ tolerance is to include PP that may have some small degrees of degradation from the turtle digestive tract as well as account for small variations $(<10\\%)$ that were observed in HT-SEC analysis of consumer-grade PP (stamped resin code 5) when compared to reagent grade PP obtained from Sigma Aldrich. Currently, there is no documentary standard that specifies what purity a consumer polymer must have to be stamped with a specific resin code. LDPE is assigned to polymers with an average branching content between HDPE and PP, or (10 to 300) $\\mathrm{CH}_{3}/1000$ total C. For the purposes of this study, no effort was made to distinguish LLDPE from LDPE in ingested plastics, as ATR FT-IR cannot make that distinction. Future studies will address distinguishing LLDPE from LDPE in unknown polymer samples and mixtures with the addition of a differential viscometer to measure long chain branching.  \n\nThree of the 17 samples were identified as PS (Fig. 6a) based on several lines of evidence. Their positive differential RI signal and minimal alkyl content lead to small IR peak areas and large standard deviations in $\\mathrm{CH}_{3}/1000$ total C determinations. Also, there was agreement between polymer number average $(\\mathbf{M}_{\\mathrm{n}})$ and mass average $(\\mathbf{M}_{\\mathrm{w}})$ molar masses determined by MALS (considered an absolute measurement technique) and those determined by relative comparison to polystyrene standards. $\\mathbf{M}_{\\mathbf{n}}$ and $\\mathbf{M}_{\\mathbf{w}}$ values were determined for all samples and are listed as information values in Table 3.  \n\nOne sample measured by HT-SEC (assigned as PETE by ATR FT-IR) was completely insoluble in 1,2,4-trichlorobenzene at $160^{\\circ}\\mathrm{C}.$ , and no peaks were not observed in HT-SEC chromatograms using any detector. The second independent assignment of this sample was instead based on survey $\\mathbf{x}$ -ray photoelectron spectroscopy (XPS) to measure the elemental composition in the sample. Elemental composition of this sample was $(80.3~\\pm~0.9)~\\%~\\mathrm{C};$ $(3.6~\\pm~0.3)\\%\\mathrm{~N~}$ , and $(15.0~\\pm~0.9)\\%$ O, plus additional trace elements (silica and calcium and sodium salts), taken as an average of three locations on the sample. A tentative assignment of PU was made on the material, as the carbon, nitrogen, and oxygen content was closest to database values for PU at $78.6\\%$ , $8.4\\%$ , and $14\\%$ , respectively. As XPS only excites photoelectrons within the first $\\approx10~\\mathrm{{nm}}$ of a material, further sampling and measurements of the material will have to be performed. While this piece of brown fabric produced an ATR FT-IR spectrum with six absorption bands matching PETE, it was a poor-quality spectrum. XPS database values for PETE are $68.9\\%$ carbon, $31.1\\%$ oxygen, and do not contain nitrogen, which are generally more different from the readings of this fabric piece than those of PU. Taking all data into account, this piece was assigned PU.  \n\nWith the exception of one misidentification, these novel inter-laboratory results support using ATR FT-IR to identify polymers of degraded and ingested plastics. Identifying the polymers comprising ingested plastic using this simple, accurate method can help us understand many aspects of the marine debris problem. The polymer type will dictate the transport and fate of marine debris and its affinity for other chemical pollutants. Furthermore, knowing the predominant polymer can inform better conservation and management practices. For example, LDPE and PP (resin code $\\#4$ and $\\#5$ ) represent large proportions of marine debris and are not commonly recycled in the Hawaiian Islands. Incentive programs for recycling these polymers and innovative post-use applications could be prioritized to help reduce the abundance of LDPE and PP in the marine environment.  \n\n# 4. Conclusion  \n\nAs the ingestion of plastic debris by threatened marine species such as sea turtles increases, the need to categorize plastic debris by polymer type and identify marine transport mechanisms and fates has become a high research priority. Here, we provide a definitive validation of ATR FT-IR to identify ingested plastic polymer types, including resin codes $\\#1$ through $\\#6$ and many polymers within code $\\#7$ without the use of a costly database. A clear, easy to follow guide of thoroughly tested criteria was presented to confidently differentiate HDPE and LDPE. Our approach has been successfully used by four additional ongoing marine debris studies with macro to microplastics found in water, on beaches, or ingested by other marine organisms. We encourage future studies to prepare ingested plastic samples by cleaning them with water or cutting rugose pieces to get a clean surface prior to ATR FT-IR analysis to produce the most accurate results. PCA can be leveraged to assign polymer types to the small proportion of pieces that present challenging ATR FT-IR spectra. This method has been used to identify the polymer composition ingested by three species of sea turtles in the pelagic Pacific. Results on polymers ingested by sea turtle species, geographical, and other comparisons will be reported in a forthcoming manuscript (Jung, 2017). The data reported in the current method development study represent only selected pieces; therefore, calculating the percentage of each polymer reported here would misrepresent the actual ingested composition.  \n\nThe accuracy of using ATR FT-IR for identifying commercial polymers in marine debris, as demonstrated in this study, has the benefit of rapid analysis and minimal destruction to the collected samples, which is ideal for high throughput analysis of large repositories of marine debris. There is, however, much more detailed information about discarded plastics that can be explored by utilizing advanced polymer metrology methods, such as HT-SEC, thermal analysis, or rheological measurements. Systematic changes in chemical composition, molar mass, molar mass distribution and viscoelastic properties in a specific polymeric resin can provide better understanding of material degradation pathways and resulting byproducts. Comprehensive understanding of the origins, transport, fate, and lifetime of marine debris will ultimately require both high-throughput and fundamental studies of discarded materials, providing ample opportunities for collaboration between the life sciences and material science communities to address the challenges in marine plastics moving forward.  \n\n# Disclaimer  \n\nCertain commercial equipment, instruments, or materials are identified in this paper to specify adequately the experimental procedure. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.  \n\n# Acknowledgements  \n\nFunding was provided by grant 60NANB15D026 from the U.S. Pacific Islands Program of the NIST Marine Environmental Specimen Bank. The ATR FT-IR instrument was supported by National Institutes of Health grant P20GM103466. We thank Stacy (Vander Pol) Schuur for providing some of the raw manufactured polymers. We thank Tracy Schock for her advice on principal component analyses. We thank the fishermen and fisheries observers for carefully assessing, storing, and transporting the sea turtle specimens. We thank Shandell Brunson, Irene Nurzia Humburg, Devon Franke, Emily Walker, Sarah Alessi, T. Todd Jones (PIFSC), Bob Rameyer (USGS), Katherine Clukey, Jessica Jacob, Frannie Nilsen, Julia Smith, Adam Kurtz, Angela Hansen,  \n\nStephanie Shaw, Jennette VanderJagt, and Jessica Kent (Hawaii Pacific University) and numerous other volunteers for help in sample collection and processing. We thank the entire NIST Marine Environmental Specimen Bank team, especially Rebecca Pugh and Paul Becker, for sample archival. Finally, we thank Chris Stafford, Amanda Forster and Rebecca Pugh for comments on the draft manuscript. Mention of products and trade names does not imply endorsement by the U.S. Government.  \n\n# Appendix A. Supplementary data  \n\nSupplementary data to this article can be found online at https:// doi.org/10.1016/j.marpolbul.2017.12.061.  \n\n# References  \n\nAndrady, A.L., 2011. Microplastics in the marine environment. Mar. Pollut. Bull. 62 (8), 1596–1605.   \nAndrady, A.L., 2017. The plastic in microplastics: a review. Mar. Pollut. 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Handbook of Polyethylene: Structures, Properties, and Applications. Marcel Dekker, Inc., New York, pp. 534.   \nProvencher, J.F., Bond, A.L., Avery-Gomm, S., Borrelle, S.B., Rebolledo, E.L.B., Hammer, S., Kühn, S., Lavers, J.L., Mallory, M.L., Trevail, A., van Franeker, J.A., 2017. Quantifying ingested debris in marine megafauna: a review and recommendations for standardization. Anal. Methods 9 (9), 1454–1469.   \nRios, L.M., Moore, C., Jones, P.R., 2007. Persistent organic pollutants carried by synthetic polymers in the ocean environment. Mar. Pollut. Bull. 54 (8), 1230–1237.   \nRochman, C.M., Hoh, E., Hentschel, B.T., Kaye, S., 2013. Long-term field measurement of sorption of organic contaminants to five types of plastic pellets: implications for plastic marine debris. Environ. Sci. Technol. 47 (3), 1646–1654.   \nRotter, G., Ishida, H., 1992. FTIR separation of nylon6 chain conformations: clarification of the mesomorphous and γcrystalline phases. J. Polym. Sci. B Polym. Phys. 30 (5), 489–495.   \nRyan, P.G., Moore, C.J., van Franeker, J.A., Moloney, C.L., 2009. Monitoring the abundance of plastic debris in the marine environment. Philos. Trans. R. Soc. Lond. B Biol. Sci. 364 (1526), 1999–2012.   \nSingh, B., Sharma, N., 2008. Mechanistic implications of plastic degradation. Polym. Degrad. Stab. 93 (3), 561–584.   \nTomás, J., Guitart, R., Mateo, R., Raga, J.A., 2002. Marine debris ingestion in loggerhead sea turtles, Caretta caretta, from the Western Mediterranean. Mar. Pollut. Bull. 44 (3), 211–216.   \nUnger, B., Rebolledo, E.L.B., Deaville, R., Gröne, A., IJsseldijk, L.L., Leopold, M.F., Siebert, U., Spitz, J., Wohlsein, P., Herr, H., 2016. Large amounts of marine debris found in sperm whales stranded along the North Sea coast in early 2016. Mar. Pollut. Bull. 112 (1), 134–141.   \nVerleye, G.A., Roeges, N.P., De Moor, M.O., 2001. Easy Identification of Plastics and Rubbers. Rapra Technology Limited, Shropshire, pp. 174.  ",
    "Target": null
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  {
    "id": "10.1016_j.cpc.2018.05.010",
    "DOI": "10.1016/j.cpc.2018.05.010",
    "DOI Link": "http://dx.doi.org/10.1016/j.cpc.2018.05.010",
    "Relative Dir Path": "mds/10.1016_j.cpc.2018.05.010",
    "Article Title": "BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients",
    "Authors": "Madsen, GKH; Carrete, J; Verstraete, MJ",
    "Source Title": "COMPUTER PHYSICS COMMUNICATIONS",
    "Abstract": "BoltzTraP2 is a software package for calculating a smoothed Fourier expression of periodic functions and the Onsager transport coefficients for extended systems using the linearized Boltzmann transport equation. It uses only the band and k-dependent quasi-particle energies, as well as the intra-band optical matrix elements and scattering rates, as input. The code can be used via a command-line interface and/or as a Python module. It is tested and illustrated on a simple parabolic band example as well as silicon. The positive Seebeck coefficient of lithium is reproduced in an example of going beyond the constant relaxation time approximation. Program summary Program Title: Bolt zTraP2 Program Files doi: http://dx.doLorg/10.17632/bzb9byx8g8.1 Licensing provisions: GPLv3 Programming language: Python and C++ External routines/libraries: NumPy, SciPy, Matplotlib, spglib, ase, fftw, VTK, netCDF4, Eigen Nature of problem: Calculating the transport coefficients using the linearized Boltzmann transport equation within the relaxation time approximation. Solution method: Smoothed Fourier interpolation (C) 2018 Elsevier B.V. All rights reserved.",
    "Times Cited, WoS Core": 960,
    "Times Cited, All Databases": 1006,
    "Publication Year": 2018,
    "Research Areas": "Computer Science; Physics",
    "UT (Unique WOS ID)": "WOS:000437964200011",
    "Markdown": "# BoltzTraP2, a program for interpolating band structures and calculating semi-classical transport coefficients✩  \n\nGeorg K.H. Madsen a,\\*, Jesús Carrete a, Matthieu J. Verstraete b,c  \n\na Institute of Materials Chemistry, TU Wien, A-1060 Vienna, Austria   \nb nanomat/QMAT/CESAM and Department of Physics, Université de Liège, allée du 6 août, 19, B-4000 Liège, Belgium   \nc European Theoretical Spectroscopy Facility, Belgium  \n\n# a r t i c l e i n f o  \n\nArticle history:   \nReceived 19 December 2017   \nReceived in revised form 30 April 2018   \nAccepted 4 May 2018   \nAvailable online 17 May 2018  \n\nKeywords:   \nBoltzmann transport equation   \nBoltzTraP  \n\n# a b s t r a c t  \n\nBoltzTraP2 is a software package for calculating a smoothed Fourier expression of periodic functions and the Onsager transport coefficients for extended systems using the linearized Boltzmann transport equation. It uses only the band and $k$ -dependent quasi-particle energies, as well as the intra-band optical matrix elements and scattering rates, as input. The code can be used via a command-line interface and/or as a Python module. It is tested and illustrated on a simple parabolic band example as well as silicon. The positive Seebeck coefficient of lithium is reproduced in an example of going beyond the constant relaxation time approximation.  \n\n# Program summary  \n\nProgram Title: BoltzTraP2   \nProgram Files doi: http://dx.doi.org/10.17632/bzb9byx8g8.1   \nLicensing provisions: GPLv3   \nProgramming language: Python and $C{+}{+}$   \nExternal routines/libraries: NumPy, SciPy, Matplotlib, spglib, ase, fftw, VTK, netCDF4, Eigen   \nNature of problem: Calculating the transport coefficients using the linearized Boltzmann transport equation within the relaxation time approximation.   \nSolution method: Smoothed Fourier interpolation  \n\n$\\mathfrak{C}$ 2018 Elsevier B.V. All rights reserved.  \n\n# 1. Introduction  \n\nThe first BoltzTraP program [1] provided a numerically stable and efficient method for obtaining analytic representations of quasi-particle energies. It has found broad adoption for the application of the Boltzmann transport equation (BTE) to such diverse fields as superconductors [2], transparent conductors [3] intermetallic phases [4] as well as thermoelectrics. Its application has been especially widespread for thermoelectrics research [5–12] for which it was originally conceived [13,14]. Furthermore, it has served as a reference for other methods for obtaining transport coefficients, such as maximally localized Wannier functions [15,16]. The numerical stability means that the approach can be automated [14] and the methodology has subsequently been used in several high-throughput studies [3,14,17–19].  \n\nThe usefulness of the BoltzTraP approach can partly be attributed to the numerical efficiency of the procedure when the quasi-particle energies are approximated by the Kohn–Sham (KS) eigenvalues [20]. Once the multiplicative potential has been selfconsistently calculated, calculating eigenvalues on a fine k-mesh is a comparatively simple computational step that can be trivially parallelized. An alternative approach calculates the derivatives necessary for the BTE directly from the intra-band momentum matrix elements [21]. However, within KS theory, it is often simpler to calculate a finer mesh of eigenvalues than to calculate the momentum matrix elements. When the quasi-particle energies cannot be calculated using a fixed multiplicative KS potential, as in beyond-KS methods such as hybrid functionals [22] or the GW approach [23], this argument no longer holds and calculating the momentum matrix elements [21] or using alternative interpolation methods [16,24,25] could be an advantage.  \n\nWith the release of BoltzTraP2 we wish to achieve three objectives. First of all, a method to use both the eigenvalues and momentum matrix elements is introduced. This ensures that the interpolated manifolds exactly reproduce both the value and derivative at the calculated points. The advantages of the interpolation scheme of the original BoltzTraP approach [1], and the advantages of using the intra-band momentum matrix elements [21] are combined. Thereby the method becomes more suited for beyond-KS approaches. Secondly, we wish to make it more straightforward to avoid the constant relaxation time approximation (RTA) and handle e.g. a temperature-dependent transport distribution function due to electron–phonon coupling [26,27]. Finally, a further motivation for rewriting and rereleasing BoltzTraP is to provide a modular code based on the modern scripting language Python 3. While BoltzTraP is mainly thought of as a tool to evaluate the transport coefficients, the underlying algorithm can be generally useful for interpolating any periodic function. We hope that the new code can also serve as a library for further developments in this domain.  \n\nThe paper is built as follows. First we present the interpolation scheme as well as the RTA-BTE. We discuss the interface to the code and provide an outlook and finally we use three examples to illustrate the methodology and results of the code.  \n\n# 2. Background  \n\n# 2.1. Band interpolation  \n\nThe method is based on writing the quasi-particle energies and their derivatives, for each band, as Fourier sums  \n\n$$\n\\tilde{\\varepsilon}_{\\mathbf{k}}=\\sum_{\\varLambda}c_{\\varLambda}\\sum_{R\\in\\varLambda}\\exp(i\\mathbf{k}\\cdot\\mathbf{R})\n$$  \n\n$$\n\\nabla\\tilde{\\varepsilon}_{\\mathbf{k}}=i\\sum_{\\varLambda}c_{\\varLambda}\\sum_{R\\in\\varLambda}\\mathbf{Rexp}(i\\mathbf{k}\\cdot\\mathbf{R})\n$$  \n\nwhere $\\varLambda$ are so-called stars representing a set of symmetryequivalent lattice vectors. BoltzTraP was based on the idea by Shankland [28–30] that the coefficients should be obtained by minimizing a roughness function under the constraints that calculated quasi-particle energies should be exactly reproduced. This in turn means that the number of coefficients should be larger than the number of calculated points.  \n\nThe derivatives can also be obtained from the intra-band optical matrix elements [21,31]  \n\n$$\n\\nabla\\varepsilon_{\\mathbf{k}}=-\\langle\\psi_{\\mathbf{k}}|\\mathbf{p}|\\psi_{\\mathbf{k}}\\rangle.\n$$  \n\nIn BoltzTraP2 the Shankland algorithm [28–30] is extended so that the coefficients ensure that both the quasi-particle energies and their derivatives, Eq. (3), are exactly reproduced. This corresponds to minimizing the Lagrangian  \n\n$$\nI=\\frac{1}{2}\\sum_{\\bf R}c_{\\bf R}\\rho_{\\bf R}+\\sum_{\\bf k}\\left[\\lambda_{\\bf k}\\left(\\varepsilon_{\\bf k}-\\tilde{\\varepsilon}_{\\bf k}\\right)+\\sum_{\\alpha}\\lambda_{\\alpha,\\bf k}\\nabla_{\\alpha}\\left(\\varepsilon_{\\bf k}-\\tilde{\\varepsilon}_{\\bf k}\\right)\\right]\n$$  \n\nwith respect to the Fourier coefficient $\\left(c_{R}\\right)$ , and choosing the Lagrange multipliers $\\big(\\lambda_{\\mathbf{k}}\\mathrm{and}\\lambda_{\\alpha,\\mathbf{k}}\\big)$ so that the constraints are fulfilled. The index $\\alpha$ labels the three Cartesian directions and indicates that each calculated derivative, Eq. (3), adds three Lagrange multipliers. Like in the BoltzTraP code, we use the roughness function provided by Pickett et al. [32]  \n\n$$\n\\rho_{\\mathrm{R}}=\\left(1-c_{1}\\frac{R}{R_{m i n}}\\right)^{2}+c_{2}\\biggl(\\frac{R}{R_{m i n}}\\biggr)^{6}.\n$$  \n\n# 2.2. Boltzmann transport equation  \n\nBoltzTraP2 calculates transport coefficients based on the rigid-band approximation (RBA), which assumes that changing the temperature, or doping a system, does not change the band  \n\nstructure. In the RBA the carrier concentration, for a given $T$ and $\\mu$ , in a semiconductor can be obtained directly from the density of states (DOS)  \n\n$$\nn(\\varepsilon)=\\int\\sum_{b}\\delta(\\varepsilon-\\varepsilon_{b,{\\bf k}})\\frac{\\mathrm{d}{\\bf k}}{8\\pi^{3}},\n$$  \n\nwhere the subscript $b$ runs over bands, by calculating the deviation from charge neutrality  \n\n$$\nc(\\mu,T)=N-\\int n(\\varepsilon)f^{(0)}(\\varepsilon;\\mu,T)\\mathrm{d}\\varepsilon.\n$$  \n\nIn Eq. (7), $N$ is the nuclear charge and $f^{(0)}$ is the Fermi distribution function. In a semiconductor where charge neutrality would place the Fermi level in the band-gap, one can thus imagine how (at $T\\ =\\ 0\\dot{.}$ ) moving $\\mu$ into the conduction bands would produce a $n$ -type material and moving $\\mu$ into the valence bands would produce a $p$ -type material.  \n\nThe BTE describes the behavior of an out-of-equilibrium system in terms of a balance between scattering in and out of each possible state, with scalar scattering rates [33]. We have implemented the linearized version of the BTE under the RTA, where the transport distribution function  \n\n$$\n\\sigma(\\varepsilon,T)=\\int\\sum_{b}\\mathbf{v}_{b,\\mathbf{k}}\\otimes\\mathbf{v}_{b,\\mathbf{k}}\\tau_{b,\\mathbf{k}}\\delta(\\varepsilon-\\varepsilon_{b,\\mathbf{k}}){\\frac{\\mathrm{d}\\mathbf{k}}{8\\pi^{3}}}\n$$  \n\nis used to calculate the moments of the generalized transport coefficients  \n\n$$\n\\mathcal{L}^{(\\alpha)}(\\mu;T)=q^{2}\\int\\sigma(\\varepsilon,T)(\\varepsilon-\\mu)^{\\alpha}\\left(-\\frac{\\partial f^{(0)}(\\varepsilon;\\mu,T)}{\\partial\\varepsilon}\\right)\\mathrm{d}\\varepsilon\n$$  \n\nwhich give the charge and heat currents  \n\n$$\n\\begin{array}{l}{{\\displaystyle j_{e}=\\mathcal{L}^{(0)}{\\bf E}+\\frac{\\mathcal{L}^{(1)}}{q T}(-\\nabla T)}}\\\\ {{\\displaystyle j_{Q}=\\frac{\\mathcal{L}^{(1)}}{q}{\\bf E}+\\frac{\\mathcal{L}^{(2)}}{q^{2}T}(-\\nabla T)}.}\\end{array}\n$$  \n\nIdentifying the two experimental situations of zero temperature gradient and zero electric current, we obtain the electrical conductivity, the Peltier coefficient, the Seebeck coefficient and the charge carrier contribution to the thermal conductivity as  \n\n$$\n\\begin{array}{l}{{\\displaystyle\\sigma=\\mathcal{L}^{(0)}}}\\\\ {{\\displaystyle\\pi=\\frac{\\mathcal{L}^{(1)}}{q\\mathcal{L}^{(0)}}}}\\\\ {{\\displaystyle S=\\frac{1}{q T}\\frac{\\mathcal{L}^{(1)}}{\\mathcal{L}^{(0)}}}}\\\\ {{\\displaystyle\\kappa_{e}=\\frac{1}{q^{2}T}\\left[\\frac{(\\mathcal{L}^{(1)})^{2}}{\\mathcal{L}^{(0)}}-\\mathcal{L}^{(2)}\\right].}}\\end{array}\n$$  \n\nThe main advantage of the BoltzTraP procedure for evaluating the transport coefficients is that it is straightforward to obtain the group velocities from the $\\mathbf{k}$ -space derivatives of the quasi-particle energies, Eq. (2).  \n\nBoltzTraP is often associated with the constant relaxation time approximation (CRTA). The CRTA means that the Seebeck coefficient and Hall coefficient become independent of the scattering rate [34]. Therefore, they can be obtained on an absolute scale as a function of doping and temperature in a single scan. The CRTA in combination with the RBA, which makes the group velocities independent of $\\mu$ and $T$ , also has a computational advantage as it makes the transport distribution function, Eq. (8) independent of temperature and doping. The temperature and doping dependence of the transport coefficients ${\\mathcal{L}}^{(\\alpha)}$ , Eq. (9), is solely due to the Fermi distribution function, and can be obtained via a scan over a fixed transport distribution function.  \n\nClearly the CRTA will have limitations. It only delivers $\\sigma$ and $\\kappa_{e}$ dependent on $\\tau$ as a parameter. Furthermore, the independence of S and $R_{H}$ from $\\tau$ is known to break down, even qualitatively, for specific cases [26]. While it is possible to run the original BoltzTraP with a temperature-, momentum- and band-dependent relaxation time, the structure of the code makes it inconvenient, and the functional form is quite limited. BoltzTraP2 makes it much more straightforward. The interpolated quasi-particle energies are usually considered to be independent from parameters such as temperature and Fermi level, and hence the interpolation does not need to be repeated (only the integration), for instance, to estimate thermoelectric coefficients for a different doping level or temperatures. The direct interface to the interpolation routines make it straightforward to interpolate the quasi-particle energies once and for all, and to avoid duplication of work when interpolating a temperature dependent $\\tau$ .  \n\n# 3. Implementation and interface  \n\n# 3.1. General implementation aspects  \n\nBoltzTraP2 is implemented in Python 3, using syntax and standard library features that make it incompatible with Python 2. The Fortran code base of the original BoltzTraP was taken as a reference, but the new version was written from scratch. Numerical data is handled internally in the form of arrays, making extensive use of the NumPy and SciPy libraries [35,36]. Matplotlib [37] is used for plotting.  \n\nEfficiency is an important goal of this new implementation. Despite being implemented in a higher-level language, BoltzTraP2 achieves speeds comparable to the original BoltzTraP. There are several factors contributing to this. First, many of the most expensive operations are vectorized to be performed at a lower level by NumPy, avoiding expensive loops. Second, the symmetry-related code, heavy with loops over long lists of lattice points, is written in $C{+}{+}$ and calls routines from the C API of the spglib library [38]. The $C{+}{+}$ components are interfaced to Python by a Cython layer [39]. Third, fast Fourier transforms are delegated to optimized low-level libraries; specifically, the pyFFTW wrapper around FFTW [40] is used if available, with the default NumPy wrapper around FFTPACK [41] as a fallback. Finally, certain ‘‘embarrassingly parallel’’ loops can be run on several cores thanks to the multiprocessing module in the Python standard library.  \n\nBoltzTraP2 allows users to save results to files in JSON format, which is both human readable and parseable with a virtually limitless variety of tools and programming libraries. Specifically, there are two different sorts of JSON-formatted files at play. The first kind, bt2 files, contain the DFT input, all information about $k$ points, the interpolation coefficients and a dictionary of metadata. The second category, btj files, contain the DFT input, the DOS, the thermoelectric coefficients for all values of temperature and chemical potential, and another metadata dictionary. Those dictionaries comprise several pieces of version information, a creation timestamp, and information about the scattering model used, if applicable. All JSON files are processed using the LZMA-based xz compressor, to drastically reduce the overhead of the text-based format.  \n\nThe decision to stick with standard formats also affects other outputs of the calculations, stored in plain text with a column layout very similar to the one created by the original BoltTraP [1]. Most existing post-processing and analysis scripts can be adapted with minimal effort.  \n\nRegarding input formats, the current version of the code can read the native output of Wien2k [42,43], VASP [44] and ABINIT [45]. In the case of a VASP calculation, only the vasprun.xml file is required, while for Wien2k the necessary pieces of information are read from case.struct, case.energy and case.scf. If derivatives of the bands are to be used, the output-file case.mommat2 from the OPTIC [46] program is used. The code is modular enough that support for other DFT codes can be easily implemented. Alternatively, any DFT code can be adapted (or a translation script written) to create output in BoltzTraP2’s own GENE format, designed for maximum simplicity. Examples of files in this format are provided with the source distribution, and it is only slightly modified compared to the original BoltTraP code GENE format.  \n\n![](images/55b4f64c38575c2aab84ecdb2e274603da32c5df24bf8435e8725cac57faadae.jpg)  \nFig. 1. Typical BoltzTraP2 workflow taking the user from the results of a DFT calculation to estimates of the thermoelectric coefficients for the system under study, and other related results, using the btp2 command-line interface.  \n\nBoltzTraP2 relies on Python’s setuptools module for its build system. On many platforms, the program can be installed from source with a python setup.py install command, and no previous manual configuration. Moreover, we have uploaded it to the Python Package Index, so that even the downloading step can be bypassed in favor of a simple pip install BoltzTraP2. A copy of spglib is bundled with the BoltzTraP2, to avoid a separate installation process. Naturally, a C compiler and a $C{+}{+},$ -compliant $C{+}{+}$ compiler are still needed when building from source.  \n\n# 3.2. Command-line interface  \n\nThe most typical use case of BoltzTraP2 is the calculation of transport coefficients. This can be done directly through the btp2 command-line front-end, which implements the general workflow depicted in Fig. 1. It is designed to be self-documenting and controllable through command-line arguments, without the need for configuration files.  \n\nThe process starts with a set of DFT results, typically from a nonself-consistent calculation using a dense $k$ -point grid. The user first calls btp2 in ‘‘interpolate’’ mode to generate a representation of the bands interpolated to an even denser grid, which is stored in a JSON file. Optional command-line parameters to the ‘‘interpolate’’ step can be used to control the grid density, the number of bins used to compute the density of states, the minimum and maximum energies, etc. By saving the result of the interpolation to a file, we avoid repeating the calculation even if the user then wants to generate results for different situations.  \n\nTo obtain a set of thermoelectric coefficients, the user needs to invoke btp2 a second time, now in ‘‘integrate’’ mode. In this mode of operation, the command-line script accepts parameters such as the range of temperatures to scan, the dependence of scattering on electron energy, and so on. It then generates output in the form of text files, plus another compressed JSON file as described in the previous section.  \n\nThe detailed list of command-line parameters, with their meanings, can be obtained by invoking btp2 or one of its subcommands with the -h or --help flag.  \n\nIn addition to the ‘‘integrate’’ and ‘‘interpolate’’ subcommands, Fig. 1 illustrates the use of the ‘‘fermisurface’’, ‘‘plotbands’’ and ‘‘plot’’ modes of the btp2 command-line interface. Their job is to generate graphical representations of the BoltzTraP2 output: an interactive 3D plot of the Fermi surface for different values of the energy, a plot of the interpolated electron energies along a specific path in reciprocal space, and plots of the thermoelectric coefficients as functions of temperature and chemical potential, respectively. 3D plotting will only be available if the optional vtk module is detected.  \n\nThe code is documented, and a set of unit tests covering all the basic functionality is provided with the source distribution. The whole battery of tests can be run with pytest.  \n\n# 3.3. Using BoltzTraP2 as a module  \n\nAdvanced users may prefer to skip the command-line interface and access the full feature set of BoltzTraP2 more directly. Those wanting to use the interpolation capabilities of BoltzTraP2 in their own code, or using it as part of an automated workflow, will probably fall in this category. Furthermore, the btp2 commandline interface only allows choosing between a uniform-relaxationtime model and a uniform-mean-free-path one. Users requiring custom parameterizations of the electronic scattering rates will need to bypass the interface. This is easily accomplished by calling the API of the BoltzTraP2 Python module, either from a script or from an interactive Python shell, such as the Jupyter notebook [47]. Crystal structures are represented as ase atoms objects [48], which allows for easy interfacing with many other Python libraries and external software.  \n\nThe best reference about the API of the BoltzTraP2 is the source code of the btp2 interface itself, and a set of documented examples that are provided with the source distribution of BoltzTraP2. The examples illustrate how to accomplish specific tasks and reproduce several results obtained with the original BoltzTraP code as well as the three examples in the following section.  \n\n# 4. Examples  \n\nThe API of BoltzTraP2 is illustrated through three examples. These represent ‘‘non-standard’’ uses of the code, which cannot be accessed through the command line interface. The python source codes reproducing the plots shown below and several others can be found in the examples directory.  \n\n# 4.1. Isotropic parabolic band model  \n\nThe simplest way to illustrate the library functionality of BoltzTraP2 is the parabolic band model. Consider a dispersion relation expressed as,  \n\n$$\n\\varepsilon(k)=\\frac{\\hbar^{2}k^{2}}{2m^{*}}\n$$  \n\nwhere $m^{*}$ is the effective mass of the electrons in the band. For an isotropic parabolic band model we can replace the outer product of group velocities, Eq. (8), by $k/m^{*}$ and the volume element dk in the  \n\n![](images/e706a4316c5041ac40220bbe9b73db87cb73443754945e72f2c3495e23e22f8b.jpg)  \nFig. 2. S, $\\sigma$ , and the thermoelectric $P F$ as a function of carrier concentration. The transport coefficients have been evaluated using a parabolic band model with $m^{*}=$ $m_{e}$ . The temperature and relaxation time were set to $T=500\\mathrm{K}$ and $\\tau=10^{-14}s$ respectively.  \n\nthree dimensional volume integral in Eq. (8) by $4\\pi k^{2}{\\mathrm d}k$ , thereby obtaining analytic expressions for $n(\\varepsilon)$ and $\\sigma(\\varepsilon)$  \n\n$$\n\\begin{array}{l}{{n(\\varepsilon)={\\displaystyle\\frac{1}{4\\pi^{2}}}\\left(\\frac{2m^{*}}{\\hbar^{2}}\\right)^{3/2}\\varepsilon^{1/2}}}\\\\ {{\\sigma(\\varepsilon)={\\displaystyle\\frac{1}{3\\pi^{2}}}\\frac{\\sqrt{2m^{*}}}{\\hbar^{3}}}\\tau\\varepsilon^{3/2}.}}\\end{array}\n$$  \n\nEvaluating the carrier concentration, Eq. (7), and the transport coefficients, Eqs. (9)–(14), leads directly to the famous plot Fig. 2. For comparison we have created a single parabolic band numerically on a $25\\times25\\times25{\\bf k}$ -mesh for a cubic unit cell with $a=5\\mathring{\\mathsf{A}}.$ The band was interpolated onto a mesh containing 5 times the points. The resulting transport coefficients are indistinguishable from the analytic in Fig. 2.  \n\nFig. 2 can be reproduced with the parabolic.py script in the examples directory of the BoltzTraP2 distribution.  \n\n# 4.2. Inclusion of momentum matrix elements. Silicon  \n\nThe band structure of silicon makes one of the simplest examples that is not trivial for an interpolation scheme. The conduction band minimum (CBM) of Si is made up of pockets found along the six-fold degenerate $\\boldsymbol{{\\cal T}}-\\boldsymbol{X}$ line. Furthermore, it has a nonsymmorphic space group so that the bands can cross at the zone boundary. A crossing at the zone boundary will mean that the bands will not necessarily touch the zone-boundary ‘‘horizontally’’. A purely Fourier-based interpolation scheme, as the one used in BoltzTraP2 can give false derivatives at these points, meaning that a very fine $k$ -mesh can be necessary to obtain converged results.  \n\nThe CBM pocket found along the $\\boldsymbol{{\\cal{T}}}{-}\\boldsymbol{{\\cal{X}}}$ line, which will dominate the transport properties of $n$ -doped Si, is illustrated in Fig. 3. Fig. 3 compares the result of a usual DFT calculation of a band structure, with a fine set of $k$ -points along a specific direction, with that obtained by the analytic interpolation of a coarse $9\\times9\\times9k$ -point mesh, Eq. (1). $\\mathbf{A9}\\times\\mathbf{9}\\times\\mathbf{9}$ -mesh corresponds to only $35k$ -points in the irreducible part of the Brillouin zone (IBZ). A $9\\times9\\times9$ -mesh corresponds to a typical $k$ -mesh used for a self-consistent DFT calculation and is obviously not a fine $k$ -mesh, that would typically be calculated non-self-consistently for transport calculations [14] Furthermore, as the lowest conduction bands are degenerate at the $X$ -point, Fig. 3, the non-symmorphic space group does result in the derivatives of interpolated bands being incorrect at this point. However, Fig. 3 illustrates how the modified Lagrangian, Eq. (4), forces the fit to reproduce the exact derivatives at the calculated points. Thereby, both the position and derivatives of the pocket are well reproduced. On the other hand, if only the eigenvalues are included in the fit, this mesh is obviously too coarse and the algorithm fails to reproduce either the position or the derivatives at the pocket (purple dashed line in Fig. 3).  \n\n![](images/292caf5e24b44e9f50e5bb6ebad4e96ad2c5692f4dfc76e600eb1ee1575e29f1.jpg)  \nFig. 3. Silicon band edges along the $\\boldsymbol{{\\cal T}}-\\boldsymbol{X}$ line. The black points are calculated points along this specific direction. The colored lines correspond to the interpolated bands based on a coarse ${\\mathfrak{s}}\\times{\\mathfrak{s}}\\times{\\mathfrak{s}}k.$ -point mesh. The points belonging to this mesh are marked with larger colored points. The full lines are obtained by including the momentum matrix elements in the fit and the dashed use only the eigenvalues. Thin dashed line: chemical potential used below in Fig. 4.  \n\n![](images/0dd377bad29c9e336d91edcbf7bca9a726ee84942d0bcd48e0a521cef7c33ae9.jpg)  \nFig. 4. Convergence of the Seebeck coefficient and thermoelectric power factor as a function of number $k$ -points in the irreducible Brillouin zone. The full lines are obtained by including the momentum matrix elements in the fit whereas the dashed correspond to including only the eigenvalues.  \n\n![](images/7fd3380e056ef70fd927fb7432889ef67f96a18252e69ac166239cef4ef3cdb2.jpg)  \nFig. 5. Calculated conductivity of bcc-Li using a constant relaxation time, band and momentum dependent relaxation times due to electron–phonon coupling, and an energy dependent model $\\tau^{-1}(\\varepsilon)=c n(\\varepsilon)$ , relaxation time.  \n\nThe impression obtained graphically in Fig. 3 is quantified in Fig. 4. The Seebeck coefficient and the thermoelectric power factor, $S^{2}\\sigma/\\tau$ , are calculated at a chemical potential close to the CBM (marked by the thin dashed line in Fig. 3) using the CRTA, Eqs. (8)–(14). It is seen how the results obtained by the modified Lagrangian show both a faster and more systematic trend, reaching convergence at about half the number of $k$ -points needed when the derivatives are not included in the fit.  \n\nFigs. 3 and 4 can be reproduced with the Si_pocket.py and Si_conv.py scripts in the examples directory of the BoltzTraP2 distribution. Before running these two scripts, the script Si_btp.py which does the Fourier interpolation and stores the results in JSON formatted bt2 files.  \n\n# 4.3. State dependent relaxation time in Lithium  \n\nIn BoltzTraP2 the interpolation and integration steps are more explicitly decoupled than in BoltzTraP. This allows the interpolation capabilities of the code to be used to match quantities represented on different grids. We illustrate this possibility by considering the transport distribution function of bcc-Lithium.  \n\nFirst the KS eigenvalues were obtained on a $36\\times36\\times36\\mathbf{k}$ - point mesh and interpolated onto a grid containing 60 times the number of $\\mathbf{k}$ -points. Fig. 5 illustrates how this leads to a positive slope of the CRTA transport distribution function at the Fermi level. Consequently, Eqs. (9) and (14), we find a negative Seebeck coefficient of $S=-1.9\\upmu\\mathrm{V}/\\mathrm{K}$ at $300\\mathrm{K}$ as in Ref. [26]. The obtained CRTA transport distribution function, Fig. 5, is in good agreement with a more ‘‘usual’’ BoltzTraP type calculation where the KS eigenvalues were calculated on $58\\times58\\times58$ -grid and interpolated onto a grid containing four times as many $\\mathbf{k}$ -points. This lends credibility to the interpolation of the scattering rates that we will now perform.  \n\nAs a by-product of the calculations in $\\mathtt{X u}$ et al. [26], we can obtain the computationally costly relaxation times due to electron– phonon scattering, $\\tau_{n\\mathbf{k}}^{e p}$ , on a relatively coarse $24\\times24\\times24\\mathbf{k}$ -point mesh. Using our interpolation scheme, the calculated band- and momentum-dependent relaxation times were interpolated onto the same fine grid as used for the KS eigenvalues. The inclusion of $\\tilde{\\tau}_{n\\mathbf{k}}^{e p}$ leads to a change of slope at the Fermi level and consequently a positive Seebeck coefficient $(S=+2.9\\upmu\\mathrm{V}/\\mathrm{K}$ at $300\\mathrm{K}$ ).  \n\nAs pointed out in the original work [26], the change of sign can be understood by a simple model where the scattering rate is proportional to the DOW. This is illustrated in the inset in Fig. 5 where a peak in the DOS is found above the Fermi level. Consequently, we fit a $\\tau^{-1}(\\varepsilon)\\ =\\ c n(\\varepsilon)$ model to the calculated $(\\tau_{n\\mathbf k}^{e p})^{-1}$ , thereby obtaining an energy dependent relaxation time. As seen in Fig. 5 a very good agreement, especially around the Fermi level, is obtained with the transport distribution calculated using the full $\\tau_{n\\mathbf{k}}^{e p}$ .  \n\nFig. 5 can be reproduced with the Li_bcc.py script in the examples directory of the BoltzTraP2 distribution.  \n\n# 5. Conclusion  \n\nWe have presented a new a software package, BoltzTraP2, based mainly on Python 3. The methodology is based on a smoothed Fourier expression for periodic functions and uses only the band and $k$ -dependent quasi-particle energies as well as the intra-band optical matrix elements and scattering rates as input. The Onsager transport coefficients have been evaluated for a simple periodic band, as well as Silicon and Lithium using the linearized Boltzmann transport equation. The code can be used via a command-line interface and as a Python module.  \n\n# Acknowledgments  \n\nMJV acknowledges support from the Communauté française de Belgique through an ARC grant (AIMED 15/19-09) and the Belgian Fonds National de la Recherche Scientifique FNRS, under Grant Number PDR T.1077.15-1/7. GKHM and JC acknowledge support from M-era.net through the ICETS project (DFG: MA 5487/4-1) and the EU Horizon 2020 Grant No. 645776 (ALMA). Computational resources have been provided by the Consortium des Equipements de Calcul Intensif en Fédération Wallonie Bruxelles (CECI), funded by FRS-FNRS G.A. 2.5020.11; the Tier-1 supercomputer of the Fédération Wallonie-Bruxelles, funded by the Walloon Region under G.A. 1117545; the PRACE-3IP DECI grants, on ARCHER and Salomon (ThermoSpin, ACEID, OPTOGEN, and INTERPHON 3IP G.A. RI-312763) and the Vienna Scientific Cluster (project number 70958: ALMA).  \n\n# References  \n\n[1] G.K.H. Madsen, D.J. Singh, Comput. Phys. 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    "Target": null
  },
  {
    "id": "10.1126_sciadv.aar4206",
    "DOI": "10.1126/sciadv.aar4206",
    "DOI Link": "http://dx.doi.org/10.1126/sciadv.aar4206",
    "Relative Dir Path": "mds/10.1126_sciadv.aar4206",
    "Article Title": "nullophotonic particle simulation and inverse design using artificial neural networks",
    "Authors": "Peurifoy, J; Shen, YC; Jing, L; Yang, Y; Cano-Renteria, F; DeLacy, BG; Joannopoulos, JD; Tegmark, M; Soljacic, M",
    "Source Title": "SCIENCE ADVANCES",
    "Abstract": "We propose a method to use artificial neural networks to approximate light scattering by multilayer nulloparticles. We find that the network needs to be trained on only a small sampling of the data to approximate the simulation to high precision. Once the neural network is trained, it can simulate such optical processes orders of magnitude faster than conventional simulations. Furthermore, the trained neural network can be used to solve nullophotonic inverse design problems by using back propagation, where the gradient is analytical, not numerical.",
    "Times Cited, WoS Core": 681,
    "Times Cited, All Databases": 748,
    "Publication Year": 2018,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000443175500028",
    "Markdown": "# C O M P U T E R S C I E N C E  \n\n# Nanophotonic particle simulation and inverse design using artificial neural networks  \n\nCopyright $\\circledcirc$ 2018 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).  \n\nJohn Peurifoy,1\\* Yichen Shen,1\\* Li Jing,1 Yi Yang,1,2 Fidel Cano-Renteria,3 Brendan G. DeLacy,4   \nJohn D. Joannopoulos,1 Max Tegmark,1 Marin Soljačić1  \n\nWe propose a method to use artificial neural networks to approximate light scattering by multilayer nanoparticles. We find that the network needs to be trained on only a small sampling of the data to approximate the simulation to high precision. Once the neural network is trained, it can simulate such optical processes orders of magnitude faster than conventional simulations. Furthermore, the trained neural network can be used to solve nanophotonic inverse design problems by using back propagation, where the gradient is analytical, not numerical.  \n\n# INTRODUCTION  \n\nInverse design problems are pervasive in physics (1–4). Quantum scattering theory (1), photonic devices (2), and thin film photovoltaic materials (3) are all problems that require inverse design. A typical inverse design problem requires optimization in high-dimensional space, which usually involves lengthy calculations. For example, in photonics, where the forward calculations are well understood with Maxwell’s equations, solving one instance of an inverse design problem can often be a substantial research project.  \n\nThere are many different ways to solve inverse design problems, which can be classified into two main categories: the genetic algorithm (5, 6) (searching the space step by step) and adjoint method (7) (mathematically reversing the equations). For problems with many parameters, solving these with genetic algorithms takes a lot of computation power and time, and this time grows exponentially as the number of parameters increases. On the other hand, the adjoint method is far more efficient than the genetic algorithms; however, setting up the adjoint method often requires a deep knowledge in photonics and can be quite nontrivial, even with such knowledge.  \n\nNeural networks (NNs) have previously been used to approximate many physics simulations with high degrees of precision. Recently, Carleo et al. (8) used NNs to solve many-body quantum physics problems, and Faber et al. (9) used NNs to approximate density functional theory. Here, we propose a novel method to further simulate light interaction with nanoscale structures and solve inverse design problems using artificial NNs. In this method, an NN is first trained to approximate a simulation; thus, the NN is able to map the scattering function into a continuous, higher-order space where the derivative can be found analytically, based on our earlier work presented in the study of Peurifoy et al. (10). The “approximated” gradient of the figure of merit with respect to input parameters is then obtained analytically with standard back propagation $(l l)$ . The parameters are then optimized efficiently with the gradient descent method. Finally, we compare our performance with the standard gradient-free optimization method and find that our method can be more effective and orders of magnitude faster than traditional methods.  \n\nWhile we focus here on a particular problem of light scattering from nanoparticles, the approach presented here can be fairly easily generalized to many other nanophotonic problems. This approach offers both the generality present in numerical optimization schemes (where only the forward calculation must be found) and the speed of an analytical solution (owing to the use of an analytical gradient). Conceptually, there are a number of reasons why the approach used here is useful for a myriad of branches of physics. After the NN is trained, there are three key uses discussed here.  \n\n# Approximate  \n\nOnce the NN is trained to approximate a complex physics simulation (such as density functional theory or finite-difference time-domain simulation), it can approximate the same computation in orders of magnitude less time.  \n\n# Inverse design  \n\nOnce trained, the NN can solve inverse design problems more quickly than its numerical counterpart because the gradient can be found analytically instead of numerically. Furthermore, the series of calculations for inverse design can be computed more quickly due to the faster backward calculation. Finally, the NN can search more easily for a global minimum possibly because the space might be smoothed in the approximation.  \n\n# Optimization  \n\nSimilarly to inverse design, the network can be asked to optimize for a desired property. This functionality can be implemented simply by changing the cost function used for the design and without retraining the NN.  \n\n# RESULTS  \n\n# NNs can learn and approximate Maxwell interactions  \n\nWe evaluate this method by considering the problem of light scattering from a multilayer (denoting the nanoparticle layer by shell from here on) dielectric spherical nanoparticle (Fig. 1). Our goal is to use an NN to approximate this simulation. For definiteness, we choose a particle that has a lossless silica core $\\begin{array}{r}{(\\epsilon=5.913+\\frac{0.2441}{\\lambda^{2}-0.0803})}\\end{array}$ and lossless silica shells. Specifically, we con- ${\\bf\\Psi}(\\epsilon=2.04\\$ ) and then alternating lossless $\\mathrm{TiO}_{2}$ sider eight shells between 30- and $70\\mathrm{-nm}$ thicknesses per shell. Thus, the smallest particle we consider is $480\\mathrm{nm}$ in diameter, and the largest is $1120~\\mathrm{nm}$ .  \n\nThis problem can be solved analytically or numerically with the Maxwell equations, although for multiple shells, the solution becomes involved. The analytical solution is well known (12). We used the simulation to generate 50,000 examples from these parameters with the Monte Carlo sampling.  \n\n![](images/48613275eec0dd21867e5c296fbb7819521424da2379cef00efaf9df6c19eccf.jpg)  \nFig. 1. The NN architecture has as its inputs the thickness of each shell of the nanoparticle, and as its output the scattering cross section at different wavelengths of the scattering spectrum. Our actual NN has four hidden layers.  \n\nNext, we trained the NN using these examples. We used a fully connected network, with four layers and 250 neurons per layer, giving us 239,500 parameters. The input was the thickness of each nanomaterial shell (the materials were fixed), and the output was the spectrum sampled at points between 400 and $800\\mathrm{nm}$ . The training error is graphed in Fig. 2A, and a table of cross-validation responses for various particle configurations is presented in Table 1. For each nanoparticle configuration, a hyperparameter search (that is, changing learning rates and the number of neurons per layer) was performed to minimize the validation error. In our experience, changing the architecture of the model, such as the number of neurons, by a small amount did not affect the mean relative error (MRE) significantly. Additional details about the network architecture, training data, and loss computation are discussed in the Methods section, and all codes used to generate the simulations and results, as well as implement the model discussed here for a general problem, can be accessed at https://github.com/iguanaus/ScatterNet. Once the training was complete, the weights of the NN were fixed and saved into files, which can be easily loaded and used. Next, we began to experiment with applications and uses of this NN.  \n\nThe first application was to test the forward computation of the network to see how well it approximates the spectra it was not trained on (for an example, see Fig. 2B). Impressively, the network matches the sharp peaks and high Q features with much accuracy, although the model was only trained with 50,000 examples, which is equivalent to sampling each shell thickness between 30 and $70\\mathrm{nm}$ only four times.  \n\nTo study whether the network learned anything about the system and can produce features it was not trained on, we also graphed the closest examples in the training set. The results show that the network is able to match quite well spectra even outside of the training set. Furthermore, the results from Fig. 2B visually demonstrate that the network is not simply interpolating, or averaging together the closest training spectra. This suggests that the NN is not simply fitting to the data, but instead discovering some underlying pattern and structure to the input and output data such that it can solve problems it had not encountered and, to some extent, generalize the physics of the system.  \n\nThis method is similar to the well-known surrogate modeling (13), where it creates an approximation to solve the computationally expensive problem, instead of the exact solution. However, the result indicates that NNs can be very powerful in approximating linear optical phenomena (such as nanoparticle scattering phenomena shown here).  \n\n# NNs solve nanophotonic inverse design  \n\nFor an inverse design, we want to be able to draw any arbitrary spectrum and find the geometry that would most closely produce this spectrum. Results demonstrate that NNs are able to efficiently solve inverse design problems. With the weights fixed, we set the input as a trainable variable and used back propagation to train the inputs of the NN. In simple terms, we run the NN “backward.” To do this, we fix the output to the desired output and let the NN slowly iterate the inputs to provide the desired result. After a few iterations, the NN suggests a geometry to reproduce the spectrum.  \n\nWe test this inverse design on the same problem as above—an eightshell nanoparticle made of alternating shells of $\\mathrm{TiO}_{2}$ and silica. We choose an arbitrary spectrum and have the network learn what inputs would generate a similar spectrum. We can see an example optimization in Fig. 3. To ensure that we have a physically realizable spectra, the desired spectrum comes from a random valid nanoparticle configuration.  \n\nWe also compare our method to state-of-the-art numerical nonlinear optimization methods. We tested several techniques and found that interior-point methods (14) were most effective for this problem. We then compared these interior-point methods to our results from the NN, shown in Fig. 3. Visually, we can see that the NN is able to find a much closer minimum than the numerical nonlinear optimization method. This result is consistent across many different spectra, as well as for particles with different materials and numbers of shells.  \n\nWe found that for a few parameters to design over (for three to five dielectric shells), the numerical solution presented a more accurate inverse design than the NN. However, as more parameters were added (regimes of 5 to 10 dielectric shells), the numerical solution quickly became stuck in local minima and was unable to solve the problem, while the NN still performed well and found quite accurate solutions to inverse design. Thus, for difficult inverse design problems involving many parameters, NNs can often solve inverse design easily. We believe  \n\n![](images/bbf3cc13d59b05f9cc84bf4af0615599f20e0cbfc080c6bcdb73aae65f165584.jpg)  \nFig. 2. NN results on spectrum approximation. (A) Training loss for the eight-shell case. The loss has sharp declines occasionally, suggesting that the NN is finding a pattern about the data at each point. (B) Comparison of NN approximation to the real spectrum, with the closest training examples shown here. One training example is the most similar particle larger than desired, and the other is the most similar particle smaller than desired. These results were consistent across many different spectra.  \n\nTable 1. Network architecture and cross-validation results for various sizes of nanoparticles. The common architecture throughout is a fourlayer densely connected network. The errors are presented as the mean percent off per point on the spectrum (subtracting the output by desired and dividing by the magnitude). The validation set was used to select the best model; the test was never seen until final evaluation. The errors are close, suggesting that not much overfitting is occurring, although the effects become more pronounced for more shells.  \n\n<html><body><table><tr><td>Nanoparticle shells</td><td>Neurons per layer</td><td>MRE (train)</td><td>MRE (validation)</td><td>MRE (test)</td></tr><tr><td>8</td><td>250</td><td>1.4%</td><td>1.5%</td><td>1.5%</td></tr><tr><td colspan=\"5\"></td></tr><tr><td>7</td><td>225 225</td><td>0.98% 0.97%</td><td>1.0% 1.0%</td><td>1.0% 1.0%</td></tr><tr><td colspan=\"5\">6</td></tr><tr><td>5</td><td>200</td><td>0.45%</td><td>0.46%</td><td>0.46%</td></tr><tr><td colspan=\"5\"></td></tr><tr><td>4</td><td>125</td><td>0.60%</td><td>0.60%</td><td>0.60%</td></tr><tr><td colspan=\"5\"></td></tr><tr><td>3</td><td>100</td><td>0.32%</td><td>0.33%</td><td>0.32%</td></tr><tr><td colspan=\"5\"></td></tr><tr><td>2</td><td>100</td><td>0.29%</td><td>0.30%</td><td>0.29%</td></tr></table></body></html>  \n\nthat this might be because the optimization landscape might be smoothed in the approximation.  \n\nWe further studied how the NN behaves in regions where D has a strong dependence on w, such as the case of J aggregates (15). This material produced complex and sharp spectra, and it is interesting to study how well the NN approximated these particles, particularly for particles that it had not trained on. Results demonstrate that the network was able to behave fine in these situations (see the Supplementary Materials for more details).  \n\n# NNs can be used as an optimization tool for broadband and specific-wavelength scattering  \n\nFor optimization, we want to be able to give the boundary conditions for a model (for instance, how many shells, how thick of a particle, and what materials it could be) and find the optimal particle to produce $\\upsigma(\\lambda)$ as close as possible to the desired $\\left[\\upsigma_{\\mathrm{desired}}(\\lambda)\\right]$ . Now that we can design an arbitrary spectrum using our tool with little effort, we can further use this as an optimization tool for more difficult problems. Here, we consider two: how to maximize scattering at a single wavelength while minimizing the rest, and how to maximize scattering across a broad spectrum while minimizing scattering outside of it.  \n\nTo do this, we fix the weights of the NN and create a cost function that will produce the desired results. We simply compute the average of the $\\upsigma(\\lambda)$ inside of the range of interest and compute the average of the points outside the range and then maximize this ratio. This cost function $J$ is given by  \n\n$$\nJ=\\frac{\\overline{{\\upsigma_{\\mathrm{in}}}}}{\\overline{{\\upsigma_{\\mathrm{out}}}}}\n$$  \n\nIdeally, this optimization would be performed using metals and other materials with plasmonic resonances (15) in the desired spectrum range. These materials are well suited for having sharp, narrow peaks and, as such, can generate spectra that are highly efficient at scattering at precisely a single wavelength. Our optimization here uses alternating layers of silver and silica, although we also found that using solely dielectric materials, we were able to force the NN to find a total geometry that still scatters at a single peak, despite the underlying materials being unable to. A figure showing the results of this for a narrow set of wavelengths close to $465\\mathrm{nm}$ can be seen in Fig. 4A.  \n\nNext, we consider the case of broadband scattering, where we want a flat spectrum across a wide array of wavelengths. For this case, we allow the optimization to consider metal layers as well (modeled by silver, with the inner core still silica). In this case, we choose the same $J$ as above, maximizing the ratio of values inside to outside. After training the network for a short number of iterations, we achieve a geometry that will broadband scatter across the desired wavelengths. A figure of this can be seen in Fig. 4B.  \n\n# Comparison of NNs with some conventional inverse design algorithms  \n\nAs mentioned, we tested several techniques and found that interiorpoint methods (14) were most suited for nanoparticle inverse design.  \n\n![](images/a1f0dc517b7a195fe7fa636ee2a5fe87da1186012e500733444f7cd945c7ca79.jpg)  \nFig. 3. Inverse design for an eight-shell nanoparticle. The legend gives the dimensions of the particle, and the blue is the desired spectrum. The NN is seen to solve he inverse design much more accurately.  \n\nTo compare this numerical nonlinear optimization method to our NN, we use the same cost function for both, namely, that of the mean square distance between points on the spectra. For definiteness, we code both the NN and simulation in Matlab. This allows for reasonably fair comparisons of speed and computation resources.  \n\nWe train a different NN on each number of particle shells from 2 to 10. The networks’ size increased as we increased the number of shells, and the training can often require substantial time. However, once the networks were trained, the runtime of these was significantly less than the forward computation time of the simulation. We tested this by running 100 spectra and then finding the average time required for the computation. These were run on a 2.9-GHz Intel Core i5 processor, and all were parallelized onto two CPUs. A plot of these results is shown in Fig. 5. Once fitting with lines, it is evident that if the problem becomes complex, then the simulation would struggle to run more than a few shells, while the NN would be able to handle more. Thus, the NN approach has much to offer to physics and inverse design even in just speeding up and approximating simulations.  \n\nNext, we looked at the optimization runtime versus the complexity of the problem, once again comparing our method against interiorpoint algorithms. To find the speed of this optimization, we chose a spectrum, set a threshold cost, and timed how long it took for the methods to find a spectrum that is below that cost or converged into a local minimum. On a number of spectra, we found that both methods were often sensitive to initialization points. To investigate these results rigorously, and not be influenced by the choice of initial conditions, we took 50 starting points for each spectrum and tested three spectra for each number of shells. Results demonstrate that inverse design using the NN was able to handle more complex problems than the numerical inverse design (see the Supplementary Materials for more details).  \n\n# DISCUSSION  \n\nThe results of this method suggest that it can be easily used and implemented, even for complex inverse design problems. The architecture used in the examples above—a fully connected layer—was chosen without much optimization and still performs quite well. Our preliminary testing with other architectures (convolutions, dropouts, and residual networks) appeared to have further promise as well.  \n\nPerhaps the two most surprising results were how few examples it takes for the network to approximate the simulation, as well as how complex the approximation can really be. For instance, in the eight-shell case, the NN only saw 50,000 examples over eight independent inputs. This means that, on average, it sampled only four times per shell thickness and yet could reproduce the entire range of 30- to $70\\mathrm{-nm}$ shell thickness continuously. The approximation was even able to handle quite sharp features in the spectrum that it otherwise had not seen.  \n\nPromising and effective results have been seen by applying this method to other nanophotonic inverse design problems. Recently, Liu et al. (16) demonstrated that by using a bidirectional NN (17), optimization and inverse design can be performed for one-dimensional shells of dielectric mediums. The approach was to first train the network to approximate the forward simulation and then do a second iteration of training (in the inverse direction) to further improve the accuracy of the results. By using a second iteration of training, Liu et al. (16) was able to overcome degeneracy problems wherein the same spectrum can be generated by particles of different geometrical arrangements. Overall, this and similar work are promising to the idea that experimenting with different architectures, and adding more training data, can allow these NNs to be useful for solving inverse design in many more scenarios.  \n\nAnother interesting aspect of this method is to study the smoothness and robustness of these networks. The validation results show the network to be likely smooth—in particular that there is not much difference in the cross-validation errors. On the other hand, the optimization methods and results can be used to investigate the robustness of the networks. In particular, the optimization presented here was done by an ensemble optimization, wherein the network was initialized several times at different starting points, and each time allowed to converge to a single point. In a typical run, several different initialization points were found to converge to similar error amounts (with possibly different parameters), the lowest of which was chosen as global minimum for the inverse design problem. This finding is consistent with other findings in the field of machine learning, where almost all local minima have similar values to the global optimum of the NN (18). These experiments showed that the NN was not entirely accurate—still getting stuck in local minima on some trial—but preliminary testing suggested that this performs more robustly than the numerical counterpart, as depicted above.  \n\n![](images/8673ab4862024d78ca9d3f1344d5a737e5aea43bc073f32af6512719959cefa4.jpg)  \nFig. 4. Spectra produced by using our approach as an optimization tool. (A) Scattering at a narrow range close to a single wavelength. Here, we force the NN to find a total geometry that scatters around a single peak, using alternating layers of silver and silica. (B) Scattering across a broadband of wavelengths. The legend specifies the thickness of each shell in nanometers, alternating silica and silver shells. The network here was restricted to fewer layers of material (only five shells) but given a broader region of shell sizes than previously (from 10 to $70~\\mathsf{n m}$ ).  \n\n![](images/9b2b1159c5f4ddc5bfbadbc9b0b788bb7c0f25a9b4624b7ddda804bd84da22ca.jpg)  \nFig. 5. Comparison of forward runtime versus complexity of the nanoparticle. The simulation becomes infeasible to run many times for large particles, while the NN’s time increases much more slowly. Conceptually, this is logical as the NN is using pure matrix multiplication—and the matrices do not get much bigger—while the simulation must approximate higher and higher orders. The scale is log-log. The simulation was fit with a quadratic fit, while the NN was a linear fit. See the Supplementary Materials for more details and inverse design speed comparison.  \n\nOne clear concern with the method is that we still have to generate the data for each network, and this takes up time for each inverse design problem. It is true that generating the data takes significant effort, but there are two reasons why this method is still very useful. First, hardware is cheap, and the generation of data can be done easily in parallel across machines. This is not true for inverse design. Inverse design must often be done in a serial approach as each step gets a little closer to the optimal, so the time cannot be reduced significantly by parallel computation. The second reason this method is highly valuable is while the forward propagation is linear in complexity, the optimization is often polynomial. Specifically, by looking at Fig. 5 and the inverse design runtimes (see the Supplementary Materials), we can see that the inverse design speed is growing much faster than the forward runtime. This is important because it means that for complex simulations, the numerical inverse design could take an infeasible amount of time (especially when one needs to solve many inverse design problems for the same physical system), while the NN inverse design may not take long; it will simply have many variables.  \n\nThis method could be used in many other fields of computational physics; it would allow us to approximate physics simulations in fractions of the time. Furthermore, owing to the robustness of back propagation, this method allows us to solve many inverse design problems without having to manually calculate the inverse equations. Instead, we simply have to write a simulation for the forward calculation and then train the model on it to easily solve the inverse design.  \n\n# METHODS  \n\nAnalytically solving scattering via the transfer matrix method We use the transfer matrix method, described in the study of Qiu et al. (19). We consider a multishell nanoparticle. Because of spherical symmetry, we decompose the field into two parts: transverse electric (TE) and transverse magnetic (TM). Both these potentials satisfy the Helmholtz equation, and each scalar potential can be decomposed into a discrete set of spherical modes  \n\n$$\n\\Phi_{l m}=R_{l}(r)P_{l}^{|m|}(\\cos{\\theta})e^{i m\\Phi}\n$$  \n\nFor a specific wavelength, because the dielectric constant is constant within each shell, $R_{l}(r)$ is a linear combination of the first and second kinds of spherical Bessel functions within the two respective shells  \n\n$$\nR_{l}(r)|_{i}=A_{i}j_{l}(k_{i}r)+B_{i}y_{l}(k_{i}r)\n$$  \n\nWe can solve for these coefficients with the transfer matrix of the interface. Thus, we can calculate the transfer matrix of the whole system by simply telescoping these solutions to individual interfaces  \n\n$$\n\\left[\\begin{array}{l}{A_{n+1}}\\\\ {B_{n+1}}\\end{array}\\right]=M_{n+1,n}M_{n,n-1}...M_{3,2}M_{2,1}\\left[\\begin{array}{l}{A_{1}}\\\\ {B_{1}}\\end{array}\\right]=M\\left[\\begin{array}{l}{A_{1}}\\\\ {B_{1}}\\end{array}\\right]\n$$  \n\nFor the first shell, the contribution from the second kind of Bessel function must be zero because the second kind of Bessel function is singular at the origin. Thus, $A_{1}=1$ and $B_{1}=0$ . The coefficients of the surrounding shell are given by the transfer matrix element $A_{n+1}=M_{11}$ and $B_{n+1}=M_{21}$ . To find the coefficients of this surrounding medium, we write the radical function as a linear combination of spherical Hankel functions  \n\n$$\nR_{l}(r)\\vert_{n+1}=C_{n+1}h_{l}^{1}(k_{n+1}r)+D_{n+1}h_{l}^{2}(k_{n+1}r)\n$$  \n\nHere, $h_{l}^{1}(k_{n+1}r)$ and $h_{l}^{2}(k_{n+1}r)$ are the outgoing and incoming waves, respectively, using the convention that fields vary in time as $e^{-i\\omega t}$ . The reflection coefficients $r_{l}$ are given by  \n\n$$\nr_{l}=\\frac{C_{n+1}}{D_{n+1}}=\\frac{M_{11}-i M_{21}}{M_{11}+i M_{21}}\n$$  \n\nBy solving for the reflection coefficients $r_{l},$ we can find the scattered power in each channel  \n\n$$\nP_{l,m=\\pm1}^{\\mathrm{sca}}=\\frac{\\uplambda^{2}}{16\\pi}(2l+1)I_{0}\\lvert1-r_{l}\\rvert^{2}\n$$  \n\nLast, by summing overall channel contributions of the TE and TM polarization (both of the $\\upsigma$ terms), we find the total scattering cross section  \n\n$$\n\\upsigma_{\\mathrm{sca}}=\\sum_{\\upsigma}\\sum_{l=1}^{\\infty}\\frac{\\uplambda^{2}}{8\\pi}(2l+1)|1-r_{\\upsigma,l}|^{2}\n$$  \n\nFor practical reasons, the $l$ summation did not go to $\\infty$ . Instead, before the training data were generated, the order of $l$ was slowly increased until the spectrum had converged, and adding more orders would not change the result. For a typical calculation here, the order ranged from $4l$ terms to $18~l$ terms.  \n\n# Inverse design with NNs  \n\nThe arrangement of the network was a fully connected dense feedforward network. This smallest network we used had four layers, with 100 neurons per layer, which gave the network around 50,300 parameters. The network size was increased as the number of layers increased, with the maximum size being four layers with 300 neurons each for the particle with 10 alternating shells. The input to this network was the normalized (subtracting the mean and dividing by the SD) thickness of each shell of the particle (with the materials fixed), and the output was an unnormalized spectrum sampled at 200 points between 400 and $800\\ \\mathrm{nm}.$ . As a common practice, we found that normalizing the inputs helped training, but equivalent results were found with unnormalized inputs—just taking longer to converge. We intentionally did not normalize the output to not give any outside knowledge of what the range of outputs should be.  \n\nBetween each layer was an activation function of a rectified linear unit (20). There was also one last matrix multiplication after the final layer of the network to map the output to the 200 dimensional desired output. This transformation had no nonlinearity. To initialize the weights, we used a simple normal distribution around 0 with an SD of 0.1 for all weights and biases.  \n\nWe trained the network using a batch size of 100 and a root mean square prop optimizer (21). We split the data into three categories: train, validation, and test (80, 10, and 10, respectively). The train loss was used to generate the gradients, while we stopped training when the validation loss stopped improving. Several different architectures and models (for example, neuron counts) were tested, and the model with the lowest validation loss was chosen. The test loss was used as a final marker that was never trained to ensure cross-validation accuracy. Note that all figures in the paper are from the validation set, so the model was never trained on these particular examples, but we did optimize the model to ensure suitable performance. Most trials took around 1000 to 2000 epochs of 50,000 data points to train, using a learning rate of 0.0006 and decay of 0.99. These parameters were not heavily optimized, and more efficient schemes can certainly be found.  \n\nThere were two cost functions used for training. One was used in actual training and back propagation, and the other was used for illustrative purposes in this paper. The first cost function that we used is the mean square error between each point on the spectrum and the 200 dimensional output of the NN. This cost function was consistent between the training and inverse design; for the training data set, each input had a unique and different output, but for the inverse design, we fixed what we wanted the output to be and modified the input.  \n\nThe other cost function used for illustrative purposes, and presented in Table. 1, was the mean percent off per point on the spectrum. This meant that we found the error between the output of the NN and the desired spectrum, then normalized by the value of the desired spectrum, and found the mean over the whole spectrum. The idea of this function is to offer a more physically meaningful interpretation of how the network is performing—in giving how much each “average point on the spectrum is off by.” All codes can be found at https://github.com/ iguanaus/ScatterNet.  \n\n# SUPPLEMENTARY MATERIALS  \n\nSupplementary material for this article is available at http://advances.sciencemag.org/cgi/ content/full/4/6/eaar4206/DC1   \nsection S1. Details for the comparison of NNs with inverse design algorithms   \nsection S2. J aggregates   \nfig. S1. Comparison of inverse design runtime versus complexity of the nanoparticle. fig. S2. Comparison of NN approximation to the real spectrum for a particle made with a J-aggregate material.   \nfig. S3. Optimization of scattering at a particular wavelength using the J-aggregate material.  \n\n# REFERENCES AND NOTES  \n\n1. B. Apagyi, G. Endredi, P. Levay, Inverse and Algebraic Quantum Scattering Theory (Springer-Verlag, 1996).   \n2. A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, J. Vučković, Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer. Nat. Photonics 9, 374–377 (2015).   \n3. L. Yu, R. S. Kokenyesi, D. A. Keszler, A. Zunger, Inverse design of high absorption thin-film photovoltaic materials. Adv. Energy Mater. 3, 43–48 (2013).   \n4. E. Martín, M. Meis, C. Mourenza, D. Rivas, F. 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D. Joannopoulos, M. Tegmark, M. Soljacic, Nanophotonic inverse design using artificial neural network, in Frontiers in Optics 2017 (Optical Society of America, 2017), pp. FTh4A.4.   \n11. D. E. Rumelhart, G. E. Hinton, R. J. Williams, Learning representations by back-propagating errors. Nature 323, 533–536 (1986).   \n12. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).   \n13. Y. S. Ong, P. B. Nair, A. J. Keane, Evolutionary optimization of computationally expensive problems via surrogate modeling. AIAA J. 41, 687–696 (2003).   \n14. A. S. Nemirovski, M. J. Todd, Interior-point methods for optimization. Act. Num.   \n17, 191–234 (2008).   \n15. B. G. DeLacy, O. D. Miller, C. W. Hsu, Z. Zander, S. Lacey, R. Yagloski, A. W. Fountain, E. Valdes, E. Anquillare, M. Soljačić, S. G. Johnson, J. D. Joannopoulos, Coherent plasmon-exciton coupling in silver platelet-J-aggregate nanocomposites. Nano Lett.   \n15, 2588–2593 (2015).   \n16. D. Liu, Y. Tan, E. Khoram, Z. Yu, Training deep neural networks for the inverse design of nanophotonic structures. arXiv: 1710.04724 (2018).   \n17. K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators. Neural Netw. 2, 359–366 (1989).   \n18. A. Choromanska, M. Henaff, M. Mathieu, G. B. Arous, Y. LeCun, The Loss Surfaces of Multilayer Networks, in Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (AISTATS), San Diego, CA, USA, 9 to 12 May 2015.   \n19. W. Qiu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, Optimization of broadband optical response of multilayer nanospheres. Opt. Express 20, 18494–18504 (2012).   \n20. V. Nair, G. E. Hinton, Rectified linear units improve restricted boltzmann machines, in Proceedings of the 27th International Conference on International Conference on Machine Learning (ICML’10), Haifa, Israel, 21 to 24 June 2010.   \n21. S. Ruder, An overview of gradient descent optimization algorithms, arXiv:1609.04747 (2016).  \n\nAcknowledgments: We thank S. Kim for code review and suggestions to improve the code base, and we furthermore thank S. Peurifoy for reviewing and revising this work. Funding: This material is based on work supported in part by the NSF under grant no. CCF-1640012 and in part by the Semiconductor Research Corporation under grant no. 2016-EP-2693-B. This work is also supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under contract numbers W911NF-18-2-0048 and W911NF-13-D-0001, and in part by the MRSEC (Materials Research Science and Engineering Center) Program of the NSF under award number DMR-1419807. Author contributions: M.S., J.D.J., Y.S., and L.J. conceived the method of using NNs to solve photonics problems. Y.Y. suggested studying the scattering spectra design of nanoparticles. J.P. performed the network modeling and data analysis. F.C.-R. and J.P. analyzed different architectures and particle sizes. Y.Y. developed the mathematical models and theoretical background for the nanoparticle solutions. M.T. and B.G.D. gave technical support and conceptual assistance with directions on how the research should proceed. J.P. prepared the manuscript. M.S. and Y.S. supervised the project. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.  \n\nSubmitted 9 November 2017   \nAccepted 23 April 2018   \nPublished 1 June 2018   \n10.1126/sciadv.aar4206  \n\nCitation: J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, M. Soljačić, Nanophotonic particle simulation and inverse design using artificial neural networks. Sci. Adv. 4, eaar4206 (2018).  \n\n# ScienceAdvances  \n\n# Nanophotonic particle simulation and inverse design using artificial neural networks  \n\nJohn Peurifoy, Yichen Shen, Li Jing, Yi Yang, Fidel Cano-Renteria, Brendan G. DeLacy, John D. Joannopoulos, Max Tegmark and Marin Soljacic  \n\nSci Adv 4 (6), eaar4206. DOI: 10.1126/sciadv.aar4206  \n\nARTICLE TOOLS  \n\nSUPPLEMENTARY MATERIALS  \n\nREFERENCES  \n\nThis article cites 13 articles, 1 of which you can access for free http://advances.sciencemag.org/content/4/6/eaar4206#BIBL  \n\nPERMISSIONS  \n\nUse of this article is subject to the Terms of Service  ",
    "Target": null
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  {
    "id": "10.1038_s41560-019-0356-8",
    "DOI": "10.1038/s41560-019-0356-8",
    "DOI Link": "http://dx.doi.org/10.1038/s41560-019-0356-8",
    "Relative Dir Path": "mds/10.1038_s41560-019-0356-8",
    "Article Title": "Data-driven prediction of battery cycle life before capacity degradation",
    "Authors": "Severson, KA; Attia, PM; Jin, N; Perkins, N; Jiang, B; Yang, Z; Chen, MH; Aykol, M; Herring, PK; Fraggedakis, D; Bazan, MZ; Harris, SJ; Chueh, WC; Braatz, RD",
    "Source Title": "NATURE ENERGY",
    "Abstract": "Accurately predicting the lifetime of complex, nonlinear systems such as lithium-ion batteries is critical for accelerating technology development. However, diverse aging mechanisms, significant device variability and dynamic operating conditions have remained major challenges. We generate a comprehensive dataset consisting of 124 commercial lithium iron phosphate/graphite cells cycled under fast-charging conditions, with widely varying cycle lives ranging from 150 to 2,300 cycles. Using discharge voltage curves from early cycles yet to exhibit capacity degradation, we apply machine-learning tools to both predict and classify cells by cycle life. Our best models achieve 9.1% test error for quantitatively predicting cycle life using the first 100 cycles (exhibiting a median increase of 0.2% from initial capacity) and 4.9% test error using the first 5 cycles for classifying cycle life into two groups. This work highlights the promise of combining deliberate data generation with data-driven modelling to predict the behaviour of complex dynamical systems.",
    "Times Cited, WoS Core": 1569,
    "Times Cited, All Databases": 1690,
    "Publication Year": 2019,
    "Research Areas": "Energy & Fuels; Materials Science",
    "UT (Unique WOS ID)": "WOS:000467965700011",
    "Markdown": "# Data-driven prediction of battery cycle life before capacity degradation  \n\nKristen A. Severson $\\textcircled{10}1,5$ , Peter M. Attia $\\textcircled{10}2,5$ , Norman Jin $\\textcircled{10}2$ , Nicholas Perkins $\\oplus2$ , Benben Jiang $\\oplus1$ , Zi Yang $\\textcircled{10}2$ , Michael H. Chen $\\textcircled{10}2$ , Muratahan Aykol $\\textcircled{10}3$ , Patrick K. Herring $\\textcircled{10}3$ , Dimitrios Fraggedakis $\\oplus1$ , Martin Z. Bazant $\\textcircled{15}$ 1, Stephen J. Harris $\\textcircled{10}2,4$ , William C. Chueh $\\textcircled{10}2\\star$ and Richard D. Braatz   1\\*  \n\nAccurately predicting the lifetime of complex, nonlinear systems such as lithium-ion batteries is critical for accelerating technology development. However, diverse aging mechanisms, significant device variability and dynamic operating conditions have remained major challenges. We generate a comprehensive dataset consisting of 124 commercial lithium iron phosphate/ graphite cells cycled under fast-charging conditions, with widely varying cycle lives ranging from 150 to 2,300 cycles. Using discharge voltage curves from early cycles yet to exhibit capacity degradation, we apply machine-learning tools to both predict and classify cells by cycle life. Our best models achieve $9.1\\%$ test error for quantitatively predicting cycle life using the first 100 cycles (exhibiting a median increase of $0.2\\%$ from initial capacity) and $4.9\\%$ test error using the first 5 cycles for classifying cycle life into two groups. This work highlights the promise of combining deliberate data generation with data-driven modelling to predict the behaviour of complex dynamical systems.  \n\nithium-ion batteries are deployed in a wide range of applications due to their low and falling costs, high energy densities and long lifetimes1–3. However, as is the case with many chemical, mechanical and electronic systems, long battery lifetime entails delayed feedback of performance, often many months to years. Accurate prediction of lifetime using early-cycle data would unlock new opportunities in battery production, use and optimization. For example, manufacturers can accelerate the cell development cycle, perform rapid validation of new manufacturing processes and sort/ grade new cells by their expected lifetime. Likewise, end users could estimate their battery life expectancy4–6. One emerging application enabled by early prediction is high-throughput optimization of processes spanning large parameter spaces (Supplementary Figs. 1 and 2), such as multistep fast charging and formation cycling, which are otherwise intractable due to the extraordinary time required. The task of predicting lithium-ion battery lifetime is critically important given its broad utility but challenging due to nonlinear degradation with cycling and wide variability, even when controlling for operating conditions7–11.  \n\nMany previous studies have modelled lithium-ion battery lifetime. Bloom et al.12 and Broussely et al.13 performed early work that fitted semi-empirical models to predict power and capacity loss. Since then, many authors have proposed physical and semi-empirical models that account for diverse mechanisms such as growth of the solid–electrolyte interphase14,15, lithium plating16,17, active material loss18,19 and impedance increase20–22. Predictions of remaining useful life in battery management systems, summarized in these reviews5,6, often rely on these mechanistic and semi-empirical models for state estimation. Specialized diagnostic measurements such as coulombic efficiency23,24 and impedance spectroscopy25–27 can also be used for lifetime estimation. While these chemistry and/or mechanism-specific models have shown predictive success, developing models that describe full cells cycled under relevant operating conditions (for example, fast charging) remains challenging, given the many degradation modes and their coupling to thermal28,29 and mechanical28,30 heterogeneities within a cell30–32.  \n\nApproaches using statistical and machine-learning techniques to predict cycle life are attractive, mechanism-agnostic alternatives. Recently, advances in computational power and data generation have enabled these techniques to accelerate progress for a variety of tasks, including prediction of material properties33,34, identification of chemical synthesis routes35 and material discovery for energy storage36–38 and catalysis39. A growing body of literature6,40,41 applies machine-learning techniques for predicting the remaining useful life of batteries using data collected in both laboratory and online environments. Generally, these works make predictions after accumulating data corresponding to degradation of at least $25\\%$ along the trajectory to failure42–48 or using specialized measurements at the beginning of life11. Accurate early prediction of cycle life with significantly less degradation is challenging because of the typically nonlinear degradation process (with negligible capacity degradation in early cycles) as well as the relatively small datasets used to date that span a limited range of lifetimes49. For example, Harris et al.10 found a weak correlation $(\\rho=0.1)$ between capacity values at cycle 80 and capacity values at cycle 500 for 24 cells exhibiting nonlinear degradation profiles, illustrating the difficulty of this task. Machine-learning approaches are especially attractive for high-rate operating conditions, where first-principles models of degradation are often unavailable. In short, opportunities for improving upon state-of-the-art prediction models include higher accuracy, earlier prediction, greater interpretability and broader application to a wide range of cycling conditions.  \n\nIn this work, we develop data-driven models that accurately predict the cycle life of commercial lithium iron phosphate (LFP)/ graphite cells using early-cycle data, with no prior knowledge of degradation mechanisms. We generated a dataset of 124 cells with cycle lives ranging from 150 to 2,300 using 72 different fast-charging conditions, with cycle life (or equivalently, end of life) defined as the number of cycles until $80\\%$ of nominal capacity. For quantitatively predicting cycle life, our feature-based models can achieve prediction errors of $9.1\\%$ using only data from the first 100 cycles, at which point most batteries have yet to exhibit capacity degradation. Furthermore, using data from the first 5 cycles, we demonstrate classification into low- and high-lifetime groups and achieve a misclassification test error of $4.9\\%$ . These results illustrate the power of combining data generation with data-driven modelling to predict the behaviour of complex systems far into the future.  \n\n![](images/c2d039e0ea934f4326cf9a2a09491e74d2f8beb3253238333d4da8f40c730c80.jpg)  \nFig. 1 | Poor predictive performance of features based on discharge capacity in the first 100 cycles. a, Discharge capacity for the first 1,000 cycles of LFP/ graphite cells. The colour of each curve is scaled by the battery’s cycle life, as is done throughout the manuscript. b, A detailed view of a, showing only the first 100 cycles. A clear ranking of cycle life has not emerged by cycle 100. c, Histogram of the ratio between the discharge capacity of cycle 100 and that of cycle 2. The cell with the highest degradation $(90\\%)$ is excluded to show the detail of the rest of the distribution. The dotted line indicates a ratio of 1.00. Most cells have a slightly higher capacity at cycle 100 relative to cycle 2. d, Cycle life as a function of discharge capacity at cycle 2. The correlation coefficient of capacity at cycle 2 and log cycle life is $-0.06$ (remains unchanged on exclusion of the shortest-lived battery). e, Cycle life as a function of discharge capacity at cycle 100. The correlation coefficient of capacity at cycle 100 and log cycle life is 0.27 (0.08 excluding the shortest-lived battery). f, Cycle life as a function of the slope of the discharge capacity curve for cycles 95–100. The correlation coefficient of this slope and log cycle life is 0.47 (0.36 excluding the shortest-lived battery).  \n\n# Data generation  \n\nWe expect the space that parameterizes capacity fade in lithiumion batteries to be high dimensional due to their many capacity fade mechanisms and manufacturing variability. To probe this space, commercial LFP/graphite cells (A123 Systems, model  \n\nAPR18650M1A, 1.1 Ah nominal capacity) were cycled in a temperature-controlled environmental chamber $\\left(30^{\\circ}\\mathrm{C}\\right)$ under varied fast-charging conditions but identical discharging conditions ( $4C$ to $2.0\\mathrm{V},$ where $1C$ is $1.1\\mathrm{A}$ ; see Methods for details). Since the graphite negative electrode dominates degradation in these cells, these results could be useful for other lithium-ion batteries based on graphite32,50–54. We probe average charging rates ranging from $3.6C_{\\mathrm{;}}$ the manufacturer’s recommended fast-charging rate, to $6C$ to probe the performance of current-generation power cells under extreme fast-charging conditions $\\mathord{\\sim}10\\mathrm{min}$ charging), an area of significant commercial interest55. By deliberately varying the charging conditions, we generate a dataset that captures a wide range of cycle lives, from approximately 150 to 2,300 cycles (average cycle life of 806 with a standard deviation of 377). While the chamber temperature is controlled, the cell temperatures vary by up to $10^{\\circ}\\mathrm{C}$ within a cycle due to the large amount of heat generated during charge and discharge. This temperature variation is a function of internal impedance and charging policy (Supplementary Figs. 3 and 4). Voltage, current, cell can temperature and internal resistance are continuously measured during cycling (see Methods for additional experimental details). The dataset contains approximately 96,700 cycles; to the best of the authors’ knowledge, our dataset is the largest publicly available for nominally identical commercial lithium-ion batteries cycled under controlled conditions (see Data availability section for access information).  \n\n![](images/d1eca78c24e09eb054c27bf474e84d200f6462643ae296b000c149741dad74b0.jpg)  \nFig. 2 | High performance of features based on voltage curves from the first 100 cycles. a, Discharge capacity curves for 100th and 10th cycles for a representative cell. b, Difference of the discharge capacity curves as a function of voltage between the 100th and 10th cycles, $\\Delta Q_{100-10}(V).$ , for 124 cells. c, Cycle life plotted as a function of the variance of $\\Delta Q_{100-10}(V)$ on a log–log axis, with a correlation coefficient of $-0.93$ . In all plots, the colours are determined based on the final cycle lifetime. In c, the colour is redundant with the y-axis. In b and c, the shortest lived battery is excluded.  \n\n![](images/f796de77dedac78c191b9a6fce2eda97ae1d156a50beb63eb6f4f55d45409c60.jpg)  \nFig. 3 | Observed and predicted cycle lives for several implementations of the feature-based model. The training data are used to learn the model structure and coefficient values. The testing data are used to assess generalizability of the model. We differentiate the primary test and secondary test datasets because the latter was generated after model development. The vertical dotted line indicates when the prediction is made in relation to the observed cycle life. The inset shows the histogram of residuals (predicted – observed) for the primary and secondary test data. a, ‘Variance’ model using only the log variance of $\\Delta Q_{100-10}(V).{\\bf b}$ , ‘Discharge’ model using six features based only on discharge cycle information, described in Supplementary Table 1. c, ‘Full’ model using the nine features described in Supplementary Table 1. Because some temperature probes lost contact during experimentation, four cells are excluded from the full model analysis.  \n\nFig. 1a,b shows the discharge capacity as a function of cycle number for the first 1,000 cycles, where the colour denotes cycle life. The capacity fade is negligible in the first 100 cycles and accelerates near the end of life, as is often observed in lithium-ion batteries. The crossing of the capacity fade trajectories illustrates the weak relationship between initial capacity and lifetime; indeed, we find weak correlations between the log of cycle life and the discharge capacity at the second cycle $(\\rho=-0.06,$ , Fig. 1d) and the 100th cycle $_{\\mathscr{\\rho}=0.27}$ , Fig. 1e), as well as between the log of cycle life and the capacity fade rate near cycle 100 ( $_{\\mathscr{S}}\\mathscr{\\mathrm{~\\tiny~(~2~)~}}$ , Fig. 1f ). These weak correlations are expected because capacity degradation in these early cycles is negligible; in fact, the capacities at cycle 100 increased from the initial values for $81\\%$ of cells in our dataset (Fig. 1c). Small increases in capacity after a slow cycle or rest period are attributed to charge stored in the region of the negative electrode that extends beyond the positive electrode56,57. Given the limited predictive power of these correlations based on the capacity fade curves, we employ an alternative data-driven approach that considers a larger set of cycling data including the full voltage curves of each cycle, as well as additional measurements including cell internal resistance and temperature.  \n\n# Machine-learning approach  \n\nWe use a feature-based approach to build an early-prediction model. In this paradigm, features, which are linear or nonlinear transformations of the raw data, are generated and used in a regularized linear framework, the elastic net58. The final model uses a linear combination of a subset of the proposed features to predict the logarithm of cycle life. Our choice of a regularized linear model allows us to propose domain-specific features of varying complexity while maintaining high interpretability. Linear models also have low computational cost; the model can be trained offline, and online prediction requires only a single dot product after data preprocessing.  \n\nWe propose features from domain knowledge of lithium-ion batteries (though agnostic to chemistry and degradation mechanisms), such as initial discharge capacity, charge time and cell can temperature. To capture the electrochemical evolution of individual cells during cycling, several features are calculated based on the discharge voltage curve (Fig. 2a). Specifically, we consider the cycle-to-cycle evolution of $Q(V)_{\\sun}$ , the discharge voltage curve as a function of voltage for a given cycle. As the voltage range is identical for every cycle, we consider capacity as a function of voltage, as opposed to voltage as a function of capacity, to maintain a uniform basis for comparing cycles. For instance, we can consider the change in discharge voltage curves between cycles 20 and 30, denoted $\\Delta Q_{30-20}(V){=}Q_{30}(V)\\ -\\ Q_{20}(V)$ , where the subscripts indicate cycle number. This transformation, $\\Delta Q(V)$ , is of particular interest because voltage curves and their derivatives are a rich data source that is effective in degradation diagnosis50,51,53,59–64.  \n\nTable 1 | Model metrics for the results shown in Fig. 3   \n\n\n<html><body><table><tr><td rowspan=\"2\"></td><td colspan=\"3\">RMSE (cycles)</td><td colspan=\"3\"> Mean percent error (%)</td></tr><tr><td> Train</td><td>Primary test</td><td>Secondary test</td><td> Train</td><td>Primary test</td><td>Secondary test</td></tr><tr><td>'Variance' model</td><td>103</td><td>138 (138)</td><td>196</td><td>14.1</td><td>14.7 (13.2)</td><td> 11.4</td></tr><tr><td>'Discharge' model</td><td>76</td><td>91(86)</td><td>173</td><td>9.8</td><td>13.0 (10.1)</td><td>8.6</td></tr><tr><td>'Full' model</td><td>51</td><td>118 (100)</td><td>214</td><td>5.6</td><td>14.1 (7.5)</td><td>10.7</td></tr></table></body></html>\n\nTrain and primary/secondary test refer to the data used to learn the model and evaluate model performance, respectively. One battery in the test set reaches $80\\%$ state-of-health rapidly and does not match other observed patterns. Therefore, the parenthetical primary test results correspond to the exclusion of this battery.  \n\nThe $\\varDelta Q(V)$ curves for our dataset are shown in Fig. 2b using the 100th and 10th cycles, that is, $\\Delta Q_{100-10}(V)$ . We discuss our selection of these cycle numbers at a later point. Summary statistics, for example minimum, mean and variance, were then calculated for the $\\Delta Q(V)$ curves of each cell. Each summary statistic is a scalar quantity that captures the change in voltage curves between two cycles. In our data-driven approach, these summary statistics are selected for their predictive ability, not their physical meaning. Immediately, a clear trend emerges between cycle life and a summary statistic, specifically variance, applied to $\\Delta\\dot{Q}_{100-10}(V)$ (Fig. 2c).  \n\nBecause of the high predictive power of features based on $\\Delta Q_{100-10}(V)$ , we investigate three different models using (1) only the variance of $\\Delta Q_{100-10}(V)$ , (2) additional candidate features obtained during discharge and (3) features from additional data streams such as temperature and internal resistance. In all cases, data were taken only from the first 100 cycles. These three models, each with progressively more candidate features, were chosen to evaluate both the cost–benefit of acquiring additional data streams and the limits of prediction accuracy. The training data (41 cells) are used to select the model features and set the values of the coefficients, and the primary testing data (43 cells) are used to evaluate the model performance. We then evaluate the model on a secondary testing dataset (40 cells) generated after model development. Two metrics, defined in the ‘Machine-learning model development’ section, are used to evaluate our predictive performance: root-mean-squared error (RMSE), with units of cycles, and average percentage error.  \n\n# Performance of early prediction models  \n\nWe present three models to predict cycle life using increasing candidate feature set sizes; the candidate features are detailed in Supplementary Table 1 and Supplementary Note 1. The first model, denoted as the ‘variance’ model, does not consider subset selection and uses only the log variance of $\\Delta Q_{100-10}(V)$ for prediction. Surprisingly, using only this single feature results in a model with approximately $15\\%$ average percentage error on the primary test dataset and approximately $11\\%$ average percentage error on the secondary test dataset. We stress the error metrics of the secondary test dataset, as these data had not been generated at the time of model development and are thus a rigorous test of model performance. The second, ‘discharge’ model, considers additional information derived from measurements of voltage and current during discharge in the first 100 cycles (row blocks 1 and 2 of Supplementary Table 1). Of 13 features, 6 were selected. Finally, the third, ‘full’ model con­ si­ders all available features (all rows blocks of Supplementary Table 1). In this model, 9 out of 20 features were selected (Supplementary Fig. 5). As expected, by adding additional features, the primary test average percentage error decreases to $7.5\\%$ and the secondary test average percentage error decreases slightly to $10.7\\%$ . The error for the secondary test set is slightly higher for the full model when compared with the discharge model (Supplementary Note 2 and Supplementary Figs. 6–7). In all cases, the average percentage error is less than $15\\%$ and decreases to as little as $7.5\\%$ in the full model, excluding an anomalous cell. Table 1 and Fig. 3 display the performance of the ‘variance’, ‘discharge’ and ‘full’ models applied to our three datasets.  \n\nTable 2 | Model metrics for the classification setting with a cycle life threshold of 550 cycles   \n\n\n<html><body><table><tr><td colspan=\"4\">Classification accuracy (%)</td></tr><tr><td></td><td> Train</td><td>Primary test</td><td>Secondary test</td></tr><tr><td>Variance classifier</td><td>82.1</td><td>78.6</td><td>97.5</td></tr><tr><td>Full classifier</td><td>97.4</td><td>92.7</td><td>97.5</td></tr></table></body></html>\n\nTrain and primary/secondary test refer to the data used to learn the model and evaluate model performance, respectively.  \n\nWe benchmark the performance of our cycle life prediction using early-cycle data against both prior literature and naïve models. A relevant metric is the extent of degradation that has to occur before an accurate prediction can be made. In our work, accurate prediction was achieved using voltage curves from early cycles corresponding to a capacity increase of $0.2\\%$ (median) relative to initial values (with the first and third quartile percentiles being $0.06\\%$ and $0.34\\%$ , respectively; see Fig. 1c). We are unaware of previous early-prediction demonstrations that do not require degradation in the battery capacity or specialized measurements. In fact, published models42–48 generally require data corresponding to at least $25\\%$ capacity degradation before making predictions at an accuracy comparable to that of this work. We also benchmark our model performance using naïve models, for example univariate models and/or models that only utilize information from the capacity fade curve (Supplementary Note 3, Supplementary Figs. 8–13 and Supplementary Tables 2–3). Notably, if the average cycle life of the training data is used for prediction, the average percentage error is approximately $30\\%$ and $36\\%$ for the primary and secondary test sets, respectively. Using data from the first 100 cycles, the most complex benchmark model using only features from the discharge capacity fade curve has errors of $23\\%$ and $50\\%$ for the primary and secondary test sets, respectively. In fact, a similar model that uses discharge capacity fade curve data from the first 300 cycles achieves comparable performance $27\\%$ and $46\\%$ for the primary and secondary test data, respectively), highlighting the difficulty of prediction without using voltage features.  \n\nWe also consider contexts in which predictions are required at very low cycle number but the accuracy requirements are less stringent, such as sorting/grading and pack design applications. As an example, we develop a logistic regression model to classify cells into either a ‘low-lifetime’ or a ‘high-lifetime’ group, using only the first five cycles for various cycle life thresholds. For the ‘variance classifier’, we use only the $\\Delta Q(V)$ variance feature between the fourth and fifth cycles, $\\operatorname{var}(\\Delta Q_{5-4}(V))$ , and attain a test classification accuracy of $88.8\\%$ . For the ‘full classifier’, we use regularized logistic regression with 18 candidate features to achieve a test classification accuracy of $95.1\\%$ . These results are summarized in Table 2 and detailed in Supplementary Note 4, Supplementary Fig. 14–17 and Supplementary Tables 4–6. This approach illustrates the predictive ability of $\\Delta Q(V)$ even if data from the only first few cycles are used, and, more broadly, showcases our flexibility to tailor data-driven models to various use cases.  \n\n![](images/dcf4c6a75db6002c44d3c4aea3451e514b61901cbb36e123d471c6386bcd37ea.jpg)  \nFig. 4 | Transformations of voltage–capacity discharge curves for three fast-charged cells that were tested with periodic slow diagnostic cycles. a–c, dQ/dV at C/10; d–f, dV/dQ at C/10; g–i, dQ/dV at $4C;$ j–l, $\\Delta Q(V)$ at 4 C. a,d,g,j, 4 C/4 C; b,e,h,k, $6C/4C;$ c,f,i,l, $8C/4C$ The solid black line is the first cycle (cycle 10 for fast cycling), the dotted grey line is cycle 101 or 100 (fast and slow, respectively) and the coloured thick line is the end-of-life cycle $80\\%$ state-of-health). The colour of the end-of-life cycle is consistent with the colour scale in Figs. 1 and 2. For $\\Delta Q(V),$ , a thin dotted grey line is added every 100 cycles. The patterns observed using slow cycling are consistent with $\\mathsf{L A M}_{\\mathrm{deNE}}$ and loss of lithium inventory (Supplementary Fig. 18). The features are smeared during fast charging. The log variance $\\Delta Q(V)$ model dataset predicts the lifetime of these cells within $15\\%$ .  \n\n# Rationalization of predictive performance  \n\nWhile models that include features from all available data streams generally have the lowest errors, our predictive ability primary comes from features based on transformations of the voltage curves, as evidenced by the performance of the single-feature ‘variance’ model. This feature is consistently selected in both models with feature selection (‘discharge’ and ‘full’). Other transformations of the voltage curves can also be used to predict cycle life; for example, the full model selects both the minimum and variance of $\\Delta Q_{100-10}(V)$ . In particular, the physical meaning of the variance feature is associated with the dependence of the discharged energy dissipation on voltage, which is indicated by the grey region between the voltage curves in Fig. 2a. The integral of this region is the total change in energy dissipation between cycles under galvanostatic conditions and is linearly related to the mean of $\\Delta Q(V)$ . Zero variance would indicate energy dissipations that are independent of voltage. Thus, the variance of $\\Delta Q(V)$ reflects the extent of non-uniformity in the energy dissipation with voltage, due to either open-circuit or kinetic processes, a point that we return to later.  \n\n![](images/f8acb35d04e4215cf970c6ad5e6dd605042b65503ea41c84614661073077152e.jpg)  \nFig. 5 | Prediction error as a function of cycle indices. RMSE error, in units of cycles, is presented for training (a) and testing $(\\pmb{\\ b})$ datasets using only the log variance of $\\Delta Q_{i-j}(V),$ , where indices i and j are varied. These errors are averaged over 20 random partitions of the data into equal training and testing datasets. The errors are relatively flat after cycle 80. The increases in error around cycles $\\scriptstyle{j=55}$ and $i{=}70$ are due to temperature fluctuations of the environmental chamber (see Supplementary Fig. 25).  \n\nWe observe that features derived from early-cycle discharge voltage curves have excellent predictive performance, even before the onset of capacity fade. We rationalize this observation by investigating degradation modes that do not immediately result in capacity fade yet still manifest in the discharge voltage curve and are also linked to rapid capacity fade near the end of life.  \n\nWhile our data-driven approach has successfully revealed predictive features from early-cycle discharge curves, identification of degradation modes using only high-rate data is challenging because of the convolution of kinetics with open-circuit behaviour. Thus, we turn to established methods for mechanism identification using low-rate cycling data. Dubarry et al.61 mapped degradation modes in LFP/graphite cells to their resultant shift in $\\mathrm{d}Q/\\mathrm{d}V$ and dV/dQ derivatives for diagnostic cycles at $C/20$ . One degradation mode—loss of active material of the delithiated negative electrode $\\left(\\mathrm{LAM}_{\\mathrm{deNE}}\\right)$ —results in a shift in discharge voltage with no change in capacity. This behaviour is observed when the negative electrode capacity is larger than that of the positive electrode, as is the case in these LFP/graphite cells. Thus, a loss of delithiated negative electrode material changes the potentials at which lithium ions are stored without changing the overall capacity50,61. As proposed by Anséan et al.50, at high rates of $\\mathrm{LAM}_{\\mathrm{deNE}}.$ , the negative electrode capacity will eventually fall below the remaining lithium-ion inventory. At this point, the negative electrode will not have enough sites to accommodate lithium ions during charging, inducing lithium plating50. Since plating is an additional source of irreversibility, the capacity loss accelerates. Thus, in early cycles, $\\mathrm{LAM}_{\\mathrm{deNE}}$ shifts the voltage curve without affecting the capacity fade curve and induces rapid capacity fade at high cycle number. This degradation mode, in conjunction with loss of lithium inventory, is widely observed in commercial LFP/graphite cells operated under similar conditions32,50–54. We note that the graphitic negative electrode is common to nearly all commercial lithium-ion batteries in use today.  \n\nTo investigate the contribution of $\\mathrm{LAM}_{\\mathrm{deNE}}$ we perform additional experiments for cells cycled with varied charging rates (4 C, $6C$ and $8C$ ) and a constant discharge rate $(4C)$ , incorporating slow cycling at the 1st, 100th and end-of-life cycles. Derivatives of diagnostic discharge curves at $C/10$ (Fig. 4, rows 1 and 2) are compared with these, and $\\Delta Q(V)$ , at $4C$ at the 10th, 101st and end-of-life cycles (rows 3 and 4). The shifts in $\\mathrm{d}Q/\\mathrm{d}V$ and dV/dQ observed in diagnostic cycling correspond to a shift of the potentials at which lithium is stored in graphite during charging and are consistent with $\\mathrm{LAM}_{\\mathrm{deNE}}$ and loss of lithium inventory operating concurrently (Supplementary Fig. 18)50,51,61. The magnitude of these shifts from the 1st to 100th cycle increases with charging rate (Supplementary Note 5 and Supplementary Fig. 19). These observations rationalize why models using features based on discharge curves have lower errors than models using only features based on capacity fade curves, since $\\mathrm{LAM}_{\\mathrm{deNE}}$ does not manifest in capacity fade in early cycles. We note that $\\mathrm{LAM}_{\\mathrm{deNE}}$ alters a fraction of, rather than the entire, discharge voltage curve, consistent with the strong correlation between the variance of $\\Delta Q(V)$ and cycle life (Fig. 2c). In summary, we attribute the success of our predictive models to features that capture changes in both the capacity fade curves and voltage curves, since degradation may be silent in discharge capacity but present in voltage curves.  \n\nAs noted above, differential methods such as dQ/dV and dV/dQ are used extensively to pinpoint degradation mechanisms50,51,53,59–61. These approaches require low-rate diagnostic cycles, as higher rates smear out features due to heterogeneous de(intercalation)32, as seen by comparing row 1 with row 3 in Fig. 4. However, these diagnostic cycles often induce a temporary capacity recovery, commonly observed in cells when the geometric area of the negative electrode exceeds that of the positive electrode56,57. As such, they interrupt the trajectory of capacity fade (Supplementary Fig. 20). Therefore, by applying summary statistics to $\\Delta Q(V)$ collected at high rate, we simultaneously avoid both low-rate diagnostic cycles and numerical differentiation, which decreases the signalto-noise ratio65. However, these high-rate discharge voltage curves can additionally reflect both kinetic degradation modes and heterogeneities that are not observed in $\\mathrm{d}Q/\\mathrm{d}V$ and $\\mathrm{d}V/\\mathrm{d}Q$ curves at $C/10$ . We consider the influence of kinetic degradation modes in Supplementary Note 6, Supplementary Fig. 21 and Supplementary Tables 7–8); briefly, we estimate that low-rate modes such as $\\mathrm{LAM}_{\\mathrm{deNE}}$ primarily contribute $(50-80\\%)$ to $\\Delta Q(V)$ . We also mention that low-rate degradation modes such as $\\mathrm{LAM}_{\\mathrm{deNE}}$ influence the kinetics at high rate, in this case by increasing the local current density of the active regions.  \n\nFinally, additional analysis was performed to understand the impact of the cycle indices chosen for $\\Delta Q(V)$ features in the regression setting. Univariate linear models using only the variance of $Q_{i}(V)\\ –\\ Q_{j}(V)$ for the training and primary testing datasets were investigated and are displayed in Fig. 5. We find that the model is relatively insensitive to the indexing scheme for $i>60$ , suggesting that quantitative cycle life prediction using even earlier cycles is possible. This trend is further validated by the model coefficients shown in Supplementary Fig. 22. We hypothesize that the insensitivity of the model to the indexing scheme implies linear degradation with respect to cycle number, which is often assumed for LAM modes50,61. Relative indexing schemes based on cycles in which a specified capacity fade was achieved were also investigated and did not result in improved predictions. Furthermore, because the discharge capacity initially increases, specified decreases in capacity require more cycles to develop than fixed indexing (Supplementary Note 7 and Supplementary Figs. 23–25).  \n\n# Conclusions  \n\nData-driven modelling is a promising route for diagnostics and prognostics of lithium-ion batteries and enables emerging applications in their development, manufacturing and optimization. We develop cycle life prediction models using early-cycle discharge data yet to exhibit capacity degradation, generated from commercial LFP/graphite batteries cycled under fast-charging conditions. In the regression setting, we obtain a test error of $9.1\\%$ using only the first 100 cycles; in the classification setting, we obtain a test error of $4.9\\%$ using data from the first 5 cycles. This level of accuracy is achieved by extracting features from high-rate discharge voltage curves as opposed to only from the capacity fade curves, and without using data from slow diagnostic cycles or assuming prior knowledge of cell chemistry and degradation mechanisms. The success of the model is rationalized by demonstrating consistency with degradation modes that do not manifest in capacity fade during early cycles but impact the voltage curves. In general, our approach can complement approaches based on physical and semi-empirical models and on specialized diagnostics. Broadly speaking, this work highlights the promise of combining data generation and data-driven modelling for understanding and developing complex systems such as lithium-ion batteries.  \n\n# Methods  \n\nCell cycling and data generation. 124 commercial high-power LFP/graphite A123 APR18650M1A cells were used in this work. The cells have a nominal capacity of 1.1 Ah and a nominal voltage of $3.3\\mathrm{V}.$ The manufacturer’s recommended fast-charging protocol is $3.6C$ constant current–constant voltage (CC-CV). The rate capability of these cells during charge and discharge is shown in Supplementary Fig. 27.  \n\nAll cells were tested in cylindrical fixtures with four-point contacts on a 48-channel Arbin LBT battery testing cycler. The tests were performed at a constant temperature of $30^{\\circ}\\mathrm{C}$ in an environmental chamber (Amerex Instruments). Cell can temperatures were recorded by stripping a small section of the plastic insulation and contacting a type T thermocouple to the bare metal casing using thermal epoxy (OMEGATHERM 201) and Kapton tape.  \n\nThe cells were cycled with various candidate fast-charging policies (Supplementary Table 9) but identically discharged. Cells were charged from $0\\%$ to $80\\%$ state-of-charge (SOC) with one of 72 different one-step and two-step charging policies. Each step is a single $C$ rate applied over a given SOC range; for example, a two-step policy could consist of a $6C$ charging step from $0\\%$ to $50\\%$ SOC, followed by a $4C$ step from $50\\%$ to $80\\%$ SOC. The 72 charging polices represent different combinations of current steps within the $0\\%$ to $80\\%$ SOC range. The charging time from $0\\%$ to $80\\%$ SOC ranged from 9 to $13.3\\mathrm{min}$ . An internal resistance measurement was obtained during charging at $80\\%$ SOC by averaging 10 pulses of $\\pm3.6C$ with a pulse width of 30 or $33\\mathrm{ms}$ , where $1C$ is $1.1\\mathrm{A}$ , or the current required to fully (dis)charge the nominal capacity (1.1 Ah) in 1 h. All cells then charged from $80\\%$ to $100\\%$ SOC with a uniform 1 C CC-CV charging step to $3.6\\mathrm{V}$ and a current cutoff of $C/50$ . All cells were subsequently discharged with a CC-CV discharge at $4C$ to $2.0\\mathrm{V}$ with a current cutoff of $C/50$ . The voltage cutoffs used in this work follow those recommended by the manufacturer.  \n\nOur dataset is described in Supplementary Table 9. In total, our dataset consists of three ‘batches’, or cells run in parallel. Each batch has slightly different testing conditions. For the $^{<}2017–05–12^{,}$ batch, the rests after reaching $80\\%$ SOC during charging and after discharging were $1\\mathrm{min}$ and 1 s, respectively. For the $^{<}2017–06–30^{3}$ batch, the rests after reaching $80\\%$ SOC during charging and after discharging were both $5\\mathrm{{min}}$ . For the ‘2018-04-12’ batch, 5 s rests were placed after reaching $80\\%$ SOC during charging, after the internal resistance test and before and after discharging.  \n\nA histogram of cycle life for the three datasets is presented in Supplementary Fig. 28. We note that four cells had unexpectedly high measurement noise and were excluded from analysis.  \n\nTo standardize the voltage–capacity data across cells and cycles, all $4C$ discharge curves were fitted to a spline function and linearly interpolated (Supplementary Fig. 29). Capacity was fitted as a function of voltage and evaluated at 1,000 linearly spaced voltage points from $3.5\\mathrm{V}$ to $2.0\\mathrm{V}.$ These uniformly sized vectors enabled straightforward data manipulations such as subtraction.  \n\nMachine-learning model development. This study involved both model fitting (setting the coefficient values) and model selection (setting the model structure). To perform these tasks simultaneously, a regularization technique was employed. A linear model of the form  \n\n$$\n\\hat{y}_{i}=\\hat{\\mathbf{w}}^{\\mathrm{T}}\\mathbf{x}_{i}\n$$  \n\nwas proposed, where $\\hat{y}_{i}$ is the predicted number of cycles for battery i, $\\mathbf{X}_{i}$ is a $\\boldsymbol{p}$ -dimensional feature vector for battery i and $\\hat{\\mathbf{w}}$ is a $p$ -dimensional model coefficient vector. When applying regularization techniques, a penalty term is added to the least-squares optimization formulation to avoid overfitting. Two regularization techniques, the lasso66 and the elastic net58, simultaneously perform model fitting and selection by finding sparse coefficient vectors. The formulation is  \n\n$$\n\\begin{array}{r}{\\hat{\\mathbf{w}}=\\operatorname*{argmin}_{\\mathbf{w}}\\left\\|\\mathbf{y}-\\mathbf{X}\\mathbf{w}\\right\\|_{2}^{2}+\\lambda P(\\mathbf{w})}\\end{array}\n$$  \n\nwhere the argmin function represents finding the value of w that minimizes the argument, $\\mathbf{y}$ is the $n$ -dimensional vector of observed battery lifetimes, $\\mathbf{X}$ is the $n\\times p$ matrix of features, and $\\lambda$ is a non-negative scalar. The first term  \n\n$$\n\\begin{array}{r}{\\left\\|\\mathbf{y}-\\mathbf{Xw}\\right\\|_{2}^{2}}\\end{array}\n$$  \n\nis found in ordinary least squares. The formulation of the second term, $P(\\mathbf{w})$ , depends on the regularization technique being employed. For the lasso,  \n\n$$\nP\\left(\\mathbf{w}\\right)=\\left\\|\\mathbf{w}\\right\\|_{1},\n$$  \n\nand for the elastic net,  \n\n$$\nP\\left(\\mathbf{w}\\right)=\\frac{1-\\alpha}{2}\\ \\left\\|\\mathbf{w}\\right\\|_{2}^{2}+\\alpha\\ \\left\\|\\mathbf{w}\\right\\|_{1}\n$$  \n\nwhere $\\alpha$ is a scalar between 0 and 1. Both formulations will result in sparse models. The elastic net has been shown to perform better when $p\\gg n^{58}$ , as is often the case in feature engineering applications, but requires fitting an additional hyperparameter ( $\\mathbf{\\chi}_{(\\alpha}$ and $\\lambda$ , as opposed to only λ in the lasso). The elastic net is also preferred when there are high correlations between the features, as is the case in this application. To choose the values of the hyperparameters, we apply four-fold cross-validation and Monte Carlo sampling.  \n\nThe model development dataset is divided into two equal sections, referred to as the training and primary testing sets. These two sections are chosen such that each spans the range of cycle lives (see Supplementary Table 9). The training data are used to choose the hyper-parameters $\\alpha$ and $\\lambda$ and determine the values of the coefficients, w. The training data are further subdivided into calibration and validation sets for cross-validation. The primary test set is then used as a measure of generalizability. The secondary test dataset was generated after model development.  \n\nRMSE and average percentage error are chosen to evaluate model performance. RMSE is defined as  \n\n$$\n\\mathrm{RMSE}=\\sqrt{\\frac{1}{n}\\sum_{i=1}^{n}\\left(y_{i}-\\hat{y}_{i}\\right)^{2}}\n$$  \n\nwhere $\\boldsymbol{y}_{i}$ is the observed cycle life, $\\hat{y}_{i}$ is the predicted cycle life and $n$ is the total number of samples. The average percentage error is defined as  \n\n$$\n\\%\\ \\mathrm{err}=\\frac{1}{n}\\sum_{i=1}^{n}\\frac{|y_{i}-\\hat{y}_{i}|}{y_{i}}\\times100\\\n$$  \n\nwhere all variables are defined as above.  \n\nTo summarize our procedure, we first divide the data into training and test sets. We then train the model on the training set using the elastic net, yielding a linear model with downselected features and coefficients. Finally, we apply the model to the primary and secondary test sets.  \n\nThe data processing and elastic net prediction is performed in MATLAB, while the classification is performed in Python using the NumPy, pandas and sklearn packages.  \n\n# Data availability  \n\nThe datasets used in this study are available at https://data.matr.io/1.  \n\n# Code availability  \n\nCode for data processing is available at https://github.com/rdbraatz/data-drivenprediction-of-battery-cycle-life-before-capacity-degradation. 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Gaussian process regression for in-situ capacity estimation of lithium-ion batteries. IEEE Trans. Ind. Inform. 15, 127–138 (2019).   \n66.\tTibshirani, R. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. B 58, 267–288 (1996).  \n\n# Author contributions  \n\n# Acknowledgements  \n\nP.M.A., N.J., N.P., M.H.C. and W.C.C. conceived of and conducted the experiments. K.A.S., Z.Y. and B.J. performed the modelling. M.A., Z.Y. and P.K.H. performed data management. P.M.A., K.A.S., N.J., B.J., D.F., M.Z.B., S.J.H., W.C.C. and R.D.B. interpreted the results. All authors edited and reviewed the manuscript. W.C.C. and R.D.B. supervised the work.  \n\nThis work was supported by Toyota Research Institute through the Accelerated Materials Design and Discovery programme. P.M.A. was supported by the Thomas V. Jones Stanford Graduate Fellowship and the National Science Foundation Graduate Research Fellowship under Grant No. DGE-114747. N.P. was supported by SAIC Innovation Center through Stanford Energy 3.0 industry affiliates programme. S.J.H. was supported by the Assistant Secretary for Energy Efficiency, Vehicle Technologies Office of the US Department of Energy under the Advanced Battery Materials Research Program. We thank E. Reed, S. Ermon, Y. Li, C. Bauemer, A. Grover, T. Markov, D. Deng, A. Baclig and H. Thaman for discussions.  \n\n# Competing interests  \n\nK.A.S., R.D.B., W.C.C., P.M.A., N.J., S.J.H. and N.P. have filed a patent related to this work: US Application No. 62/575,565, dated 16 October 2018.  \n\n# Additional information  \n\nSupplementary information is available for this paper at https://doi.org/10.1038/ s41560-019-0356-8.  \n\nReprints and permissions information is available at www.nature.com/reprints.  \n\nCorrespondence and requests for materials should be addressed to W.C.C. or R.D.B.  \n\nPublisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\n$\\circledcirc$ The Author(s), under exclusive licence to Springer Nature Limited 2019  ",
    "Target": null
  },
  {
    "id": "10.1021_acs.chemmater.9b01294",
    "DOI": "10.1021/acs.chemmater.9b01294",
    "DOI Link": "http://dx.doi.org/10.1021/acs.chemmater.9b01294",
    "Relative Dir Path": "mds/10.1021_acs.chemmater.9b01294",
    "Article Title": "Graph Networks as a Universal Machine Learning Framework for Molecules and Crystals",
    "Authors": "Chen, C; Ye, WK; Zuo, YX; Zheng, C; Ong, SP",
    "Source Title": "CHEMISTRY OF MATERIALS",
    "Abstract": "Graph networks are a new machine learning (ML) paradigm that supports both relational reasoning and combinatorial generalization. Here, we develop universal MatErials Graph Network (MEGNet) models for accurate property prediction in both molecules and crystals. We demonstrate that the MEGNet models outperform prior ML models such as the SchNet in 11 out of 13 properties of the QM9 molecule data set. Similarly, we show that MEGNet models trained on similar to 60 000 crystals in the Materials Project substantially outperform prior ML models in the prediction of the formation energies, band gaps, and elastic moduli of crystals, achieving better than density functional theory accuracy over a much larger data set. We present two new strategies to address data limitations common in materials science and chemistry. First, we demonstrate a physically intuitive approach to unify four separate molecular MEGNet models for the internal energy at 0 K and room temperature, enthalpy, and Gibbs free energy into a single free energy MEGNet model by incorporating the temperature, pressure, and entropy as global state inputs. Second, we show that the learned element embeddings in MEGNet models encode periodic chemical trends and can be transfer-learned from a property model trained on a larger data set (formation energies) to improve property models with smaller amounts of data (band gaps and elastic moduli).",
    "Times Cited, WoS Core": 796,
    "Times Cited, All Databases": 903,
    "Publication Year": 2019,
    "Research Areas": "Chemistry; Materials Science",
    "UT (Unique WOS ID)": "WOS:000468242300054",
    "Markdown": "# Graph Networks as a Universal Machine Learning Framework for Molecules and Crystals  \n\nChi Chen, Weike Ye, Yunxing Zuo, Chen Zheng, and Shyue Ping Ong\\*  \n\nDepartment of NanoEngineering, University of California San Diego, 9500 Gilman Dr, Mail Code 0448, La Jolla, California 92093-0448, United States  \n\n$\\otimes$ Supporting Information  \n\nABSTRACT: Graph networks are a new machine learning (ML) paradigm that supports both relational reasoning and combinatorial generalization. Here, we develop universal MatErials Graph Network (MEGNet) models for accurate property prediction in both molecules and crystals. We demonstrate that the MEGNet models outperform prior ML models such as the SchNet in 11 out of 13 properties of the QM9 molecule data set. Similarly, we show that MEGNet models trained on ${\\sim}60~000$ crystals in the Materials Project substantially outperform prior ML models in the prediction of the formation energies, band gaps, and elastic moduli of crystals, achieving better than density functional theory accuracy over a much larger data set. We present two new strategies to address data limitations common in materials science and chemistry. First, we demonstrate a physically intuitive approach to unify four separate molecular MEGNet models for the internal energy at $0\\mathrm{~K~}$ and room temperature, enthalpy, and Gibbs free energy into a single free energy MEGNet model by incorporating the temperature, pressure, and entropy as global state inputs. Second, we show that the learned element embeddings in MEGNet models encode periodic chemical trends and can be transfer-learned from a property model trained on a larger data set (formation energies) to improve property models with smaller amounts of data (band gaps and elastic moduli).  \n\n![](images/9010cd819a0e506d11633eaeba82aa74df0d41917f51ad56d01414233dae067e.jpg)  \n\n# INTRODUCTION  \n\nMachine learning $(\\mathbf{ML})^{1,2}$ has emerged as a powerful new tool in materials science,3−14 driven in part by the advent of large materials data sets from high-throughput electronic structure calculations15−18 and/or combinatorial experiments.19,20 Among its many applications, the development of fast, surrogate ML models for property prediction has arguably received the most interest for its potential in accelerating materials design21,22 as well as accessing larger length/time scales at near-quantum accuracy.11,23−28  \n\nThe key input to any ML model is a description of the material, which must satisfy the necessary rotational, translational, and permutational invariances as well as uniqueness. For molecules, graph-based representations29 are a natural choice. This graph representation concept has been successfully applied to predict molecular properties.30,31 Recently, Faber et al.32 have benchmarked different features in combination with models extensively on the QM9 data set.33 They showed that the graph-based deep learning models34,35 generally outperform classical ML models with various features. Furthermore, graph-based models are generally less sensitive to the choice of atomic descriptors, unlike traditional feature engineering-based ML models. For example, Schütt et al.10,36 achieved state-of-the-art performance on molecules using only the atomic number and atom coordinates in a graph-based neural network model. Gilmer et al.37 later proposed the message passing neural network (MPNN) framework that includes the existing graph models with differences only in their update functions.  \n\nUnlike molecules, descriptions of crystals must account for lattice periodicity and additional space group symmetries. In the crystal graph convolutional neural networks (CGCNNs) proposed by Xie and Grossman,9 each crystal is represented by a crystal graph, and invariance with respect to permutation of atomic indices and unit cell choice are achieved through convolution and pooling layers. They demonstrated excellent prediction performance on a broad array of properties, including formation energy, band gap, Fermi energy, and elastic properties.  \n\nDespite these successes, current ML models still suffer from several limitations. First, it is evident that most ML models have been developed on either molecular or crystal data sets. A few notable exceptions are the recently reported SchNet36 and an update of the MPNN,38 which have been tested on both molecules and crystals, respectively, although in both cases performance evaluation on crystals is limited to formation energies only. Second, current models lack a description of global state (e.g., temperature), which are necessary for predicting state-dependent properties such as the free energy. Last but not least, data availability remains a critical bottleneck for training high-performing models for some properties. For example, while there are ${\\sim}69$ 000 computed formation energies in the Materials Project,15 there are only $\\mathord{\\sim}6000$ computed elastic constants.  \n\n![](images/fd37bc702c211cd5ac804ab73a3df072db57a3abef7ad957380c4e91d297836d.jpg)  \nFigure 1. Overview of a MEGNet module. The initial graph is represented by the set of atomic attributes $V=\\{\\mathbf{v}_{i}\\}_{i=1:N}^{\\mathrm{v}},$ bond attributes $E=\\left\\{\\left(\\mathbf{e}_{k},r_{k},\\right.\\right.$ $s_{k})\\bar{\\}_{k=1:N}^{\\mathrm{e}},$ and global state attributes u. In the first update step, the bond attributes are updated. Information flows from atoms that form the bond, the state attributes, and the previous bond attribute to the new bond attributes. Similarly, the second and third steps update the atomic and global state attributes, respectively, by information flow among all three attributes. The final result is a new graph representation.  \n\nIn this work, we aim to address all these limitations. We propose graph networks39 with global state attributes as a general, composable framework for quantitative structure− state−property relationship prediction in materials, that is, both molecules and crystals. Graph networks can be shown to be a generalization/superset of previous graph-based models such as the CGCNN and MPNN; however, because graph networks are not constrained to be neural network-based, they are different from the aforementioned models. We demonstrate that our MatErials Graph Network (MEGNet) models outperform prior ML models in the prediction of multiple properties on the ∼131 000 molecules in the QM9 data set33 and ${\\sim}69~000$ crystals in the Materials Project.15 We also present a new physically intuitive strategy to unify multiple free energy MEGNet models into a single MEGNet model by incorporating state variables such as temperature, pressure, and entropy as global state inputs, which provides for multifold increase in the training data size with minimal increase in the number of model parameters. Finally, we demonstrate how interpretable chemical trends can be extracted from elemental embeddings trained on a large data set, and these elemental embeddings can be used in transfer learning to improve the performance of models with smaller data quantities.  \n\n# METHODS  \n\nMEGNet Formalism. Graph networks were recently proposed by Battaglia et al.39 as a general, modular framework for ML that supports both relational reasoning and combinatorial generalization. Indeed, graph networks can be viewed as a superset of the previous graph-based neural networks, though the use of neural networks as function approximators is not a prerequisite. Here, we will outline the implementation of MEGNet models for molecules and crystals, with appropriate modifications for the two different material classes explicitly described. Throughout this work, the term “materials” will be used generically to encompass molecules to crystals, while the more precise terms “molecules” and “crystals” will be used to refer to collections of atoms without and with lattice periodicity, respectively.  \n\nLet $V,E,$ and u denote the atomic (node/vertex), bond (edge), and global state attributes, respectively. For molecules, bond information (e.g., bond existence, bond order, and so on) is typically provided as part of the input definition. For crystals, a bond is loosely defined between atoms with distance less than certain cutoff. Following the notation of Battaglia et al.,39 $V$ is a set of $\\mathbf{v}_{i},$ which is an atomic attribute vector for atom i in a system of $N^{\\mathrm{v}}$ atoms. $E=\\left\\{\\left(\\mathbf{e}_{k},\\ r_{k},\\right.\\right.$ $s_{k})\\}_{k=1:N}^{\\mathrm{e}}$ are the bonds, where $\\mathbf{e}_{k}$ is the bond attribute vector for bond $k,r_{k}$ and $s_{k}$ are the atom indices forming bond $k,$ and $N^{\\mathrm{e}}$ is the total number of bonds. Finally, $\\mathbf{u}$ is a global state vector storing the molecule/crystal level or state attributes (e.g., the temperature of the system).  \n\nA graph network module (Figure 1) contains a series of update operations that map an input graph $G=\\left(E,V,\\mathbf{u}\\right)$ to an output graph $\\bar{G^{\\prime}}=\\left(E^{\\prime},\\ V^{\\prime},\\ \\mathbf{u}^{\\prime}\\right)$ . First, the attributes of each bond $(\\mathbf{e}_{k},\\bar{r}_{k},s_{k})$ are updated using attributes from itself, its connecting atoms (with indices $r_{k}$ and $s_{k},$ , and the global state vector ${\\bf u},$ as follows  \n\n$$\n\\mathbf{e}_{k}^{\\prime}=\\phi_{\\mathrm{e}}(\\mathbf{v}_{s_{k}}\\oplus\\mathbf{v}_{r_{k}}\\oplus\\mathbf{e}_{k}\\oplus\\mathbf{u})\n$$  \n\nwhere $\\phi_{\\mathrm{e}}$ is the bond update function and $\\oplus$ is the concatenation operator. Next, the attributes of each atom $\\mathbf{v}_{i}$ are updated using attributes from itself, the bonds connecting to it, and the global state vector $\\mathbf{u}$ , as follows  \n\n$$\n\\overline{{\\mathbf{v}}}_{i}^{\\mathrm{e}}=\\frac{1}{N_{i}^{\\mathrm{e}}}\\sum_{k=1}^{N_{i}^{\\mathrm{e}}}\\left\\{\\mathbf{e}_{k}^{\\prime}\\right\\}_{r_{k}=i}\n$$  \n\n$$\n\\mathbf{v}_{i}^{\\prime}=\\phi_{\\mathrm{v}}(\\overline{{\\mathbf{v}}}_{i}^{\\mathrm{e}}\\oplus\\mathbf{v}_{i}\\oplus\\mathbf{u})\n$$  \n\nwhere $N_{i}^{\\mathrm{e}}$ is the number of bonds connected to atom i and $\\phi_{\\mathrm{v}}$ is the atom update function. The aggregation step (eq 2) acts as a local pooling operation that takes the average of bonds that connect to the atom i.  \n\nThe first two update steps contain localized convolution operations that rely on the atom-bond connectivity. One can imagine that if more graph network modules are stacked, atoms and bonds will be able to “see” longer distances, and hence, longer-range interactions can be incorporated even if the initial distance cutoff is small to reduce the computational task.  \n\nFinally, the global state attributes u are updated using information from itself and all atoms and bonds, as follows  \n\n$$\n\\begin{array}{l}{{\\displaystyle{{\\overline{{\\mathbf{u}}}}^{\\mathrm{e}}}=\\frac{1}{N^{\\mathrm{e}}}\\sum_{k=1}^{N^{\\mathrm{e}}}\\left\\{\\mathbf{e}_{k}^{\\prime}\\right\\}}}\\\\ {{\\displaystyle{\\overline{{\\mathbf{u}}}^{\\mathrm{v}}}=\\frac{1}{N^{\\mathrm{v}}}\\sum_{i=1}^{N^{\\mathrm{v}}}\\left\\{\\mathbf{v}_{i}^{\\prime}\\right\\}}}\\end{array}\n$$  \n\n$$\n\\mathbf{u}^{\\prime}=\\phi_{\\mathrm{u}}(\\bar{\\mathbf{u}}^{\\mathrm{e}}\\oplus\\bar{\\mathbf{u}}^{\\mathrm{v}}\\oplus\\mathbf{u})\n$$  \n\nwhere $\\phi_{\\mathrm{u}}$ is the global state update function. In addition to providing a portal to input state attributes (e.g., temperature), $\\mathbf{u}$ also acts as the global information placeholder for information exchange on larger scales.  \n\nThe choice of the update functions $\\phi_{\\mathrm{e}},\\ \\phi_{\\mathrm{v}},$ and $\\phi_{\\mathrm{u}}$ largely determines the model performance in real tasks. In this work, we choose the $\\phi\\mathbf{s}$ to be multilayer perceptrons with two hidden layers (eq 7), given their ability to be universal approximators for nonlinear functions.40  \n\n$$\n\\boldsymbol{\\Phi}(\\mathbf{x})=\\mathbf{W}_{3}(\\zeta(\\mathbf{W}_{2}(\\zeta(\\mathbf{W}_{1}\\mathbf{x}+\\mathbf{b}_{1}))+\\mathbf{b}_{2}))+\\mathbf{b}_{3}\n$$  \n\nwhere $\\zeta$ is the modified softplus function10 acting as a nonlinear activator, Ws are the kernel weights, and bs are the biases. Note that the weights for atom, bond, and state updates are different. Each fully connected layer will be referred to as a “dense” layer using keras terminology.  \n\nTo increase model flexibility, two dense layers are added before each MEGNet module to preprocess the input. This approach has been found to increase model accuracy. We define the combination of the two dense layers with a MEGNet module as a MEGNet block, as shown in Figure 2. The block also contains residual netlike42 skip connections to enable deeper model training and reduce over-fitting. Multiple MEGNet blocks can be stacked to make more expressive models. In the final step, a readout operation reduces the output graph to a scalar or vector. In this work, the order-invariant set2set model43 that embeds a set of vectors into one vector is applied on both atomic and bond attribute sets. After the readout, the atomic, bond, and state vectors are concatenated and passed through multilayer perceptrons to generate the final output. The overall model architecture is shown in Figure 2. If the atom features are only the integer atomic numbers, an embedding layer is added after the atom inputs $V.$ .  \n\n![](images/541bcfc25d724d1167e12d124fa10021954dee2401f80f53d51e719365edae91.jpg)  \nFigure 2. Architecture for the MEGNet model. Each model is formed by stacking MEGNet blocks. The embedding layer is used when the atom attributes are only atomic numbers. In the readout stage, a set2set neural network is used to reduce sets of atomic and bond vectors into a single vector. The numbers in brackets are the number of hidden neural units for each layer. Each MEGNet block contains a MEGNet layer as well as two dense layers. The “add” arrows are skip connections to enable deep model training.  \n\nAtomic, Bond, and State Attributes. Table 1 summarizes the full set of atomic, bond, and state attributes used as inputs to the MEGNet models. The molecule attributes are similar to the ones used in the benchmarking work by Faber et al.32 For crystals, only the atomic number and spatial distance are used as atomic and bond attributes, respectively.  \n\nTable 1. Atomic, Bond, and State Attributes Used in the Graph Network Models   \n\n\n<html><body><table><tr><td> system</td><td>level</td><td>attributes name</td><td> description</td></tr><tr><td>molecule</td><td>atom</td><td>atom type</td><td>H, C, O, N, F (one-hot)</td></tr><tr><td></td><td></td><td>chirality ring sizes</td><td>R or S (one-hot or null) for each ring size (3-8), the number</td></tr><tr><td></td><td></td><td></td><td>of rings that include this atom. If the atom is not in a ring, this field</td></tr><tr><td rowspan=\"8\"></td><td></td><td>hybridization</td><td>is null sp,sp², sp3 (one-hot or null)</td></tr><tr><td>donor</td><td>acceptor</td><td>whether the atom is an electron acceptor (binary) whether the atom donates electrons</td></tr><tr><td></td><td></td><td>(binary) whether the atom belongs to an</td></tr><tr><td>bond</td><td>aromatic bond type</td><td>aromatic ring (binary) single, double, triple, or aromatic</td></tr><tr><td></td><td></td><td>(one-hot or null) whether the atoms in the bond are in</td></tr><tr><td></td><td>same ring graph distance</td><td>the same ring (binary) shortest graph distance between atoms (1-7). This is a topological</td></tr><tr><td></td><td></td><td>distance. For example, a value of 1 means that the two atoms are nearest neighbors, whereas a value of 2 means they are second nearest neighbors, etc. distance r valued on Gaussian basis</td></tr><tr><td>state</td><td>distance</td><td>exp(-(r - ro)²/o²), where ro takes values at 20 locations linearly placed between O and 4, and the width o = 0.5</td></tr><tr><td></td><td></td><td>average atomic weight bonds per</td><td>molecular weight divided by the number of atoms (float) average number of bonds per atom</td></tr><tr><td> system</td><td>attributes</td><td>atom</td><td>(float)</td></tr><tr><td>crystal</td><td>level atom Z</td><td>name</td><td>description the atomic number of element (1-94)</td></tr><tr><td rowspan=\"2\"></td><td>bond spatial</td><td></td><td>expanded distance with Gaussian basis exp(-(r - ro)2/o2) centered at 100 points</td></tr><tr><td></td><td>distance 0 = 0.5</td><td>linearly placed between O and 5 and</td></tr><tr><td></td><td>state</td><td>two zeros</td><td>placeholder for global information exchange</td></tr></table></body></html>  \n\nData Collections. The molecule data set used in this work is the QM9 data set33 processed by Faber et al.32 It contains the B3LYP/6- $31\\mathrm{G}(2\\mathrm{df,p})$ -level density functional theory (DFT) calculation results on $130~462$ small organic molecules containing up to 9 heavy atoms.  \n\nThe crystal data set comprises the DFT-computed energies and band gaps of $69640$ crystals from the Materials Project15 obtained via the Python Materials Genomics (pymatgen)44 interface to the Materials Application Programming Interface (API)45 on June 1, 2018. We will designate this as the MP-crystals-2018.6.1 data set to facilitate future benchmarking and comparisons as data in the Materials Project is constantly being updated. The crystal graphs were constructed using a radius cutoff of $\\mathbf{\\hat{4}}\\ \\hat{\\mathbf{A}}.$ Using this cutoff, 69 239 crystals do not form isolated atoms and are used in the models. All crystals were used for the formation energy model and the metal against the nonmetal classifier, while a subset of 45 901 crystals with a finite band gap was used for the band gap regression. A subset of 5830 structures have elasticity data that do not have calculation warnings and will be used for elasticity models.  \n\nModel Construction and Training. A customized Python version of MEGNet was developed using the keras $\\mathrm{API}^{41}$ with the tensorflow backend.46 Because molecules and crystals do not have the same number of atoms, we assemble batches of molecules/crystals  \n\nTable 2. Comparison of MAEs of 13 Properties in the QM9 Data Set for Different Modelsa,b   \n\n\n<html><body><table><tr><td colspan=\"3\">MEGNet-full°</td><td colspan=\"5\"> MEGNet-simpled</td></tr><tr><td> property</td><td>units</td><td>(this work)</td><td>(this work)</td><td> SchNet36</td><td>enn-s2s37</td><td> benchmark32</td><td> target</td></tr><tr><td>EHOMO</td><td>eV</td><td>0.038±0.001</td><td>0.043</td><td>0.041</td><td>0.043</td><td>0.055℃</td><td>0.043</td></tr><tr><td>ELUMO</td><td>eV</td><td>0.031±0.000</td><td>0.044</td><td>0.034</td><td>0.037</td><td>0.064e</td><td>0.043</td></tr><tr><td>△</td><td>eV</td><td>0.061±0.001</td><td>0.066</td><td>0.063</td><td>0.069</td><td>0.087e</td><td>0.043</td></tr><tr><td>ZPVE</td><td>meV</td><td>1.40±0.06</td><td>1.43</td><td>1.7</td><td>1.5</td><td>1.9g</td><td>1.2</td></tr><tr><td>μ</td><td>D</td><td>0.040 ± 0.001</td><td>0.050</td><td>0.033</td><td>0.030</td><td>0.101℃</td><td>0.1</td></tr><tr><td>α</td><td>bohr3</td><td>0.083±0.001</td><td>0.081</td><td>0.235</td><td>0.092</td><td>0.161</td><td>0.1</td></tr><tr><td>(R²)</td><td>bohr²</td><td>0.265 ± 0.001</td><td>0.302</td><td>0.073</td><td>0.180</td><td></td><td>1.2</td></tr><tr><td>U。</td><td>eV</td><td>0.009±0.000</td><td>0.012</td><td>0.014</td><td>0.019</td><td>0.025g</td><td>0.043</td></tr><tr><td>U</td><td>eV</td><td>0.010±0.000</td><td>0.013</td><td>0.019</td><td>0.019</td><td></td><td>0.043</td></tr><tr><td>H</td><td>eV</td><td>0.010±0.000</td><td>0.012</td><td>0.014</td><td>0.017</td><td></td><td>0.043</td></tr><tr><td>G</td><td>eV</td><td>0.010±0.000</td><td>0.012</td><td>0.014</td><td>0.019</td><td></td><td>0.043</td></tr><tr><td>C,</td><td>cal (mol K)-1</td><td>0.030±0.001</td><td>0.029</td><td>0.033</td><td>0.040</td><td>0.044%</td><td>0.05</td></tr><tr><td></td><td>cm-1</td><td>1.10±0.08</td><td>1.18</td><td></td><td>1.9</td><td>2.71h</td><td>10</td></tr></table></body></html>  \n\naThe “benchmark” column refers to the best model in the work by Faber et al.,32 and the “target” column refers to the widely accepted thresholds for “chemical accuracy”.32 The standard deviations in the MAEs for the MEGNet-full models over three randomized training:validation:test splits are also provided. $^{b}\\epsilon_{\\mathrm{HOMO}}$ : highest occupied molecular orbital; $\\epsilon_{\\mathrm{LUMO}}$ : lowest unoccupied molecular orbital; $\\Delta\\epsilon$ : energy gap; ZPVE: zero-point vibrational energy; $\\mu$ : dipole moment; $\\alpha$ : isotropic polarizability; $\\big<R^{2}\\big>$ : electronic spatial extent; $U_{0}$ : internal energy at $0\\mathrm{K};$ U: internal energy at 298 $\\operatorname{K};H$ : enthalpy at $298\\ \\mathrm{K}$ ; G: Gibbs free energy at 298 K; $C_{\\nu}$ : heat capacity at $298\\ \\mathrm{K};$ ; $\\omega_{1}$ : highest vibrational frequency. cFull MEGNet models using all listed features in Table 1. The optimized models for ZPVE, $\\big<R^{2}\\big>,\\mu,$ and $\\omega_{1}$ contain five, five, three, and one MEGNet blocks, respectively, while the optimized models for all other properties use two MEGNet blocks. dSimple MEGNet models using only the atomic number as the atomic feature, expanded distance as bond features, and no dummy state features. All models contain three MEGNet blocks. eGraph convolution with a molecular graph feature.34 fGated-graph neural network with a molecular graph feature.35 gKernel-ridge regression with histograms of distances, angles, and dihedrals. hRandom forest model with bonds, angles, ML feature.  \n\ninto a single graph with multiple targets to enable batch training. The Adam optimizer47 was used with an initial learning rate of 0.001, which is reduced to 0.0001 during later epochs for tighter convergence.  \n\nEach data set is divided into three partstraining, validation, and test. For the molecule models, $90\\%$ of the data set was used for training and the remaining were divided equally between validation and test. For the crystal formation energy models, $60~000$ crystals were used for training and the remaining were divided equally between validation and test for direct comparison to the work of Schütt et al.36 For the band gap classification models and elastic moduli models, an 80:10:10 split was applied. All models were trained on the training set, and the configuration and hyperparameters with the lowest validation error were selected. Finally, the test error is calculated. During training, the validation error is monitored and the training is stopped when the validation error does not improve for 500 consecutive epochs. The models were trained on Nvidia GTX 1080Ti GPUs. On average, it takes $80~\\mathrm{s}$ and $110\\mathrm{~s~}$ per epoch for each molecular and crystal model, respectively. Most models reach convergence within 1000 epochs. However, models for $U_{0},U,H,G,$ and $\\langle\\check{R}^{2}\\rangle$ require 2000−4000 epochs. The embedding dimension is set to 16. The elemental embeddings trained on the formation energy using one MEGNet block were transferred to the band gap regression model and kept fixed. We use the same architecture featuring three MEGNet blocks in the models for crystals.  \n\nData and Model Availability. To ensure reproducibility of the results, the MP-crystals-2018.6.1 data set used in this work has been made available as a JavaScript Object Notation file at https://figshare. com/articles/Graphs_of_materials_project/7451351. The graph network modules and overall models have also been released as opensource code in a Github repository at https://github.com/ materialsvirtuallab/megnet.  \n\n# RESULTS  \n\nPerformance on QM9 Molecules. Table 2 shows the comparison of the mean absolute errors (MAEs) of 13 properties for the different models, and the convergence plots with the number of training data are shown in Figure S1. It can be seen that the MEGNet models using the full set of attributes (“full” column in Table 2) outperform the state-of-art SchNet36 and MPNN enn-s2s models37 in all but two of the propertiesthe norm of dipole moment $\\mu$ and the electronic spatial extent $R^{2}$ . Out of 13 properties, only the errors on zeropoint vibrational energy (ZPVE) $\\left(1.40\\mathrm{\\meV}\\right)$ and band gap $\\bar{(}\\Delta\\epsilon)$ $\\left(0.060~\\mathrm{eV}\\right)$ exceed the thresholds for chemical accuracy. The errors of various properties follow Gaussian distributions, as shown in Figure S2.  \n\nWe note that the atomic and bond attributes in Table 1 encode redundant information. For example, the bond type can usually be inferred from the bonding atoms and the spatial distance. We therefore developed “simple” MEGNet models that utilize only the atomic number and spatial distance as the atomic and bond attributes, respectively. These are the same attributes used in the crystal MEGNet models. From Table 2, we may observe that these simple MEGNet models achieve largely similar performance as the full models, with only slightly higher MAEs that are within chemical accuracy and still outperforming prior state-of-the-art models in 8 of the 13 target properties. It should be noted, however, that the convergence of the “simple” models is slower than the “full” models for certain properties (e.g., $\\mu_{\\mathrm{-}}$ ZVPE). This may be due to the models having to learn more complex relationships between the inputs and the target properties.  \n\nUnified Molecule Free Energy Model. To achieve the results presented in Table 2, one MEGNet model was developed for each target, similar to previous works.36,37 However, this approach is extremely inefficient when multiple targets are related by a physical relationship and should share similar features. For instance, the internal energy at 0 K $\\left(U_{0}\\right)$ and room temperature $(U)$ , enthalpy $\\left(H=U+{\\dot{P}}V\\right)$ , and Gibbs free energy $\\ ^{\\prime}G=U+P V-T S)$ are all energy quantities that are related to each other by temperature $(\\bar{T})$ , pressure $(P)$ , volume $(V)$ , and entropy (S). To illustrate this concept, we have developed a combined free energy model for $U_{0},U,H,$ and $G$ for the QM9 data set by incorporating the temperature, pressure (binary), and entropy (binary) as additional global state attributes in $\\mathbf{u},$ , that is, $(0,0,0)$ , (298, 0, 0), (298, 1, 0), and (298, 1, 1) for $U_{0},U,H,$ and $G_{;}$ , respectively. Using the same architecture, this combined free energy model achieves an overall MAE of $0.010\\ \\mathrm{~eV}$ for the four targets, which is comparable to the results obtained using the separate MEGNet models for each target.  \n\nIn principle, the combined free energy model should be able to predict free energies at any temperature given sufficient training data. Indeed, the predicted $U$ at 100 and $200\\mathrm{~K~}$ matches well with our DFT calculations (see Figure S3), even though these data points were not included in the training data. However, the predicted $H$ and $G$ at the same temperatures show large deviations from the DFT results. We hypothesize that this is due to the fact that only one temperature data for these quantities exist in the training data and that the addition of $H$ and $G$ data at multiple temperatures into the training data would improve the performance of the unified free energy MEGNet model.  \n\nPerformance on Materials Project Crystals. Table 3 shows the comparison of the performance of the MEGNet models against the SchNet36 and CGCNN models.9 The convergence of formation energy model is shown in Figure S4. We may observe that the MEGNet models outperform both the SchNet and CGCNN models in the MAEs of the formation energies $E_{\\mathrm{f}}$ band gap $E_{\\mathrm{g}},$ bulk modulus $K_{\\mathrm{VRH}},$ and shear modulus $G_{\\mathrm{VRH}}.$ . It should be noted that these results especially the prediction of $E_{\\mathrm{g}}$ and the metal/nonmetal classifiersare achieved over much diverse data sets than previous works, and the prediction error in $E_{\\mathrm{f}},E_{\\mathrm{g}},K_{\\mathrm{VRH}},$ and GVRH is well within the DFT errors in these quantities.48−52 The MEGNet models, similar to the SchNet models, utilize only one atomic attribute (atomic number) and one bond attribute (spatial distance), while nine attributes were used in the CGCNN model. We also found that transferring the elemental embeddings from the $E_{\\mathrm{f}}$ model, which was trained on the largest data set, significantly accelerates the training and improves the performance of the $E_{\\mathrm{g}},$ $K_{\\mathrm{VRH}},$ and $G_{\\mathrm{VRH}}$ models. For example, an independently trained model (without transfer learning) for $E_{\\mathrm{g}}$ has a higher MAE of $0.38\\ \\mathrm{eV}$ .  \n\nTable 3. Comparison of the MAEs in the Formation Energy $E_{\\mathrm{f}}$ Band Gap $E_{\\mathrm{g}},$ Bulk Modulus $K_{\\mathrm{VRH}},$ Shear Modulus ${\\cal G}_{\\mathrm{VRH}},$ and Metal/Nonmetal Classification between MEGNet Models and Prior Works on the Materials Project Data Seta   \n\n\n<html><body><table><tr><td></td><td>units</td><td>MEGNet</td><td> SchNet36</td><td>CGCNN9</td></tr><tr><td>elements</td><td></td><td>89</td><td>89</td><td>87</td></tr><tr><td>Ef</td><td>eV atom-1</td><td>0.028 ± 0.000 (60 000)</td><td>0.035 (60 000)</td><td>0.039 (28 046)</td></tr><tr><td></td><td>eV</td><td>0.33 ± 0.01 (36 720)</td><td></td><td>0.388 (16 485)</td></tr><tr><td>KVRH</td><td>log1o (GPa)</td><td>0.050 ± 0.002 (4664)</td><td></td><td>0.054 (2041)</td></tr><tr><td>GVRH</td><td>log1o (GPa)</td><td>0.079 ± 0.003 (4664)</td><td></td><td>0.087 (2041)</td></tr><tr><td>metal classifier</td><td></td><td>78.9% ± 1.2% (55391)</td><td></td><td>80% (28 046)</td></tr><tr><td>nonmetal classifier</td><td></td><td>90.6% ± 0.7% (55 391)</td><td></td><td>95% (28 046)</td></tr></table></body></html>\n\naThe number of structures in the training data is in parentheses. The standard deviations in the MAEs for the MEGNet models over three randomized training:validation:test splits are also provided.  \n\nWe note that the data set used in the development of the CGCNN model is significantly smaller than that of MEGNet or SchNet, despite all three models having obtained their data from the Materials Project. The reason is that crystals with warning tags or without band structures were excluded from the CGCNN model training. Using this exclusion strategy and a similar training data size, the MEGNet models for formation energy and band gap have MAEs of $0.032\\ \\mathrm{eV}$ atom−1 and 0.35 eV, respectively. The accuracies for metal and nonmetal classifiers are increased to 82.7 and $93.1\\%$ , respectively.  \n\nThere are also nongraph-based crystal ML models such as the JARVIS-ML model53 and the AFLOW-ML model.54 The MAEs of the JARVIS-ML models53 for formation energy, band gap, bulk moduli, and shear moduli are $0.12~\\mathrm{eV~atom}^{-1}$ , 0.32 eV, $10.5\\ \\mathrm{GPa},$ , and $9.5\\ \\mathrm{GPa}$ , respectively, while the MAEs of AFLOW-ML models54 for band gap, bulk moduli, and shear moduli are 0.35 eV, $8.68\\ \\mathrm{GPa}_{,}$ , and $10.62~\\mathrm{GPa}_{.}$ , respectively. However, these ML models are developed with very different data sets (e.g., the JARVIS-DFT database contains formation energies, elastic constants, and band gaps for bulk and 2D materials computed using different functionals) and are therefore not directly comparable to the MEGNet, SchNet, or CGCNN models, which are all trained using Materials Project data.  \n\nFigure 3a,b provides a detailed analysis of the MEGNet model performance on $E_{\\mathrm{f}}$ The parity plot (Figure 3a) shows that the training and test data are similarly well-distributed, and consistent model performance is achieved across the entire range of $E_{\\mathrm{f}}$ We have performed a sensitivity analysis of our MEGNet $E_{\\mathrm{f}}$ model to various hyperparameters. Increasing the radius cutoff to $6\\textup{\\AA}$ slightly increases the MAE to $0.03\\ \\mathrm{eV}$ atom−1. Using one or five MEGNet blocks instead of three results in MAEs of 0.033 and $0.027~\\mathrm{eV~atom}^{-1}$ , respectively. Hence, we can conclude that our chosen radius cutoff of 4 Å and model architecture comprising three MEGNet blocks are reasonably well-optimized. Figure 3b plots the average test MAEs for each element against the number of training structure containing that element. In general, the greater the number of training structures, the lower the MAE for structures containing that element. Figure 3c shows the receiver operating characteristic (ROC) curve for the metal/ nonmetal classifier. The overall test accuracy is $86.9\\%$ , and the area under curve for the receiver operation conditions is 0.926.  \n\n![](images/158413294f023af0d634dc8d97928a27522d0f05025b939f18a47f2fa71bbcb0.jpg)  \nFigure 3. Performance of MEGNet models on the Materials Project data set. (a) Parity plots for the formation energy of the training and test data sets. (b) Plot of average MAE for each element against the number of training structures containing that element. (c) ROC curve for test data for the MEGNet classifier trained to distinguish metals against nonmetals.  \n\n# DISCUSSION  \n\nIt is our belief that the separation of materials into molecules and crystals is largely arbitrary, and a true test of any structured representation is its ability to achieve equally good performance in property prediction in both domains. We have demonstrated that graph networks, which provide a natural framework for representing the attributes of atoms and the bonds between them, are universal building blocks for highly accurate prediction models. Our MEGNet models, built on graph network concepts, show significantly improved accuracies over prior models in most properties for both molecules and crystals.  \n\nA key advance in this work is the demonstration of the incorporation of global state variables to build unified models for related properties. A proof of concept is shown in our unified molecule free energy MEGNet model, which can successfully predict the internal energy at multiple temperatures, enthalpy, and Gibbs free energy with temperature, entropy, and pressure as global state variables. This stands in sharp contrast to the prevailing approach in the materials ML community of building single-purpose models for each quantity, even if they are related to each other by wellknown thermodynamic relationships. The unification of related models has significant advantages in that one can achieve multifold increases in training data with minimal increase in model complexity, which is particularly important given the relatively small data sets available in materials science.  \n\nInterpretability. For chemistry and materials science applications, a particularly desirable feature for any representation is interpretability and reproduction of known chemistry intuition.55 To this end, we have extracted the elemental embeddings from the MEGNet model for crystal formation energy. As shown in Figure 4, the correlations between the elemental embeddings correctly recover the trends in the periodic table of the elements. For example, the alkaline, alkali, chalcogen, halogen, lanthanoid, transition metals, posttransition metals, metalloid, and actinoid show highest similarities within their groups. It is important to note that the extracted trends reproduce well-known “exceptions” in the periodic arrangement of atoms as well. For example, the fact that Eu and Yb do not follow the lanthanoids but are closer to alkaline earth elements (Figure S6) is in good agreement with chemical intuition and matches well with the structure graphs proposed by Pettifor.56 Furthermore, these trends are obtained from the diverse Materials Project dataset encompassing most known crystal prototypes and 89 elements, rather than being limited to specific crystal systems.57,58  \n\nSuch embeddings obtained from formation energy models are particularly useful for the development of models to predict stable new compounds or as features for other ML models. Hautier et al.59 previously developed an ionic substitution prediction algorithm using data mining, which has been used successfully in the discovery of several new materials.60,61 The ion similarity metric therein is purely based on the presence of ions in a given structural prototype, a slightly coarse-grained description. Here, the MEGNet models implicitly incorporate the local environment of the site and should in principle better describe the elemental properties and bonding relationships. We note that with more MEGNet blocks, the contrast of the embeddings between atoms is weaker, as shown in Figure S5. The two-dimensional t-SNE plots62 confirm these conclusions, as shown in Figure S6. This is because with more blocks, the environment seen by the atom spans a larger spatial region, and the impact of geometry becomes stronger, which obscures the chemical embeddings.  \n\n![](images/bfe6afefeb19eb5d6e03f0e6b54dc9a13f1cadff9406cd3b38c2d4117a297cd5.jpg)  \nFigure 4. Pearson correlations between elemental embedding vectors. Elements are arranged in order of increasing Mendeleev number56 for easier visualization of trends.  \n\nComposability. A further advantage of the graph networkbased approach is its modular and composable nature. In our MEGNet architecture, a single block captures the interactions between each atom and its immediate local environment (defined via specified bonds in the molecule models and a radius cutoff in the crystal models). Stacking multiple blocks allows for information flow and hence capturing of interactions, across larger spatial distances.  \n\nWe can see this effect in the MEGNet models for the QM9 data set, where different numbers of blocks are required to obtain good accuracy for different properties. For most properties, two blocks are sufficient to achieve MAEs within chemical accuracy. However, more blocks are necessary for the ZPVE (five), electronic spatial extent (five), and dipole moment (three), which suggests that it is important to capture longer-ranged interactions for these properties. In essence, the choice of number of MEGNet blocks for a particular property model boils down to a consideration of the range of interactions necessary for accurate prediction, or simply increasingly the number of blocks until convergence in accuracy is observed.  \n\nData Limitations and Transfer Learning. The critical bottleneck in building graph networks models, like all other ML models, is data availability. For instance, we believe that the inability of the unified free energy MEGNet model to accurately predict $H$ and $G$ at 100 and $200~\\mathrm{K}$ is largely due to the lack of training data at those temperatures. Similarly, a general inverse relationship can be seen between the number of training structures and the average MAE in formation energies of the crystals in Figure 3b.  \n\nBesides adding more data (which is constrained by computational cost as well as chemistry considerations), another avenue for improvement is to use ensemble models. We tested this hypothesis by training two independent three block MEGNet models and used the average as the ensemble prediction for the formation energies of the Materials Project data set. The MAE reduces from $\\bar{0}.028\\ \\mathrm{eV\\atom^{-1}}$ for a single MEGNet model to 0.024 eV atom−1 for the ensemble MEGNet model.  \n\nYet, another approach to address data limitations is transfer learning,63,64 and we have demonstrated an instructive example of how this can be applied in the case of the crystal MEGNet models. Data quantity and quality is a practical problem for many materials properties. Using the Materials Project as an example, the formation energy data set comprises ${\\sim}69\\ 000$ crystals, that is, almost all computed crystals in the database. However, only about half of these have nonzero band gaps. Less than $10\\%$ crystals in Materials Project have computed elastic constants because of the high computational effort in obtaining these properties. By transferring the elemental embeddings, which encode the learned chemical trends from the much larger formation energy data set, we were able to efficiently train the band gap and elastic moduli MEGNet models and achieve significantly better performance than prior ML models. We believe this to be a particularly effective approach that can be extended to other materials properties with limited data availability.  \n\n# CONCLUSIONS  \n\nTo conclude, we have developed MEGNet models that are universally high performing across a broad variety of target properties for both molecules and crystals. Graphs are a natural choice of representation for atoms and the bonds between them, and the sequential update scheme of graph networks provides a natural approach for information flow among atoms, bonds, and global state. Furthermore, we demonstrate two advancesincorporation of global state inputs and transfer learning of elemental embeddingsin this work that extend these models further to state-dependent and data-limited properties. These generalizations address several crucial limitations in the application of ML in chemistry and materials science and provide a robust foundation for the development of general property models for accelerating materials discovery.  \n\n# ASSOCIATED CONTENT  \n\n# $\\otimes$ Supporting Information  \n\nThe Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.9b01294.  \n\nConvergence of MEGNet models for the QM9 data set; MEGNet error distributions on the QM9 dataset; QM9 energy predictions at different temperatures; convergence of the MEGNet model for Materials Project formation energy; elemental embeddings for one and five MEGNet block models; and t-SNE visualization of elemental embeddings for one, three, and five MEGNet block models (PDF)  \n\n# AUTHOR INFORMATION  \n\nCorresponding Author   \n$^{*}\\mathrm{E}$ -mail: ongsp@eng.ucsd.edu.   \nORCID   \nChi Chen: 0000-0001-8008-7043   \nShyue Ping Ong: 0000-0001-5726-2587   \nNotes   \nThe authors declare no competing financial interest.  \n\n# ACKNOWLEDGMENTS  \n\nThis work is supported by the Samsung Advanced Institute of Technology (SAIT)’s Global Research Outreach (GRO) Program. The authors also acknowledge data and software resources provided by the Materials Project, funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05-CH11231: Materials Project program KC23MP, and computational resources provided by Triton Shared Computing Cluster (TSCC) at the University of California, San Diego, the National Energy Research Scientific Computing Centre (NERSC), and the Extreme Science and Engineering Discovery Environment (XSEDE) supported by National Science Foundation under grant no. ACI-1053575.  \n\n# REFERENCES  \n\n(1) Michalski, R. S.; Carbonell, J. G.; Mitchell, T. M. Machine Learning: An Artificial Intelligence Approach; Springer Science & Business Media, 2013.   \n(2) LeCun, Y.; Bengio, Y.; Hinton, G. Deep learning. Nature 2015, 521, 436.   \n(3) Mueller, T.; Kusne, A. G.; Ramprasad, R. Machine Learning in Materials Science. 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  {
    "id": "10.1017_S0885715619000812",
    "DOI": "10.1017/S0885715619000812",
    "DOI Link": "http://dx.doi.org/10.1017/S0885715619000812",
    "Relative Dir Path": "mds/10.1017_S0885715619000812",
    "Article Title": "The Powder Diffraction File: a quality materials characterization database",
    "Authors": "Gates-Rector, S; Blanton, T",
    "Source Title": "POWDER DIFFRACTION",
    "Abstract": "The ICDD' s Powder Diffraction File (TM) (PDF (R)) is a database of inorganic and organic diffraction data used for phase identification and materials characterization by powder diffraction. The PDF has been available for over 75 years and finds application in X-ray, synchrotron, electron, and neutron diffraction analyses. With entries based on powder and single crystal data, the PDF is the only crystallographic database where every entry is editorially reviewed and marked with a quality mark that alerts the user to the reliability/quality of the submitted data. The editorial processes of ICDD' s quality management system are unique in that they are ISO 9001:2015 certified. Initially offered as text on paper cards and books, the PDF evolved to a computer-readable database in the 1960s and today is both computer and web accessible. With data mining and phase identification software available in PDF products, and the databases' compatibility with vendor (third party) software, the 1 000 000+ published PDF entries serve a wide range of disciplines covering academic, industrial, and government laboratories. Details describing the content of database entries are presented to enhance the use of the PDF. (C) 2019 International Centre for Diffraction Data.",
    "Times Cited, WoS Core": 961,
    "Times Cited, All Databases": 1003,
    "Publication Year": 2019,
    "Research Areas": "Materials Science",
    "UT (Unique WOS ID)": "WOS:000500163500010",
    "Markdown": "# CRYSTALLOGRAPHY EDUCATION ARTICLE  \n\n# The Powder Diffraction File: a quality materials characterization database  \n\nStacy Gates-Rector $\\textcircled{1}$ ,a) and Thomas Blanton $\\textcircled{1}$ International Centre for Diffraction Data, 12 Campus Blvd, Newtown Square, Pennsylvania 19073-3273, USA (Received 10 May 2019; accepted 9 September 2019)  \n\nThe ICDD’s Powder Diffraction File $(\\mathrm{PDF}^{\\mathfrak{P}})$ is a database of inorganic and organic diffraction data used for phase identification and materials characterization by powder diffraction. The PDF has been available for over 75 years and finds application in X-ray, synchrotron, electron, and neutron diffraction analyses. With entries based on powder and single crystal data, the PDF is the only crystallographic database where every entry is editorially reviewed and marked with a quality mark that alerts the user to the reliability/quality of the submitted data. The editorial processes of ICDD’s quality management system are unique in that they are ISO 9001:2015 certified. Initially offered as text on paper cards and books, the PDF evolved to a computer-readable database in the 1960s and today is both computer and web accessible. With data mining and phase identification software available in PDF products, and the databases’ compatibility with vendor (third party) software, the $1\\ 000\\ 000+$ published PDF entries serve a wide range of disciplines covering academic, industrial, and government laboratories. Details describing the content of database entries are presented to enhance the use of the PDF. $\\mathfrak{C}$ 2019 International Centre for Diffraction Data. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.   \n[doi:10.1017/S0885715619000812]  \n\nKey words: powder X-ray diffraction, XRD database, quality mark, subfiles, PDF card  \n\n# I. INTRODUCTION  \n\nThe Powder Diffraction File (PDF) is a database produced and maintained by the International Centre for Diffraction Data $\\mathrm{(ICDD^{\\circledast})}$ ), a non-profit scientific organization committed to meeting the needs of the scientific community through the collection, editing, publishing, and distribution of powder X-ray diffraction (PXRD) data for the identification of materials (Fawcett et al., 2017). The primary purpose of the PDF is to serve as a quality reference tool for the powder diffraction community. This tool provides insight into the structural and crystallographic properties of a material, which allows for phase identification using powder diffraction techniques. The PDF has been the primary qualitative crystalline phase identification reference for powder diffraction data since 1941 (Jenkins et al., 1987) and, in recent years, has expanded its coverage to include semi-crystalline and amorphous materials (Gates et al., 2014).  \n\nDatabases, like the PDF, that provide structural details, such as lattice parameters, space group, atomic coordinates, and thermal parameters, can be used for a range of tasks, including (but not limited to) structure modeling, phase identification, and quantification [Belsky et al., 2002 (ICSD); Downs and Hall-Wallace, 2003 (AMCSD); Gražulis et al., 2012 (COD); Groom et al., 2016 (CSD); Villars and Cenzual, 2018 (PCD)]. As a result, structural databases are one of the key tools used in the crystallographic community (Kuzel and Danis, 2007). Though these databases do tend to have some common applications, they often differ in content, format, and functionality. In PDF-4 products, structure details, when available, are provided on a designated tab of the PDF card, as will be discussed later in the text.  \n\n# A. Creation of the PDF  \n\nIn 1941, the American Society for Testing and Materials (ASTM) published the first official set (Set 1) of the PDF; each entry was printed on a $3^{\\prime\\prime}\\times5^{\\prime\\prime}$ paper card [Figure 1(a)]. The Joint Committee on Powder Diffraction Standards (JCPDS) was evolved from ASTM in 1969 and renamed to ICDD in 1978. The inaugural PDF database consisted of 978 cards displaying a collection of $d{-}I$ data pairs, where the $d.$ -spacing $(d)$ was determined from the angle of diffraction, and the relative peak intensity $(I)$ was obtained experimentally under the best possible conditions for a phase pure material. The list of $d{-}I$ data pairs is often described as the diffraction “fingerprint” of a compound. Each $3^{\\prime\\prime}\\times5^{\\prime\\prime}$ PDF file card was formatted so that the $d{-}I$ pairs of the three most intense peaks were placed at the top-left edge of the card, which was ideal when using the Hanawalt search method (Hanawalt et al., 1938) for phase identification. The full $d{-}I$ list and supporting information were presented on the remaining space of each card. In 1967, a computer-based version of the PDF (PDF-1) became available on magnetic tape, with limited data provided for search-match only. However, it was not until 1985 that the first digital version containing all of the data that appeared on a PDF card image was made available in computer-readable format on CD-ROM (PDF-2) (Messick, 2011).  \n\nThe increased use of the computer-based PDF database, and the problems associated with the storage of the classic cards, prompted the ICDD to discontinue the production of  \n\n<html><body><table><tr><td colspan=\"9\">(a)</td></tr><tr><td>d</td><td></td><td>5.52.853.03</td><td></td><td></td><td>##</td><td></td><td></td></tr><tr><td>#</td><td>1.001.00</td><td></td><td>0.80</td><td></td><td>0.50 1.00</td><td></td><td></td></tr><tr><td></td><td>5050</td><td></td><td>40</td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"4\"></td><td>2.85</td><td></td><td></td><td>50</td></tr><tr><td colspan=\"4\"></td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"4\"></td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"4\"></td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"4\">Z=</td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"4\">a.= b= c.=</td><td>3.15 2.0Y</td><td></td><td></td><td></td></tr><tr><td colspan=\"4\">A= C=</td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"4\">D=</td><td></td><td></td><td></td><td></td></tr><tr><td colspan=\"4\">n= w= =</td><td></td><td></td><td></td><td>N</td></tr></table></body></html>  \n\n![](images/72c6e6ee5534a0db5716745cbe859f2ed084000d3115483abe2536c96b5a0fc3.jpg)  \nFigure 1. (Color online) PDF card image for $\\mathrm{Ce}_{2}(\\mathrm{SO}_{4})_{3}$ from PDF Set 1 as issued (a) in 1941 and (b) in 2019.  \n\ncards in 1987, leaving books as the only hardcopy form published (Smith and Jenkins, 1996). Over the years, various seminal events have transformed the original card catalog/ index styled database into an electronic resource, having a relational database format (Faber and Fawcett, 2002). The initial design of the PDF “card” supported the primary purpose of the database, phase identification by search-match processing. Thus, the core design was retained upon conversion from printed to digital cards and has progressively been enhanced over time [Figure 1(b)]. With reduced limitations on presentation space, pivotal information pertaining to the phase(s) of interest were added to PDF card entries, which greatly enriched the comprehensiveness of the database. By 2015, benefits associated with the computer-based PDF significantly outweighed those of hardcopy versions. Thus, as of 2017, all printed PDF database products were discontinued.  \n\nHaving a comprehensive database that houses all PDF patterns allows users to carry out single and multi-phase identification. The capacity of the database has evolved over time in order to support innovative analyses methods and instrumentation advances that have been developed in the field of diffraction. Currently, the PDF contains data to support analysis pertaining to X-ray, electron, and neutron diffraction, which may include entries for crystalline, semi-crystalline, amorphous, modulated, disordered clays, and nanomaterials (Bruno et al., 2017).  \n\n# II. DATABASE DEVELOPMENT  \n\nThe PDF is continuously enhanced, and maintained, by ICDD staff and members. An updated, revised, and new version of the database is released annually providing users a contemporary reference tool that helps them to obtain “correct” answers – not just “any” answer. Each new PDF release comes with an increased number of entries (Figure 2), as well as enhanced functionality via new software features. In recent years, there have been expectations for reference data to be freely available, but often times this comes at the sacrifice of quality.  \n\n![](images/afd96bae8dd6844a2dcad21132e95e327cf7cb5e0334aa49afa79d3ef778f21c.jpg)  \nFigure 2. (Color online) Graphical representation of the lifetime growth of PDF-4, with the number of new entries for the specified annual release shown on top (red). The spike in newly published entries observed in 1998 and 2005 reflect the onset of collaborations with FIZ (ICSD) and MPDS (LPF), respectively.  \n\nThe overall quality of the PDF improves progressively as higher quality entries are included in the database. Subsequently, the number of low-quality patterns has decreased by ${\\sim}10\\%$ in the last 15 years. Though some believe that low-quality entries can undesirably influence the overall statistics of the database and they should be removed, these patterns are editorially reviewed and included with significant details. The mindset being, even a partial match of data may be crucial in assisting users in proper phase identification, or even elimination of certain phases/possibilities.  \n\n# A. Data acquisition (data sources)  \n\nThe PDF is a compilation of diffraction data from various sources, including ICDD and contributions from collaborative databases: Inorganic Crystal Structure Database (ICSD), the Linus Pauling File (LPF), the National Institute of Standards & Technology (NIST) Crystal Data, and Cambridge Structural Database (CSD). As a result, much of the diffraction data originates from the literature and originally lacked uniformity. The quality of the data is generally limited by both the diffraction technique used during experimentation and the instrument capabilities at the time of publication. Therefore, when utilizing data from multiple sources, ICDD processes entries in a manner that converts data into a common format and requires every entry to undergo editorial review and be classified based on their level of quality. Various databases offer calculated PXRD data, including the PDF. However, a unique feature of the PDF is the availability of actual experimental raw powder data acquired primarily through ICDD’s grant-in-aid program. This type of data is useful during materials’ characterization as the calculated PXRD data may not always clearly explain the observed diffraction pattern and can be significantly different from what is observed during the experiment. Therefore, theoretical, or calculated, patterns may not always properly represent what is truly observed during experimental data collection. Some examples of where this discrepancy is evident are materials of small crystallite size, polymeric materials, and clays. When the phase of interest is amorphous or semi-crystalline, a d–I list and/or atomic coordinates will not adequately define the amorphous profile observed in a diffraction pattern (Figure 3). To allow for improved whole pattern analysis, particularly when amorphous phases are present, the PDF also includes digital raw data patterns, referred to as PDF experimental patterns (PD3s), for amorphous, semi-crystalline, and crystalline phases (Gates et al., 2014).  \n\n# B. Data standardization (quality system)  \n\nAnother distinct attribute of the PDF is its quality system. Once the data are obtained from a source, prior to inclusion in the PDF, each entry is reviewed for appropriateness and quality. In the past, the evaluation of quality was somewhat subjective until 1965 when the $d$ -values and intensities were first entered into computer-readable files (Smith and Jenkins, 1996). This allowed data to be checked based on crystallographic principles in a timely and efficient manner. Over time, important criteria could be added, or removed, based on the knowledge of the crystallographic field and experimental capabilities of the time. As a result, the review process has become more rigorous, with multiple computer algorithms utilized to aid in pattern assessment and quality determination. The benefit being an objective uniformity of quality mark assignments.  \n\n![](images/39e9bd6e45602ffba0cddc98a6a4b36a36c232a3ae47df58c60f4a7601d4ae7d.jpg)  \nFigure 3. (Color online) (a) Stick pattern $\\scriptstyle(d-I$ list) and (b) raw data (PD3) representation of cellulose Iβ. This example demonstrates how raw data patterns fo polymers cannot always be accurately represented from a $d{-}I$ list alone.  \n\nICDD has developed a multi-tiered editorial process that enables the categorization of each entry based on the quality and comprehensiveness of the data. Subsequently, a quality mark (QM) is assigned to each entry to distinguish for PDF users the high-quality $\\mathrm{\\nabla{QM}=^{*}}$ or G), medium-quality (QM ${\\bf\\varepsilon}={\\bf I}$ , C, P, or M), low-quality $\\mathbf{\\mathrm{QM}}=\\mathbf{B}$ or O), or hypothetical $\\mathrm{(QM=H)}$ patterns. The original criteria for quality mark assignment were somewhat broad and designed for single phase crystalline patterns. As the database grew and evolved in both content and format, so did the quality system. The original five quality marks and criteria, shown in Table I (Jenkins and Smith, 1987), have expanded in order to reflect processed/refined data patterns, as well amorphous and semicrystalline materials (Bruno et al., 2017). The current quality marks and major criteria are shown in Table II, along with a brief description of any commonly associated warnings.  \n\nAnother benefit resulting from ongoing editorial review of the PDF is the designation of subfile(s) and subclass(es). The PDF subfiles are generated based on (1) chemical definitions and/or (2) field application (via expert opinion). Subfiles are powerful tools in the database, as they allow users to target their searches in order to save time and reduce the frequency of false matches during the identification process. Some of the editorial improvements, not available in other databases, result from specialized task groups. These expert-lead groups focus on specific classes of materials (i.e. Ceramics, Metals & Alloys, Minerals, Zeolites, etc.) and make detailed recommendations based on the field of study (subfile), for specific PDF entries. These recommendations are then submitted for final review before being incorporated into the database(s). The  \n\nTABLE I. Original major criteria in the assignment of quality marks for PDF entries.   \n\n\n<html><body><table><tr><td>Mark</td><td>Average △ 2-Theta</td><td>Crystallographic information</td><td>Significant figures in “d\"</td><td>Other</td></tr><tr><td>★</td><td>0.03</td><td>Cell known, no unindexed lines</td><td>d<2.5 (3), d<1.2 (4)</td><td> I's measured quantitatively chemistry confirmed</td></tr><tr><td>I</td><td>0.06</td><td>Cell known, two unindexed lines</td><td>d<2 (3)</td><td>I's measured quantitatively</td></tr><tr><td>C</td><td>一</td><td>Cell known</td><td>d<2.5 (3), d<1.2 (4)</td><td>Structure factor R<0.1</td></tr></table></body></html>\n\nNotes: (1) An “O” indicates: low precision, no cell quoted, poorly chemically characterized, and possible mixture (or a combination of the above). (2) A “Blank” indicates patterns which do not meet the criteria for a $\\star^{,,}$ , an “I”, or an “O” or patterns for which no cell is known and, therefore, cannot be assessed fo consistency and line indexing.  \n\n<html><body><table><tr><td colspan=\"6\">TABLEI. Major criteria for PDF quality mark assignment.</td></tr><tr><td>Mark</td><td>20</td><td>Crystallographic information</td><td>Warnings</td><td>Other</td></tr><tr><td>Sor (★)</td><td>Average ≤ 0.03°</td><td>·High-quality diffractometer or Guinier data · Known unit cell · Complete indexing</td><td>· None</td><td>Specific to experimentally based patterns</td></tr><tr><td>G</td><td>n/a</td><td>· Significant amorphous component present · Good signal-to-noise ratio in digital diffraction pattern (PD3) provided</td><td>n/a</td><td>(i.e. spectroscopy, pair distribution functions, commercial source, etc.)</td></tr><tr><td>I</td><td>Average ≤ 0.06° Absolute ≥ 0.20° for individual reflection</td><td>composition of a material · Indexed pattern · Known unit cell · Reasonable range and uniform distribution in intensities</td><td>· No serious systematic errors · Maximum of two unindexed, space group extinct, or impurity reflections; none of these</td><td>Completeness of the pattern is sensible. Reflections with d-value less than or equal to 2.000 A have at least three significant figures after the decimal point</td></tr><tr><td>C</td><td></td><td>· Pattern calculated from single crystal structural parameters for which the structural refinement R-factor was <0.10 ·lF(calc)l data have been checked against the corresponding IF(obs)l -OR— ·A complete check of the bond</td><td>· If the calculated pattern does not meet the“S” quality mark parameters, it is assigned QM=B</td><td>If the structure is derived by X-ray Rietveld methods, the calculated pattern is accepted only in unusual cases; the original powder pattern is preferred.</td></tr><tr><td>M</td><td></td><td>· The number of required significant digits is the same as for an “S\" quality mark · Amorphous component present · Good signal-to-noise ratio in digital</td><td></td><td>No chemical analysis data to support the materials composition provided</td></tr><tr><td> Blank</td><td></td><td>diffraction pattern (PD3） provided No cell No indexing</td><td></td><td>Does not meet criteria for higher quality mark (\"T' or “O\")</td></tr><tr><td>0</td><td></td><td>data are known (or suspected) to be of low precision · Number of unindexed, space group extinct, or impurity</td><td>· Poorly characterized material or the· Usually, the editor has inserted a comment to explain why the“O” was assigned</td><td>A low-precision quality mark means that the diffraction data remain questionable and user's should evaluate closely if used</td></tr></table></body></html>  \n\n![](images/35654f892ece0846a562da3a351291a983a340f7347275337959215d17e4c716.jpg)  \nFigure 4. (Color online) Digital PDF card 00-045-0338: calcium iron phosphate, $\\mathrm{Ca_{9}F e(P O_{4})_{7}}$ with designators corresponding to descriptions in Tables III and IV  \n\nTABLE III. Overview of contents of a PDF entry.   \n\n\n<html><body><table><tr><td>1</td><td>PDFID</td><td colspan=\"2\">Entry number</td></tr><tr><td>2</td><td>Diffraction data</td><td colspan=\"2\">Diffraction type (X-ray, electron, and neutron), wavelength apertures, intensity variables</td></tr><tr><td>3</td><td> d-I list</td><td colspan=\"2\">Interplanar spacings (d) and intensities (I). Miller indices are listed when available</td></tr><tr><td>4</td><td>Tick marks</td><td colspan=\"2\">Graphical representation of peak position</td></tr><tr><td>5</td><td> Diffractogram(s)</td><td colspan=\"2\">Experimental profile (raw diffraction data) or simulated profile diffractogram</td></tr><tr><td>6</td><td>Function keys</td><td colspan=\"2\"> Tools and simulations associated withthe PDF entry.Gray icons indicate the toolor simulation is not available</td></tr><tr><td rowspan=\"6\">７</td><td rowspan=\"6\">Supplemental information tabs</td><td>for this entry</td><td></td></tr><tr><td>Provides additional details pertaining to: (A)“PDF\"- chemistry & general info</td><td>(F) “Classification\"- subfile(s), structure type(s)</td></tr><tr><td>(B）“Experimental”- diffraction experiment</td><td>(G)“Cross-references”- correlated PDF entries</td></tr><tr><td></td><td></td></tr><tr><td>(C)“Physical’- unit cell data (from the author)</td><td>(H)“Reference”-- bibliographic references</td></tr><tr><td>(D)“Crystal”-- ICDD calculated unit cell data (E)“Structure”- atomic structure</td><td>(I)“Comments\"- database comments</td></tr></table></body></html>  \n\nPDF subfiles are continuously edited, reviewed, and classified by ICDD, and can be used with any software system that recognizes ICDD subfile designations, to improve the efficiency and accuracy of the identification process when using the PDF.  \n\n# III. THE PDF CARD  \n\nIndividual entries of the PDF are often referred to as “PDF cards”, which is a term carried over from the original “hardcopy” format of the database. The “cards” are categorically numbered (AA-BBB-XXXX) to indicate the data source (where AA means “00” – ICDD; “01” – ICSD; “02” – CSD; “03” – NIST; $^{\\cdot\\cdot}04^{,3}\\mathrm{~-~}\\mathrm{LPF}$ ; and $\\mathrm{\\ddot{\\mathrm{05^{,9}}}-I C D D_{\\mathrm{(cry}}}$ stal data)). Each data source has set numbers, BBB, corresponding to the annual publication, and a pattern number (XXXX). As a result, one can surmise from the PDF card in Figure 4 (00-045-0338) that this entry contains data from an ICDD pattern (data source $=00^{\\cdot}$ ) that was first included in set 45 (published in 1995) and is pattern number 0338. The PDF entries also contain extensive chemical, physical, bibliographic, and crystallographic data. The red labels in Figure 4 highlight the different components of the PDF card and corresponding details can be found in Tables III and IV.  \n\nWhen available, much of the crystallographic data resides on the “Physical” tab of the PDF card (Figure 5). This is where the basic information pertaining to the unit cell can be found (i.e. crystal symmetry, lattice parameters, space group, etc.).  \n\nTABLE IV. Details for fields on the “PDF” supplemental information tab (Tab A in Figure 4).   \n\n\n<html><body><table><tr><td></td><td>Editorial designations</td><td>Status- primary, alternate,or deletedQuality mark- See Table II: Major criteria for PDFquality mark assgnment</td></tr><tr><td>A1 A2</td><td>Sample conditions</td><td>Specifies the environment, temperature, and pressure of data collection</td></tr><tr><td>A3</td><td> Chemistry</td><td>Indicatesthe polymorphicdesignation of agiven phase,as wellas the formula(e),weight percent, and atomic percent</td></tr><tr><td></td><td></td><td>associate with entry, as available</td></tr><tr><td>A4 A5</td><td>Compound identifier(s) History</td><td>Names and registry numbers used to identify the compound Specifics pertaining to the entries initial publication, and recent modifications, if applicable</td></tr><tr><td></td><td></td><td></td></tr></table></body></html>  \n\n![](images/76b9d0fe3f2c07562a195c70dc9aca787eb865037f46f8d1cefc794b44e44326.jpg)  \nFigure 5. (Color online) “Physical” tab with designators showing where to locate (C1) unit cell settings, (C2) lattice parameters, (C3) unit cell volume, (C4) axial ratio(s), (C5) density values (calculated, measured, and structural), and (C6) data validation values.  \n\n![](images/c30aaeb0939fc634a1a918259bda0b5f6ef14101e0301a3b37709d287dc35e1d.jpg)  \nFigure 6. (Color online) “Structure” tab with designations pertaining to (E1) the original source of the atomic coordinates (if cross-referenced), (E2) symmetry operators, (E3) thermal parameter type, (E4) fractional/atomic coordinates, (E5) anisotropic displacement parameters, (E6) and (E7) unit cell information derived from the single crystal experiment (which can be different from powder unit cell information on the “Physical” tab).  \n\nAtomic coordinates are included for a significant number of PDF entries and can be located on the “Structure” tab, shown in Figure 6. The information provided on this tab enables users to perform qualitative and/or semi-quantitative analyses using the PDF-4 products. Additionally, users have the option to export the crystallographic information in several different formats for use in third party pattern fitting software including quantitative analysis techniques.  \n\n# IV. AVAILABILITY  \n\nThe PDF is available for individual or collaborative use. ICDD provides numerous database products that are specifically designed to meet the needs of those in a variety of areas in the diffraction community, ranging from phase identification (PDF-2) to semi-quantification or full pattern fitting using atomic coordinates (PDF-4). Product summaries, licensing info, and operating specifications for each member of the PDF product line are available on the ICDD website (www.icdd.com).  \n\nCurrently, ICDD collaborates with licensed software developers, including equipment manufacturers and independent developers, and works diligently to cultivate commercial data analysis programs that work seamlessly with our databases. PDF products include their own front-end software that allows the use of the stored PDF entry data for data mining, compound and structure visualization, and data simulations. Use of the PDF front-end software offers valuable features, and information that may not always be seen through vendor’s software. This factor can be of benefit to users looking for additional details or supplemental information pertaining to a material, or group of materials. Also available is ICDD’s search indexing programs, SIeve or SIeve+, that are operable through the PDF front-end software. SIeve/SIeve+ is designed to search and identify unknown materials by engaging data mining interfaces, searches, sorting, and then applying various algorithms to optimize the phase identification process.  \n\n![](images/77e5a46bf1a055742b525a60853fad5d4e0ae0b41ceefb7901de1eec458e5b37.jpg)  \nFigure 7. (Color online) Composition graph displaying PDF entries as points on the phase diagram of Zn, $\\mathtt{C u}$ , and Ni.  \n\n# V. SIGNIFICANCE OF UPDATING PDF (CONCLUSION)  \n\nTechnological advances in instrumentation over the years have undoubtedly influenced the manner in which PXRD data are collected, reviewed, stored, and presented in PDF products. These advances play a role in the quality of diffraction data inserted into the database. The data collection capabilities of newer instruments, combined with meticulous editorial processes, have greatly improved the overall quality of the PDF database. Though the initial creation the PDF was intended for phase identification, over the years the database has transitioned into a high-quality, comprehensive, materials’ identification tool whose increased size and functionality has also improved its quality and usefulness. Recent developments include the addition of composition graphing, 2D diffraction pattern overlay capability, 2D diffraction phase identification analysis $(i n\\ S I e\\nu e+)$ , and microanalysis (XRF) searches. The phase composition graph/search feature allows the user the ability to data mine based on binary or ternary compositions, and generate a resultant plot (Figure 7) where the user can select any data point on the graph to open the corresponding PDF entry. This new functionality has reduced the necessary steps it takes to compare and contrast binary and/or ternary phases in the PDF and has proven quite useful for individuals working with metals and alloys. With significant changes made annually, it is of benefit to all PDF users to keep their PDF databases up-to-date in order to ensure comprehensiveness and relevancy.  \n\n![](images/a8373f27b396e2fce60768d3840703520945e6f8ecfa6d865d085db57d7c825b.jpg)  \nFigure 8. (Color online) Lifetime growth of the PDF- $^{4+}$ (teal) and PDF-4/Organics (purple) databases.  \n\nThe current release of PDF products (Release 2019) contains a combined total of 1 004 568 published entries (Figure 8) that can be used in analyses of powder diffraction data. Using PDF product that are several years old omits recently added entries and prevents users from taking advantage of the latest features and capabilities of the PDF. The Powder Diffraction File continues to grow and evolve concurrently with the community and remains the world’s most comprehensive source of inorganic and organic diffraction data for phase identification and materials’ characterization.  \n\nBelsky, A., Hellenbrandt, M., Karen, V. L., and Luksch, P. (2002). “New developments in the inorganic crystal structure database (ICSD):  \n\naccessibility in support of materials research and design,” Acta Crystallogr. Sect. A. 58(3), 364–369.   \nBruno, I., Gražulis, S., Helliwell, J. R., Kabekkodu, S. N., McMahon, B., and Westbrook, J. (2017). “Crystallography and databases,” Data Sci. J. 16, 1–17.   \nDowns, R. and Hall-Wallace, M. (2003). “The American Mineralogist crystal structure database,” Am. Mineral. 88, 247–250.   \nFaber, J. and Fawcett, T. (2002). “The powder diffraction file: present and future,” Acta Crystallogr. Sect. B. 58, 325–332.   \nFawcett, T. G., Kabekkodu, S. N., Blanton, J. R., and Blanton, T.™N. (2017). “Chemical analysis by diffraction: the powder diffraction file ,” Powder Diffr. 32, 63–71.   \nGates, S. D., Blanton, T. N., and Fawcett, T. G. (2014)™. “A new ‘Chain’ of events: polymers in the powder diffraction file $(\\mathrm{PDF}^{\\mathfrak{B}})$ ,” Powder Diffr. 29, 102–107.   \nGražulis, S., Daškevič, A., Merkys, A., Chateigner, D., Lutterotti, L., Quirós, M., Serebryanaya, N. R., Moeck, P., Downs, R. T., and Le Bail, A. (2012). “Crystallography open database (COD): an open-access collection of crystal structures and platform for World-Wide Collaboration,” Nucleic Acids Res. 40, D420–D427.   \nGroom, C. R., Bruno, I. J., Lightfoot, M. P., and Ward, S. C. (2016). “The Cambridge structural database,” Acta Crystallogr. Sect. B. 72, 171–179.   \nHanawalt, J. D., Rinn, H. W., and Frevel, L. K. (1938). “Chemical analysis by X-ray diffraction,” Ind. Eng. Chem. Anal. Ed. 10, 457–512.   \nJenkins, R. and Smith, D. (1987). “Powder Diffraction File,” in Crystallographic Databases, edited by F. H. Allen, G. Bergerhoff, and R. Sievers (Data Commission of the International Union of Crystallography, UK), pp. 159-174.   \nJenkins, R., Holomany, M., and Wong-Ng, W. (1987). “On the need for users of the powder diffraction file to update regularly,” Powder Diffr. 2, 84–87.   \nKuzel, R. and Danis, S. (2007). “Structural databases of inorganic materials,” Mater. Struct. Chem., Biol., Phys. Technol. 14, 89–96.   \nMessick, J. (2011). “The history of the international centre for diffraction data,” Powder Diffr. 27(1), 36–44.   \nSmith, D. and Jenkins, R. (1996). “The powder diffraction file: past, present, and future,” J. Res. Natl. Inst. Stand. Technol. 101(3), 259–271.   \nVillars, P. and Cenzual, K. (2018). Pearson’s Crystal Data: Crystal Structure Database for Inorganic Compound (ASM International and Material Phases Data System, Materials Park, OH).  ",
    "Target": null
  },
  {
    "id": "10.1038_s41586-019-1481-z",
    "DOI": "10.1038/s41586-019-1481-z",
    "DOI Link": "http://dx.doi.org/10.1038/s41586-019-1481-z",
    "Relative Dir Path": "mds/10.1038_s41586-019-1481-z",
    "Article Title": "Quantifying inactive lithium in lithium metal batteries",
    "Authors": "Fang, CC; Li, JX; Zhang, MH; Zhang, YH; Yang, F; Lee, JZ; Lee, MH; Alvarado, J; Schroeder, MA; Yang, YYC; Lu, BY; Williams, N; Ceja, M; Yang, L; Cai, M; Gu, J; Xu, K; Wang, XF; Meng, YS",
    "Source Title": "NATURE",
    "Abstract": "Lithium metal anodes offer high theoretical capacities (3,860 milliampere-hours per gram)(1), but rechargeable batteries built with such anodes suffer from dendrite growth and low Coulombic efficiency (the ratio of charge output to charge input), preventing their commercial adoption(2,3). The formation of inactive ('dead') lithium- which consists of both (electro)chemically formed Li+ compounds in the solid electrolyte interphase and electrically isolated unreacted metallic Li-0 (refs(4,5))-causes capacity loss and safety hazards. Quantitatively distinguishing between Li+ in components of the solid electrolyte interphase and unreacted metallic Li-0 has not been possible, owing to the lack of effective diagnostic tools. Optical microscopy(6), in situ environmental transmission electron microscopy(7,8), X-ray microtomography(9) and magnetic resonullce imaging(10) provide a morphological perspective with little chemical information. Nuclear magnetic resonullce(11), X-ray photoelectron spectroscopy(12) and cryogenic transmission electron microscopy(13,14) can distinguish between Li+ in the solid electrolyte interphase and metallic Li-0, but their detection ranges are limited to surfaces or local regions. Here we establish the analytical method of titration gas chromatography to quantify the contribution of unreacted metallic Li-0 to the total amount of inactive lithium. We identify the unreacted metallic Li-0, not the (electro)chemically formed Li+ in the solid electrolyte interphase, as the dominullt source of inactive lithium and capacity loss. By coupling the unreacted metallic Li-0 content to observations of its local microstructure and nullostructure by cryogenic electron microscopy (both scanning and transmission), we also establish the formation mechanism of inactive lithium in different types of electrolytes and determine the underlying cause of low Coulombic efficiency in plating and stripping (the charge and discharge processes, respectively, in a full cell) of lithium metal anodes. We propose strategies for making lithium plating and stripping more efficient so that lithium metal anodes can be used for next-generation high-energy batteries.",
    "Times Cited, WoS Core": 817,
    "Times Cited, All Databases": 903,
    "Publication Year": 2019,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000482219600042",
    "Markdown": "# Quantifying inactive lithium in lithium metal batteries  \n\nChengcheng Fang1,6, Jinxing $\\mathrm{Li}^{2,6}$ , Minghao Zhang2, Yihui Zhang1, Fan $\\mathrm{Yang^{3}}$ , Jungwoo Z. Lee2, Min-Han Lee1, Judith Alvarado1,4, Marshall A. Schroeder4, Yangyuchen Yang1, Bingyu $\\mathrm{Lu^{2}}$ , Nicholas Williams3, Miguel $\\mathrm{Ceja^{2}}$ , Li Yang5, Mei $\\mathrm{Cai}^{5}$ , Jing $\\mathrm{Gu}^{3}$ , Kang $\\mathrm{{Xu^{4}}}$ Xuefeng Wang2 & Ying Shirley Meng1,2\\*  \n\nLithium metal anodes offer high theoretical capacities (3,860 milliampere-hours per gram)1, but rechargeable batteries built with such anodes suffer from dendrite growth and low Coulombic efficiency (the ratio of charge output to charge input), preventing their commercial adoption2,3. The formation of inactive (‘dead’) lithium— which consists of both (electro)chemically formed ${{\\bf{L i}}^{+}}$ compounds in the solid electrolyte interphase and electrically isolated unreacted metallic ${\\bf{L i}}^{0}$ (refs 4,5)—causes capacity loss and safety hazards. Quantitatively distinguishing between ${{\\bf{L i}}^{+}}$ in components of the solid electrolyte interphase and unreacted metallic ${\\bf L i^{0}}$ has not been possible, owing to the lack of effective diagnostic tools. Optical microscopy6, in situ environmental transmission electron microscopy7,8, $\\mathbf{X}$ -ray microtomography9 and magnetic resonance imaging1 provide a morphological perspective with little chemical information. Nuclear magnetic resonance11, $\\mathbf{X}$ -ray photoelectron spectroscopy12 and cryogenic transmission electron microscopy13,14 can distinguish between ${{\\bf{L i}}^{+}}$ in the solid electrolyte interphase and metallic ${\\bf\\cal L i}^{0}$ , but their detection ranges are limited to surfaces or local regions. Here we establish the analytical method of titration gas chromatography to quantify the contribution of unreacted metallic ${\\bf{L i}}^{0}$ to the total amount of inactive lithium. We identify the unreacted metallic ${\\bf L i^{0}}$ , not the (electro) chemically formed ${{\\bf{L i}}^{+}}$ in the solid electrolyte interphase, as the dominant source of inactive lithium and capacity loss. By coupling the unreacted metallic ${\\bf L i}^{0}$ content to observations of its local microstructure and nanostructure by cryogenic electron microscopy (both scanning and transmission), we also establish the formation mechanism of inactive lithium in different types of electrolytes and determine the underlying cause of low Coulombic efficiency in plating and stripping (the charge and discharge processes, respectively, in a full cell) of lithium metal anodes. We propose strategies for making lithium plating and stripping more efficient so that lithium metal anodes can be used for next-generation highenergy batteries.  \n\nInactive lithium consists of diverse ${\\mathrm{Li}}^{+}$ compounds within the solid electrolyte interphase (SEI), such as LiF, $\\mathrm{Li}_{2}\\mathrm{CO}_{3}$ , $\\mathrm{Li}_{2}\\mathrm{O};$ , $\\mathrm{ROCO}_{2}\\mathrm{Li}$ (refs 15,16), and of unreacted metallic ${\\mathrm{Li}}^{0}$ which is isolated by the SEI from the electronic conductive pathway. It is generally assumed that low Coulombic efficiency mostly arises from continuous repair of SEI fractures, which consumes both the electrolyte and active Li metal17, although some researchers have suggested that unreacted metallic $\\mathrm{Li}^{0}$ may increase the tortuosity at the electrode/electrolyte interface and decrease the Coulombic efficiency in this way18,19. These assumptions and hypotheses are mostly based on observation, and the contribution to capacity loss from SEI formation has not been successfully quantified. Consequently, efforts to make Li metal a valid anode material may be misdirected. Differentiating and quantifying the $\\mathrm{Li^{+}}$ and $\\mathrm{Li}^{0}$ remaining on the electrode after stripping is key to understanding the mechanisms leading to capacity decay.  \n\nIn our work, the pivotal difference exploited between the SEI $\\mathrm{Li^{+}}$ compounds and metallic ${\\mathrm{Li}^{0}}$ is their chemical reactivity: only the metallic ${\\mathrm{Li}}^{0}$ reacts with protic solvents (such as $\\mathrm{H}_{2}\\mathrm{O}\\dot{}$ ) and generates hydrogen gas $\\left(\\operatorname{H}_{2}\\right)$ . The solubility and reactivity of known SEI species with $\\mathrm{H}_{2}\\mathrm{O}$ are listed in Extended Data Table 1. The possible presence of LiH (refs 20–22) in inactive Li might affect the quantification of metallic $\\mathrm{Li}^{0}$ because LiH also reacts with water and produces $\\mathrm{H}_{2}$ , so it was important to exclude this possibility in our results (see Methods for details). We combine $\\mathrm{H}_{2}\\mathrm{O}$ titration (the step in which all metallic ${\\mathrm{Li}}^{0}$ is reacted) and gas chromatography (the subsequent step to quantify the $\\mathrm{H}_{2}$ generated in the reaction) into a single analytical tool, hereafter referred to as titration gas chromatography (TGC; schematic process in Extended Data Fig. 1), which is able to quantify the content of metallic ${\\mathrm{Li}}^{0}$ based on the reaction  \n\n$$\n2\\mathrm{Li}+2\\mathrm{H}_{2}\\mathrm{O}\\longrightarrow2\\mathrm{LiOH}+\\mathrm{H}_{2}\\uparrow\n$$  \n\nWhen this is coupled with an advanced barrier ionization $\\mathrm{H}_{2}$ detector, the measurement of metallic $\\mathrm{Li}^{0}$ in the designed system is accurate to $10^{-7}\\mathrm{g}$ . The complete TGC methodologies are illustrated in Methods.  \n\nWe then applied TGC to correlate the origin of inactive Li with the Coulombic efficiency in $\\mathrm{Li}||\\mathrm{Cu}$ half-cells. As the Coulombic efficiency of Li metal varies greatly with electrolyte properties and current density, we compared two representative electrolytes, a high-concentration electrolyte (HCE; 4 M lithium bis(fluoro sulfonyl)imide (LiFSI) and $2\\mathrm{M}$ lithium bis(trifluoromethane sulfonyl)imide (LiTFSI) in 1,2-dimethoxyethane (DME))23 and a commercial carbonate electrolyte (CCE; $1\\mathrm{MLiPF}_{6}$ in ethylene carbonate/ethyl methyl carbonate (EC/EMC)), at three stripping rates $(0.5\\operatorname{mA}\\mathrm{cm}^{-2}$ , $2.5\\mathrm{m}\\dot{\\mathrm{A}}\\mathrm{cm}^{-2}$ and $5.0\\mathrm{mA}\\mathrm{cm}^{-2}$ ; all plating at $0.5\\mathrm{\\mA}\\mathrm{cm}^{-2}$ for 2 hours) . In addition, we examined six other electrolytes with a variety of salts, solvents and additives that frequently appear in the literature: 2 M LiFSI in dimethyl carbonate (DMC), $0.5{\\mathrm{~M~}}$ LiTFSI in DME/1,3-dioxolane (DOL), 1 M LiTFSI– DME/DOL, 1 M LiTFSI–DME/DOL plus $2\\%$ $\\mathrm{LiNO}_{3}$ , CCE plus ${{\\mathrm{Cs}}^{+}}$ , and CCE plus fluoroethylene carbonate (FEC). Figure 1a shows that their first-cycle average Coulombic efficiencies have a broad range of values, from $17.2\\%$ to $97.1\\%$ . Representative voltage profiles are shown in Extended Data Fig. 2a, b. The total amount of inactive Li is equal to the capacity loss between the plating and stripping processes, displaying a linear relationship with Coulombic efficiency (Fig. 1d). The content of metallic $\\mathrm{Li}^{0}$ was directly measured by the TGC method. Once the amount of unreacted metallic ${\\mathrm{Li}^{0}}$ has been determined, the SEI $\\mathrm{Li^{+}}$ amount can be calculated, as the total amount of inactive Li $\\left({\\mathrm{known}}\\right)=$ unreacted metallic $\\mathrm{Li}^{0}$ (measured) $+\\operatorname{SEI}\\operatorname{Li}^{+}$ .  \n\nThe average capacity utilization under all conditions was quantified by the TGC, as summarized in Fig. 1b. The reversible capacity increases with increasing Coulombic efficiency. Interestingly, the unreacted metallic $\\mathrm{Li}^{0}$ amount increases significantly with the decrease of Coulombic efficiency, whereas the SEI ${\\mathrm{Li}^{+}}$ amount remains at a constantly low level under all testing conditions. Further analysing the data, we found to our surprise that the amount of unreacted metallic $\\mathrm{Li}^{0}$ exhibits a linear relationship with loss of Coulombic efficiency (Fig. 1e), and this relationship is almost independent of the testing conditions. This implies that the Coulombic efficiency loss is governed by the formation of unreacted metallic $\\mathrm{Li}^{0}$ . Meanwhile, the SEI ${\\mathrm{Li}}^{+}$ amount (Fig. 1f), as deduced from the total inactive Li and unreacted metallic $\\mathrm{Li}^{0}$ , remains low and relatively constant under various testing conditions. The ratio of SEI ${\\mathrm{Li}^{+}}$ and unreacted metallic $\\mathrm{Li}^{0}(\\mathrm{Li}^{+}/\\mathrm{Li}^{0})$ (Fig. 1c) reveals that the unreacted metallic $\\mathrm{Li}^{0}$ dominates the content of inactive L $\\mathrm{(Li^{+}/L i^{0}<1)}$ as well as the capacity loss when Coulombic efficiency is under about $95\\%$ in the first cycle. Once the Coulombic efficiency is higher than about $95\\%$ , the amount of SEI ${\\mathrm{Li}}^{+}$ starts to dominate. The ratios of unreacted metallic ${\\mathrm{Li}^{0}}$ to total inactive Li, and of SEI ${\\mathrm{Li}^{+}}$ to total inactive Li, are further shown in Extended Data Fig. 2c, d.  \n\n![](images/51a26f823d628f99c95357579ddd88985a59a98be7c017bd7108ac166a1380c6.jpg)  \nFig. 1 | Quantitative differentiation of inactive Li by the TGC method. the standard deviation of the average values of Coulombic efficiency. a, Average first-cycle Coulombic efficiency of $\\mathrm{Li}||\\mathrm{Cu}$ cells under different b, Analysis of capacity usage (SEI $\\bar{\\mathrm{Li^{+}}}$ , unreacted metallic ${\\mathrm{Li}}^{0}$ and reversible testing conditions. Eight electrolytes (HCE, CCE, 2 M LiFSI–DMC, $0.5{\\mathrm{~M~}}$ Li) under different testing conditions by the TGC method. c, The ratio of LiTFSI–DME/DOL, 1 M LiTFSI–DME/DOL, 1 M LiTFSI–DME/DOL $+$ SEI ${\\mathrm{Li}^{+}}$ to unreacted metallic ${\\mathrm{Li}}^{0}$ based on TGC quantification results. The $2\\%\\mathrm{LiNO}_{3}$ , $\\mathrm{CCE}+\\mathrm{Cs}^{+}$ and $\\mathrm{CCE}+\\mathrm{FEC})$ and three stripping rates ( $\\mathrm{\\cdot}0.5\\mathrm{mA}$ blue line indicates exponential fitting. d, Total capacity loss as a function $c\\mathrm{m}^{-2}$ , $2.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ and $5.0\\mathrm{mA}\\mathrm{cm}^{-2}$ to $1\\mathrm{V},$ are used. In all electrolytes, of Coulombic efficiency. (For unit conversion between milliampere-hours Li was plated at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for 2 hours $\\mathrm{`1~mAh~cm^{-2}}.$ ). HCE and CCE and milligrams of Li, see Extended Data Fig. 2f.) e, Amount of unreacted were selected for the three stripping rates study. Li formed in the rest metallic $\\bar{\\mathrm{Li}^{0}}$ measured by the TGC method as a function of Coulombic electrolytes were stripped at $0.{\\overset{-}{5}}\\operatorname*{mA}\\operatorname{cm}^{-2}$ to $1\\mathrm{V}.$ At each condition, three efficiency. f, Calculated SEI ${\\mathrm{Li}^{+}}$ amount as a function of Coulombic to five cells were tested to obtain better statistics. The error bar represents efficiency.  \n\nIncreasing Li deposition capacity is reported to improve the firstcycle Coulombic efficiency24. To extend this method under different electrochemical conditions, we performed TGC tests on the CCE with Li plating capacities increased to $2\\mathrm{mAh}\\mathrm{cm}^{-2}$ , $3\\mathrm{mAh}\\mathrm{cm}^{-2}$ and $5\\mathrm{mAh}\\mathrm{cm}^{-2}$ . The TGC results (Extended Data Fig. 3a, b) show that the SEI $\\mathrm{Li^{+}}$ amount increases with the extended deposition capacity; the improvement in Coulombic efficiency with increased Li deposition capacity is due to the reduction in the amount of unreacted metallic $\\mathrm{Li}^{\\hat{0}}$ . At $\\dot{3}\\mathrm{mAh}\\mathrm{cm}^{-2}$ , the Coulombic efficiency reaches $95.21\\%$ , while the ratio of SEI ${\\mathrm{Li}}^{+}$ to unreacted metallic ${\\mathrm{Li}^{0}}$ is measured to be 1.43, consistent with the above results.  \n\nBesides the first cycle, we also investigated the ratio of SEI $\\mathrm{Li^{+}}$ to unreacted metallic $\\mathrm{Li}^{\\mathrm{\\dot{0}}}$ after multiple cycles (two, five and ten) until the Coulombic efficiency is stabilized around $90\\%$ in CCE. As shown in Extended Data Fig. 3c, d, the $\\mathrm{Li^{+}/L i^{0}}$ ratio after one, two, five and ten cycles remains 0.27, 0.30, 0.27 and 0.34, respectively, indicating that the main capacity loss is from the unreacted metallic $\\mathrm{Li}^{0}$ . The TGC results also reveal that the unreacted metallic ${\\mathrm{Li}^{0}}$ amount accumulates during extended cycles, indicating continuous consumption of active Li in Li metal batteries. These experiments, with varying electrolytes, additives, deposition capacities and cycles, all validate the TGC method as a reliable tool in studying the inactive Li.  \n\nFurther examining the SEI components in HCE and CCE by X-ray photoelectron spectroscopy (XPS), we found that stripping rates have negligible impact on the relative contributions from SEI components (see Extended Data Fig. 4a, b). The TGC quantification analysis and XPS results establish that the contribution from the SEI ${\\mathrm{Li}}^{+}$ to the global content of inactive Li is not as large as commonly believed from previous studies25–27.  \n\nTo elucidate the formation mechanism of inactive Li, we use cryogenic focused ion beam–scanning electron microscopy (cryo-FIB–SEM) to explore the microstructures of inactive Li. HCE and CCE samples under different stripping rates are chosen for the morphological study. Cryogenic protection is critical here, because the highly reactive Li metal is not only sensitive to the electron beam but is also apt to react with the incident Ga ion beam to form a $\\operatorname{Li}_{x}\\operatorname{Ga}_{y}$ alloy at room temperature28. Completely different morphologies are generated by variations in stripping rates, even though all samples start from the same chunky Li deposits after plating at $\\bar{0}.5\\operatorname{mA}{\\mathrm{cm}^{-2}}$ (Extended Data Fig. 5a–c). As the stripping rate increases, the morphology of inactive Li in HCE evolves from uniform sheets to local clusters (Fig. 2a–c) with a thickness increased from $500\\mathrm{nm}$ to $2\\upmu\\mathrm{m}$ (Fig. 2d–f). For the CCE, the individual whisker-like Li deposits (Extended Data Fig. 5d–f) become thinner after stripping (Fig. $2\\mathrm{g-i})$ , but the whole inactive Li layer becomes thicker in cross-section with the increased stripping rates (Fig. 2j–l), corresponding to the increased loss of Coulombic efficiency at high rates. It is worth noting that these residues exhibit poor connection to the current collector, indicating the loss of electronic conductive pathways.  \n\n![](images/0530e89acc4694bdb27071901333bcbfc7326328d31838749fc6ac1290d0f80a.jpg)  \nig. 2 | Microstructures of inactive Li generated in HCE and CCE sections obtained by cryo-FIB. Each column represents a different maged by cryo-FIB–SEM. a–f, Results for HCE. $\\mathbf{g}\\mathbf{-}\\mathbf{l}$ , Results for CCE. stripping rate: 0.5 mA $\\dot{\\mathrm{cm}}^{-2}$ (a, d, g, j); 2.5 $\\mathrm{m}\\mathrm{A}\\mathrm{\\dot{c}m}^{-2}\\left(\\mathbf{b},\\mathbf{e},\\mathbf{h},\\mathbf{k}\\right)$ ; or a–c, $\\mathbf{g-i},$ Top view of the inactive Li at $52^{\\circ}$ tilted stage. d–f, j–l, Cross- $5.0\\mathrm{\\mA}\\mathrm{cm}^{-2}$ (c, f, i, l).  \n\nWe further used cryogenic transmission electron microscopy (cryoTEM) to investigate the nanostructure of the inactive Li in HCE and CCE after stripping at $0.5\\operatorname{mA}{\\mathrm{cm}^{-2}}$ . Sheet-like inactive Li appears in the HCE sample (Fig. 3a), whereas inactive Li in the CCE retains a whisker-like morphology (Fig. 3e). Based on the (110) lattice plane distance of body-centered cubic Li, the region that contains crystalline metallic $\\mathrm{Li}^{\\dot{0}}$ is highlighted in green in the high-resolution TEM (HRTEM) images for both electrolytes (Fig. 3b, f). Compared with the inactive Li obtained from CCE, a much smaller area of metallic $\\mathrm{Li}^{0}$ component is observed in HCE. This indicates that most of the deposited metallic ${\\mathrm{Li}}^{0}$ in HCE has been successfully stripped, corresponding to the high Coulombic efficiency. Whisker-like unreacted metallic $\\mathrm{Li}^{\\smash{\\breve{0}}}$ up to about $80\\mathrm{nm}$ in length remains in the CCE sample and is well isolated by the surrounding SEI. The SEI components were determined by matching the lattice spacing in HRTEM images with their fast Fourier transform (FFT) patterns (Fig. 3c, g). The SEI components from more than 50 different sample positions have been analysed and are provided in Extended Data Fig. 5g, h for better statistics. Consistent with the XPS results (Extended Data Fig. 4), $\\mathrm{Li}_{2}\\mathrm{CO}_{3}$ and $\\mathrm{Li}_{2}\\mathrm{O}$ constitute the majority of the SEI, which also contains LiF as well as other amorphous organic species for both electrolytes. The above observations from cryo-TEM are summarized in the schematic plot (Fig. 3d, h), which shows the form of inactive Li with two different morphologies at the nanoscale.  \n\nCorrelating the inactive metallic ${\\mathrm{Li}^{0}}$ content with the micro- and nanostructures of inactive Li formed under different conditions, we propose mechanisms for the formation of inactive Li and for the stripping of Li metal. Two processes are involved in the stripping. The first of these is ${\\mathrm{Li}}^{+}$ dissolution: under the electric field, metallic ${\\mathrm{Li}}^{0}$ is oxidized to $\\mathrm{Li^{+}}$ , which diffuses through the SEI layers and dissolves into the electrolyte. The second is SEI collapse: when the Li is removed, the SEI simultaneously shrinks and collapses towards the current collector. During these two dynamic processes, we emphasize an ignored but crucial aspect, the structural connection, which is defined as the capability of the active Li to maintain an electronic conductive network. The cryo-FIB–SEM and cryo-TEM images show that inactive $\\mathrm{Li}^{0}$ was either disconnected from the current collector or encapsulated by the insulating SEI, leading to the loss of structural connection. Obviously, for a Li deposit with whisker morphology and large tortuosity (Fig. 4a, taking the Li deposits formed in CCE as an example), the undesired microstructure can easily produce both ways of losing structural connection, leaving more unreacted metallic $\\mathrm{Li}^{\\dot{0}}$ during the stripping process. In contrast, dense Li with chunky morphology and low tortuosity (Fig. 4b, from HCE) has bulk integrity to maintain its structural connection and intimate contact with the current collector, resulting in a reduced presence of unreacted metallic $\\mathrm{Li}^{0}$ and high Coulombic efficiency. This is further evidenced by an advanced electrolyte with columnar microstructure and minimum tortuosity, which can deliver a first-cycle Coulombic efficiency as high as $96.2\\%$ (Extended Data Fig. 6a, b).  \n\n![](images/5511ad2f6b4e24838c4581f481b81f3cd846ac67e47ebe874bb5412b2fc95af9.jpg)  \nFig. 3 | Nanostructures of inactive Li generated in HCE and CCE imaged by cryo-TEM. a–c, Results for HCE. $\\mathbf{e}{\\mathbf{-}}\\mathbf{g}$ , Results for CCE. a, e, Inactive Li morphology at low magnifications for both electrolytes. b, f, HRTEM shows that a different amount of metallic $\\mathrm{Li}^{0}$ is wrapped by SEI in the two types of electrolyte. The highlighted metallic $\\mathrm{Li}^{0}$ region in green is identified through an inverse FFT process by applying mask filter on the  \n\n![](images/3d4fed0b7b8c5c8cfdf234630074bf6ebd0ea9da702ef755963d1080e73a405f.jpg)  \norigin FFT patterns. c, g, FFT patterns of corresponding HRTEM indicate the SEI component, which contains crystalline $\\mathrm{Li}_{2}\\mathrm{CO}_{3}$ , $\\mathrm{Li}_{2}\\mathrm O$ and LiF. ${\\bf d},{\\bf h}$ , Schematic of inactive Li nanostructure in HCE (d) and CCE (h). A small area of metallic $\\mathrm{Li}^{0}$ is embedded in a sheet-like SEI layer for HCE, whereas a large bulk of metallic $\\mathrm{Li}^{0}$ is isolated in a whisker-like SEI layer in CCE.   \nsmall amounts of metallic $\\mathrm{Li}^{0}$ are stuck in tortuous SEI edges. c, An ideal Li deposit should have a columnar microstructure with a large granular size, minimum tortuosity and homogeneous distribution of SEI components, facilitating a complete dissolution of metallic $\\mathrm{Li}^{0}$ . d, A general correlation of morphology of Li deposits, Coulombic efficiency and the ratio of SEI ${\\mathrm{Li}^{+}}$ to unreacted metallic $\\mathrm{Li}^{0}$ .  \n\nFig. 4 | Schematic of inactive Li formation mechanism in different electrolytes, based on TGC quantification, cryo-FIB–SEM and cryo-TEM observation. a, Li deposits with whisker morphology and high tortuosity are more likely to lose electronic connection and maintain poor structural connection, leaving large amounts of unreacted metallic $\\mathrm{Li}^{\\hat{0}}$ trapped in SEI. b, Li deposits with large granular size and less tortuosity tend to maintain a good structural electronic connection, in which only  \n\nBased on the above observations and discussion, we propose the following strategies to improve Coulombic efficiency. An ideal architecture of deposited Li would promote structural connection and mitigate inactive Li formation, especially the formation of unreacted metallic $\\mathrm{Li}^{0}$ . The ideal architecture includes the following. (1) The Li deposits should retain a columnar microstructure with a large granular size and minimum tortuosity, to minimize the unreacted metallic ${\\mathrm{Li}}^{0}$ residue (Fig. 4c, d). (2) The SEI should be both chemically and spatially homogeneous so that uniform ${\\mathrm{Li}}^{+}$ dissolution occurs. It should be mechanically elastic enough to accommodate the volume change. The SEI could be refilled during extended cycles, as schematized in Extended Data Fig. 6g. Using advanced electrolytes and artificial SEI may help to meet these requirements, while three-dimensional (3D) hosts that maintain electronic pathway and low tortuosity can contribute to constructing a durable structural connection and guiding the Li plating and stripping. To test this hypothesis, we compared 2D Cu foil and 3D Cu foam as the current collectors (Extended Data Fig. 6c, d). The initial Coulombic efficiency of 2D Cu foil and 3D Cu foam is $82\\%$ and $90\\%$ , respectively (Extended Data Fig. 6e). The increased Coulombic efficiency in the latter is attributed to the reduced amount of unreacted metallic ${\\dot{\\mathrm{Li}}}^{0}$ (Extended Data Fig. 6f), despite the fact that the amount of SEI ${\\mathrm{Li}^{+}}$ increases from $21.5\\%$ to $62.7\\%$ owing to the higher surface area of 3D Cu foam. Therefore, although the 3D current collector helps in maintaining a good electronic conductive network, it is necessary to control its surface properties to minimize SEI formation. The structural connection can be further enforced by applying external pressure. Slight stacking pressure can improve cycling performance29,30. In our proposed model, we believe that pressure promotes structural collapse towards the current collector, thus leading to better structural connection which mitigates the generation of unreacted metallic $\\mathrm{Li}^{0}$ . We found the critical pressure in maintaining good structural connection to be as low as about 5 psi, which should not damage any SEI (Extended Data Fig. 6h). A fast stripping rate could accelerate the ${\\mathrm{Li}}^{+}$ dissolution but may destroy the structural connection, because Li at the tip of column or whiskers could fail to keep pace with the rapid dynamic. Overall, the tools established here can be universally extended to examine various battery chemistries under different conditions, with the aim of developing a better battery that is energy-dense and safe.  \n\n# Online content  \n\nAny methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-019-1481-z.  \n\nReceived: 21 January 2019; Accepted: 12 June 2019;   \nPublished online 21 August 2019.   \n1.\t Tarascon, J. M. & Armand, M. Issues and challenges facing rechargeable lithium batteries. Nature 414, 359–367 (2001).   \n2.\t Lin, D., Liu, Y. & Cui, Y. Reviving the lithium metal anode for high-energy batteries. Nat. Nanotechnol. 12, 194–206 (2017).   \n3.\t Xu, W. et al. Lithium metal anodes for rechargeable batteries. 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ChemSusChem 11, 3821–3828 (2018).  \n\nPublisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\n# Methods  \n\nElectrochemical testing. Coulombic efficiency was measured in Li||Cu coin cells: Li metal ( $\\mathrm{~\\chi~mm~}$ thick, 0.5 inch in diameter), two pieces of separators (Celgard) and $\\mathrm{cu}$ foil (0.5 inch in diameter) were sandwiched in CR2032 coin cells with a spacer and a spring, and crimped inside an Ar-filled glovebox. A $50\\upmu\\mathrm{l}$ amount of the electrolyte was added in each cell. HCE consists of $^{4\\mathrm{M}}$ LiFSI (battery grade; Oakwook Products, Inc.) $+2\\mathbf{M}$ LiTFSI (battery grade; Solvay) in DME (anhydrous, $>99.5\\%$ ; BASF). CCE consists of $1\\mathrm{MLiPF}_{6}$ (battery grade, BASF) in EC/EMC (battery grade, BASF) (3:7 by weight) with $2\\mathrm{wt\\%}$ of vinylene carbonate (battery grade, BASF). $\\mathrm{CCE}+\\mathrm{Cs}^{+}$ contains $50\\mathrm{mM}$ of ${\\mathrm{CsPF}}_{6}$ (Synquest Laboratory). $\\mathrm{CCE}+\\mathrm{FEC}$ contains $10\\mathrm{wt\\%}$ of FEC (anhydrous, $599\\%$ , Sigma-Aldrich). DOL (anhydrous, ${>}99.5\\%$ ) and $\\mathrm{LiNO}_{3}$ were purchased from Sigma-Aldrich. Cells for TGC are plating at $0.5\\operatorname{mA}{\\mathrm{cm}^{-2}}$ for $1\\mathrm{\\mAh}\\mathrm{cm}^{-2}$ and stripping at various rates $(0.5\\mathrm{mA}\\mathrm{cm}^{-2}$ , $\\bar{2}.5\\operatorname{mA}{\\mathrm{cm}^{-2}}$ and $5\\operatorname{mA}\\mathrm{cm}^{-2}$ ) to ${}^{1\\mathrm{V},}$ unless otherwise specified.  \n\nTitration gas chromatography. Extended Data Fig. 1 demonstrates the typical processes of the TGC method for the inactive Li quantification, including the following six main steps. (1) After plating and stripping, the Li||Cu coin cell was disassembled in an Ar-filled glovebox. (2) While still in the glovebox, both the Cu foil and separator on the Cu foil side were harvested without washing and sealed in a container with an inside pressure of 1 atm by a rubber septum which is stable against water. (3) After transferring the sample container out of the glovebox, we injected $0.5\\mathrm{ml}$ of $_\\mathrm{H_{2}O}$ into the container to react with the inactive Li completely. (4) A gas-tight syringe was used to transfer $30\\upmu\\mathrm{l}$ of the resultant gas from the container into the gas chromatography (GC) system. (5) The amount of $\\mathrm{\\ddot{H}}_{2}$ was measured by the GC. (6) The content of the metallic $\\mathrm{Li}^{0}$ was determined by converting the corresponding $\\mathrm{H}_{2}$ amount according to a pre-established standard calibration curve (Extended Data Fig. 7b). All the processes minimize the potential damage and contamination during sample transfer, to obtain reliable results.  \n\nIn the GC column, the stationary phase has a different affinity with different species, so that gas species can be differentiated by retention time. Extended Data Fig. 8a shows the GC chromatograms of background gases from a well-sealed empty container in the Ar-filled glovebox. The peaks at 1.45 min, 2.42 min and $3.21\\mathrm{min}$ represent Ar, $\\Nu_{2}$ and $\\mathrm{CH}_{4},$ , respectively. When an $\\mathrm{H}_{2}\\mathrm{O}$ titration process is applied to a small piece of pure Li metal, $\\mathrm{H}_{2}$ will be generated. The $\\mathrm{H}_{2}$ characteristic peak appears at $1.05\\mathrm{min}$ , as shown in Extended Data Fig. 8b.  \n\nSource of $\\cdot_{N_{2}}$ in the gas chromatograms. Note that ${\\bf N}_{2}$ detected from the GC comes from the gas sampling process, instead of existing in the reaction container. As schematized in the TGC process in Extended Data Fig. 1, the inactive Li samples were loaded and sealed into the reaction container in an Ar-filled and $\\Nu_{2}$ -free glovebox. After the samples were taken out of the glovebox, $0.5{\\mathrm{ml}}$ of pure $\\mathrm{H}_{2}\\mathrm{O}$ was injected into the container and reacted with the inactive Li sample. This is an air-free process. A gas-tight syringe was then used to take the gas sample for GC injection and measurement. There is a small amount of air left in the needle space of the gas-tight syringe. Moreover, in the GC injection process, when the needle breaks the septum, a tiny amount of air might be introduced into the GC column. Even though the air peak is inevitable for the injection gas sampling method, the inactive Li samples have indeed all been reacted in advance and were never exposed to $\\Nu_{2}$ .  \n\nWe carried out the following tests to demonstrate that the $\\Nu_{2}$ comes from the air during sampling and that the presence of air has negligible impact on the inactive Li $\\left(\\mathrm{H}_{2}\\right)$ quantification. First, we measured the gas in the blank container without any inactive Li, which has been well sealed in the Ar-filled glovebox. If the $\\Nu_{2}$ comes from the reaction container, the intensity of the $\\Nu_{2}$ peaks will vary proportionally to the injected gas amount into the GC. We took different amounts of gas sample from the blank container and performed GC measurement. As shown in Extended Data Fig. 8c, the $\\mathrm{N}_{2}$ peak intensities remain almost identical for injection amounts varying from 5 to $30\\upmu\\mathrm{l}$ . The same result was obtained when $\\mathrm{H}_{2}\\mathrm{O}$ titration was performed on inactive Li. The measured $\\mathrm{H}_{2}$ content increase as a function of injected gas amount while the $\\Nu_{2}$ content remains almost constant (Extended Data Fig. 8d). Noting that the reaction container is sealed in the Ar-filled glovebox, the Ar peak is saturated even if only ${5\\upmu\\mathrm{l}}$ of sample gas is injected and remains unchanged in all measurements. The invariability of $\\mathrm{N}_{2}$ peaks is not because of saturation in the previous measurements, since the intensity of the $\\Nu_{2}$ peak increases significantly after purposely injecting $10\\upmu\\mathrm{l}$ of air (Extended Data Fig. 8e). Therefore, we have confirmed that the $\\Nu_{2}$ comes from the injection sampling process and it will not have any chemical reaction with the inactive Li samples. Moreover, the $\\mathrm{H}_{2}$ quantification is not influenced by the injection sampling process.  \n\nPossible existence of LiH. Besides the SEI species listed in Extended Data Table 1, there have been mixed reports regarding the existence of LiH in Li metal electrodes20–22. There are two possible scenarios in which LiH may exist in the Li metal electrodes: (1) LiH may exist within the SEI as an electrochemical reduction product at excessively negative potentials21,22; (2) LiH may largely exist in the bulk electrode as mossy dendrites20. To examine the possible influence from LiH in SEI, we repeatedly polarized the current collectors above $0\\mathrm{V},$ so that only SEI forms without metallic ${\\mathrm{Li}^{0}}$ deposition31. For the electrolytes investigated in this work, after such cyclic polarizations between $0\\mathrm{V}$ and $1\\mathrm{V}$ for ten cycles, the TGC detected no $\\mathrm{H}_{2}$ from all SEI–water reactions (Extended Data Fig. 9a-h), indicating that LiH does not exist in the SEIs. To examine the possible influence from LiH in bulk inactive Li, we changed the titration solution from $\\mathrm{H}_{2}\\mathrm{O}$ to $\\mathrm{D}_{2}\\mathrm{O},$ which can distinguish LiH and metallic ${\\mathrm{Li}}^{0}$ by producing HD and $\\mathrm{D}_{2}$ , respectively, based on the following reactions: (1) $\\mathrm{LiH}+\\mathrm{D}_{2}\\mathrm{O}=\\mathrm{LiOD}+\\mathrm{HD}\\uparrow$ ; (2) $2\\mathrm{Li}+2\\mathrm{D}_{2}\\mathrm{O}=2\\mathrm{LiOD}$ $+\\mathrm{D}_{2}\\uparrow$ . Differentiating HD and $\\mathrm{D}_{2}$ was then achieved based on partial pressure analysis by residual gas analyser $(\\mathrm{RGA})^{32}$ . From the RGA results (Extended Data Fig. 9i–n), we confirmed that LiH does not exist in the bulk inactive Li generated by the electrolyte systems of low Coulombic efficiency. The exclusion of LiH from either SEI or bulk inactive Li confirms that the conclusions drawn from the TGC analysis should be reliable and free of interference from possible LiH species.  \n\nCalibration. The $\\mathrm{H}_{2}$ concentration was calibrated and measured using a Shimadzu GC-2010 Plus Tracera equipped with a barrier ionization discharge (BID) detector. Helium $(99.9999\\%$ ) was used as the carrier gas. Split temperature was kept at $200^{\\circ}\\mathrm{C}$ with a split ratio of 2.5 (split vent flow: $2\\mathrm{0.58\\ml\\min^{-1}}$ , column gas flow: $8.22\\mathrm{ml}\\mathrm{min}^{-1}$ , purge flow: $0.5\\mathrm{{\\dot{ml}}\\mathrm{{min}^{-1}}}$ ). Column temperature (RT-Msieve 5A, $0.53\\mathrm{mm}\\dot{}$ was kept at $40^{\\circ}\\mathrm{C}$ . A BID detector was kept at $235^{\\circ}\\mathrm{C},$ and BID detector gas flow rate was $50\\mathrm{ml}\\mathrm{min}^{-1}$ . All calibration and sample gases were immediately collected via a $50\\upmu\\mathrm{l}$ Gastight Hamilton syringe before injection. For calibration of $\\mathrm{H}_{2}$ concentration, 1,500 p.p.m. of $\\mathrm{H}_{2}$ gas was produced by reacting highpurity sodium with DI water in a septum sealed glass vial. We collected ${5\\upmu\\mathrm{l}}$ $10\\upmu\\mathrm{l},$ $15\\upmu\\mathrm{l}$ $20\\upmu\\updownarrow,25\\upmu\\mathrm{l}$ and $30\\upmu\\mathrm{l}$ of the $\\mathrm{H}_{2}$ gas produced, corresponding to 250 p.p.m., 500 p.p.m., 750 p.p.m., 1,000 p.p.m., 1,250 p.p.m. and $1,500{\\mathrm{p.p.m}}.$ , respectively, and injected them into the GC. The calibration curve was plotted and fitted with $\\mathrm{H}_{2}$ concentration versus $\\mathrm{H}_{2}$ peak area as measured by the GC. The as-established $\\mathrm{H}_{2}$ calibration curve $\\mathrm{(H}_{2,\\mathrm{ppm}}$ versus detected $\\mathrm{H}_{2}$ area) and equation are shown in Extended Data Fig. 7a. To acquire the exact number of $\\mathrm{H}_{2}$ molecules within the container, the $\\mathrm{H}_{2}$ concentration calibration curve was converted to a calibration curve in terms of the mole number of $\\mathrm{H}_{2}$ as a function of detected area based on the following two conditions: (1) $1\\ {\\mathrm{p.p.m.}}=4.08\\times10^{-8}\\ {\\mathrm{mmol}}\\ {\\mathrm{ml^{-1}}}$ (1 atm, 298 K); (2) container volume $(30\\pm0.5\\mathrm{ml})$ ).  \n\nThe mole number of $\\mathrm{H}_{2}$ calibration curve established a direct relationship between $\\mathrm{H}_{2}$ area reported by the GC software and the number of $\\mathrm{\\ddot{H}}_{2}$ molecules in the fixed TGC set-up, making the following inactive Li measurement independent of slight pressure change. Based on the chemical reaction $2\\mathrm{Li}+2\\mathrm{H}_{2}\\mathrm{O}\\longrightarrow2\\mathrm{LiOH}+$ $\\mathrm{H}_{2}\\uparrow$ , the standard calibration curve and the equation for Li metal mass $(m_{\\mathrm{Li}})$ as a function of the detected $\\mathrm{H}_{2}$ area are obtained and shown in Extended Data Fig. 7b. Validation of the GC measurement. (1) The $\\mathrm{H}_{2}$ concentration in p.p.m. as a function of GC detected $\\mathrm{H}_{2}$ area (Extended Data Fig. 7a) was verified by using the certified GASCO $\\mathrm{H}_{2}$ calibration test gas. (2) We then used commercial Li metal of known mass to verify the relationship established for $m_{\\mathrm{Li}}$ versus detected $\\mathrm{H}_{2}$ area (Extended Data Fig. 7b). We carefully weighed nine pieces of commercial Li metal with mass ranging from $0.54\\mathrm{mg}$ to $1.53\\mathrm{mg}$ in the Ar-filled glovebox with a fivedigit balance $(\\bar{1}0^{-5}\\mathbf{g})$ and then performed the TGC measurement. The detected $\\mathrm{H}_{2}$ area as a function of the Li metal mass from the nine pieces of Li metal is shown in Extended Data Fig. 7c. The result shows that the mass of Li metal is linearly related $(R^{2}=99.8\\%)$ ) to the detected $\\mathrm{H}_{2}$ area, indicating the validity of the TGC system for quantifying metallic $\\mathrm{Li}^{0}$ . In reverse, we calculated the Li metal mass from the detected $\\mathrm{H}_{2}$ area using the relationship of $m_{\\mathrm{Li}}$ versus detected $\\mathrm{H}_{2}$ area. The TGC-measured and balance-measured Li metal masses are compared in Extended Data Fig. 7d. The exact values of TGC-measured and balance-measured Li metal masses are listed in Extended Data Fig. 7e. The negligible differences between the TGC quantification and balance measurement indicate the validity and accuracy of the TGC method. The significant digit of the balance is $0.01\\mathrm{img}(10^{-5}\\mathrm{g})$ , as marked in red in Extended Data Fig. 7e, whereas that of the TGC is $0.0001\\mathrm{mg}$ $(0.1\\upmu\\mathbf{g},10^{-7}\\mathbf{g})$ , which has been demonstrated in the limit of detection/limit of quantification (LOD/LOQ) analysis. Thus, the minimal difference between the two quantification methods is mainly ascribed to the inaccuracy of the balance, which has a precision two orders of magnitude smaller than the TGC.  \n\nLOD/LOQ analysis. The concentration of hydrogen in the air is $0.000053\\%$ . To get the LOD/LOQ values, $30\\upmu\\mathrm{l}$ of the air sample was injected into GC by the same gas-tight syringe as used for the hydrogen measurement and repeated for a total of 10 times. The results are listed in Extended Data Fig. 7f. Based on the definition of LOD/LOQ, the calculated LOD/LOQ from the table is $16.44\\mathrm{p.p.m}$ and 49.81 p.p.m., respectively, corresponding to $0.28\\upmu\\mathrm{g}$ and $0.84\\upmu\\mathrm{g}$ of metallic ${\\mathrm{Li}}^{0}$ in the designed TGC system.  \n\nInactive Li sample measurement. After stripping under various conditions, $\\mathrm{Li}||\\mathrm{Cu}$ cells were disassembled inside an Ar-filled glovebox $(\\mathrm{H}_{2}\\mathrm{O}<0.5\\:\\mathrm{p.p.m.}$ ). The Cu foil and separator near the Cu foil side with inactive Li residue on top were placed into a $30\\mathrm{ml}$ container without washing. The container was sealed by a rubber septum and further capped by a stainless-steel/copper ring for safety and to minimize the deformation of the rubber septum when gas was generated later. The internal pressure of the sealed container was adjusted to 1 atm by connecting the container and glovebox environment (0 mbar) with an open-ended syringe needle. After transferring the sealed container out of the glovebox, $0.5\\mathrm{ml}$ of water was injected into the container, allowing complete reaction of inactive Li residue with water. An excess amount of $_\\mathrm{H}_{2}\\mathrm{O}$ was added to react with all the inactive metallic $\\mathrm{Li}^{0}$ , leading to complete conversion to the $\\mathrm{H}_{2}$ products. The Cu foil became shiny and the separator normally became clean when reactions finished, indicating a complete reaction of the inactive Li with $\\mathrm{H}_{2}\\mathrm{O}$ The as-generated gases were then well dispersed and mixed by shaking the container to prevent $\\mathrm{H}_{2}$ accumulation on top of the container. Then a gas-tight syringe was used to quickly take $30\\upmu\\mathrm{l}$ of the well-mixed gas and to inject it into the GC for $\\mathrm{H}_{2}$ measurement. The GC-measured $\\mathrm{H}_{2}$ areas as a function of cell Coulombic efficiency are shown in Extended Data Fig. 2e. The conversion between mAh and mg of Li is shown in Extended Data Fig. 2f.  \n\nSafety considerations. Li is electrochemically inactive, but chemically hyperactive due to the high surface areas which may lead to serious potential safety hazards33. Inactive Li quantification using the TGC method should be done carefully, taking the following aspects into consideration:  \n\n(1) The proper amount of inactive Li for TGC measurement. The minimum amount of inactive metallic Li that has been measured is as low as $1\\upmu\\mathrm{g}$ $({\\sim}0.004\\ \\mathrm{mAh})$ ). The maximum amount measured in the present work is ${\\sim}1.6\\mathrm{mg}.$ corresponding to ${\\sim}6\\mathrm{mAh}$ . It is generally preferred to reduce the amount of inactive Li sample, as the GC with an advanced $\\mathrm{H}_{2}$ detector can be very sensitive (1 p.p.m.). The greater the amount of $\\mathrm{H}_{2}$ generated, the more dangerous it could be.  \n\n(2) The reaction container must be completely sealed inside the Ar-filled glovebox before it is taken out, to avoid $\\mathrm{O}_{2}$ and moisture entering the container. Moisture in air influences the measurement accuracy, and $\\mathrm{O}_{2}$ may lead to an explosion when a large amount of water reacts with inactive Li.  \n\n(3) Stainless steel/copper rings should be used to minimize the deformation of the rubber septum when $\\mathrm{H}_{2}$ is generated inside the reaction container after water titration, and to prevent potential explosion due to the increased internal pressure.  \n\n(4) Waste gas in the container after TGC measurement should be disposed of in a fume hood to avoid regional $\\mathrm{H}_{2}$ accumulation which can lead to an explosion (explosive limits of $\\mathrm{H}_{2}$ in air range from about $18\\%$ to $60\\%$ ; the flammable limits are $4\\mathrm{-}75\\%$ ).  \n\nCryogenic focused ion beam scanning electron microscopy. The inactive Li samples on Cu foil were disassembled and washed with anhydrous DME (for HCE) or DMC (for CCE) in the Ar-filled glovebox. The samples were mounted on the SEM sample holder in the glovebox, then transferred to a FEI Helios NanoLab Dualbeam. Platinum was deposited for surface protection from the ion beam: $100\\mathrm{nm}$ of Pt was deposited using the electron beam at $5\\mathrm{kV},0.8\\mathrm{nA}$ ; ${300}\\mathrm{nm}$ of Pt was deposited using the ion beam at $30\\mathrm{kV},0.1\\mathrm{nA}$ . The stage was cooled with liquid nitrogen to $-180^{\\circ}\\mathrm{C}$ or below. Sample cross-sections were exposed using a 1 nA ion beam current and 100 ns dwell time, and cleaned twice at $0.5\\mathrm{nA}$ and $0.1\\mathrm{nA}$ , respectively. SEM images were taken with an Everhart-Thornley Detector (ETD) at $5\\mathrm{kV.}$  \n\nCryogenic transmission electron microscopy. The cryo-TEM sample for HCE was directly deposited and stripped on a lacey carbon grid in the Li||Cu half-cell. The sample for CCE was prepared by peeling the inactive Li from Cu foil cycled in the half-cell, and then depositing it onto the same type of TEM grid. Both half-cells were plated at $0.5\\mathrm{mAcm}^{-2}$ for 2 hours and then stripped to 1 V at the same rate. Both TEM samples were slightly rinsed with DME/DMC in the Ar-filled glovebox to remove trace Li salt. Once dry, the samples were sealed in airtight bags and plunged directly into a bath of liquid nitrogen. The airtight bags were then cut and the TEM grids were immediately immersed in liquid nitrogen. Then the grids were mounted onto a TEM cryo-holder (Gatan) via a cryo-transfer station. In short, the whole TEM sample preparation and transfer process prevents any contact of Li metal with the air at room temperature. TEM characterizations were carried out on JEOL JEM-ARM300CF at $300\\mathrm{kV}$ and JEM-2100F at $200\\mathrm{kV}.$ HRTEM images were taken at a magnification of $\\times500{,}000$ with a Gatan OneView Camera (full $4,000\\times4,000$ pixel resolution) when the temperature of samples reached about $100\\mathrm{K}.$ . The FFT pattern and inverse FFT image after mask filtering were analysed with DigitalMicrograph software.  \n\n$\\mathbf{X}$ -ray photoelectron spectroscopy. After a plating/stripping process, cells were disassembled in an Ar-filled glovebox with $\\mathrm{H}_{2}\\mathrm{O}<0.5$ p.p.m. Cu foils with inactive Li residue were gently and thoroughly rinsed by DME (for HCE) and DMC (for CCE) to remove residual surface Li salts. The rinsed electrodes were sealed in an airtight stainless-steel container and transferred into the glovebox connected to the XPS chamber. XPS was performed with a Kratos AXIS Supra, with the Al anode source operated at $15\\mathrm{kV.}$ The chamber pressure was $<10^{-8}$ torr during all measurements. All XPS measurements were collected with a spot size $300\\upmu\\mathrm{m}$ by $700\\upmu\\mathrm{m}$ with a charger neutralizer during acquisition. Survey scans were collected with a $1.0\\mathrm{eV}$ step size, followed by high-resolution scans collected with a step size of $0.05\\mathrm{eV.}$ Fittings of the XPS spectra were performed with CasaXPS software (version 2.3.15, Casa Software Ltd) to estimate the atomic compositions and chemical species. All species (Li 1s, F 1s, O 1s and C 1s) were fitted using a Shirley type background. High-resolution spectra were calibrated using the $\\mathrm{~C~}1s$ peak at $284.6\\mathrm{eV}.$ The peak positions and areas were optimized by a Levenberg– Marquardt least-squares algorithm using $70\\%$ Gaussian and $30\\%$ Lorentzian line shapes. Quantification was based on relative sensitivity factors. The curve fit for the core peaks was obtained using a minimum number of components to fit the experimental curves.  \n\nResidual gas analyser. To exclude the potential influence on TGC $\\left(\\mathrm{H}_{2}\\right)$ quantification from LiH, which may exist in the bulk electrode as mossy dendrite, we designed an alternative approach to distinguish between LiH and metallic ${\\mathrm{Li}}^{0}$ by changing the titration solution from $\\mathrm{H}_{2}\\mathrm{O}$ to $\\mathrm{D}_{2}\\mathrm{O}$ , which reacts with LiH and Li to produce HD and $\\mathrm{D}_{2}\\mathrm{:}$ , respectively, followed by differentiating between HD and $\\mathrm{D}_{2}$ based on partial pressure analysis by RGA. The base pressure in the vacuum chamber is $\\mathrm{{\\sim}8\\times10^{-8}}$ torr. The gas mixtures were introduced into the chamber using an MKS pressure/flow control valve controlled by a computer. The partial pressures of gases in the system were measured using an SRS RGA with a detection limit down to $10^{-11}$ torr. The turbo pump and backing pump used in this vacuum system were specially designed for handling highly corrosive gases. This setup provides a wide-range partial pressure control $(\\bar{10}^{-11}$ torr to $10^{-4}$ torr) and allow the identification of the different gas molecules including hydrogen isotopes present in the system under high vacuum conditions. Before measuring each sample, the whole system was pumped down to high vacuum $10^{-8}$ torr) to minimize the possibility of contamination.  \n\n# Data availability  \n\nThe data that support the findings of this study are available from the corresponding author on reasonable request.  \n\n31.\t Wood, S. M. et al. Predicting calendar aging in lithium metal secondary batteries: the impacts of solid electrolyte interphase composition and stability. Adv. Energy Mater. 8, 1–6 (2018).   \n32.\t Drenik, A. et al. Evaluation of the plasma hydrogen isotope content by residual gas analysis at JET and AUG. Phys. Scr. T170, 014021 (2017).   \n33.\t Xu, K. Nonaqueous liquid electrolytes for lithium-based rechargeable batteries. Chem. Rev. 104, 4303–4418 (2004).  \n\nAcknowledgements This work was supported by the Office of Vehicle Technologies of the US Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium) under contract DE-EE0007764. Cryo-FIB was performed at the San Diego Nanotechnology Infrastructure, a member of the National Nanotechnology Coordinated Infrastructure, which is supported by the US National Science Foundation (NSF) (grant ECCS-1542148). We acknowledge the UC Irvine Materials Research Institute for the use of the cryo-electron microscopy and XPS facilities, funded in part by the NSF Major Research Instrumentation Program under grant CHE-1338173. The partial pressure measurements and analysis were done using a unique RGA based high vacuum gas evolution system developed under the guidance of I. K. Schuller’s laboratory at UC San Diego. The development of this system were supported by the US Department of Energy, Office of Science, Basic Energy Science (BES) under grant DE FG02 87ER-45332. C.F. thanks D. M. Davies for his suggestions on the manuscript and Shuang Bai for her assistance with the TEM experiment. J.L. thanks W. Wu for helping on figure design.  \n\nAuthor contributions C.F., J.L., X.W. and Y.S.M. conceived the ideas. C.F. designed and implemented the TGC system. C.F. designed and performed the TGC, cryoFIB–SEM, XPS experiments and data analysis. M.Z. collected the cryo-TEM data. C.F., M.Z. and B.L. interpreted TEM data. Y.Z., C.F. and M.C. prepared samples for characterizations. J.Z.L. and Y.Y. helped to set up cryo-FIB instrumentation. F.Y., N.W. and J.G. helped with GC set up and calibration. C.F. and M.-H.L. performed the RGA experiment. J.A., M.A.S. and K.X. formulated and provided the HCE electrolyte. L.Y. and M.C. formulated and provided the GM electrolyte. J.L. and C.F. wrote the manuscript. All authors discussed the results and commented on the manuscript. All authors have approved the final manuscript.  \n\nCompeting interests The authors declare no competing interests.  \n\n# Additional information  \n\nCorrespondence and requests for materials should be addressed to Y.S.M. Reprints and permissions information is available at http://www.nature.com/ reprints.  \n\n![](images/a91c75ad4e45890a3835f37c0b4a0cdfdc1f76677615e5c346affb62264040dd.jpg)  \nExtended Data Fig. 1 | Schematic working principle of the TGC method. By combining $[\\mathrm{H}_{2}\\mathrm{O}$ titration on an inactive Li sample and $\\mathrm{H}_{2}$ quantificatio y GC, the amount of metallic $\\mathrm{Li}^{0}$ is calculated based on the chemical reaction $2\\mathrm{Li}+2\\mathrm{H}_{2}\\mathrm{O}\\rightarrow2\\mathrm{LiOH}+\\mathrm{H}_{2}\\uparrow$ .  \n\n![](images/9760d009ab04910c0a8d4bad1025171d052d1b571a653b12bf3e8b525619677b.jpg)  \nExtended Data Fig. 2 | Supplementary materials for TGC analysis. a, b, Representative voltage profiles of $\\mathrm{Li}||\\mathrm{Cu}$ cells in (a) HCE and CCE, plating at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for $\\bar{1}\\mathrm{mAh}\\mathrm{cm}^{-2}$ , stripping to $1\\mathrm{V}$ at $0.5\\mathrm{mA}\\mathrm{cm}^{-2}$ , $\\bar{2}.5\\operatorname{mA}{\\mathrm{cm}^{-2}}$ and $5.0\\mathrm{mA}\\mathrm{cm}^{-2}$ (voltage profiles below $0\\mathrm{V}$ represents the plating process, while those above $0\\mathrm{V}$ represents the stripping process); (b) $2\\mathrm{M}$ LiFSI–DMC, $0.5{\\mathrm{~M~}}$ LiTFSI–DME/DOL, $1\\mathrm{M}$ LiTFSI–DME/DOL, 1 M LiTFSI–DME/DOL $+2\\%$ $\\mathrm{LiNO}_{3}$ , $\\mathrm{CCE}+\\mathrm{Cs}^{+}$ and CCE $^+$ FEC, plating at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for $1\\mathrm{mAh}\\mathrm{cm}^{-2}$ , stripping to $1\\mathrm{V}$ at $0.5\\mathrm{mA}\\mathrm{cm}^{-2}$ .  \n\nc, The isolated metallic ${\\mathrm{Li}}^{0}$ percentage in total capacity loss $\\mathrm{(Li^{0}/L i^{0}}$ $+\\mathrm{Li^{+}})$ . d, SEI ${\\mathrm{Li}}^{+}$ percentage in total capacity loss $(\\mathrm{Li^{+}/L i^{0}+L i^{+}})$ . e, Measured $\\mathrm{H}_{2}$ area as a function of Coulombic efficiency under a variety of testing conditions. Every data point is an average of three separate GC measurements. The error bars represent the standard deviation, indicating the accuracy and reproducibility of the GC measurement. f, Unit conversion between milliampere-hours and milligrams of Li.  \n\n![](images/b37599b60980348e0d2f7845db916e7c2bc368e240876151c20f7350efc96551.jpg)  \nExtended Data Fig. 3 | TGC analysis of inactive Li formed under extended electrochemical conditions. a, The voltage profiles of CCE with different deposition capacities at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for 1 mAh $c\\mathrm{m}^{-2}$ , $2\\mathrm{mAh}\\mathrm{cm}^{-2}$ , $3\\mathrm{\\mAh\\cm}^{-2}$ and 5 mAh $c\\mathrm{m}^{-2}$ . b, The corresponding TGC analysis of inactive Li with associated capacity loss and Coulombic   \nefficiency under different deposition capacities. c, The cycling performance of CCE in $\\mathrm{Li}||\\mathrm{Cu}$ half-cells at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for $\\bar{1\\mathrm{mAh}}\\mathrm{cm}^{-2}$ . d, TGC analysis showing $\\mathrm{Li}^{0}$ and ${\\mathrm{Li}^{+}}$ contents with associated capacity loss after one, two, five and ten cycles, respectively.  \n\nb  \n\n![](images/a43bdd522cef185a1dc0e28dc57921e1313f55448e693bece6d368514a4eb88d.jpg)  \nExtended Data Fig. 4 | XPS analysis of inactive Li SEI components formed in HCE and CCE for various stripping rates. a, Inactive Li formed in HCE. b, Inactive Li formed in CCE. The stripping rates show negligible impact on SEI components and contents in both electrolytes.  \n\n![](images/c2eedf77e57b81273f4cebadb630374ab8db590445e20c117efcdd444aee7af0.jpg)  \nExtended Data Fig. 5 | Supplementary materials for cryo-FIB-SEM and cryo-TEM analysis. a–c, Top view, cryo-FIB cross-section and schematic of deposited Li in HCE, respectively. The Li deposited in HCE forms large particles with several micrometres in size, with reduced porosity. d–f Top view, cryo-FIB cross-section and schematic of deposited Li in   \nCCE, respectively. The Li shows a whisker-like morphology with high porosity. All deposited at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for $0.5\\mathrm{mAh}\\mathrm{\\overline{{c}}m}^{-2}$ . g, Statistics of inactive Li SEI components formed in HCE, as detected at 50 different sample positions by cryo-TEM. h, Statistics of inactive Li SEI components formed in CCE, as detected at 50 different sample positions by cryo-TEM.  \n\n![](images/367ac969dd6e3afb958771e4d50288060c554ee4daa003c1c9a3e73989d68050.jpg)  \n\nExtended Data Fig. 6 | Strategies that may mitigate inactive Li formation. a, Cross-sectional morphology of Li deposits generated in an advanced electrolyte developed by General Motors (GM), showing a columnar structure. b, The GM electrolyte delivers a first-cycle Coulombic efficiency of $96.2\\%$ , plating at $0.5\\mathrm{mAc}\\dot{\\mathrm{m}}^{-2}$ for $1\\mathrm{mAh}\\mathrm{cm}^{-2}$ , stripping at $0.5\\mathrm{mA}\\mathrm{cm}^{-2}$ to 1 V. c–f, 3D current collector. c, SEM image of Li deposits on Cu foil. d, SEM image of Li deposits on Cu foam. Both were deposited at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for $1\\mathrm{\\dot{m}A h}\\mathrm{cm}^{-\\bar{2}}$ in CCE. e, Representative first-cycle voltage profiles of Cu foil and Cu foam, plating at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ for $1\\dot{\\operatorname*{mAh}}{\\operatorname{cm}^{-2}}$ , stripping at $0.5\\operatorname{mA}{\\mathsf{c m}}^{-2}$ to $1\\mathrm{V}$ in CCE. f, TGC quantification of inactive Li for $\\mathtt{C u}$ foil and Cu foam samples. g, Schematic of an ideal artificial SEI design. The polymer-based artificial SEI should  \n\nbe chemically stable against Li metal and mechanically elastic enough to accommodate the volume and shape change. Meanwhile, the edges of the artificial SEI should be fixed to the Li metal or the current collector, preventing the electrolyte from diffusing and making contact with fresh Li metal. The flexible polymer SEI thus can accommodate expansion and shrinkage during repeated Li plating and stripping. In this way, no Li will be consumed to form SEI during extended cycles, and we can realize anode-free Li metal batteries. h, Influence of pressure on Li plating/ stripping. The results are from the HCE, at $0.{\\overset{-}{5}}\\operatorname*{mA}\\ c\\operatorname*{m}^{-2}$ for $\\mathrm{1}\\mathrm{mA}\\mathrm{\\bar{h}}\\mathrm{cm}^{-2}$ , using a load cell. At each condition, two load cells were measured. The error bars indicate the standard deviation.  \n\n![](images/6d2566fdc057d4906e8db87e850fcca36bbd605d67b9946cef4142818502eee5.jpg)  \n\nef  \n\n<html><body><table><tr><td>Li, mg</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>TGC</td><td>0.0005</td><td>0.5820</td><td>0.6825</td><td>0.8970</td><td>0.9304</td><td>0.9472</td><td>1.1943</td><td>1.4550</td><td>1.4781</td><td>1.5098</td></tr><tr><td>Balance</td><td>Blank</td><td>0.54</td><td>0.66</td><td>0.86</td><td>0.90</td><td>0.91</td><td>1.18</td><td>1.45</td><td>1.51</td><td>1.53</td></tr></table></body></html>\n\nSignificant digits of the measurement methods are marked in red  \n\n<html><body><table><tr><td>Blank Sample</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>H2 concentration in ppm</td><td>47.20</td><td>58.27</td><td>54.39</td><td>41.07</td><td>46.54</td><td>51.02</td><td>51.03</td><td>54.65</td><td>45.49</td><td>53.98</td></tr></table></body></html>  \n\nExtended Data Fig. 7 | TGC calibration and LOD/LOQ analysis. a, $\\mathrm{H}_{2}$ concentration in ppm calibration curve as a function of detected $\\mathrm{H}_{2}$ area and verification with certified GSCO $\\mathrm{H}_{2}$ calibration gas. b, Converted metallic $\\mathrm{Li}^{0}$ mass calibration curve as a function of detected $\\mathrm{H}_{2}$ area. c, Nine pieces of Li metal with known mass were tested using the TGC set-up. The strongly linear relationship with detected $\\mathrm{H}_{2}$ area indicates the feasibility of this method. d, Comparison between the balance-measured mass and TGC-quantified mass of the commercial Li metal pieces. e, Numerical comparison between the balance-measured mass and TGC-quantified mass of the commercial Li metal pieces. As the accuracy of the balance is two orders of magnitude lower than the TGC $(10^{-5}\\mathrm{g}$ versus $10^{-7}{\\bf g}$ , the differentials should mainly come from the balance. f, $\\mathrm{H}_{2}$ concentration in the blank samples measured for LOD/LOQ analysis. A total of 10 measurements were taken for the LOD/LOQ calculation.  \n\n![](images/879d1087068c69f96af6702cbd1535ccf4492ad58a70d472cdbaad40a2b46be2.jpg)  \nExtended Data Fig. 8 | GC chromatogram and ${\\bf N}_{2}$ interference analysis. a, GC chromatogram of the background gas from glovebox. $\\mathbf{b}$ , GC chromatogram of gases with $\\mathrm{H}_{2}$ after $\\mathrm{H}_{2}\\mathrm{O}$ titration on metallic $\\mathrm{Li}^{0}$ . c, Glovebox background gas measurements with various sampling amounts. The $\\mathrm{N}_{2}$ amounts remain at the same level with various injection amounts, indicating the $\\Nu_{2}$ does not exist in the reaction container. d, Container gas measurements with various sampling amounts after   \nthe $\\mathrm{H}_{2}\\mathrm{O}$ titration. The $\\Nu_{2}$ amounts still remain identical with different injection amounts, whereas the $\\mathrm{H}_{2}$ amounts increase in proportion to the increment of injection amounts, indicating that the $\\mathrm{N}_{2}$ does not originally exist in the reaction container but comes from the gas sampling process, and thus will not have any chemical reactions with the inactive Li samples; the $\\mathrm{H}_{2}$ quantification is not influenced by the injection sampling process. e, GC chromatogram of $10\\upmu\\mathrm{l}$ of air.  \n\n![](images/2351e0959651098c4fada74791292c4e929cc2aa68bac21ff5f4ff85d3f5e270.jpg)  \nExtended Data Fig. 9 | Analysis of possible LiH presence in inactive Li. a-h, Possible influence from LiH in SEI. $\\mathbf{a-g}$ , The voltage profiles of SEI formation between $0\\mathrm{V}$ and $1\\mathrm{V}$ at $0.1\\mathrm{mA}$ for ten cycles in $2\\mathrm{M}$ LiFSI–DMC (a), $0.5{\\mathrm{M}}{}$ LiTFSI–DME/DOL ${\\bf(b)}$ , 1 M LiTFSI–DME/DOL (c), CCE (d), HCE (e), $\\mathrm{CCE}+\\mathrm{Cs}^{+}$ (f) and $\\mathbf{CCE}+\\mathbf{FEC}\\left(\\mathbf{g}\\right)$ . After the SEI formation, we performed TGC measurements on the current collectors with SEI. h, TGC results of the seven types of electrolytes. No $\\mathrm{H}_{2}$ can be detected from any of them, indicating no LiH presence in the SEI of the systems studied. i–n, Possible influence from LiH in bulk inactive Li. To   \ndifferentiate the two species, we substitute the titration solution with $\\mathrm{D}_{2}\\mathrm{O}$ instead of $\\mathrm{H}_{2}\\mathrm{O}$ The $\\mathrm{D}_{2}\\mathrm{O}$ reacts with LiH and metallic ${\\mathrm{Li}}^{0}$ to produce HD and $\\mathrm{D}_{2}$ , respectively. RGA can effectively distinguish between HD (relative molecular mass 3) and $\\mathrm{D}_{2}$ (relative molecular mass 4) by partial pressure analysis. i, The $\\mathrm{D}_{2}$ standard from the reaction between commercial pure Li metal and $\\mathrm{D}_{2}\\mathrm{O}$ . j, The HD standard from the reaction between commercial LiH powder and $\\mathrm{D}_{2}\\mathrm{O}$ k–n, Analysis of gaseous products from reactions between $\\mathrm{D}_{2}\\mathrm{O}$ and inactive Li forming in 2 M LiFSI–DMC (k), 0.5 M LiTFSI–DME/DOL (l), $1\\mathrm{M}$ LiTFSI–DME/DOL $\\mathbf{\\tau}(\\mathbf{m})$ and CCE (n).  \n\nExtended Data Table 1 | The solubility or reactivity of known SEI species with $H_{2}O$   \n\n\n<html><body><table><tr><td>SEl component</td><td>Solubility in 100 mL H2O</td></tr><tr><td>LiF</td><td>0.134 g (0.67 mg in 0.5 mL HzO)</td></tr><tr><td>LiOH</td><td>12.8 g</td></tr><tr><td>LiC204</td><td>8g</td></tr><tr><td>LiCO3</td><td>1.29 g</td></tr><tr><td>Li2O</td><td>Li2O + H2O = 2LiOH</td></tr><tr><td>CH3Li</td><td>CHLi + HO = LiOH + CH4 ↑</td></tr><tr><td>ROLi</td><td>ROLi + H2O = LiOH + ROH</td></tr><tr><td>(CH2OCO2Li)2</td><td>(CH2OCO2Li)2 + HO = LiCO3 + (CHOH)2 + CO2 ↑</td></tr><tr><td>LiOCO2R</td><td>2LiOCO2R + HO = LiCO3 + 2ROH + CO2↑</td></tr></table></body></html>  ",
    "Target": null
  },
  {
    "id": "10.1039_c9ee02020a",
    "DOI": "10.1039/c9ee02020a",
    "DOI Link": "http://dx.doi.org/10.1039/c9ee02020a",
    "Relative Dir Path": "mds/10.1039_c9ee02020a",
    "Article Title": "The impact of energy alignment and interfacial recombination on the internal and external open-circuit voltage of perovskite solar cells",
    "Authors": "Stolterfoht, M; Caprioglio, P; Wolff, CM; Márquez, JA; Nordmann, J; Zhang, SS; Rothhardt, D; Hörmann, U; Amir, Y; Redinger, A; Kegelmann, L; Zu, FS; Albrecht, S; Koch, N; Kirchartz, T; Saliba, M; Unold, T; Neher, D",
    "Source Title": "ENERGY & ENVIRONMENTAL SCIENCE",
    "Abstract": "Charge transport layers (CTLs) are key components of diffusion controlled perovskite solar cells, however, they can induce additional non-radiative recombination pathways which limit the open circuit voltage (V-OC) of the cell. In order to realize the full thermodynamic potential of the perovskite absorber, both the electron and hole transport layer (ETL/HTL) need to be as selective as possible. By measuring the photoluminescence yield of perovskite/CTL heterojunctions, we quantify the non-radiative interfacial recombination currents in pin- and nip-type cells including high efficiency devices (21.4%). Our study comprises a wide range of commonly used CTLs, including various hole-transporting polymers, spiro-OMeTAD, metal oxides and fullerenes. We find that all studied CTLs limit the V-OC by inducing an additional non-radiative recombination current that is in most cases substantially larger than the loss in the neat perovskite and that the least-selective interface sets the upper limit for the V-OC of the device. Importantly, the V-OC equals the internal quasi-Fermi level splitting (QFLS) in the absorber layer only in high efficiency cells, while in poor performing devices, the V-OC is substantially lower than the QFLS. Using ultraviolet photoelectron spectroscopy and differential charging capacitance experiments we show that this is due to an energy level mis-alignment at the p-interface. The findings are corroborated by rigorous device simulations which outline important considerations to maximize the V-OC. This work highlights that the challenge to suppress non-radiative recombination losses in perovskite cells on their way to the radiative limit lies in proper energy level alignment and in suppression of defect recombination at the interfaces.",
    "Times Cited, WoS Core": 704,
    "Times Cited, All Databases": 726,
    "Publication Year": 2019,
    "Research Areas": "Chemistry; Energy & Fuels; Engineering; Environmental Sciences & Ecology",
    "UT (Unique WOS ID)": "WOS:000486019600011",
    "Markdown": "# Energy & Environmental Science  \n\n![](images/225dbf5d9e0b11bd14085aae21ec28c8a11d2218d8579ab4ba25aea4dd26aee6.jpg)  \n\nAccepted Manuscript  \n\nThis article can be cited before page numbers have been issued, to do this please use:  M. Stolterfoht, P. Caprioglio, C. M. Wolff, J. A. Márquez Prieto, J. Nordmann, S. Zhang, D. Rothhardt, U. Hörmann, Y. Amir, A. Redinger, L. Kegelmann, F. Zu, S. Albrecht, N. Koch, T. Kirchartz, M. Saliba, T. Unold and D. Neher, Energy Environ. Sci., 2019, DOI: 10.1039/C9EE02020A.  \n\n![](images/ab4fe1141543a757e845f8befd7fa9e756ccbb133365a1af9b09673fd7d0b822.jpg)  \n\nThis is an Accepted Manuscript, which has been through the Royal Society of Chemistry peer review process and has been accepted for publication.  \n\nAccepted Manuscripts are published online shortly after acceptance, before technical editing, formatting and proof reading. Using this free service, authors can make their results available to the community, in citable form, before we publish the edited article. We will replace this Accepted Manuscript with the edited and formatted Advance Article as soon as it is available.  \n\nYou can find more information about Accepted Manuscripts in the Information for Authors.  \n\nPlease note that technical editing may introduce minor changes to the text and/or graphics, which may alter content. The journal’s standard Terms & Conditions and the Ethical guidelines still apply. In no event shall the Royal Society of Chemistry be held responsible for any errors or omissions in this Accepted Manuscript or any consequences arising from the use of any information it contains.  \n\n# The impact of energy alignment and interfacial recombination on the open-circuit voltage of perovskiteViseowlAartricle Online DOI: 10.1039/C9EE02020A cells  \n\nMartin Stolterfoht1,\\*, Pietro Caprioglio1,3, Christian M. Wolff1, José A. Márquez2, Joleik Nordmann1, Shanshan Zhang1, Daniel Rothhardt1, Ulrich Hörmann1, Yohai Amir1, Alex Redinger2, Lukas Kegelmann3, Fengshuo $\\mathsf{Z u}^{4,5}$ , Steve Albrecht3, Norbert Koch4,5, Thomas Kirchartz6, Michael Saliba7, Thomas Unold2,\\*, Dieter Neher1,\\*  \n\n1Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24-25, D-14476 PotsdamGolm, Germany.  \n\n2Department of Structure and Dynamics of Energy Materials, Helmholtz-Zentrum-Berlin, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany  \n\n3Young Investigator Group Perovskite Tandem Solar Cells, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Kekuléstraße 5, 12489 Berlin, Germany  \n\n4Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, 12489 Berlin, Germany  \n\n5Institut für Physik & IRIS Adlershof, Humboldt-Universitat zu Berlin, 12489 Berlin, Germany  \n\n6Institut für Energie- und Klimaforschung, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany  \n\n7Soft Matter Physics, Adolphe Merkle Institute, CH-1700 Fribourg, Switzerland  \n\nE-mail: stolterf@uni-potsdam.de, unold@helmholtz-berlin.de, neher@uni-potsdam.de  \n\n# Abstract  \n\nCharge transport layers (CTLs) are key components of diffusion controlled perovskite solar cells, however, they can induce additional non-radiative recombination pathways which limit the open circuit voltage $(V_{0\\mathrm{C}})$ of the cell. In order to realize the full thermodynamic potential of the perovskite absorber, both the electron and hole transport layer (ETL/HTL) need to be as selective as possible. By measuring the photoluminescence of perovskite/CTL heterojunctions, we quantify the non-radiative interfacial recombination currents in pin- and niptype cells including high efficiency devices $(21.4\\%)$ . Our study comprises a wide range of commonly used CTLs, including various hole-transporting polymers, Spiro-OMeTAD, metal oxides and fullerenes. We find that all studied CTLs limit the $V_{0\\mathrm{C}}$ by inducing an additional non-radiative recombination current that is in most cases substantially larger than the loss in the neat perovskite and that the least-selective interface sets the upper limit for the $V_{\\mathrm{0C}}$ of the device. Importantly, the $V_{\\mathrm{0C}}$ equals the internal quasi-Fermi level splitting (QFLS) in the absorber layer only in high efficiency cells while in poor performing devices, the $V_{\\mathrm{0C}}$ is substantially lower than the QFLS. Using ultraviolet photoelectron spectroscopy and differential charging experiments we show that this is due to an energy level mis-alignment at the $p$ -interface. The findings are corroborated by rigorous device simulations which outline several important considerations to maximize the $V_{\\mathrm{0C}}$ . This work highlights that the challenge to suppress non-radiative recombination losses in perovskite cells on their way to the radiative limit lies in proper energy level alignment and in suppression of defect recombination at the interfaces.  \n\n# Broader context  \n\nAs perovskite solar cells continue to improve at a rapid pace, more fundamental insights into the remaining opencircuit voltage $(V_{\\mathrm{{oc}}})$ losses are required in order to unlock power conversion efficiencies (PCEs) of $\\sim30\\%$ . Several studies highlight that the perovskite absorber exhibits an opto-electronic quality that is comparable to GaAs in terms of external fluorescence, therefore potentially allowing PCEs close to the radiative limits. However, the high internal potential in the absorber layer can often not be directly translated into an equal potential at the metal electrodes. In this work, we reveal the reasons for the discrepancy by decoupling the main $V_{\\mathsf{O C}}$ losses in the bulk, perovskite/charge transport layer (CTL) interfaces and/or metal contacts for a broad range of different perovskite compositions and several, commonly used CTLs. Undoubtedly, by introducing additional non-radiative recombination centres at the interfaces, the CTLs have the most striking impact on the device VOC. MoreVioewveArt,icle Online DOI: 10.1039/C9EE02020A interface recombination is often exponentially increased in case of an energy level mismatch between the perovskite and the CTLs. We conclude that energy level matching is of primary importance to achieve the implied $V_{\\mathrm{{oc}}}$ of the perovskite/CTL stack, followed by suppression of defect recombination at the interfaces and in the absorber layer.  \n\n# Introduction  \n\nHuge endeavours are devoted to understanding and improving the performance of perovskite solar cells, which continue to develop at a rapid pace already outperforming other conventional thin-film technologies on small cells $(<\\mathsf{1c m}^{2})$ .1 It is well established that further improvements will require suppression of non-radiative recombination losses to reach the full thermodynamic potential in terms of open-circuit voltage $(V_{\\mathrm{{0C}}})$ and fill factor (FF).2 As such, a major focus of the entire field to push the technology forward is targeted at reducing defect recombination in the perovskite bulk with numerous works highlighting the importance of grain boundaries in determining the efficiency losses.3,4 In contrast, many other studies highlight the significance of traps at the perovskite surface which is likely chemically distinct from the bulk.4–6 In many cases, performance improvements were achieved by mixing additives into the precursor solution including multiple cations and/or halides.6–9 In many studies, a slower transient photoluminescence (TRPL) decay is shown as the figure of merit to prove the suppressed trap-assisted recombination in the bulk while implying its positive impact on the overall device efficiency.3,6,10 Significantly fewer publications have focused on the importance of non-radiative recombination of charges across the perovskite/CTL interface.11–13 Until recently it has been challenging to pinpoint the origin of these free energy losses in complete cells, although there have been some studies with valuable insight.11–15 Methods that have been employed to study interfacial recombination in perovskite stacks include impedance spectroscopy,11,16 transient photoluminescence (TRPL)13,17,18 or reflection spectroscopy (TRS),14 transient microwave conductivity (TRMC),15 transient photovoltage (TPV).19 Whilst these techniques exhibit in principle the required time resolution to unveil the kinetics of the interface and bulk recombination, the interpretation of these transient measurements can be very challenging. The reasons are related to the inherent fact that extraction and recombination can both reduce the emitting species in the bulk, thus causing the signal decay.2 Previously, a more direct approach to decouple the origin of these recombination losses at each individual interface has been introduced based on steady-state photoluminescence measurements.9,20–25 In particular, measurements of the emitted photoluminescence quantum yield (PLQY) on individual perovskite/transport layer junctions have been used to explain the $V_{\\mathrm{0C}}$ through QFLS losses in the perovskite bulk and at the individual interfaces.9,20,21 However, the relation between the internal QFLS and the external $V_{\\mathrm{0C}}$ remains poorly understood today, especially for different solar cells architectures with different perovskite absorbers and/or charge transport layers. For example, in a recent study, a very high external PLQY $(15\\%)$ has been reported on a nip-stack (i.e. an electron transport layer/perovskite/hole transport layer junction) upon grain boundary passivation using potassium iodide.9 Considering, the provided external quantum efficiency (EQE), this high PLQY translates in an internal QFLS of ${\\sim}1.26\\ \\mathsf{e V}$ which is very close to the radiative limit of the given perovskite absorber layer $(\\sim1.31\\ \\mathsf{e V})$ . Yet, the $V_{\\mathrm{{oc}}}$ of the optimized cells was considerably lower (1.17 V). This suggests that interfacial recombination (which impacts the QFLS of the nip-stack) is not causing the primary $V_{\\mathrm{0C}}$ limitation and suggests that losses of ${\\sim}100~\\mathrm{mv}$ are of different nature. This raises the important question whether the external $V_{\\mathrm{0C}}$ as measured on a complete solar cell truly represents the QFLS in the perovskite bulk and how this depends on the strength of interfacial defect recombination or the energy level alignment between the perovskite and the transport layers. Until today, the importance and impact of energy level alignment remains an important, yet heavily debated topic in perovskite solar cells. For example, several studies showed the benefit or a large impact of energy level alignment between the perovskite and the transport layers,13,26–28 which was however challenged in other works.29  \n\nIn this work, we studied the relation between the QFLS and the $V_{\\mathsf{O C}}$ by means of absolute PL measurements in “regular” (nip-type)7,30,31  and “inverted” (pin-type)32–34 perovskite solar cells for a broad range of CTLs including metal oxides, conjugated polymers,6,35 small molecules, and fullerenes. First, we aim to compare the selectivity of CTLs used for nip and pin configurations in triple cation perovskite cells; i.e. for instance $\\mathsf{T i O}_{2}$ or $\\mathsf{S n O}_{2}$ vs. PTAA underneath the perovskite or doped Spiro-OMeTAD vs. ${\\sf C}_{60}$ on top. We note that in this work we define the selectivity of a CTL as its ability to maintain the QFLS of the absorber layer while providing efficient maVjieowr iAtrtyicle Online  \n\nDOI: 10.1039/C9EE carrier extraction. The results suggest that when attached to the perovskite, all studied CTL cause a reduction of the QFLS with respect to the QFLS of the neat perovskite on a fused silica substrate $\\left(1.23\\mathrm{eV}\\right)$ . The results were also generalized to different perovskite absorber layers. A comparison of the QFLS obtained on CTL/perovskite (or perovskite/CTL) bilayers and nip- or pin-stacks, suggests a simple superposition principle of non-radiative recombination currents at each individual interface. This implies that the inferior interface dominates the free energy loss in the complete cell. In efficient cells, where the QFLS matches the device $V_{\\mathrm{OC}},$ we can further estimate the parallel recombination currents in the bulk, interfaces and/or metal contacts under $V_{\\mathrm{{oc}}}$ conditions. However, in poor performing cells we find that the $V_{\\mathrm{0C}}$ is substantially lower than the corresponding QFLS of the pin-stack. Drift diffusion simulations highlight the impact of energy level offsets in causing the mismatch between the internal QFLS and the external $V_{\\mathrm{0C}}$ which we further confirmed using photoemission spectroscopy (UPS) and transient differential charging capacitance experiments. The results underline that the primary non-radiative recombination loss channel of today´s perovskite cells is interfacial recombination at (or across) the perovskite/CTL interface and that interfacial recombination is often exponentially increased in case of an energy level offset between the perovskite and the TLs. As such, our findings highlight the importance of tailoring the energetics and kinetics at the perovskite/CTL interfaces to harvest the full potential in perovskite solar cells.  \n\n# Materials  \n\nThe studied CTLs in this work belong to 3 material classes, conjugated polymers, small molecules and metal oxides. Regarding the conjugated polymers, we studied highly selective wide-band gap donors such as PolyTPD and PTAA.6,35 Poly({9,9-bis[30-({N,N-dimethyl}- $.N-$ -ethylammonium)-propyl]-2,7-fluorene}-alt-2,7-{9,9-di-noctylfluorene})dibromide (PFN) was added on top of both materials to improve the wettability. In order to draw correlations between the QFLS and the energetics of the HTL, we also investigated P3HT,36,37 as well as highly conductive composite PEDOT:PSS.12 As small molecule HTL, we tested Spiro-OMeTAD38,39 which requires doping by different ionic salts and other additives.39 For the case of small molecule ETLs, we tested the fullerene ${\\sf C}_{60}$ (with and without the interlayer LiF20) and the solution-processable fullerene derivative PCBM.34,40 Lastly, we studied the commonly used transparent metal oxides $\\mathsf{T i O}_{2}$ and $\\mathsf{S n O}_{2}$ . $\\mathsf{T i O}_{2}$ is widely considered as an ideal electron transporting layer due to its high selectivity and high charge carrier mobility,41 while $\\mathsf{S n O}_{2}$ is the preferred platform for planar efficient nip cells.11 These chemical structures of the materials are shown in Figure 1. As absorber layer we chose the so-called triple cation perovskite $(\\mathsf{C s P b l}_{3})_{0.05}[(\\mathsf{F A P b l}_{3})_{0.83}(\\mathsf{M A P b B r}_{3})_{0.17}]_{0.95}$ (see Supplementary Methods),8 while the results were extended to other perovskite systems as discussed below.  \n\n# Comparison of CTLs for pin and nip type devices  \n\nIn order to quantify the free energy losses at the CTL/perovskite interface, we measured the absolute photoluminescence (PL) yield of perovskite/transport layer heterojunctions. The absolute PL is a direct measure of the quasi-Fermi level splitting (QFLS or $\\mu$ ) in the absorber,24,42–45 and this approach has been recently applied to perovskite solar cells by various groups.20–23  The ratio of emitted $(\\phi_{\\mathrm{em}})$ and absorbed photon fluxes $(\\phi_{\\mathrm{abs}})$ defines the absolute external PL quantum yield (PLQY):  \n\n$$\n\\mathrm{PLQY}={\\frac{\\phi_{\\mathrm{em}}}{\\phi_{\\mathrm{abs}}}}={\\frac{J_{\\mathrm{rad}}/e}{J_{\\mathrm{G}}/e}}={\\frac{J_{\\mathrm{rad}}}{J_{\\mathrm{R,tot}}}}={\\frac{J_{\\mathrm{rad}}}{J_{\\mathrm{rad}}+J_{\\mathrm{non}-\\mathrm{rad}}}}={\\frac{J_{\\mathrm{rad}}}{J_{\\mathrm{rad}}+J_{\\mathrm{B}}+J_{\\mathrm{p}-\\mathrm{i}}+J_{\\mathrm{i-n}}+\\ ...}}\n$$  \n\nIf all emission is from the direct recombination of free charges, and also every absorbed photon generates a free electron-hole pair, the PLQY equals the ratio of the radiative recombination current density $U_{\\mathrm{rad}})^{42}$ and the total free charge generation current density $\\left({J_{\\mathrm{G}}}\\right)$ . At $V_{\\mathrm{0C}},$ charge extraction is zero, meaning that $J_{\\mathrm{G}}$ is equal to the total recombination current $U_{\\mathrm{R,tot}})$ of radiative and non-radiative losses $(J_{\\mathrm{rad}}+J_{\\mathrm{non-rad}})$ . Furthermore, $J_{\\mathrm{non-rad}}$ is equal to the sum of all non-radiative recombination pathways in the bulk $\\left(J_{\\mathrm{B}}\\right)$ , at the HTL/perovskite $(J_{\\mathrm{p-i}})$ and perovskite/ETL $U_{\\mathrm{i-n}})$ interfaces, and potentially other losses (e.g. recombination in the transport layers, or at the CTL/metal interfaces). Using the expression for the radiative recombination current density according to Shockley-Queisser42 and Equation 1, we can write the QFLS as a function of the radiative efficiency  \n\n𝐽rad  \n\n$$\n{\\begin{array}{r l}&{=J_{\\mathrm{0,rad}}e^{\\mu/k T}\\to\\mu=k T\\ln\\left({\\frac{J_{\\mathrm{rad}}}{J_{\\mathrm{0,rad}}}}\\right)=k T\\ln\\left(\\operatorname{PLQY}{\\frac{J_{\\mathrm{G}}}{J_{\\mathrm{0,rad}}}}\\right)=k T\\ln\\left({\\frac{J_{\\mathrm{G}}}{J_{\\mathrm{0}}}}\\right)=k T}\\\\ &{\\ln\\left({\\frac{J_{\\mathrm{rad}}}{J_{\\mathrm{rad}}+J_{\\mathrm{B}}+J_{\\mathrm{p-i}}+J_{\\mathrm{i-n}}+\\ldots..J_{\\mathrm{0,rad}}}}\\right)}\\end{array}}\n$$  \n\nwhere $J_{\\mathrm{0,rad}}$ is the radiative thermal equilibrium recombination current density in the dark and $J_{0}=J_{0,\\mathrm{rad}}/\\mathrm{PLQY}$ the dark saturation current. We note, that the PLQY depends itself on external conditions such as the illumination intensity or the internal QFLS. This originates from the fact that the non-radiative recombination pathways depend differently on the actual number of charge pairs present in the device compared to radiative recombination.46 Thus, in order to predict the QFLS under 1 sun and open-circuit, the PLQY needs to be measured under the same illumination conditions. Equation 2 also shows that the QFLS depends logarithmically on the non-radiative recombination currents in the bulk, interface etc. In order to quantify the QFLS, the generated current density under illumination $\\left({J_{\\mathrm{G}}}\\right)$ and $J_{\\mathrm{0,rad}}$ need to be known, as well as the thermal energy (we measured a temperature of ${\\sim}26{\\cdot}28^{\\circ}C$ on the sample under 1 sun equivalent illumination using a digital standard infrared sensor). $J_{\\mathrm{G}}$ and $J_{\\mathrm{0,rad}}$ are obtained from the product of the external quantum efficiency (EQE) and the solar $(\\phi_{\\mathrm{sun}})$ and the 300 K - black body spectrum $(\\phi_{\\mathrm{BB}})$ , respectively.42,43,47,48 As such, we obtained a $J_{\\mathrm{0,rad}}$ of $\\sim6.5{\\times}10^{-21}\\mathsf{A}/\\mathsf{m}^{2}$ $(\\pm1\\mathrm{x}10^{-21}\\mathsf{A}/\\mathsf{m}^{2})$ independent of the bottom CTL (Supplementary Figure S1) as it is predominantly determined by the tail absorption of the triple cation perovskite absorber layer (with Urbach energies around $15\\ \\mathsf{m e V})$ . In all cases, the QFLS was measured by illuminating the films through the perovskite (or the transparent layer in case of pin or nip stacks) in order to avoid parasitic absorption of the studied CTL (and $\\phi_{\\mathrm{abs}}$ doesn’t equal $\\boldsymbol{J_{\\mathrm{G}}}/e$ anymore, see Supplementary Figure S2). The results of the PL measurements of the different transport layers are summarized in Table 1 and plotted in Figure 1b. All results were obtained as an average of multiple fabricated films (Supplementary Figure S3) with representative PL spectra shown in Supplementary Figure S4. Details of the measurements conditions are discussed in Supplementary Methods.  \n\nTable 1. Optoelectronic quality of several tested perovskite-CTL layer junctions.   \n\n\n<html><body><table><tr><td>Film</td><td>Absorption</td><td>PLQY</td><td>Jo, nr [Am-2]</td><td>QFLS [eV] </td></tr><tr><td>ITO/Pero</td><td>0.839</td><td>2.0x10-5</td><td>3.5×10-16</td><td>1.060</td></tr><tr><td>PEDOT:PSS/Pero</td><td>0.854</td><td>7.5x10-5</td><td>9.9x10-17</td><td>1.092</td></tr><tr><td>P3HT/Pero</td><td>0.848</td><td>7.7x10-4</td><td>1.0x10-17</td><td>1.152</td></tr><tr><td>Pero/Spiro-OMeTAD</td><td>0.944</td><td>1.4x10-3</td><td>4.6x10-18</td><td>1.172</td></tr><tr><td>PTAA/PFN/Pero</td><td>0.852</td><td>5.1x10-3</td><td>1.3x10-18</td><td>1.204</td></tr><tr><td>PolyTPD/PFN/Pero</td><td>0.851</td><td>7.3x10-3</td><td>1.1x10-18</td><td>1.208</td></tr><tr><td>Pero</td><td>0.850</td><td>1.4x10-2</td><td>4.6x10-19</td><td>1.231</td></tr><tr><td>SnO2/Pero</td><td>0.854</td><td>5.9x10-3</td><td>1.5×10-18</td><td>1.201</td></tr><tr><td>TiO2/Pero</td><td>0.854</td><td>2.1x10-3</td><td>3.2x10-18</td><td>1.181</td></tr><tr><td>Pero/PCBM</td><td>0.934</td><td>5.7x10-4</td><td>1.3x10-17</td><td>1.145</td></tr><tr><td>Pero/C60</td><td>0.927</td><td>3.8x10-4</td><td>1.8×10-17</td><td>1.137</td></tr><tr><td>Pero/LiF/C60</td><td>0.892</td><td>1.3x10-3</td><td>4.9x10-18</td><td>1.170</td></tr></table></body></html>  \n\n![](images/6cc114da7bfe908eb5b4a2c60b391815155345c907b5801542d81f12fd942178.jpg)  \nFigure 1. The optoelectronic quality of triple cation perovskite/CTL layer junctions. (a) Materials studied in this paper. (b) The calculated quasi-Fermi level splitting of the studied heterojunctions with different hole and electron transporting materials and of the neat absorber layer based on equation 2 using absolute photoluminescence measurements. The absorber was spin casted from the same solution for all transport layers. The non-radiative dark saturation current is plotted on the right and was obtained from $J_{\\mathrm{0,nr}}=J_{0}-J_{0,\\mathrm{rad}}$ which allows comparing the strength of non-radiative recombination of different junctions.  \n\nFigure 1b shows that the triple cation perovskite on a fused silica substrate limits the QFLS to approximately 1.231 eV, which is ${\\sim}110$ meV below the radiative $V_{\\mathrm{0C}}$ limit (where the PLQY equals 1). We note that we cannot rule out that this value is limited by recombination at the fused silica/perovskite interface and that we observe a substantially lower QFLS $({\\sim}40\\mathsf{m e V})$ of the bare perovskite layer on a glass substrate (see Supplementary Figure S5). Moreover, significantly higher PLQY values above $20\\%$ were observed on methylammonium lead triiodide films where the top surface was passivated with tri-n-octylphosphine oxide (TOPO).49 These results highlight the high opto-electronic quality of the perovskite bulk comparable (or already better) than highly pure silicon or GaAs but also indicates substantial recombination losses at the perovskite top surface. For the HTL/perovskite junctions we also tested the influence of the underlying ITO layer, however this did not significantly influence the obtained QFLS within a small error except for samples with $\\mathsf{S n O}_{2}$ (see Supplementary Figure S3). Likewise, we tested the influence of the copper metal electrode on top of the ${\\sf C}_{60}$ in perovskite/ $\\mathsf{^{\\prime}C}_{60}$ heterojunctions and of pin stacks (Supplementary Figure S6). Overall, these tests suggest that there is an essentially lossless charge transfer between the metal electrodes and the HTL. Interestingly, Figure 1b shows that the polymers PTAA/PFN and PolyTPD/PFN performed best - even outperforming the omnipresent Spiro-OMeTAD. However, it is clear that the selectivity of a TL can be different underneath or on top of the perovskite. Therefore, we do not aim to quantify the opto-electronic quality of a CTL itself but rather assess the selectivity of the CTL in a particular configuration (i.e. either on top or underneath a particular perovskite layer). Among the studied ETLs, $\\mathsf{S n O}_{2}$ and $\\mathsf{T i O}_{2}$ outperform the organic ETLs ${\\sf C}_{60}$ and PCBM which are usually used in pin-type cells. Therefore, this data suggests that the $p$ -interface is the limiting interface for nip cells, and the $\\boldsymbol{n}$ -interface for pin cells consistent with earlier studies.21 Moreover, we observe that the capping CTLs PCBM and ${\\sf C}_{60}$ are worse than Spiro-OMeTAD. Considering that the inferior interface will dominate the final $V_{\\mathrm{0C}}$ (equation 1 and 2), this might be one reason for the superior performance of nip cells today. One approach to suppress non-radiative recombination at the perovskite/ $\\mathsf{\\Delta C}_{60}$ interface is to insert a thin LiF interlayer as demonstrated earlier20 and in Table 1.  \n\nA frequently arising question is how much the perovskite morphology, which potentially varies depending on the underlying CTL, could influence the obtained QFLS and the interpretation of the results. Thus, we performed top scanning electron microscopy and AFM measurements (see Supplementary Figure S7). Interestingly, we find the largest grains on a PEDOT:PSS bottom CTL despite it being the worst among the studied transport layers. The largest grain size distribution is visible on perovskite films on $\\mathsf{T i O}_{2}$ while the perovskite morphology on all other substrates appears, at least qualitatively, similar where we observe relatively small grains $(<10-100\\ \\mathsf{n m})$ . In addition, AFM measurements reveal root mean square surface roughnesses varying from $12-27{\\mathsf{n m}}$ , where the perovskite on PolyTPD/PFN and PTAA/PFN appears to be roughest $(>20\\mathsf{n m})$ while the perovskite film on TVieOw2Airtsicle DOI: 10.1039/C9EE0 the smoothest. We also note the similar Urbach tail of the perovskite absorber layer when processed on different CTLs (Supplementary Figure S1) which is related to the density of subgap states. This further indicates a similar opto-electronic quality of the perovskite. Considering these results, it seems unlikely that the perovskite bulk morphology can explain the changes in the non-radiative recombination loss currents which increase by orders of magnitude depending on the underlying substrate (as shown in Figure 1b). It is also worth to note that these results do not allow distinguishing whether the critical recombination loss occurs across the perovskite/CTL interface, or at the perovskite surface next to the interface. In any case, the presence of the additional CTL triggers additional (non-radiative) interfacial recombination losses, which are dominating the non-radiative recombination losses.  \n\n# Comparison of the QFLS and device $\\pmb{V_{0\\mathrm{c}}}$ and origin of free energy losses  \n\nIn the following, we aim to compare the non-radiative recombination losses at the $p$ - and $\\boldsymbol{n}$ -interfaces with the QLFS of the pin stacks and the $V_{\\mathrm{0C}}$ of the complete cells with different HTLs and $(\\mathsf{L}|\\mathsf{F}/)\\mathsf{C}_{60}$ as ETL. Figure 2a shows that the device $V_{\\mathrm{0C}}$ (black line) generally increases with the average QFLS of the pin-stack (orange line) which was taken as an average as obtained on 3-4 samples for each configuration. Importantly, for optimized cells with PolyTPD or PTAA, the $V_{\\mathrm{0C}}$ (black line) matches the QFLS of the stack (orange line) within a small error. This is also nearly identical to the QFLS of the less selective perovskite $1C_{60}$ interface (blue line). This indicates that for these particular cells, the losses determining the $V_{\\mathrm{0C}}$ occur almost entirely at the inferior interface to the perovskite while the electrodes are not causing additional $V_{\\mathrm{0C}}$ losses. On the other hand, in case of the less selective PEDOT:PSS and P3HT bottom layers, the $V_{\\mathrm{0C}}$ was found to be substantially lower than the corresponding QFLS. This will be discussed further below. The current density vs. voltage $(J V)$ characteristics of the corresponding to cells are shown in Figure 2b which highlight the large differences in the measured $V_{0\\mathrm{C}^{\\mathsf{S}}}$ . Device statistics of individually measured stacks are shown in Supplementary Figure S9. We note that our devices with LiF $1C_{60}$ as ETL reach efficiencies of up to $21.4\\%$ with a $V_{\\mathrm{0C}}$ of ${\\sim}1.2\\lor$ (for a triple cation perovskite with a bandgap of ${\\sim}1.6$ eV), which is among the highest reported values for pin-type cells (Supplementary Figure S10).50–52  \n\nNext, we compared the PLQY with the external electroluminescence quantum efficiency $(E Q E_{\\mathrm{EL}})$ as shown in Figure 2c. Under conditions where the dark injection current equals the generation current, the $E Q E_{\\mathrm{EL}}$ of PTAA and PolyTPD cells ( $3{\\times}10^{-4}$ for both devices) approaches the PLQY of the stack within a factor of two $(5.9{\\times}10^{-4}$ for PTAA and $4.6{\\times}10^{-4}$ for PolyTPD). Improving the perovskite/ETL interface by inserting LiF increases both the QFLS of the pin stack and the $V_{\\mathrm{0C}}$ to 1.17 V corresponding to a  PLQY of ${\\sim}1.3{\\times}10^{-3}$ and $E Q E_{\\mathrm{EL}}$ of ${\\sim}8.3{\\times}10^{-4}$ .20 However, for devices with PEDOT:PSS, the $E Q E_{\\mathrm{EL}}(\\sim1.4{\\times}10^{-8})$ is orders of magnitude lower than the PLQY of the stack $(\\sim$ $\\scriptstyle1\\times10^{-5})$ . We note that the measured $E Q E_{\\mathrm{EL}}$ matches roughly the expected $E Q E_{\\mathrm{EL}}$ value for a $V_{\\mathrm{0C}}$ of $0.9\\mathrm{\\:V}$ as obtained from the $J V$ scan $(3.8\\times10^{-8})$ . Therefore, we conclude that the inferior interface (PEDOT:PSS/perovskite) limits the QFLS of the stack, however, there is an additional loss which affects the $V_{\\mathrm{0C}}$ but not the QFLS. This will be addressed further below. Lastly, films with P3HT lie somewhat in between PEDOT:PSS and PTAA (PolyTPD) devices. Here, both interfaces (P3HT/perovskite and perovskite $/{\\mathsf C}_{60}]$ appear to be equally limiting the QFLS of the stack which also lies below the QFLS of the individual heterojunctions (bilayers). Similar to PEDOT:PSS devices, we observe a considerable mismatch between PLQY of the optical pin stack $(6.2\\times10^{-5})$ and the $E Q E_{\\mathrm{EL}}(\\sim9{\\times}10^{-7})$ . We note that the measure $E Q E_{\\mathrm{EL}}$ is again very close to the $E Q E_{\\mathrm{EL}}$ that is expected for a P3HT device with a $V_{\\mathrm{0C}}$ of ${\\sim}1.0$ V $(\\sim1.8{\\times}10^{-6})$ .  \n\n![](images/7fa9302ed9826007b1f4a4a9e54b7a37609437036f2afb5f8a371e2d297c7c59.jpg)  \nFigure 2. Open-circuit voltage, quasi-Fermi level splitting and electroluminescence of pin cells. (a) Average $V_{\\mathrm{0C}}$ of pin cells employing different conjugated polymers as HTLs and a $C_{60}{E T L},$ , compared to the average QFLS of the corresponding HTL/perovskite bilayers (red), and of the pin stacks (orange). The QFLS of the perovskite/C60 junction and of the neat perovskite on fused silica are shown in dashed blue and black lines, respectively. The dark saturation current $(J_{0,n r}=J_{0}-J_{0,r a d})$ as plotted on the right allows to compare the strength of non-radiative recombination of different junctions. (b) Corresponding current density vs. voltage characteristics of the pin cells with different HTLs, and (c) the external electroluminescence efficiency as a function of voltage. The dashed line shows conditions where the dark injection and light generation currents are equal for each device.  \n\nAs for the nip-cells with $\\mathsf{S n O}_{2}$ and $\\mathsf{T i O}_{2}$ as the ETL, and SpiroOMeTAD as the HTL, we observe a similar trend as in our optimized pin-type cells with PTAA or PolyTPD, that is a close match between the average device $V_{\\mathrm{0C}}\\left(\\sim1.15\\right.$ V) and the average internal QFLS (1.161 eV and $\\mathtt{1.168\\ e V}$ for $\\mathsf{T i O}_{2}$ and $\\mathsf{S n O}_{2}$ based cells, respectively) under 1 sun conditions. All results obtained on nip-cells are shown in Supplementary Figure S11. Regarding the potential impact of the perovskite morphology when the samples are prepared on different hole (electron) transport layers, it is important to note that the losses in the neat material (dashed black in Figure 2a) cannot be larger than the cumulative losses observed in the CTL/perovskite bilayers (red). Moreover, the match between the QFLS of the glass/perovskite/CTL bilayers (blue) and the pin or nip stacks (Figure 2a and Supplementary Figure S11) means that the recombination at the top CTL interface can consistently explain the overall $V_{\\mathrm{{oc}}}$ regardless, if the perovskite is deposited on glass or on the CTL (PTAA:PFN, PolyTPD:PFN, ${\\mathsf{T i O}}_{2},$ $\\mathsf{S n O}_{2}\\vert$ . This highlightVsiewt hArteicle Online I: 10. 039/C9EE02020A importance of the top interface in determining the non-radiative recombination current in perovskite solar cells.  \n\n# Quantification of parallel recombination currents at $\\mathtt{v_{o c}}$  \n\nThe absolute-PL approach allows to further estimate the parallel recombination currents at $V_{\\mathsf{O C}}$ . To this end, we successively quantify the non-radiative recombination currents in the neat material and the bottom and top interfaces from the PLQY of the corresponding perovskite/CTL films (Equation 1) and knowledge of $J_{\\mathrm{rad}}$ (Equation 2). Important to note is that the PLQY needs to be known at the $V_{\\mathrm{0C}}$ of the complete cell. Moreover, the individual recombination currents must add up to $J_{\\mathrm{G}}$ which allows to the check the consistency of the approach. This is possible in efficient cells where the QFLS in the absorber layer matches the device $e V_{O C}$ within a relatively small error $(\\approx20\\mathrm{\\meV})$ , but the procedure is prone to fail in cells where ${\\mathsf{Q F L S}}>e V_{O C}$ . Figure 3a shows the obtained recombination currents for efficient pin-type and nip-type cells. Figure 3b illustrates our optimized pin cells with $\\mathsf{L I F/C}_{60}$ as ETL at $V_{\\mathrm{{oc}}}$ by a bucket with holes which represent the recombination losses (see caption). We note again that the recombination current in the neat perovskite (green) is obtained from a film on fused silica and therefore the loss in the neat absorber layer might be slightly different when deposited on top of a CTL. However, as we detail throughout the manuscript, changes in the perovskite morphology when deposited on different CTL cannot explain the $\\mathsf{v}_{\\mathsf{o c}}$ of the final cells, and the fact that the recombination currents add up to $J_{\\mathrm{G}}$ suggests that this loss estimation provides a realistic description of the parallel recombination currents at $V_{\\mathrm{{OC}}}$ .  \n\n![](images/7399b1a53aa7440a42541c2368b785e6f7fc0c25dcad75abd76a990355200676.jpg)  \nFigure 3. (a) Bulk and Interfacial recombination currents at open-circuit as obtained on nip and pin type cells with nearly flat quasi-Fermi levels. In pin type cells, the non-radiative recombination current is dominated by the $C_{60}$ interface (blue) – even if optimized with LiF. In nip-type cells, the recombination at the upper perovskite/Spirointerface (red) dominates the recombination loss, although the recombination at the $p$ - and $\\boldsymbol{n}$ -interface are quite similar in case of cells based on $\\bar{T i O_{2}}$ . In all cases, the non-radiative recombination losses in the neat perovskite (green) are smaller than at the top interface. We note the radiative recombination current density is very small, e.g. $7.8\\mu A c m^{-2}$ in panel (a). (b) Illustrates a solar cell as bucket with holes where the water level represents the $c e I I s^{\\prime}V_{O C}$ .53 The water stream from the tap corresponds to the generation current density from the sun. The holes in the buck represent the recombination losses at $V_{O C}$ in the bulk, interfaces etc. Depending on the exact size of the holes, the water level will change so as the $V_{O C}$ of the device.  \n\n# Understanding the QFLS across the pin (nip) junction  \n\nThe experimental results in the previous sections show that $\\mathsf{Q F L S}{\\sim}V_{\\mathrm{0C}}$ in case of good performing transport layers (PTAA and PolyTPD). This indicates that interfacial recombination in these devices lowers the QFLS throughout the whole bulk equally. However, in case of PEDOT:PSS or P3HT, the device $V_{\\mathrm{0C}}$ is lower than the QFLS in the perovskite layer. In such cases, at least one QFL bends, presumably at the interfaces or contacts, causing a further reduction in the electrochemical potential of the photogenerated charges. This bending has a much larger effect on the final $V_{\\mathrm{0C}}$ than on the average QFLS in the perovskite bulk. In order to check whether this phenomenon depends on the charge carrier generation profile, we analysed all samples by illuminating the samples through the bottom glass or top using a 445 nm laser (Supplementary Figure S2) and through intensity and wavelength dependent $V_{\\mathrm{0C}}$ measurements (Supplementary Figure S12). However, we concludedV ietwhAartic DOI: 10.1039/C9EE neither the QFLS nor the $V_{\\mathrm{0C}}$ depend significantly on the charge generation profile, which we attribute to the rapid diffusion of charges through the perovskite. In order to understand the spatial distribution of the recombination losses and the QFLS, we simulated our perovskite solar cells using the well-established driftdiffusion simulator SCAPS.54 These simulations take into account previously measured interface recombination velocities and perovskite bulk lifetimes.20 The simulated electron/hole quasi-Fermi levels $\\cdot E_{\\mathrm{F,e}}$ and ${E}_{\\mathrm{F,h}})$ at opencircuit are shown along with the conduction and valence bands in Figure 4a for a PTAA/PFN/perovskite $1C_{60}$ device. Important simulation parameters listed in Supplementary Table S1. Qualitatively, these simulations confirm that $E_{\\mathrm{F,e}}$ and $E_{\\mathrm{F,h}}$ are spatially flat in the perovskite bulk and extend to the corresponding electrodes which explains that $e V_{\\mathrm{0C}}$ is nearly identical to the QFLS (of ${\\sim}1.13\\mathsf{e V}$ ) in these devices. Interestingly, to reproduce the comparatively high open-circuit voltages $(\\sim1.14\\lor)$ and FFs up to $80\\%$ of these devices, a considerable builtin voltage $(V_{\\mathrm{BI}})$ of at least $\\ensuremath{1.0\\mathrm{~V~}}$ had to be assumed considering realistic interface recombination velocities. Otherwise, a strong backfield would hinder charge extraction in forward bias but also accumulate minority carriers at the wrong contact (Supplementary Figure S13). We note that the role of the $V_{\\mathrm{BI}}$ across the absorber layer is currently an important topic in the community and further efforts need to be taken to properly consider the impact of ions on the field distribution.55 Moreover, we had to assume a small majority carrier band offset (Δ $E_{\\mathrm{maj}}<0.1\\mathrm{eV_{\\iti}}$ ) between the perovskite valance/conduction band and the HOMO/LUMO of the HTL/ETL, respectively in order to reproduce the measured device $V_{\\mathrm{{OC}}}$ .  \n\nInterestingly, the implementation of a majority carrier band offset at the $p$ -interface causes a considerable bending of the hole quasi-Fermi level close to the interface which explains the $\\mathsf{\\Omega}\\mathsf{e}F\\mathsf{L S-}V_{\\mathrm{OC}}$ mismatch (Figure 4b). Considering that $E_{\\mathrm{F,e}}$ and $E_{\\mathrm{F,h}}$ need to extend throughout the CTLs to the metal contacts in order to produce an external $V_{\\mathrm{OC}},$ it is clear that any $\\Delta E_{\\mathrm{maj}}$ will cause an exponential increase of the hole population in the HTL. This implies an exponential increase in the recombination rate. Therefore, it is expected that a finite $\\Delta E_{\\mathrm{maj}}$ will lead to an equal loss in the device $V_{\\mathsf{O C}}$ . In order to generalize the conditions under which the $V_{\\mathrm{0C}}$ deviates from the QFLS, we extended our simulations by studying a wide range of parameters (Supplementary Table S1). We found that at least two requirements must be fulfilled in order to explain the $\\mathsf{\\Omega}\\mathsf{e r c l}{\\mathsf{S}}{\\mathsf{-}}V_{\\mathrm{OC}}$ mismatch: (a) a band offset for the majority carrier of at least ${\\sim}0.2\\ \\mathsf{e V}$ , and (b) a sufficiently high recombination velocity $(>1\\mathsf{c m}/\\mathsf{s})$ , otherwise $E_{\\mathrm{F,e}}$ and $E_{\\mathrm{F,h}}$ can remain flat despite the energy offset (Supplementary Figure S14). Indeed, these simulations show that the $V_{\\mathrm{{oc}}}$ loss scales linearly with the $\\Delta E_{\\mathrm{maj}}$ offset as long as the $p$ -interface is limiting. We also note that the minority carrier band offset $\\Delta E_{\\mathrm{min}}$ (i.e. the LUMO of the HTL and the perovskite conduction band) is not influencing the results if $\\Delta E_{\\mathrm{min}}$ is larger than only $0.1\\mathsf{e V}$ which is further discussed at Supplementary Figure S15. We also simulated a pin stack with a PEDOT:PSS bottom layer which we simplified by a metal with a work function of $5\\upepsilon\\upnu$ , a high surface recombination velocity for holes and an intermediate value for electrons (Supplementary Table S1). Also, for these settings we observed that $E_{\\mathrm{F,h}}$ bends at the interface, giving rise to the experimentally observed $\\mathsf{\\Omega}\\mathsf{e r c l}{\\mathsf{S}}{\\mathsf{-}}V_{0\\mathrm{C}}$ mismatch of roughly $150\\mathrm{\\meV}$ in the PEDOT cell. All results on PEDOT:PSS cells are summarized in Supplementary Figure S16. We acknowledge that these simulations only illustrate one possible scenario of the internal device energetics using a set of plausible parameters, and thus different energetic alignments or a morphological issue at the interface cannot be excluded. However, we can conclude that energy level alignment of all layers is a crucial requirement to maximize the $V_{\\mathrm{0C}}$ while the defect density at the interface is also a critical parameter in determining the non-radiative recombination losses.  \n\n![](images/e69c8ead174b706150b67fee33847dcf08797a98eaaad090c1fc7f37b573186b.jpg)  \nFigure 4. Simulation of the QFLS and $\\pmb{V_{0\\mathrm{c}}}$ of pin-type devices using SCAPS. (a) The simulated quasi-Fermi level splitting (QFLS) in junctions with aligned transport layers (PTAA/perovskite $\\ensuremath{\\langle C_{60}\\rangle}$ is identical to $e V_{\\mathrm{0C}}$ but not in case of energetically mis-aligned transport layers (b) where the hole QFL bends at the interface to the hole transport layer which causes a $\\boldsymbol{Q F L S–V_{O C}}$ mismatch. The perovskite is represented in brown showing unoccupied states in between the conduction band minimum $(E_{\\complement})$ and valence band maximum $(E_{\\mathrm{V}}),$ while the dashed lines show the electron and hole quasi-Fermi levels $\\begin{array}{r}{(E_{\\mathrm{F,e}}a n d E_{\\mathrm{F,h}}),}\\end{array}$ the resulting QFLS in the absorber and the open-circuit voltage $(V_{0\\mathrm{C}})$ at the contacts. The HTL (red) and ETL (blue) are represented by their unoccupied states in between the highest and lowest unoccupied molecular orbitals.  \n\n# Energy Level Alignment at the HTL/perovskite interface  \n\nThe findings in the previous sections suggest that the observed mismatch between the internal QFLS and the $V_{\\mathrm{0C}}$ in cells comprising PEDOT:PSS and P3HT is due to an energy offset at the $p$ -interface. To study the energy level alignment between the perovskite and the transport layer, we first performed photoelectron yield spectroscopy measurements (PYS) on the individual layers of the solar cells (Supplementary Figure S17). However, these measurements did not allow a reliable prediction of $\\Delta E_{\\mathrm{maj}}$ which is due to the assumption of a constant vacuum level across different layers of the stack. To measure the energetic offsets between the perovskite and the transport layers with respect to the fixed Fermi level $(E_{\\mathsf{F}})$ of the ITO substrate, we performed UPS measurements with background illumination. Recently, it has been shown that the perovskite surface can be considerably ndoped,56 which will directly impact the location of the valence band onset with respect to $\\boldsymbol{E}_{\\mathsf{F}}$ when measuring the top surface of the perovskite film with a He beam (21.1 eV). However, when UPS is performed with an additional background light, the band bending at the surface can be flattened which then allows to access the bulk energy levels. This enabled a direct comparison between the energy levels of the transport layers and the perovskite bulk. Indeed, as shown in Figure 5 below, by properly taking into account the surface photovoltage (SPV) effect, we found that the valance band of the perovskite is aligned with the HOMO of PTAA and PolyTPD HTLs, while P3HT and PEDOT:PSS exhibited states close to, or at the Fermi-edge. Thus, we conclude that PTAA and PolyTPD allow maintaining the high QFLS that is generated from the perovskite upon illumination which is in agreement with the drift diffusion simulations. In contrast, in case of P3HT, and even worse in case of PEDOT:PSS, carriers will lose part of their free energy once they are transferred from the perovskite to the HTL, thereby causing the additional $\\mathsf{V}_{\\mathsf{O C}}$ -loss as numerically predicted and experimentally observed. A further confirmation of this picture comes from the measurement of the charge carrier density in the bulk $(n_{\\mathrm{bulk}})$ at a given $V_{\\mathrm{0C}}$ using differential charging capacitance measurements. $^{57,58}\\left|{\\mathfrak{n}}\\right.$ the case of proper energy alignment, $n_{\\mathrm{bulk}}$ would be a sole function of the $\\mathsf{V}_{\\mathsf{O C}},$ independent of the choice of the TL material. The results in Supplementary Figure S18 show that this is not the case. Instead, for a given $V_{\\mathrm{OC}},{n}_{\\mathrm{bulk}}$ is substantially larger for the PEDOT:PSS cell than for the P3HT and the PTAA cell with proper energy alignment. This is a direct consequence of the energy offset and the resulting  \n\n![](images/8b12660c3aeeba3906dbb54bfc4e3f5afab9402f2f38c7a018947f8ee26215cc.jpg)  \nFigure 5. (a) Ultraviolet photoelectron (UPS) spectra of PTAA, PolyTPD, P3HT and PEDOT:PSS on ITO. The corresponding signal of the perovskite film is shown above. The perovskite surface is n-doped56 resulting in an apparent valence band onset of $1.35~e V.$ Application of a background light (with a 1 sun equivalent intensity) flattens the band bending at the surface which allows accessing the valence band offset in the perovskite bulk (0.8 eV away from the Fermi level).56 The spectra of PEDOT:PSS is scaled by a factor of 60 as compared to the other films. As discussed by Hwang et al.,59 a high-bandgap PSS layer is present on top of a solution processed film which weakens the photoelectron signal of states at the Fermi-edge of the underlying PEDOT:PSS bulk as shown in several publications.59,60 The deduced energy levels are plotted in (b). As predicted from the $\\boldsymbol{Q F L S–V_{\\mathrm{OC}}}$ match in these cells, in case of PTAA and PolyTPD hole transport layers, the HOMO of the HTL is aligned with respect to the perovskite valence band. However, considerable majority carrier band offsets exist in case of P3HT and PEDOT:PSS. This causes the observed QFLS- $V_{\\mathrm{{OC}}}$ mismatch as carriers relax to the band edges during their transport to the extracting electrode.  \n\n# Recombination Losses in other Perovskite Systems  \n\nIn order to generalize the findings, we also studied QFLS and $V_{\\mathrm{{OC}}}$ losses in other currently popular perovskite materials (Supplementary Figure S19). The results further confirm our main conclusions: (i) the perovskite bulk usually allows to reach higher $V_{0\\mathrm{C}^{\\mathsf{S}}}$ than ultimately achieved in the cell. This is confirmed in a low-gap triple cation perovskite $(\\sim1.54~\\mathrm{eV})$ which is currently used in the highest efficiency solar cells,1 a hybrid vacuum/solution processed $\\mathsf{M A P b l}_{3}$ $(\\sim1.6\\ \\mathsf{e V})$ which is relevant for application on textured surfaces in tandem solar cells,61,62 a high-gap mixed perovskite with a bandgap of $1.7~\\mathsf{e V}$ which is the ideal bandgap for monolithic Si/perovskite tandem solar cells, as well as two-dimensional perovskites based on $\\boldsymbol{n}$ -butylammonium63 - a popular system which demonstrates increased stability under thermal and environmental stress.64 However, in some cases the QFLS of the optical stack is close to the QFLS of the neat absorber layer, e.g. for a solution processed CsFAPbI3 (\\~ $1.47\\ \\mathrm{eV})$ and $\\mathsf{M A P b l}_{3}$ $(\\sim1.6\\ \\mathsf{e V})$ . (ii) In most cases, the QFLS-PL technique can well describe the $V_{\\mathrm{0C}}$ of the final cell which allows to assess the inferior interface by comparing the QFLS of HTL/perovskite or perovskite/ELT junctions. However, in a high-bandgap $(\\sim1.7\\ \\mathrm{eV})$ mixed perovskite system we observe again a considerable mismatch between the QFLS of the pin-stack and the $V_{\\mathrm{{OC}}}$ . This highlights the difficulties in increasing the perovskite bandgap while maintaining aligned energy levels and further demonstrates the relevance of our findings for other perovskite systems.  \n\n# Conclusions  \n\nUsing absolute PL measurements, we were able to decouple the origin of non-radiative recombination losses for cells in pin and nip configurations fabricated from different CTLs. For a triple cation perovskite system, we found that a range of the most common CTLs induce large non-radiative recombination currents which dwarf the nonradiative losses in the neat perovskite. We identified that the most selective bottom CTLs are the polymers PTAA and PolyTPD and $\\mathsf{S n O}_{2}$ which are outperforming the omnipresent $\\mathsf{T i O}_{2}$ although this can vary depending on the exact preparation conditions and the absorber material. For pin-cells the perovskite/ $\\mathsf{\\Delta C}_{60}$ interface was fouVinedw Atrotic 10.1039/C9EE be a major issue which induces more interfacial recombination than Spiro-OMeTAD which could be one reason for the lower performance of pin-type cells with the standard electron transporter ${\\sf C}_{60}$ . A comparison between the QFLS of perovskite/CTL bilayers, optical pin- or nip-type stacks and the $V_{\\mathrm{0C}}$ of the complete device shows that the relevant energy losses happen at the top interface in efficient triple cation cells based on PTAA and PolyTPD, $\\mathsf{S n O}_{2}$ and $\\mathsf{T i O}_{2}$ . In these systems, the electron/hole QFLs are expected to be spatially flat throughout the junction to the electrodes, meaning that the QFLS in the perovskite bulk determines the $V_{\\mathrm{0C}}$ of the cells. This allows further quantification of the parallel recombination currents in the bulk and interfaces and/or metal contacts which defines the $V_{\\mathrm{{OC}}}$ of the complete cells. However, in cells with energetically misaligned HTLs such as PEDOT or $\\mathsf{P3H T}$ , the $V_{\\mathrm{0C}}$ is lower than the QFLS in the absorber layer due to an internal bending of the holeQFL. The fundamental study was validated in high-efficiency perovskite cells in pin-configuration with PCEs up to $21.4\\%$ and through rigorous device simulations. The simulations substantiated the understanding obtained from the experimental results and highlighted the importance of a high built-in voltage and negligible majority carrier band offsets between the perovskite and the transport layers. The presence of an energy level offset at the $p$ - contact was confirmed with UPS and also differential charging capacitance measurements. In order to generalize the findings, additional perovskite systems were studied which showed that the absorber layer often allows a substantially higher $V_{\\mathrm{0C}}$ than achieved by the cell. However, a $\\mathsf{\\Omega}\\mathsf{e}F\\mathsf{L S}\\mathsf{-}V_{\\mathrm{oc}}$ mismatch in complete devices appears also in other systems than those featuring a triple cation absorber with PEDOT:PSS and P3HT HTLs. Therefore, this work allows to conclude that energetic offsets are often harming the device $\\mathsf{v}_{\\mathsf{o c}}$ beyond the limitation imposed by defect recombination in the absorber layer and the interfaces. This implies that proper energy level alignment is a primary consideration to harvest the full potential of the optical pin or nip stack. Only then suppression of interfacial defect recombination will allow us to reach the potential of the perovskite absorber, while suppression of defects in the perovskite bulk or at grain boundaries and photon management will be the final goal to improve this technology to its radiative limit.  \n\n# References  \n\n1 N. J. Jeon, H. Na, E. H. Jung, T.-Y. Yang, Y. G. Lee, G. Kim, H.-W. Shin, S. Il Seok, J. Lee and J. Seo, Nat. Energy, 2018, 3, 682–689.   \n2 W. Tress, Adv. Energy Mater., 2017, 7, 1602358.   \n3 D. Bi, C. Yi, J. Luo, J.-D. Décoppet, F. Zhang, S. M. Zakeeruddin, X. Li, A. Hagfeldt and M. Grätzel, Nat. Energy, 2016, 1, 16142.   \n4 T. S. Sherkar, C. Momblona, L. Gil-Escrig, J. Ávila, M. Sessolo, H. J. Bolink and L. J. A. Koster, ACS Energy Lett., 2017, 2, 1214–1222.   \n5 X. Zheng, B. Chen, J. Dai, Y. Fang, Y. Bai, Y. Lin, H. Wei, X. C. Zeng and J. Huang, Nat. Energy, 2017, 2, 17102.   \n6 W. S. Yang, B. Park, E. H. Jung, N. J. Jeon, Y. C. Kim, D. U. Lee, S. S. Shin, J. 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Burn, D. Neher and M. Stolterfoht, Adv. Mater., 2019, 1901090. 64 H. Tsai, W. Nie, J. C. Blancon, C. C. Stoumpos, R. Asadpour, B. Harutyunyan, A. J. Neukirch, R. Verduzco, J. J. Crochet, S. Tretiak, L. Pedesseau, J. Even, M. A. Alam, G. Gupta, J. Lou, P. M. Ajayan, M. J. Bedzyk, M. G. Kanatzidis and A. D. Mohite, Nature, 2016, 536, 312–317.  \n\nAcknowledgements. We thank Lukas Fiedler and Frank Jaiser for lab assistance. Florian Dornack and Andreas Pucher for providing measurement and laboratory equipment. Philipp Tockhorn for characterization of $\\mathsf{S n O}_{2}$ based cells. This work was in part funded by HyPerCells (a joint graduate school of the Potsdam University and the HZB) and by the German Research Foundation (DFG) within the collaborative research center 951 “Hybrid Inorganic/Organic Systems for Opto-Electronics (HIOS)”.  \n\nAuthor contributions. M.S. planned the project, drafted the manuscript, fabricated cells and films, performed electrical measurements, developed the PL setup, measured absolute PL and performed simulations and analysed all data. P.C. developed the PL setup, measured absolute PL. and data analysis, measured SEM and contributed to film fabrication and to manuscript drafting. C.M.W. provided important conceptual ideas regarding the identification of the recombination losses, cell fabrication and electrical characterization. J.A.M. performed PL measurements and performed corresponding data analysis and interpretation. J.N. performed PL measurements on cells and films and contributed to electrical measurements. S.Z. performed PESA and UVVis measurements and analysis of this data. D.R. fabricated cells and films, and contributed to electrical measurements. U.H. provided important conceptual ideas regarding the development of the PL setup and corresponding data analysis. Y.A. performed TPC, TPV and differential charging measurements with C.M.W. and M.S. A.R. contributed to the analysis of PL data and development of the setup. L.K. fabricated $\\mathsf{S n O}_{2}$ based cells and films and performed corresponding electrical characterizations. F.Z. performed UPS measurements and interpreted corresponding data. S.A. developed $\\mathsf{S n O}_{2}$ based cells and films. N.K. performed UPS measurements and interpreted corresponding data. T.K. contributed to the analysis of recombination losses and analysed the simulation results. M.Sa. fabricated $\\mathsf{T i O}_{2}$ based cells and films and performed corresponding electrical characterizations. T.U. performed numerical simulations and analysed simulation results, contributed to the analysis of PL measurements and recombination losses. D.N. contributed to project planning, manuscript drafting and analysis of all electro-optical measurements. All co-authors contributed to proof reading of the manuscript.  \n\nCompeting financial interests. The authors declare no competing financial interests.  \n\nData availability. The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.  ",
    "Target": null
  },
  {
    "id": "10.1063_1.5143061",
    "DOI": "10.1063/1.5143061",
    "DOI Link": "http://dx.doi.org/10.1063/1.5143061",
    "Relative Dir Path": "mds/10.1063_1.5143061",
    "Article Title": "WIEN2k: An APW+lo program for calculating the properties of solids",
    "Authors": "Blaha, P; Schwarz, K; Tran, F; Laskowski, R; Madsen, GKH; Marks, LD",
    "Source Title": "JOURNAL OF CHEMICAL PHYSICS",
    "Abstract": "The WIEN2k program is based on the augmented plane wave plus local orbitals (APW+lo) method to solve the Kohn-Sham equations of density functional theory. The APW+lo method, which considers all electrons (core and valence) self-consistently in a full-potential treatment, is implemented very efficiently in WIEN2k, since various types of parallelization are available and many optimized numerical libraries can be used. Many properties can be calculated, ranging from the basic ones, such as the electronic band structure or the optimized atomic structure, to more specialized ones such as the nuclear magnetic resonullce shielding tensor or the electric polarization. After a brief presentation of the APW+lo method, we review the usage, capabilities, and features of WIEN2k (version 19) in detail. The various options, properties, and available approximations for the exchange-correlation functional, as well as the external libraries or programs that can be used with WIEN2k, are mentioned. References to relevant applications and some examples are also given. (C) 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).",
    "Times Cited, WoS Core": 1611,
    "Times Cited, All Databases": 1641,
    "Publication Year": 2020,
    "Research Areas": "Chemistry; Physics",
    "UT (Unique WOS ID)": "WOS:000519820500008",
    "Markdown": "#  \n\n# WIEN2k: An APW+lo program for calculating the properties of solids  \n\nCite as: J. Chem. Phys. 152, 074101 (2020); https://doi.org/10.1063/1.5143061   \nSubmitted: 19 December 2019 . Accepted: 24 January 2020 . Published Online: 19 February 2020  \n\nPeter Blaha , Karlheinz Schwarz , Fabien Tran , Robert Laskowski , Georg K. H. Madsen , and Laurence D. Marks  \n\n# COLLECTIONS  \n\nNote: This paper is part of the JCP Special Topic on Electronic Structure Software.  \n\n![](images/be6cfaa386b2338644e190a346d3fc9c389c3b37dbedb89e21f711dfe44eb544.jpg)  \n\nThis paper was selected as an Editor’s Pick  \n\n![](images/47343a55c9f8eaa5a80abdcb0181b5a77b2c5f2d27aac5f496221880cd574488.jpg)  \n\n# WIEN2k: An APW+lo program for calculating the properties of solids EP  \n\nCite as: J. Chem. Phys. 152, 074101 (2020); doi: 10.1063/1.514306 Submitted: 19 December 2019 $\\cdot\\cdot$ Accepted: 24 January 2020 • Published Online: 19 February 2020  \n\n# Peter Blaha,1,a) Karlheinz Schwarz,1 Fabien Tran,1 Robert Laskowski,2 Georg K. H. Madsen,1 and Laurence D. Marks3  \n\n# AFFILIATIONS  \n\n1 Institute of Materials Chemistry, Vienna University of Technology, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria   \n2Institute of High Performance Computing, A∗STAR, 1 Fusionopolis Way, #16-16, Connexis 138632, Singapore   \n3Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA  \n\nNote: This paper is part of the JCP Special Topic on Electronic Structure Software. a)Author to whom correspondence should be addressed: pblaha@theochem.tuwien.ac.at  \n\n# ABSTRACT  \n\nThe WIEN2k program is based on the augmented plane wave plus local orbitals $(\\mathrm{APW+lo}^{\\cdot}$ ) method to solve the Kohn–Sham equations of density functional theory. The APW+lo method, which considers all electrons (core and valence) self-consistently in a full-potential treatment, is implemented very efficiently in WIEN2k, since various types of parallelization are available and many optimized numerical libraries can be used. Many properties can be calculated, ranging from the basic ones, such as the electronic band structure or the optimized atomic structure, to more specialized ones such as the nuclear magnetic resonance shielding tensor or the electric polarization. After a brief presentation of the APW $+\\mathrm{lo}$ method, we review the usage, capabilities, and features of WIEN2k (version 19) in detail. The various options, properties, and available approximations for the exchange-correlation functional, as well as the external libraries or programs that can be used with WIEN2k, are mentioned. References to relevant applications and some examples are also given.  \n\n© 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5143061.  \n\n# I. INTRODUCTION  \n\nQuantum mechanical calculations play a central role in understanding the properties of materials and, increasingly, predicting the properties of new materials. While in the early days, the emphasis was mainly on understanding the energy, atom positions, and band structure, modern codes now calculate a large number of different properties ranging from piezoelectric response to nuclear magnetic resonance (NMR) shielding, examples of which will be given later. With the advent of increasingly sophisticated methods and the ever increasing speed of computers over the last decades, in some cases the accuracy of quantum mechanical calculations rivals or even surpasses the accuracy of experimental measurements.  \n\nThere are many different methods of theoretically modeling the behavior of electrons and atoms in materials. While earlier approaches focused on dealing with the electrons via wave functions,1 many current methods use density functional theory (DFT),2 which has significant speed advantages. Following the method outlined by Kohn and Sham3 (KS), the interacting many-body system of electrons is mapped onto a non-interacting system of quasiparticles, characterized by KS orbitals with a specific KS energy. They have many of the properties of the true electron wave functions and of particular importance is that one can fill up these KS orbitals as a function of their KS energy yielding the true electron density. The KS approach needs an exchange-correlation (XC) functional and the corresponding XC potential. However, the exact functional is unknown and approximations are needed (see Sec. II B).  \n\nA second split in terms of methods is how the atomic positions are considered, and there are two main methods: cluster calculations for a finite number of atoms, which focus on the local properties of some atomic arrangement, and those which are designed to exploit the periodic nature of most solids; the WIEN2k code is an example of the latter. We represent the solid by a unit cell, which is repeated in all three directions, corresponding to periodic boundary conditions. This assumes that the solid is perfect, ordered, and infinite; however, a real crystal differs from this ideal situation, since it is finite, may contain defects or impurities, and may deviate from its ideal stoichiometry. For these important aspects and how to handle them using supercells, see Chap. 8.2 of Ref. 4.  \n\nThere are many computational methods for solving the KS equations, for instance, linear combination of atomic orbitals (LCAO), numerical basis sets, pseudopotential schemes, or space partitioning methods. A recent comparison of these methods showed that especially all-electron codes predict essentially identical results, demonstrating a high reproducibility, whereas some pseudopotential codes lead to large deviations. One of the most accurate codes is our WIEN2k code,6 which is the focus of this paper and is based on the augmented plane wave (APW) method. Detailed descriptions including many conceptual and mathematical details are given in Ref. 7. The term all-electron (see Chap. 8.4 of Ref. 4) means that all electrons from the core (starting from the 1s shell) to the valence states are included.  \n\nTurning to some historical specifics of our approach (see also Ref. 8), Slater9 proposed the original APW method. Unfortunately, the original formulation leads to a nonlinear eigenvalue problem due to the energy-dependent radial basis functions, which is computationally expensive. An important improvement came from Andersen,10 who introduced a linearization of this energy dependency, and Koelling and Arbman11 made the linearized-APW (LAPW) method a practical computational scheme using the muffin-tin (MT) approximation (see Sec. II). This was taken a step further by Freeman and collaborators who made the LAPW method a full-potential all-electron total energy method.12,13  \n\nThis LAPW method formed the basis for the original WIEN code.14 However, the LAPW method had the drawback that only one principal quantum number per angular momentum $\\ell$ could be described and thus failed to give reliable results for all elements on the left of the periodic table because these atoms require a proper description of shallow core states (semi-core) and valence states at the same time (e.g., 1s and 2s in Li or $3s p$ and $^{4s p}$ in Ti). This problem was solved by Singh,15 who introduced local orbitals (LOs) for the description of semi-core states. He also noted that the LAPW method needed a larger plane-wave basis set than the APW method. To overcome this problem, he suggested the augmented plane wave plus local orbitals $(\\mathrm{APW+lo})$ ) method,16,17 where the linearization of the energy-dependent radial wave function was facilitated by an extra local orbital (lo, different from an LO, see Sec. II), which has a superior plane-wave convergence compared to LAPW. Last but not least, the linearization of the energy dependency can introduce some inaccuracy in high precision calculations. This problem was finally solved by introducing additional higher (second) derivative LOs (HDLOs).18,19 These latest developments form the basis of the present WIEN2k_19 code,6 while previous versions have been described in several reviews.7,8,20–23 The method of our choice could be named ( $_\\mathrm{L)APW+lo+LO+HDLO},$ but we use a shorter acronym $\\mathrm{\\APW+lo}$ . It is described in detail in Sec. II A.  \n\n# II. THEORY  \n\nIn the APW-based methods, the unit cell is decomposed into spheres centered at the nuclear sites and an interstitial region,7 as shown in Fig. 1. These atomic spheres with radii $R_{\\mathrm{MT}}$ must not overlap, but should be chosen for computational efficiency as large as possible with the additional constraint that $R_{\\mathrm{MT}}$ for $d$ -elements should be chosen to be about $10\\%{-}20\\%$ bigger than for $s p$ -elements, while $f$ -elements should get even larger spheres because for identical sphere sizes the number of plane-waves (PWs) to reach convergence is largest for the localized $4f$ (5f ) electrons, medium for $3d$ (4d, 5d)-electrons, and much smaller for $\\boldsymbol{s p}$ -states. An exception is the H atom, whose sphere with short $\\mathrm{C-H}$ or $_\\mathrm{O-H}$ bonds should be chosen approximately half the size of $R_{\\mathrm{MT}}({\\mathrm{C}})$ or $R_{\\mathrm{MT}}(\\mathrm{O})$ . In WIEN2k, these sphere radii can be set automatically in an optimal way using the setrmt utility. Note that non-optimal sphere sizes may lead to poor convergence (eventually only for one particular atom) and significantly longer computing time or suffer from truncated Fourier or spherical-harmonic expansions. In the worst case, they can even produce “ghost-states” (unphysical eigenvalues in the occupied spectrum) if the $R_{\\mathrm{MT}}$ of an $\\boldsymbol{s p}$ -element is much larger than that of the other atoms.  \n\nThe electron density $\\rho$ and KS potential $\\nu^{\\mathrm{KS}}$ (defined in Sec. II B) are expanded as a Fourier series in the interstitial $(I)$ region (K denotes a reciprocal lattice vector in units of inverse bohr) and as lattice harmonics (symmetry adapted combinations of spherical harmonics $Z_{L M})^{24}$ times radial functions $\\rho_{L M}(r)\\ [\\nu_{L M}^{\\mathrm{KS}}(r)$ for the potential] inside the spheres $\\cdot S_{t}.$ , where $t$ is the atom index),  \n\n$$\n\\begin{array}{r}{\\rho(\\mathbf{r})=\\left\\{\\begin{array}{l l}{\\displaystyle\\sum_{L,M}\\rho_{L M}\\big(r\\big)Z_{L M}\\big(\\hat{\\mathbf{r}}\\big),}&{\\mathbf{r}\\in S_{t}}\\\\ {\\displaystyle\\sum_{\\mathbf{K}}\\rho_{\\mathbf{K}}e^{i\\mathbf{K}\\cdot\\mathbf{r}},}&{\\mathbf{r}\\in I,}\\end{array}\\right.}\\end{array}\n$$  \n\n$$\n\\nu^{\\mathrm{KS}}(\\mathbf{r})=\\left\\{\\begin{array}{l l}{\\displaystyle\\sum_{L,M}\\nu_{L M}^{\\mathrm{KS}}(r)Z_{L M}(\\hat{\\mathbf{r}}),}&{\\mathbf{r}\\in S_{t}}\\\\ {\\displaystyle\\sum_{\\mathbf{K}}\\nu_{\\mathbf{K}}^{\\mathrm{KS}}e^{i\\mathbf{K}\\cdot\\mathbf{r}},}&{\\mathbf{r}\\in I.}\\end{array}\\right.\n$$  \n\nBy default, the Fourier expansion runs up to $\\mathbf{\\left|K\\right|}=12$ for large $R_{\\mathrm{MT}}$ (16 for $R_{\\mathrm{MT}}<1.2$ ; 20 for $R_{\\mathrm{MT}}<0.7$ bohr), while the angular momentum expansion truncates at $L=6$ . Note that the old “MT” approximation uses a constant value in the interstitial (i.e., only $\\mathbf{K}=0\\mathbf{\\dot{\\Omega}}$ ) and a spherically symmetric density/potential inside the spheres (i.e., only $L=0\\mathrm{\\dot{\\Omega}}$ ).  \n\n![](images/946f69df82ef0f0dba84a867d0fbcdd8f3564d68505f34f9edec2468823bab4c.jpg)  \nFIG. 1. Schematic unit cell with large transition metal (TM), medium O and small H spheres, and the interstitial region in between.  \n\nThis space decomposition plays a crucial role in the definition of core and valence electrons, which are treated differently in APWtype methods. Core states are defined as having wave functions (densities) completely confined inside the atomic spheres. Thus, we do not use the standard definitions of core and valence, but, e.g., in $3d$ transition metals (TMs), the 3s and $3p$ states are also considered as valence, since a couple of percent of their charge leaks out of the atomic sphere. To distinguish them from the conventional definition, we call them semi-core states. Typically, these states are treated using LOs (see below) and their energies are less than 6 Ry below the Fermi energy, but in special situations (small spheres due to short nearest neighbor distances or high pressure), even lower lying states (such as Al- ${2p}$ ) have to be included. Relativistic effects are important for the core states, and thus, they are calculated by numerically solving the radial Dirac equation in the spherical symmetric part of the potential $\\nu^{\\mathrm{KS}}$ . Core states are constrained to be localized and not hybridized with states at the neighboring atoms, but we use a thawed core (no frozen core approximation), i.e., the core states are recalculated in each self-consistent field cycle.7 The semi-core and valence electrons are commonly treated scalar relativistically, i.e., including mass velocity and Darwin $s$ -shift corrections, but neglecting spin– orbit (SO) interactions.7,25 The SO effects can later on be included in a second variational step using the scalar-relativistic orbitals as a basis.7,26 Since $p_{1/2}$ radial wave functions differ considerably from scalar relativistic (or ${{p}_{3/2}}$ ) orbitals, one can also enrich the basis set with additional $p_{1/2}$ local orbitals, specifically, an LO (see below) with a $p_{1/2}$ radial wave function, which is added in the second-variational SO calculation.27  \n\n# A. The APW+lo method as implemented in WIEN2k  \n\nThe basis functions for the valence electrons consist of APWs, which are plane waves in the interstitial region augmented with radial wave functions $u_{t\\ell}(r,E_{t\\ell})$ defined at a fixed energy $E_{t\\ell}$ , and lo.7,16,17 An APW is given by  \n\n$$\n\\phi_{\\mathbf{k}+\\mathbf{K}}^{\\mathrm{APW}}\\left(\\mathbf{r}\\right)=\\left\\{\\begin{array}{l l}{\\sum_{\\ell,m}A_{t\\ell m}^{\\mathbf{k}+\\mathbf{K}}u_{t\\ell}\\big(r,E_{t\\ell}\\big)Y_{\\ell m}\\big(\\hat{\\mathbf{r}}\\big),}&{\\mathbf{r}\\in S_{t}}\\\\ {\\frac{1}{\\sqrt{\\Omega}}e^{i(\\mathbf{k}+\\mathbf{K})\\cdot\\mathbf{r}},}&{\\mathbf{r}\\in I,}\\end{array}\\right.\n$$  \n\nwhere $\\mathbf{k}$ is a point in the first Brillouin zone (BZ), $Y_{\\ell m}(\\hat{\\mathbf{r}})$ are spherical harmonics, and $\\boldsymbol{u}_{t\\ell}$ are solutions of the scalar-relativistic radial KS equation7 inside the sphere $S_{t}$ . Note that these radial functions $\\boldsymbol{u}_{t\\ell}$ are recalculated in each self-consistent-field (SCF) cycle, allowing for an expansion/contraction corresponding to the given charge state (ionicity) of the atom. These adaptive basis functions are part of the reason for the high accuracy of APW-based methods. The coefficients $A_{t\\ell m}^{\\mathbf{k+K}}$ are chosen such that the interstitial and sphere parts of the APW match at the sphere boundary. However, these APWs allow no variations of the radial functions for eigenvalues different than $E_{t\\ell}$ and thus would be a poor basis. To overcome this constraint, the energy dependency is handled by a lo, which is nonzero only inside a MT sphere, and given by  \n\n$$\n\\phi_{t\\ell m}^{\\mathrm{lo}}(\\mathbf{r})=\\left\\{\\begin{array}{l l}{\\left[A_{t\\ell m}^{\\mathrm{lo}}u_{t\\ell}(r,E_{t\\ell})+B_{t\\ell m}^{\\mathrm{lo}}\\dot{u}_{t\\ell}(r,E_{t\\ell})\\right]Y_{\\ell m}(\\hat{\\mathbf{r}}),}&{\\mathbf{r}\\in S_{t}}\\\\ {0,}&{\\mathbf{r}\\in I,}\\end{array}\\right.\n$$  \n\nwhere $\\dot{u}_{t\\ell}$ is the first energy derivative of $\\boldsymbol{u}_{t\\ell}$ . The coefficients $A_{t\\ell m}^{\\mathrm{lo}}$ and $B_{t\\ell m}^{\\mathrm{lo}}$ are chosen such that $\\phi_{t\\ell m}^{\\mathrm{lo}}$ is zero at $R_{\\mathrm{MT}}$ and normalized.  \n\nThe APW $+\\mathrm{lo}$ basis set has the advantage of a superior PW convergence as compared to the standard LAPW method,7,17 reducing the number of PWs by almost $50\\%$ , but it needs additional lo basis functions. Thus, in WIEN2k, the default is to restrict the $\\ell_{\\mathrm{max}}$ to the chemical $\\ell$ -values ${\\mathfrak{s p}}(d,f)$ , which are hard to converge, but use a standard LAPW basis set inside the spheres for the higher $\\ell$ values (by default up to $\\ell_{\\mathrm{max}}=10$ ),  \n\n$$\n\\phi_{\\mathbf{k}+\\mathbf{K}}^{\\mathrm{LAPW}}(\\mathbf{r})=\\left\\{\\begin{array}{l l}{\\sum_{\\ell,m}\\left[A_{t\\ell m}^{\\mathbf{k}+\\mathbf{K}}u_{t\\ell}\\big(r,E_{t\\ell}\\big)+B_{t\\ell m}^{\\mathbf{k}+\\mathbf{K}}\\dot{u}_{t\\ell}\\big(r,E_{t\\ell}\\big)\\right]Y_{\\ell m}\\big(\\hat{\\mathbf{r}}\\big),}&{\\mathbf{r}\\in S_{t}}\\\\ {\\frac{1}{\\sqrt{\\Omega}}e^{i(\\mathbf{k}+\\mathbf{K})\\cdot\\mathbf{r}},}&{\\mathbf{r}\\in I,}\\end{array}\\right.\n$$  \n\nwhere the coefficients $A_{t\\ell m}^{\\mathbf{k+K}}$ and $B_{t\\ell m}^{\\mathbf{k}+\\mathbf{K}}$ are chosen such that $\\phi_{\\mathbf{k}+\\mathbf{K}}^{\\mathrm{LAPW}}$ and its first derivative are continuous at the sphere boundary.  \n\nAs mentioned before, semi-core states (or also high-lying empty states) cannot be described accurately by $\\mathrm{\\APW+lo}$ . For these states, the basis set has to be improved, and this can be done by adding another type of local orbitals, the LOs, containing radial functions $\\boldsymbol{u}_{t\\ell}$ calculated at the appropriate (e.g., semi-core) energy $E_{t\\ell}^{\\mathrm{LO},i}$ ,  \n\n$$\n\\phi_{t\\ell m}^{\\mathrm{LO},i}(\\mathbf{r})=\\left\\{\\begin{array}{l l}{\\big[A_{t\\ell m}^{\\mathrm{LO},i}u_{t\\ell}\\big(r,E_{t\\ell}\\big)+C_{t\\ell m}^{\\mathrm{LO},i}u_{t\\ell}\\big(r,E_{t\\ell}^{\\mathrm{LO},i}\\big)\\big]Y_{\\ell m}\\big(\\hat{\\mathbf{r}}\\big),}&{\\mathbf{r}\\in S_{t}}\\\\ {0,}&{\\mathbf{r}\\in I.}\\end{array}\\right.\n$$  \n\nFor instance, for $\\mathrm{TiO}_{2}$ , one would use the Ti- ${3p}$ energy to calculate $u_{\\mathrm{Ti},1}(r,E_{\\mathrm{Ti},1}^{\\mathrm{LO},i})$ and then add some Ti- $^{4p}$ radial function $u_{\\mathrm{Ti,1}}(r,E_{\\mathrm{Ti,1}})$ , choo(sing the) coefficients $A_{t\\ell m}^{\\mathrm{LO},i}$ and $C_{t\\ell m}^{\\mathrm{LO},i}$ such that the LO is zero at the $R_{\\mathrm{MT}}$ and is normalized. By adding such LOs (representing Ti- $3p$ states), a consistent and accurate description of both the Ti- $3p$ semi-core and Ti- $^{4p}$ valence states is possible, retaining orthogonality, which is not assured when the multiple-window approach is used (see Ref. 7). Cases where this improvement is essential is the electric field gradient (EFG) calculation of rutile $\\mathrm{TiO}_{2}{}^{28}$ or lattice parameter calculations of compounds with such elements. Note how lo [Eq. (4)] and LO [Eq. (6)] differ in their respective second terms.  \n\nA clever choice of energy parameters $E_{t\\ell}$ in Eqs. (3)–(6) is essential for accurate results, and WIEN2k has several automatic ways to make an optimal choice in most cases.6 $E_{t\\ell}$ of semi-core states (actually, of all states whose energy in the free atom is more than $0.5~\\mathrm{Ry}$ below the highest occupied atomic orbital, e.g., also C-2s or Ar-3s states) are determined by taking the average of the two energies $E_{\\mathrm{{bottom}}}$ and $E_{\\mathrm{top}}$ , where the corresponding $u_{t\\ell}(R_{\\mathrm{MT}})$ is zero or has zero slope. For localized $d$ or $f$ valence electrons, the same procedure is used, but $E_{\\mathrm{top}}$ is searched only 0.5 Ry above $E_{F}$ to ensure that the energy parameters are set below $E_{F}$ . The energy parameters of all other valence states are set to $0.2\\mathrm{Ry}$ below $E_{F}$ $[0.2\\mathrm{Ry}$ above $E_{F}$ if there is a high lying semi-core LO). Thus, all our energy parameters are dynamically updated during the SCF cycle and not fixed by input.  \n\nImplicit in this approximation is a linearization of the energy dependency of the radial wave functions. Since the true $u_{t\\ell}(r,\\ E_{t\\ell}\\ =\\ \\varepsilon_{i})$ varies most for more localized states (e.g., $3d$ or $\\left|4f\\right|$ at larger distances from the nucleus, large spheres and a large valence bandwidth could cause a poor description of the variations of $u_{t\\ell}(r,\\varepsilon_{i})$ with energy, leading to a significant dependency of the results on $R_{\\mathrm{MT}}$ (where smaller $R_{\\mathrm{MT}}$ yield more correct results but with a larger computational effort). This can be solved by adding an LO, which involves the second energy derivative of $u_{t\\ell}$ , called an HDLO,18,19  \n\n$$\n\\phi_{t\\ell m}^{\\mathrm{HDLO}}(\\mathbf{r})=\\left\\{\\begin{array}{l l}{\\left[A_{t\\ell m}^{\\mathrm{HDLO}}u_{t\\ell}\\left(r,E_{t\\ell}\\right)+C_{t\\ell m}^{\\mathrm{HDLO}}\\ddot{u}_{t\\ell}\\left(r,E_{t\\ell}\\right)\\right]Y_{\\ell m}(\\hat{\\mathbf{r}}),}&{\\mathbf{r}\\in S_{t}}\\\\ {0,}&{\\mathbf{r}\\in I.}\\end{array}\\right.\n$$  \n\nFigure 2 illustrates the effect of adding HDLOs for the lattice parameter of fcc-La as a function of $R_{\\mathrm{MT}}$ . The lattice parameter becomes independent of $R_{\\mathrm{MT}}$ when both $d$ and $f$ -HDLOs are added to the basis set, while the standard $_{\\mathrm{APW+lo+LO}}$ basis (LOs for 5s, $5p$ states) produces an error of 0.04 bohr for the largest $R_{\\mathrm{MT}}$ .  \n\nThe KS orbitals are expanded using the combined basis set described above ( $n$ is the band index),  \n\n$$\n\\psi_{n\\mathbf{k}}=\\sum_{i}c_{n\\mathbf{k}}^{i}\\phi_{i},\n$$  \n\nand the coefficients $c_{n\\mathbf{k}}^{i}$ are determined by the Rayleigh–Ritz variational principle. The number of APW (or LAPW) basis functions Eq. $(3)$ [or Eq. (5)] is determined by the cutoff value $K_{\\mathrm{max}}$ for the reciprocal lattice vectors $\\mathbf{K}$ such that $\\left\\vert\\mathbf{k}+\\mathbf{K}\\right\\vert\\leq K_{\\operatorname*{max}}$ and depends on the smallest of the atomic radii $R_{\\mathrm{MT}}^{\\mathrm{min}}$ a∣nd the∣ t≤ype of atom. Typically, the necessary $R_{\\mathrm{MT}}^{\\mathrm{min}}K_{\\mathrm{max}}$ values range from 3 (for small H-spheres) to 7 for $\\boldsymbol{s p}$ -elements, 8 for TM- $\\cdot d$ elements, and 9 for $4f$ lanthanides. These values can be reduced by 0.5–1 for low quality screening calculations and increased by 0.5–2 for highest precision. It should be mentioned that the efficiency of the $\\mathrm{\\APW+lo}$ method depends crucially on the possible $R_{\\mathrm{MT}}$ values. For instance, the $_{\\mathrm{O-}2p}$ states converge well with $R_{\\mathrm{MT}}^{\\mathrm{O}}K_{\\mathrm{max}}=7$ . In $\\mathrm{MgO}$ , one can use $R_{\\mathrm{MT}}^{\\mathrm{O}}=2$ bohrs, leading to a very small PW cutoff energy of $170\\ \\mathrm{eV}$ . However, in $\\mathrm{Mg(OH)}_{2}$ , one has to use $R_{\\mathrm{MT}}^{\\mathrm{O}}=1.1$ bohrs due to the short $_\\mathrm{O-H}$ distances, leading to a PW cutoff of $550~\\mathrm{eV}$ , i.e., an order of magnitude larger effort.  \n\n![](images/6c40490a95f2cc390536ac54b4f183c6d3f3b829255c78f069431875580e759c.jpg)  \nFIG. 2. Lattice parameter (bohr) using PBE of fcc La as a function of $R_{\\mathrm{MT}}$ using the standard $A P W+10+10$ basis set, or with additional f -HDLO or $d{+}f$ -HDLO.  \n\nThe specific setup of all these basis functions can, of course, be selected manually by experts, but one of the great strengths of WIEN2k is that the default input usually works quite well and is fairly robust.  \n\n# B. Available DFT approximations  \n\nIn DFT, the total energy of the system is given by  \n\n$$\nE_{\\mathrm{tot}}=T_{s}+E_{\\mathrm{en}}+E_{\\mathrm{H}}+E_{\\mathrm{xc}}+E_{\\mathrm{nn}}.\n$$  \n\nThe terms on the right-hand side represent the noninteracting kinetic, electron–nucleus, Hartree, XC, and nucleus–nucleus energies, respectively. The variational principle leads to the KS [or generalized $\\mathrm{{KS^{29}\\left(g K S\\right)}}]$ equations (in this section, the orbital index $i$ is a shorthand notation for valence and core orbitals),  \n\n$$\n\\biggl(-\\frac{1}{2}{\\boldsymbol{\\nabla}}^{2}+\\nu^{\\mathrm{KS}}(\\mathbf{r})\\biggr)\\psi_{i}(\\mathbf{r})=\\varepsilon_{i}\\psi_{i}(\\mathbf{r}),\n$$  \n\nwhere $\\nu^{\\mathrm{KS}}$ is the KS potential,  \n\n$$\n\\nu^{\\mathrm{KS}}(\\mathbf{r})=\\nu_{\\mathrm{en}}(\\mathbf{r})+\\nu_{\\mathrm{H}}(\\mathbf{r})+\\nu_{\\mathrm{xc}}(\\mathbf{r}),\n$$  \n\nwhich is the sum of the electron–nucleus, Hartree, and XC potentials. Choosing an appropriate functional $E_{\\mathrm{xc}}$ in Eq. (9) [and potential $\\nu_{\\mathrm{xc}}$ in Eq. (11)] for the XC term is crucial in order to obtain reliable results for the problem at hand.30–32 Several hundred33,34 different functionals are available in the literature; some of them were proposed as general-purpose functionals, while others were devised for a specific property (e.g., bandgap) or types of systems (e.g., van der Waals). Numerous functionals have been implemented in the WIEN2k code, and below, we provide a brief overview of the different families of functionals. Note that for XC functionals that depend explicitly on the electron density $\\rho$ , e.g., the local density approximation (LDA) or the generalized gradient approximation (GGA), the XC potential $\\nu_{\\mathrm{xc}}$ is multiplicative, while for functionals that depend implicitly on $\\rho$ , e.g., meta-GGA (MGGA) or hybrids, the XC potential is non-multiplicative when implemented in the $\\mathrm{gKS}$ scheme.  \n\n# 1. LDA, GGA, and MGGA  \n\nThe LDA, GGA, and MGGA represent the first three rungs of Jacob’s ladder of XC functionals.35 These approximations are semilocal, since $E_{x c}$ is defined as  \n\n$$\nE_{\\mathrm{xc}}=\\int\\varepsilon_{\\mathrm{xc}}({\\bf r})d^{3}r,\n$$  \n\nand the XC energy density $\\varepsilon_{\\mathrm{xc}}$ depends only locally on some properties of the system. In the LDA, $\\varepsilon_{\\mathrm{xc}}$ depends on the electron density $\\begin{array}{r}{\\rho=\\sum_{i=1}^{N}\\bigl|\\psi_{i}\\bigr|^{2}}\\end{array}$ , while in the GGA, $\\varepsilon_{\\mathrm{xc}}$ depends also on the first derivative $\\boldsymbol{\\nabla}\\rho$ . At the MGGA level, the functionals depend additionally on the L∇aplacian of the electron density $\\boldsymbol{\\nabla}^{2}\\rho$ and/or the kinetic-energy density $t=\\left(1/2\\right)\\sum_{i=1}^{N}\\boldsymbol{\\nabla}\\psi_{i}^{*}\\cdot\\boldsymbol{\\nabla}\\psi_{i}.$ . Semi∇local functionals are the most commonly used m∑et=h ods n the solid-state community for the calculation of properties depending on the total energy such as the geometry, cohesive energy, or the adsorption energy of a molecule on a surface. The main reason is that they are faster than all other types of approximations and therefore allow calculations of larger systems.  \n\nThere is a huge literature on the performance of semilocal functionals, concerning the geometry and cohesive energy of solids. Extensive benchmark studies have been conducted by us36–39 and others (see, e.g., Refs. 40 and 41). The results of these works showed that among the GGA functionals, those with a small enhancement factor such as AM05,42 PBEsol,43 or a few others44–46 are the most accurate for the lattice constant and bulk modulus, while the standard $\\mathrm{PBE}^{47}$ is the best choice for the cohesive energy. At the MGGA level, the SCAN functional48 is becoming increasingly popular and has been shown to be simultaneously as good as the best GGAs for the geometry (e.g., PBEsol) and the cohesive energy (PBE).39,41 However, it should be mentioned that SCAN can be quite problematic for iterant magnetic systems49,50 or alkali metals.51  \n\nMany semilocal functionals have been implemented directly in the WIEN2k code, but basically all existing semilocal functionals can be used because WIEN2k is interfaced to the Libxc33,34 library of XC functionals. One current limitation is that the MGGA functionals are not yet implemented self-consistently (by default, the GGA PBE potential is used for generating the orbitals although the user can choose another potential).  \n\n# 2. Hybrid functionals  \n\nBeginning in the 21st century, hybrid functionals,52 which belong to the fourth rung of Jacob’s ladder, started to be extensively used for calculations of solids.53–55 The one that is currently the most popular is HSE06,56–58 which is a screened version of the other wellknown PBE0.59,60 In (screened) hybrid functionals, the exchange energy is a linear combination of a semilocal (SL) functional and the Hartree–Fock (HF) expression,  \n\n$$\nE_{\\mathrm{xc}}^{\\mathrm{hybrid}}=E_{\\mathrm{xc}}^{\\mathrm{SL}}+\\alpha_{\\mathrm{x}}\\Big(E_{\\mathrm{x}}^{(\\mathrm{scr})\\mathrm{HF}}-E_{\\mathrm{x}}^{(\\mathrm{scr})\\mathrm{SL}}\\Big),\n$$  \n\nwhere  \n\n$$\n\\begin{array}{l}{{\\displaystyle E_{\\mathrm{x}}^{(\\mathrm{scr})\\mathrm{HF}}=-\\frac{1}{2}\\sum_{i=1}^{N}\\sum_{j=1}^{N}\\delta_{\\sigma_{i}\\sigma_{j}}\\iint\\psi_{i}^{*}(\\mathbf{r})\\psi_{j}(\\mathbf{r})}}\\\\ {{\\displaystyle~\\times\\nu\\big(\\left|\\mathbf{r}-\\mathbf{r}^{\\prime}\\right|\\big)\\psi_{j}^{*}(\\mathbf{r}^{\\prime})\\psi_{i}(\\mathbf{r}^{\\prime})d^{3}r d^{3}r^{\\prime}}.}\\end{array}\n$$  \n\nIn Eq. (14), $\\nu$ is either the bare Coulomb potential $\\nu=1/\\big|\\mathbf{r}-\\mathbf{r}^{\\prime}\\big|$ for unscreened hybrids or a potential that is screened at=sh/o∣rt−or l∣ong range for screened hybrids. For solids, it is computationally advantageous to use a potential that is short range, for instance, the Yukawa potential $\\nu=\\left.e^{-\\lambda\\left|\\mathbf{r}-\\mathbf{r}^{\\prime}\\right|}/\\left|\\mathbf{r}-\\mathbf{r}^{\\prime}\\right|$ (Ref. 61) or $\\nu=\\mathrm{erfc}(\\mu|\\mathbf{r}-\\mathbf{r}^{\\prime}|)/\\big|\\mathbf{r}-\\mathbf{r}^{\\prime}\\big|$ (Ref. 56), where erfc is the complementary error function. Although hybrid functionals are also used for the total energy (geometry optimization, and cohesive energy), they are particularly interesting for properties derived from the electronic band structure such as the bandgap, for which they significantly improve upon standard GGA functionals such as PBE (see Refs. 62–65 for recent extensive benchmarking).  \n\nIn WIEN2k, unscreened and screened hybrid functionals are implemented66 according to the scheme of Massidda et al.,67 which is based on the pseudo-charge method for calculating the Coulomb potential.12 The treatment of the Coulomb singularity is done by multiplying the Coulomb potential by a step function,68 which is very efficient compared to other methods.69 The screened hybrid functionals use the Yukawa potential, and in Ref. 66, it was shown that the results obtained with the PBE-based hybrid YS-PBE0 for the bandgap are almost identical to those obtained with HSE06 (which uses the erfc screened potential), provided that the screening parameter is chosen appropriately $~[\\lambda~=~(3/2)\\mu$ , see Ref. 70]. Because of the double integral and summations over orbitals in the HF exchange [Eq. (14)], the calculations are much more expensive (between 10 and 1000 times) than semilocal methods; however, there are a couple of ways to speed-up such calculations significantly. For instance, one can first use a rather crude $\\mathbf{k}$ -mesh and later on improve the k convergence in a few additional iterations continuing the previous calculations. Furthermore, a reduced $\\mathbf{k}$ -mesh for the internal loop in the HF potential is possible,71 and finally, often a one-shot procedure72,73 is sufficient, where the hybrid orbitals and eigenvalues are calculated perturbatively on top of a calculation with the semilocal functional on which the hybrid functional is based.  \n\nCalculations using hybrid functionals in WIEN2k can be found in Refs. 74–77 for applications and in Refs. 39, 64, and 78–80 for various benchmark studies.  \n\n# 3. On-site methods for strongly correlated electrons  \n\nThe high computational cost of hybrid methods discussed in Sec. II B 2 limits the size of the systems that can be treated. Alternatively, one can use an on-site method, namely, $\\mathrm{DFT}+U,^{81}$ exact exchange for correlated electrons (EECE),82 or on-site hybrids,83 which can be viewed as approximate but cheap versions of the hybrid or HF methods. In these methods, a hybrid/HF treatment is applied only to the electrons of a particular angular momentum belonging to a selected atom. However, using such an on-site scheme only makes sense when the considered electrons are well localized around the atom, which is, in general, the case for strongly correlated electrons. The on-site methods are mostly applied to open $3d.$ -, 4f -, or $5f$ -shells in strongly correlated materials in order to improve the description of the electronic and magnetic properties. For such systems, the standard GGA methods provide results that are often even qualitatively inaccurate.84  \n\nDifferent versions of $\\mathrm{DFT}{+}U$ exist in the literature, and those available in WIEN2k are the following: (1) the original version,81 called Hubbard in mean field (HMF) in WIEN2k, (2) the fully localized limit version,85,86 called self-interaction correction (SIC), and (3) the around mean-field (AMF) version.86 The details of the implementation of $\\mathrm{DFT}+U$ in the LAPW method can be found in the work of Shick et al.,87 while a very good summary and discussion of the $\\mathrm{DFT}+U$ flavors is given in Ref. 88. Note that since the on-site term is applied only inside the sphere surrounding the atom of interest,87 the results may depend on the radius $R_{\\mathrm{MT}}$ of this sphere (see Refs. 89 and 90 for illustrations), which is a drawback of the on-site methods. Among our works reporting DFT $+U$ calculations, we mention Refs. 77 and 91–93.  \n\nFor many technical aspects, the EECE and on-site hybrid methods are quite similar to $\\mathrm{DFT}{+}U$ ; however, there are two conceptual differences. The first one concerns the double-counting term. While in $\\mathrm{DFT}+U_{:}$ , the double-counting term is derived using concepts from the Hubbard model (see Ref. 88 for a summary of the various expressions), in EECE and on-site hybrids, it is given by the semilocal expression of $E_{\\mathrm{xc}}$ (e.g., PBE) evaluated with the density of the strongly correlated electrons.82 The second difference is the calculation of the Slater integrals in the Coulomb and Hartree– Fock terms. In $\\mathrm{DFT}+U_{:}$ , they are parameterized with screened intraatomic Coulomb $(U)$ and exchange $(J)$ interactions, which are usually chosen empirically. However, in EECE and on-site hybrids, the Slater integrals are calculated explicitly using the orbitals of the strongly correlated electrons.82 The results obtained with the on-site hybrid methods can be found in Refs. 83 and 94–98.  \n\nThe results obtained with $\\mathrm{DFT}+U$ and on-site hybrids should be qualitatively similar in many (but not necessary all) cases.83,99–101 Actually, both methods contain empirical parameters: $U$ and $J$ (or only $U_{\\mathrm{eff}}=U-J)$ in $\\mathrm{DFT}+U$ and $\\alpha_{\\mathrm{x}}$ [Eq. (13)] in on-site hybrids. In both cases t−he results will depend crucially on the value of the parameters $(U,J)$ or $\\alpha_{\\mathrm{x}}$ . For applications, the EECE method82 is less interesting since it consists of $100\\%$ of unscreened Hartree– Fock exchange applied to correlated electrons, which usually is not accurate.  \n\nTechnically, we mention that calculations with the on-site methods can only be done in a spin-polarized mode, i.e., with runsp_lapw. However, it is possible to apply on-site methods to non-magnetic systems by using the script runsp_c_lapw, which constrains the system to have no spin polarization.  \n\n# 4. Methods for bandgaps  \n\nIt is well known that the GGA functionals that are commonly used for total-energy calculations, such as PBE, provide bandgaps that are much smaller than experiment.57 Thus, one has to resort to other methods to get reliable results for the bandgap. Hybrid functionals and the GW method102,103 (see Sec. III I 4) provide much more accurate values; however, they are also significantly more expensive than semilocal methods and cannot be applied easily to very large systems. Therefore, fast semilocal methods have been proposed specifically intended for bandgap calculations, and those which are available in WIEN2k are discussed below. Note that a more detailed discussion of the DFT methods for bandgaps is provided in Ref. 104.  \n\nThe Tran–Blaha modified Becke–Johnson (TB-mBJ) potential105 consists of a modified version of the BJ potential106 for exchange (which reproduces the exact KS potential of atoms very well106,107) and $\\mathrm{{LDA}^{\\hat{1}08}}$ for correlation. The exchange part, which is a MGGA since it depends on the kinetic-energy density $t$ , is  \n\n$$\n\\nu_{\\mathrm{x}}^{\\mathrm{mBJ}}(\\mathbf{r})=c\\nu_{\\mathrm{x}}^{\\mathrm{BR}}(\\mathbf{r})+(3c-2)\\frac{1}{\\pi}\\sqrt{\\frac{5}{6}}\\sqrt{\\frac{t(\\mathbf{r})}{\\rho(\\mathbf{r})}},\n$$  \n\nwhere $\\nu_{\\mathrm{x}}^{\\mathrm{BR}}$ is the Becke–Roussel (BR) potential109 and  \n\n$$\nc=\\alpha+\\beta g^{p}\n$$  \n\nwith  \n\n$$\ng={\\frac{1}{V_{\\mathrm{cell}}}}\\int_{\\mathrm{cell}}{\\frac{\\left|\\nabla\\rho(\\mathbf{r}^{\\prime})\\right|}{\\rho(\\mathbf{r}^{\\prime})}}d^{3}r^{\\prime}\n$$  \n\nbeing the average of $|\\nabla\\rho|/\\rho$ in the unit cell. The parameters in Eq. (16) are $\\alpha=-0.012$ ∇, $\\beta=1.023~\\mathrm{bohrs}^{1/2}$ , and $\\boldsymbol{p}=1/2$ and were determined by mi−nimizing the mean absolute error of the bandgap for a set of solids.105 As shown in benchmark studies,64,65,80 the TB-mBJ potential is currently the most accurate semilocal method for bandgap prediction. Other parameterizations of Eq. (16) were proposed in Refs. 110 and 111 and are also available in WIEN2k.  \n\nThe GLLB-SC potential112,113 is given by  \n\n$$\n\\begin{array}{l}{{\\displaystyle\\nu_{\\mathrm{{xc}}}^{\\mathrm{GLLB-SC}}({\\bf r})=2e_{\\mathrm{x}}^{\\mathrm{PBEsol}}({\\bf r})+K_{\\mathrm{x}}^{\\mathrm{LDA}}}\\ ~}\\\\ {{\\displaystyle~\\times~\\sum_{n,{\\bf k}}\\sqrt{\\varepsilon_{\\mathrm{{H}}}-\\varepsilon_{n{\\bf k}}}\\frac{\\left|\\psi_{n{\\bf k}}({\\bf r})\\right|^{2}}{\\rho({\\bf r})}+\\nu_{\\mathrm{{c}}}^{\\mathrm{PBEsol}}({\\bf r})},}\\end{array}\n$$  \n\nwhere exPBEsol is the PBEsol exchange-energy density per electron, $\\nu_{c}^{\\mathrm{PBEsol}}=\\delta E_{c}^{\\mathrm{PBEsol}}/\\delta\\rho$ is the PBEsol correlation potential, and $K_{\\mathrm{x}}^{\\mathrm{LDA}}$ $=8\\sqrt{2}/\\left(3\\pi^{2}\\right)$ . Since the GLLB-SC potential depends on the orbital energies $\\varepsilon_{\\mathrm{H}}$ is the one at the valence band maximum), a non-zero derivative discontinuity114,115 can be calculated and added to the KS bandgap for comparison with the experimental value.113,116 Similar to TB-mBJ, the GLLB-SC potential is significantly more accurate than traditional GGA functionals, as shown in Refs. 80, 104, 113, and 117. On the other hand, we note that these potentials are not obtained as the functional derivative of an energy functional.  \n\nAmong other DFT methods, which have been shown to provide bandgaps more accurately than PBE, are the GGAs EV93PW91,118,119 AK13,120,121 and HLE16,122 as well as the LDAtype functional Sloc.123 As mentioned in Sec. II B 1, the potential of MGGA energy functionals is not implemented in WIEN2k; however, it is still possible to calculate bandgaps non-self-consistently using the total energy (see Ref. 124 for details). Such MGGAs that are particularly interesting for bandgaps are HLE17125 and TASK.126  \n\nFigure 3 shows results for the bandgap of 76 solids, which we considered in our previous works.64,80 Compared to the standard PBE functional, the results are much improved when TB-mBJ, GLLB-SC, or HSE06 is used, since the mean absolute error (MAE) drops from $1.99\\mathrm{eV}$ with PBE to 0.47 eV, $0.64\\:\\mathrm{eV}$ , or $0.82\\mathrm{eV}$ , respectively, for the other methods. Among all methods considered in Refs. 64 and 80, TB-mBJ leads not only to the smallest MAE but also to a slope (b = 0.97) of the linear fit that is closest to 1 (the AK13120 functional also leads to a slope of 0.97).  \n\nFinally, we also mention that the Slater127 and Krieger– Li–Iafrate128 potentials have been implemented in the WIEN2k code.129,130 However, these ab initio potentials, which are as expensive as the HF/hybrid methods, are not really intended for bandgap calculations, but may be interesting for more fundamental studies or as a better starting point for approximating the exact KS exchange potential.  \n\n# 5. Methods for van der Waals systems  \n\nThe semilocal and hybrid functionals are, in general, quite inaccurate for describing weak interactions.131,132 This is mainly due to the London dispersion forces that are not included properly in these approximations. Nevertheless, much better results can be obtained by adding to the semilocal/hybrid functional a correlation term $(E_{\\mathrm{c,disp}})$ accounting for the dispersion forces. There are essentially two types of dispersion corrections. The first one is of the atom-pairwise (at-pw) type,  \n\n![](images/d74a5a0277820fb0646bc7580ceb135afca7b1b1c11c243ffb264e2d0e91ddfd.jpg)  \nFIG. 3. Bandgaps of 76 solids calculated with the PBE, TB-mBJ, GLLB-SC, and HSE06 methods compared to experiment. The MAE compared to experiment, a linear fit $(y=a+b x.$ , dashed lines) of the data, and the corresponding root mean square deviation (RMSD) are also shown. The calculated results are from Refs. 64 and 80.  \n\n$$\nE_{\\mathrm{c,disp}}^{\\mathrm{at\\mathrm{-}p w}}=-\\sum_{A<B}\\sum_{n=6,8,10,...}f_{n}^{\\mathrm{damp}}(R_{A B})\\frac{C_{n}^{A B}}{R_{A B}^{n}},\n$$  \n\nwhere $C_{n}^{A B}$ are the dispersion coefficients for atom pair $A{-}B$ separated by the distance $R_{A B}$ and $f_{n}^{\\mathrm{damp}}$ is a damping function. Among the at-pw dispersion methods, Grimme’s DFT- ${\\bf\\cdot}{\\bf D}2^{133}$ and DFT$\\mathrm{D}3^{134}$ can be used with WIEN2k through the separate software DFT$\\mathrm{D}3^{135}$ that supports periodic boundary conditions.136 A feature of the DFT-D3 method is that the dispersion coefficients are not precomputed and fixed but depend on the coordination number of the system.  \n\nThe second type of dispersion methods available in WIEN2k are the nonlocal van der Waals (NL-vdW) functionals137  \n\n$$\nE_{\\mathrm{c,\\disp}}^{\\mathrm{NL}}={\\frac{1}{2}}\\iint\\rho(\\mathbf{r})\\Phi\\bigl(\\mathbf{r},\\mathbf{r}^{\\prime}\\bigr)\\rho(\\mathbf{r}^{\\prime})d^{3}r d^{3}r^{\\prime},\n$$  \n\nwhere the kernel $\\Phi$ is a function of $\\rho,\\nabla\\rho$ , and the interelectronic distance $\\left|\\mathbf{r}-\\mathbf{r}^{\\prime}\\right|$ . Compared to the single i∇ntegral for semilocal functionals [Eq. (12)], evaluating such a double integral is clearly more involved computationally. In order to make the calculations affordable, Román-Pérez and Soler138 proposed a very efficient method based on fast Fourier transformation (FFT) for evaluating Eq. (20). Their method is used for the implementation of NL-vdW functionals in WIEN2k. However, to apply the FFT-based method efficiently in an all-electron method, it is necessary to make $\\rho$ smoother by removing the high-density region close to the nucleus. This is done with the following formula:  \n\n$$\n\\begin{array}{r}{\\rho_{\\mathrm{s}}(\\mathbf{r})=\\left\\{\\begin{array}{l l}{\\rho(\\mathbf{r}),}&{\\rho(\\mathbf{r})\\leqslant\\rho_{\\mathrm{c}}}\\\\ {\\frac{\\rho(\\mathbf{r})+A\\rho_{\\mathrm{c}}(\\rho(\\mathbf{r})-\\rho_{\\mathrm{c}})}{1+A(\\rho(\\mathbf{r})-\\rho_{\\mathrm{c}})},}&{\\rho(\\mathbf{r})>\\rho_{\\mathrm{c}},}\\end{array}\\right.}\\end{array}\n$$  \n\nwhere $A=1{\\mathrm{~bohr}}^{3}$ and $\\rho_{\\mathrm{c}}$ is the density cutoff that determines how smooth the new density $\\rho_{\\mathrm{s}}$ should be. More details can be found in Ref. 139, where it is shown that converged benchmark results can easily be obtained at a relatively modest cost.  \n\nIn our previous works,39,140 a plethora of DFT-D3 and NL-vdW functionals were assessed on solids. The test set consists of strongly bound solids and van der Waals solids such as rare gases or layered compounds. The results showed that among the at-pw methods, PBE-D3/D3(BJ)134,141 seems to be a pretty good choice, while among the nonlocal methods, rev-vdW-DF2142 is the most balanced and actually more or less the best among all tested functionals.  \n\n# C. SCF convergence, total energies, forces, and structure optimization  \n\nThe total energy $E_{\\mathrm{tot}}$ of a periodic solid (with frozen nuclear positions) is given by Eq. (9). The individual terms are of opposite sign and, in an all-electron method, very large. In order to cancel the Coulomb singularity, we follow the algorithm of Weinert et al.,143 where the kinetic and potential-energy terms are combined and a numerically stable method is obtained.  \n\nThe force exerted on an atom $t$ residing at position $\\mathbf{R}^{t}$ , defined as  \n\n$$\n\\mathbf{F}^{t}=-\\nabla_{\\mathbf{R}^{t}}E_{\\mathrm{tot}},\n$$  \n\nis calculated from the Hellman–Feynman theorem and includes Pulay corrections,144 thereby taking into account that parts of the basis set used in WIEN2k depend upon the position of atoms (see Refs. 17 and 145–147 for the derivation specific to APW-based methods).  \n\nWIEN2k exploits the self-consistency of the KS equations, running through a sequence of calculations where the target is a density (and other parameters), which when passed through these SCF calculations yield the same density. This is equivalent to finding the solution to a set of simultaneous equations and is, in the most general case, referred to mathematically as a “fixed-point problem,” although it goes under the different name of “mixing” in the DFT literature. The general method used for just the electron density $\\rho$ and other density-like variables (for instance, the density matrix) is discussed in Ref. 148, while the extension to include atomic positions is described in Ref. 149. At any given iteration $n$ , the relevant variables can be written as a vector $(\\rho_{n},\\nu_{n}^{\\mathrm{orb}})$ , including the density $\\rho$ at the start of an iteration as a function   the Cartesian coordinates r and an orbital potential $\\nu^{\\mathrm{orb}}$ (if used as in $\\mathrm{DFT}+U$ or on-site hybrids, see Sec. II B 3), as well as other relevant variables. After running through the SCF sequence, a new density [symbolized by the SCF mapping $\\mathrm{KS}(\\rho_{n},\\nu_{n}^{\\mathrm{orb}})]$ is produced. The fixed-point for just the density is when the two are equal, i.e., for all of the variables, the set of simultaneous equations  \n\n$$\n{\\mathrm{KS}}(\\rho_{n},\\nu_{n}^{\\mathrm{orb}})-(\\rho_{n},\\nu_{n}^{\\mathrm{orb}})=D(\\rho_{n},\\nu_{n}^{\\mathrm{orb}})=0,\n$$  \n\nwhere $D(\\rho_{n},\\nu_{n}^{\\mathrm{orb}})$ is the density residual. Since the total energy with respect to he position of the atoms also has to be minimal, when these are allowed to vary, this can be expanded to include the forces $\\mathbf{F}_{n}\\big(\\rho_{n},\\mathbf{R}_{n}^{t},\\nu_{n}^{\\mathrm{orb}}\\big)\\ =\\ -\\nabla_{\\mathbf{R}_{n}^{t}}E\\big(\\rho_{n},\\mathbf{R}_{n}^{t},\\nu_{n}^{\\mathrm{orb}}\\big)$ , i.e., solve the larger problem  \n\n$$\n\\begin{array}{r l}&{\\big(D\\big(\\rho_{n},\\nu_{n}^{\\mathrm{orb}}\\big),\\mathbf{F}_{n}\\big(\\rho_{n},\\mathbf{R}_{n}^{t},\\nu_{n}^{\\mathrm{orb}}\\big)\\big)=\\mathbf{G}\\big(\\rho_{n},\\mathbf{R}_{n}^{t},\\nu_{n}^{\\mathrm{orb}}\\big)=0.}\\end{array}\n$$  \n\nThis is equivalent to solving the SCF problem for an extended KS-equation, finding the variational minimum of both the atomic positions and densities.  \n\nThe Hellman–Feynman forces due to the input density $\\rho_{n}$ are calculated within lapw0, while the Pulay corrections are calculated in lapw2 using the new density $\\mathrm{KS}(\\rho_{n},\\mathbf{R}_{n}^{t},\\nu_{n}^{\\mathrm{orb}})$ , so the forces above are, in general, not true derivative(s of the  )ergy, rather pseudoforces that converge to them as the density converges. The general method is to expand to first order, i.e., write for the next value of the variables,  \n\n$$\n\\left(\\rho_{n+1},\\mathbf{R}_{n+1}^{t},\\nu_{n+1}^{\\mathrm{orb}}\\right)=\\left(\\rho_{n},\\mathbf{R}_{n}^{t},\\nu_{n}^{\\mathrm{orb}}\\right)+\\mathbf{H}_{n}\\mathbf{G}\\left(\\rho_{n},\\mathbf{R}_{n}^{t},\\nu_{n}^{\\mathrm{orb}}\\right),\n$$  \n\nwhere $\\mathbf{H}_{n}$ is an approximation to the inverse Jacobian, which is constructed as a Simplex gradient, i.e., a multi-dimensional numerical derivative using some number of the prior steps. The approximate inverse Jacobian has two components:  \n\n1. A predicted component where the density, positions, and other variables have changed in a way that maps onto the prior steps in the SCF iterations, so some information is already available.   \n2. An unpredicted component where the changes in the variables are new, so no prior information is available.  \n\nEquation (25) ignores higher-order terms in the expansion, which can break down far from the fixed point. In addition, because it is generated by a type of numerical differentiation, it can have limited accuracy if the step sizes are inappropriate. In addition, the unpredicted component can lead to instabilities. The approach taken is to control the algorithm greed for the unpredicted step and also use trust regions for both the predicted and unpredicted steps. The general idea of trust regions is that Eq. (25) is only reasonably accurate for changes of the variables, which are smaller than some value, which is called the trust radius, as illustrated in Fig. 4. Starting from some initial defaults, at each iteration, the algorithm will check to see if the step the mixer proposes to use is small enough; if it is too large, then the step is reduced. In the next iteration, if the step used led to an adequate improvement when bounded by the trust radius, then this radius is increased; if the step was not good, then the trust radius is decreased. In addition to changing the trust radii based upon improvement (or not), the most recent algorithm150 also looks at the last step to see how large it should have been for both the predicted and unpredicted parts. This approach is significantly more stable and often leads to much smaller steps than earlier versions of the mixer used.  \n\n![](images/df24261b6374ae8c575e32e3ec2b80f952812282a70416302e434c60b93ace91.jpg)  \nFIG. 4. Illustration of a trust-region approach. With contours shown dashed, from the initial point, the best route following the gradient of the contours is shown in red. However, only a linear step shown in blue is predicted by the multisecant expansion. The trust region (brown) limits the step along this direction so it makes adequate progress downhill and not too far, which is less efficient and can diverge.  \n\nBy design, the algorithm requires minimal user input beyond an estimate for the initial step to take for the unpredicted step; the most recent version of the algorithm150 automatically controls all the internal parameters. No algorithm is perfect, and the convergence of the mixing depends significantly upon the nature of the physical problem being considered. The better the description of the underlying quantum mechanical problem, primarily the XC potential, the more rapidly it will converge. A very badly posed problem, in contrast, may converge only very slowly or not at all.  \n\nTwo main algorithms are used within WIEN2k. The first is MSEC3, which is an updated version of a multisecant Broyden method148 with trust region controls. This is a conservative algorithm, which uses the least greedy approach at every iteration. It is recommended for problems that converge badly. The more pushy MSR1149 uses a more aggressive algorithm, which is significantly better for problems with soft modes that may converge only very slowly with MSEC3.  \n\nOne unique feature of WIEN2k is that it can simultaneously converge both the density and atomic positions by solving the fixedpoint problem of Eq. (24) using a multisecant approach.149 This is often considerably faster than converging them independently as done in many other DFT codes and different from molecular dynamics approaches such as Car–Parrinello.151 The convergence rate depends upon the number and width of the eigenvalue clusters149 of the combined electron and atomic position Jacobian. This approach does not follow the Born–Oppenheimer surface, which is the energy surface when the density is converged, rather some other surface, which is a balance between having converged densities and pseudo-forces as illustrated in Fig. 5. As such it can be somewhat confusing to the user, particularly as the pseudoforces can vary in a strange fashion. This mode can be used with both the more conservative MSEC3 and the more aggressive MSR1 algorithm.  \n\n![](images/9f4bfcf15a9725a5425f663373040c66b5275596a708fcc337f820e382613739.jpg)  \nFIG. 5. The combined density and position algorithm does not follow the Born– Oppenheimer surface (indigo) where the density is converged or the surface where the pseudo-forces are zero (orange), instead it finds a fixed point of a combination (green contours) following the red path.  \n\nFor the optimization of lattice parameters, WIEN2k offers a couple of workflows and utilities to generate structures with different lattice parameters, running the corresponding SCF calculations and analyzing the results. The optimized lattice parameters, however, are found only from the lowest total energy since there is no stress tensor in WIEN2k yet. This makes the optimization tedious for low symmetry cases and practically impossible for triclinic lattices.  \n\n# D. User interface and utilities  \n\nWIEN2k consists of a large set of individual programs (mostly written in Fortran 90), which are linked together via tcsh-shell scripts representing a particular workflow. With this modular structure, WIEN2k is, on the one hand, very flexible and one can run a dedicated program for a particular task. On the other hand, there is not just one program and the specific task will be determined by directives in the input file, but a user has to know which program performs this specific task.  \n\nWIEN2k can be driven either from the command line or using a web-based graphical user interface (GUI), called $w2w e b$ , which can be accessed by any web browser. Most likely, an experienced user will use the command line and explore all advanced features of WIEN2k, but for the beginner, the web-based GUI provides a very good starting point and it also teaches the user the corresponding command line.  \n\n# 1. Structure generation  \n\nThe first task of every calculation is to define the structural data. As an example, the StructGen@w2web page is shown in Fig. 6 for the case of TiCoSb. The necessary basic input consists of the following:  \n\nthe lattice type (P, B, F, and H for primitive, body centered, face centered, and hexagonal, respectively) or, if already known, one of the 230 space groups; for the Heusler compound TiCoSb, we can select F lattice or space group 216 $\\bar{(F43m)}$ ;   \nthe lattice parameters $a,b,$ and $c$ (in bohr or $\\mathring\\mathrm{A}$ ) and the angles $\\alpha,\\beta,$ and $\\gamma$ (in degree);   \nthe atoms and their positions; if the space group is given, only one of the equivalent atoms has to be specified.  \n\nWhen the new structure is saved, the setrmt utility determines the nearest neighbor distances and automatically sets optimized atomic sphere sizes $R_{\\mathrm{MT}}$ for this structure. The choice of $R_{\\mathrm{MT}}$ has nothing to do with ionic radii but depends on the convergence properties of the atoms as discussed before (see the end of Sec. II A). It is important to note that if one wants to compare total energies for a series of calculations (e.g., for volume optimization), the $R_{\\mathrm{MT}}$ should be kept constant.  \n\nAn alternative on the command line is the makestruct utility, which works analogous to StructGen@w2web. More complex structures can be converted from cif or xyz files using the cif2struct or xyz2struct utilities. The generated structures can be conveniently visualized using XCrysDen152 or VESTA.153  \n\nStarting from a basic (simple) structure, WIEN2k has powerful tools to generate supercells and manipulate them. supercell generates quickly $h\\times k\\times l$ supercells (with/without B- or F-centering so that the supercell size can be increased by factors of two) and can add vacuum for surface slab generation. The structeditor, a collection of GNU Octave scripts, is even more powerful, since it can create arbitrary supercells (e.g., ${\\sqrt{3}}\\times{\\sqrt{3}}\\times l)$ , rotate or merge structures, and delete or add atoms.  \n\n![](images/fc4ff1246267e909d7bed5ae89d01024283b423c8a8a110429a7577cccef4218.jpg)  \nFIG. 6. Screenshot of the StructGen@w2web page of the w2web GUI of WIEN2k for TiCoSb.  \n\n# 2. Input generation  \n\nAs mentioned above, WIEN2k consists of many individual programs and most of them have their own input file. Although this sounds very tedious at first, there are default inputs for all programs and several tools for changing the most important parameters on the fly. In w2web, the next step would be to check the symmetry of the newly generated structure and generate the input files for the SCF calculation (initialize@w2web). The user can provide a couple of parameters (only needed if one wants to change the defaults, see below) and run the following steps in batch mode or step by step:  \n\nnn: Determines the distances between all atoms up to twice the nearest neighbor distance. In addition, it checks for overlapping spheres and will issue an error message if the spheres overlap. It also checks if identical elements have the same environment and eventually regroups them into equivalent sets. sgroup: Checks the structure and determines the spacegroup. It will group the atoms into sets of equivalent ones according to the Wyckoff positions of the corresponding space-group. In addition, it will check and determine the smallest possible (primitive) cell and create the corresponding structure file if necessary. For instance, if one enters the NaCl structure as a primitive cubic structure with four Na and four Cl atoms, it will automatically create a primitive FCC cell with only one Na and Cl atom.   \nsymmetry: Finds the symmetry operations of the space group as well as the point group symmetry of each atom and the corresponding LM expansion for the density/potential [Eqs. (1) and (2)].   \nlstart: Solves numerically the radial Dirac equation for free atoms and creates atomic densities. Using the eigenvalues (or the localization within the atomic spheres) of all atomic states, it groups them into core and valence states. It selects automatically LOs for semi-core states and writes the starting energy parameters $E_{\\ell}$ to case.in1 (during the SCF cycle, they are searched and adapted automatically to ensure best possible settings in all cases). kgen: Generates a shifted or non-shifted equidistant $\\mathbf{k}$ -mesh with a user specified density in the irreducible part of the BZ.   \ndstart: Superposes the atomic densities and creates the starting density for the SCF cycle.  \n\nOn the command line, a corresponding script is called, which optionally allows us to specify various parameters (the most important ones are given below with their default values for reference):  \n\nnit_lapw [-b -vxc PBE -ecut -6.0 -rkmax 7.0 -numk 1000]  \n\nThe switches are described as follows: - $\\cdot b$ indicates batch mode (instead of step by step), -vxc selects the DFT functional, -ecut gives the core–valence separation energy (in Ry), -rkmax determines the plane wave cut-off parameter $R_{\\mathrm{MT}}^{\\mathrm{min}}K_{\\mathrm{max}}$ , and -numk determines the total number of $\\mathbf{k}$ -points in the full BZ.  \n\nThe most critical parameter is $R_{\\mathrm{MT}}^{\\mathrm{min}}K_{\\mathrm{max}}$ , which determines not only the quality but also the required computing time. The type of atom with the smallest $R_{\\mathrm{MT}}$ determines this value because the $R_{\\mathrm{MT}}$ for other atoms are set such that when the smallest atom is converged with the number of PWs, all others are also converged with the number of PWs. If the smallest sphere is a H atom (for instance, in short $_\\mathrm{O-H}$ bonds), $R_{\\mathrm{MT}}^{\\mathrm{min}}K_{\\mathrm{max}}=3$ is sufficient, most $s p/d/f$ -elements converge with $R_{\\mathrm{MT}}^{\\mathrm{min}}K_{\\mathrm{max}}=7/8/9$ . For lower (higher) precision, one can decrease (increase) th=ese values by $10\\%{-}20\\%$ .  \n\nOf similar importance is the selection of a $\\mathbf{k}$ -mesh. Generally speaking, small unit cells and metallic character require a large number of k-points (typical starting values for the SCF cycle would be a $10\\times10\\times10\\mathrm{mesh},$ ), while large cells (100 atoms) and insulators can be started with only one $\\mathbf{k}$ -point. In any case, after the first SCF cycle, the k-mesh should be increased and the results (e.g., the forces on the atoms) should be checked. Certain properties (DOS, optics, and NMR) may need an even denser mesh, which in WIEN2k can be easily done and is fairly cheap, since it is only used for the property of interest.  \n\n# 3. SCF cycle  \n\nThe SCF cycle consists in WIEN2k of a complex workflow using several different programs. The main steps are as follows:  \n\nlapw0: Calculates the Coulomb and XC potential from the density.   \nlapw1: Calculates the valence and semi-core eigenvalues and eigenvectors at all requested k-points   \nlapw2: Calculates the valence electron density   \nlcore: Calculates the core eigenvalues and the core density   \nmixer: Adds up the core and valence densities and mixes the total density with densities from previous iterations. In addition, it may update the atomic positions according to the calculated forces (see Sec. II C) and also the density matrices or orbital potentials when $\\mathrm{DFT}+U$ or on-site hybrid methods are used.  \n\nAdditional programs may be called depending on the requested options to include SO coupling or one of the specialized functionals discussed in Sec. II B (DFT-D3, NL-vdW, $\\mathrm{DFT}+U,$ on-site hybrid/EECE, or hybrid-DFT/HF).  \n\nIn $w2w e b$ , the SCF cycle can be started by clicking on $S C F@w{2w e b}$ . In this interface, one can then specify several parameters such as convergence criteria, parallelization, simultaneous optimization of internal atomic positions, or adding SO coupling.  \n\nThe most important parameters for the corresponding command line script are as follows:  \n\nrun_lapw $\\left[-\\mathrm{ec}0.0001-\\mathrm{cc}0.0001\\right.$ -fc $1.0\\mathrm{-p}\\mathrm{-so-min}]$  \n\nThe SCF cycle will stop when the (optional) convergence criteria -ec (energy in Ry), - $\\cdot c c$ (charge in $\\mathrm{e}^{-}$ ), and $-f c$ (forces in mRy/bohr) are fulfilled three times in a row. SO coupling (only possible after a previous init_so_lapw step) is switched on using -so, -min relaxes the atomic positions simultaneously with the electron density (Sec. II C), and $-\\ensuremath{p}$ switches on parallelization (Sec. II E).  \n\nThe basic summary of the SCF cycle is written into the case.scf file and all relevant quantities are labeled :LABEL: and can be searched/monitored using analyse@Utils@w2web or the Linux grep command. If the desired convergence has been reached, it is advisable to save all relevant input/output files using either save@Utils@w2web or the save_lapw utility. One can now either check results using more k-points (kgen@single_prog@Execution@ $w2w e b)$ or modify other inputs (input files@Files@w2web) such as $R_{\\mathrm{MT}}^{\\mathrm{min}}K_{\\mathrm{max}}$ or the XC functional and then continue with the SCF cycle. Later on, it is always possible to come back to a previously saved calculation using restore@Utils@w2web (restore_lapw).  \n\n# 4. Tasks  \n\nOnce these steps have been finished, one could, for instance, optimize the lattice parameters (optimize@w2web) or perform various other tasks (Tasks@w2web) such as Bandstructure@Tasks@ w2web, DOS@Tasks@w2web, ElectronDensities@Tasks@w2web, XSPEC@Tasks@w2web, TELNES@Tasks@w2web, or $O P T I C@$ Tasks@w2web (see Fig. 6). Each of these tasks consists of a guided workflow and let the user prepare the necessary inputs, executes various small programs, and visualizes the results.  \n\n# E. Software requirements and parallelization  \n\n# 1. Software  \n\nWIEN2k runs on any Linux platform and also on Mac. It is written mainly in Fortran 90 (a few programs are written in C), and the workflows are managed by tcsh scripts. Most of the timecritical parts use libraries such as BLAS and LAPACK, and efficient libraries are therefore mandatory. There is direct installation support for the standard Linux tools GFortran $\\cdot+$ OpenBLAS (at least gcc 6.x) and Intel IFORT $^{\\cdot}+$ MKL. The latter still gives the best performance. w2web is a specialized web server written in perl and listens on a user-defined high port. Its access is, of course, password protected and can be limited to specific IP addresses.  \n\nFor the optional installation of the MPI-parallel version (useful only on clusters with InfiniBand network or larger shared memory workstations with at least 16 cores), one needs obviously MPI (e.g., Open MPI or Intel MPI) and also ScaLAPACK (included in the MKL), FFTW,154 and, optional but highly recommended, ELPA.155,156  \n\nThe following Linux tools are necessary (not all of them are always installed by default): tcsh, Perl 5, Ghostscript, gnuplot, GNU Octave, and Python $2.7.\\mathrm{x}\\mathrm{+NumPy}$ .  \n\nOptional, but highly recommended, programs for certain tasks include the following:  \n\nXCrysDen152 and VESTA153 for structure and electron density visualization but also generation of band structure kmeshes or plotting Fermi-surfaces.   \nLibxc33,34 for XC functionals not directly implemented in WIEN2k.   \nDFT- ${\\bf\\cdot D3}^{135}$ for $\\mathrm{DFT}+\\mathrm{D}3$ calculations of van der Waals systems.   \nWannier90157,158 for constructing Wannier functions using the wien2wannier utility.   \nphonopy,159,160 Phonon,161,162 or PHON163,164 for phonon calculations.   \nBoltzTraP2165 for transport calculations (see Sec. III I 1).   \n● fold2bloch166 to fold supercell band structures back to the primitive BZ. $\\mathrm{\\dot{S}K E A F^{167}}$ to extract de Haas-van Alphen frequencies from WIEN2k.   \nCritic2168,169 is an alternative program to the WIEN2k program aim to analyze 3D scalar fields such as the electron density by using the “atoms in molecules” (AIM) theory of Bader.170,171  \n\n# 2. Parallelization  \n\nWIEN2k is a highly accurate all-electron code based on the APW method and thus certainly not as fast as some other (pseudopotential or minimal basis set) codes. However, it takes advantage of inversion symmetry and when present it will automatically use the “real” instead of the “complex” version of the code, thus saving half of the memory and running almost four times as fast. In addition, it is highly optimized and efficiently parallelized at three different levels, which can be optimally chosen depending on the size of the problem and the available hardware.  \n\nExcept for OpenMP parallelization (see below), parallelization is activated by a $-\\boldsymbol{p}$ switch in our scripts and needs a .machines file as listed and described below:  \n\n# .machines file for parallelization   \n# OpenMP parallelization:   \nomp_global:4   \nomp_lapw0:16   \n# k-point parallelization (speed:hostname):   \n1:host1   \n1:host2   \n# MPI parallelization:   \n1:host1:16 host2:16 .   \nlapw0:host1:16 host2:16 . OpenMP parallelization: The main (time consuming) programs are all parallelized using OpenMP and can use the corresponding threaded BLAS, LAPACK, and FFTW libraries. It is activated by either setting the OMP_NUM_THREADS variable globally or using omp_prog:N directives in .machines. While many parts of the code scale very well with the number of parallel threads on a multi-core shared memory machine, unfortunately, the scaling of the matrix diagonalization is, at present, limited to 2–4 cores due to performance bottlenecks in the corresponding OpenBLAS or MKL libraries. k-point parallelization: This together with OpenMP is a very simple and highly efficient parallelization, which works even on a loosely coupled cluster of simple PCs with a slow network for small to medium sized cases (up to 100 atoms/cell) where the eigenvalue problem needs to be solved for several $\\mathbf{k}$ -points. It requires a common (NFS) filesystem on all machines and password-less ssh (private/public keys). $N$ lines speed:hostname in .machines will split the list of kpoints into $N$ junks, and $N$ jobs will be started in parallel on the corresponding hosts, followed by a summation step of the partial densities. Since WIEN2k can use temporary (local) storage for the eigenvectors, we are not limited in the number of $\\mathbf{k}$ -points and our personal record is a NMR chemical shift calculation for fcc Al with $10^{6}\\mathbf{k}$ -points.  \n\nMPI-parallelization:  \n\nWith a sufficiently powerful hardware (at least 16 cores or a cluster with InfiniBand network) and for medium to large sized problems (more than 50 atoms/cell), it is possible, and actually necessary, to parallelize further using MPI. Besides a tremendous speedup that can be achieved by parallelization over atoms and, in particular, over basis functions, this version will distribute the necessary memory on all requested computers, thus allowing calculations for unit cells with more than 1000 atoms.172–174 Such cells require basis sets of about ${10}^{5}$ APWs, and the resulting Hamiltonian, overlap, and eigenvector matrices may need about $500~\\mathrm{GB}$ of memory, which are distributed over the nodes in the standard ScaLAPACK block-cyclic distribution. For such large systems, the solution of the general eigenvalue problem can become the time-limiting $(\\breve{N}^{3})$ step (depending on the number, type, and $R_{\\mathrm{MT}}$ of the atoms, the setup of the complicated matrix elements can take a comparable fraction of the total time), but the ELPA library provides a highly efficient and scalable (1000 cores) diagonalization. In cases with fewer atoms in large cells (isolated molecules in a large box or surface slabs with sufficient vacuum), we can use an iterative diagonalization175 using the previous eigenvectors as start. Depending on the requested number of eigenvalues, this method may be up to 100 times faster than full diagonalization and still scales very well with the number of cores. The MPI version of the code is used when lines with speed:hostname:N (or lines with more than one hostname) in .machines are specified. Of course, coupling of $\\mathbf{k}-$ and MPI-parallelization (and/or OpenMP) is possible.  \n\n# III. PROPERTIES AND FEATURES  \n\n# A. Energy bands, density of states, electron densities  \n\nOnce a self-consistent solution for a chosen atomic structure is done, one can focus on the electronic structure. The energy eigenvalues as a function of the k-vector obtained at the end of the KS calculation define the band structure. The $\\mathbf{k}$ -path along high symmetry lines in the irreducible BZ (see the Bilbao Crystallographic Server176) can be either obtained from WIEN2k default templates or generated graphically using XCrysDen.152 In WIEN2k, one can plot the energy bands by using the program spaghetti, indicating that its interpretation is difficult. However, there are some tools to help. A first tool is a symmetry analysis, which determines the irreducible representation (of the corresponding point group) for each KS eigenvalue. With this knowledge, one can connect the KS eigenstates to bands by using compatibility relations and satisfying the non-crossing rule. The chemical bonding information of state nk is contained in the corresponding wave function $\\psi_{n\\mathbf{k}}$ , which is complex and three dimensional. However, when computing the square of its modulus, one obtains an electron density, which is a real function and easy to visualize. Integrating this (normalized) electron density, one obtains a charge $q$ , which can be decomposed into contributions from the interstitial region $I$ and the atomic spheres $S_{t}$ (labeled by the atom number $t$ and the quantum number $\\ell$ according to the atomic-like basis set),  \n\n$$\n1=\\sum_{t,\\ell}q_{t\\ell}+q_{I}.\n$$  \n\nThis allows us to compress the detailed information contained in the wave function of a single eigenstate state $\\psi_{n\\mathbf{k}}$ to a few numbers that can be stored and analyzed. In addition, WIEN2k decomposes the $q_{t\\ell}$ according to the symmetry of the corresponding point group. For example, the five $d$ -orbitals of a TM atom surrounded by ligands in octahedral symmetry are split into the $t_{2g}$ and $e_{g}$ manifold (crystal field splitting), while for lower symmetry, a splitting into five different $d$ -orbitals is obtained. A review paper8 (Sec. 6.2) illustrates these tools for TiC, a refractory metal (crystallizing in the sodium chloride structure) that is almost as hard as diamond but has metallic, covalent, and ionic bonding contributions. These data are very useful to analyze the electronic structure, and we illustrate this for the band structure of the Heusler compound TiCoSb (see Ref. 177). If one wants to know which atomic states (e.g., Co- $d$ , Ti- $d,$ , or $S{\\mathsf{b}}{-}P$ states) contribute most to a certain band, one can show the character of the bands, sometimes called “fat bands.” For each eigenvalue $\\varepsilon_{n\\mathbf{k}}$ , the size of the circle represents the weight of the chosen character (e.g., a particular $q_{t\\ell m}\\rangle$ . Figure 7 shows which band states originate mainly from Co- $d_{\\mathrm{{i}}}$ , Ti- $d$ , or $S{\\mathsf{b}}{-}P$ states, giving the band structure a chemical interpretation.  \n\nFrom the KS eigenvalues calculated on a sufficiently fine k-grid in the irreducible BZ, one can obtain the density of states (DOS), usually by means of the (modified) tetrahedron method.178 By using the partial charges [Eq. (26)], one can decompose the total DOS into partial DOS (PDOS), which are useful for understanding chemical bonding and interpreting various spectroscopic data. Figure 8(a) shows how much each region (atomic spheres of Co, Ti, Sb, and the interstitial) contributes to the total DOS. We show the dominating valence contributions from the $C o{-}3d$ and Ti- $\\cdot3d$ electrons in Fig. 8(b) and those from $S\\mathrm{b}{-}5s/5p$ in Fig. 8(c). Unfortunately, the interstitial PDOS cannot be decomposed into atomic and $\\ell$ -like contributions uniquely. However, by analyzing the atomic orbitals in the free atom, we see (Table I) that only a fraction of the related electron density resides inside the corresponding atomic sphere, e.g., $81\\%$ for the $C o{-}3d$ but only $15\\%$ for the Co-4s orbital. Therefore, a significant part of the density lies outside the atomic sphere, leading to a non-negligible PDOS from the interstitial region [Fig. 8(a)]. A simple scheme to eliminate this interstitial part is to renormalize the partial DOS with a factor $q_{t\\ell}^{\\mathrm{ren}}$ (determined by a least squares fit) such that the sum of the renormalized PDOS contributions yields the total DOS,  \n\n![](images/228fe188be47facfb81fa847594d3921f6ef8264f033452fcdfab4b6d2e7f0b5.jpg)  \nFIG. 7. Band structure of TiCoSb with emphasis on Ti-d (blue), Co-d (red), and $S_{b-p}$ (black). The size of the circles in this fat band plot is proportional to the corresponding partial charge.  \n\n![](images/871561c65e8cd16f5b6b4ab11b8ffe94653d62bf8243a1dbf71a7ecfcd2dfbb4.jpg)  \nFIG. 8. Total, partial, and renormalized partial DOS of TiCoSb. (a) Total DOS decomposed into atoms and interstitial. (b) Ti- $\\cdot d$ and Co-d (renormalized) PDOS. (c) Sb-s, $p$ (renormalized) PDOS. (d) Ti-s, $p$ (renormalized) PDOS. (e) Co-s, $p$ (renormalized) PDOS.  \n\nTABLE I. Fraction $q_{t\\ell}^{\\mathsf{f r e e}}$ of the charge density of atomic orbitals (atom t and momentum $\\ell)$ that resides inside the corresponding atomic sphere of the free atom and the renormalized charge $q_{t\\ell}^{\\mathsf{r e n}}$ in the solid.   \n\n\n<html><body><table><tr><td></td><td>qfre</td><td>qre</td></tr><tr><td>Co-4s</td><td>0.34</td><td>0.48</td></tr><tr><td>Co-4p</td><td></td><td>0.45</td></tr><tr><td>Co-3d</td><td>0.95</td><td>0.90</td></tr><tr><td>Ti-4s</td><td>0.15</td><td>0.30</td></tr><tr><td>Ti-4p</td><td></td><td>0.43</td></tr><tr><td>Ti-3d</td><td>0.81</td><td>0.86</td></tr><tr><td>Sb-5s</td><td>0.60</td><td>0.61</td></tr><tr><td>Sb-5p</td><td>0.35</td><td>0.40</td></tr></table></body></html>  \n\n$$\n\\mathrm{DOS}=\\sum_{t,\\ell}\\mathrm{PDOS}_{t\\ell}+\\mathrm{PDOS}_{I}=\\sum_{t,\\ell}\\mathrm{PDOS}_{t\\ell}/q_{t\\ell}^{\\mathrm{ren}}.\n$$  \n\nThis sum runs only over the “chemical” $\\ell$ , which are the main contributions. In Table I, we see that the $q_{t\\ell}^{\\mathrm{ren}}$ are close to the free atom situation for more localized orbitals but differs significantly, for example, for Co-4s, which is more localized in the solid than in the free atom. For the importance of this effect, see Sec. III B.  \n\nThe fundamental variable in DFT is the electron density $\\rho$ , which can be compared to experimental data. The total $\\rho$ , which is obtained by summing over all occupied states, can be decomposed into its contributions coming from the core, semi-core, and valence states. A variety of tools (such as XCrysDen152 or VESTA153) allows one to visualize the density $\\rho$ along a line, in a plane (2D), or in the unit cell (3D). One can easily compute the density corresponding to a selected energy window of electronic states in order to visualize their bonding character. By taking the difference between the crystalline density and a superposition of atomic densities (placed at the atomic position of the crystal), one obtains a difference density $\\Delta\\rho=\\rho_{\\mathrm{crystal}}-\\rho_{\\mathrm{atoms}}$ , which shows chemical bonding effects much more clearly than the total or valence density. Figure 9 provides an illustration of $\\Delta\\rho$ for TiCoSb within the (110) plane, where we can observe the strong asphericities in the electron density around the Ti and $\\scriptstyle\\mathrm{Co}$ atoms originating from different occupations of the five 3d orbitals, as well as the charge transfer (discussed below using Bader charges).  \n\nWhen we want to compare the computed electron densities or the related $\\mathrm{\\DeltaX}$ -ray structure factors (computed using the lapw3 module) with experimental data, we must take into account the motion of the nuclei. In DFT calculations, we assume that the nuclei are at rest, whereas in an experiment, this motion must be considered, for example, by means of the Debye–Waller factors, which can also be calculated by phonon calculations.  \n\n![](images/1123365d7e55a68629218a99bac59e5b0bb927bf956c69b3134f02a8ceca48e8.jpg)  \nFIG. 9. Difference density $\\Delta\\rho=\\rho_{\\sf c r y s t a l}-\\rho_{\\sf a t o m s}$ $(e^{-}/\\mathsf{b o h r}^{3})$ ) of TiCoSb in the (110) plane.  \n\nIt is a strength of theory to allow various decompositions (of the DOS or electron density, for instance), which are often useful for interpreting properties, but these may depend on the basis set used in a calculation, for example, when deriving atomic charges. In a LCAO scheme, one takes the weights of all atomic orbitals centered at a given atom to determine how much charge corresponds to that atom (Mulliken’s population analysis). In an APW scheme, the charge inside the related atomic sphere would give an atomic charge, but this value clearly depends on the chosen atomic radius and lacks the interstitial contribution. However, the renormalized partial charges ${Q}_{t}^{\\mathrm{ren}}$ obtained from an integral over the renormalized PDOS gives a meaningful measure of charge transfer, as shown in Table II.  \n\nA basis-set independent alternative is the AIM procedure proposed by Bader,170,171 which is based on a topological analysis of the density. It uniquely defines volumes (called “atomic basins”) that contain exactly one nucleus by enforcing a zero-flux boundary: $\\nabla\\rho\\cdot{\\hat{n}}=0$ . Inside such an atomic basin, this scheme uniquely defines t∇he Ba=der charge for a given density independently of the basisset method that was used to calculate the electron density.170,171 An example of the application of the AIM method is given in Table II, which shows the charge inside the atomic basins of TiCoSb determined with the aim module of WIEN2k. $Q_{t,\\mathrm{crystal}}^{\\mathrm{Bader}}$ is the nuclear charge $Z_{t}$ minus the number of electrons (a positive value indicates a depletion of electrons) using the SCF density, and $Q_{t,\\mathrm{super}}^{\\mathrm{Bader}}$ is the same quantity but using a density from a superposition of the free neutral atoms. In this crystal structure, according to a Bader analysis, even the superposition of neutral densities leads to a significant charge transfer from Ti to $\\scriptstyle\\mathrm{Co}$ and Sb, which is enhanced for $\\scriptstyle\\mathrm{Co}$ and Ti but reduced for Sb during the SCF cycle. These Bader charges can be compared to the $\\boldsymbol{Q}_{t}^{\\mathrm{ren}}$ . As we can see, in both methods, there is a transfer of electrons from the Ti to the Co atom, but the specific amount and, in particular, the charge state of Sb differs significantly depending on the way it is calculated. An inspection of the difference density (Fig. 9) shows a negative $\\Delta\\rho$ around Sb and thus indicates a positive charge of the Sb atom in contrast to the Bader charges, which seems to pick up a lot of charges in the interstitial region leading to a negative Sb charge. In essence, one should be careful with quantitative charge state assignments.  \n\nAn alternative to aim is the Critic2 package,168,169 which determines Bader charges using a pre-calculated 3D mesh of densities. It is very fast; however, the integration of total charges on such a crude mesh is inaccurate and one should restrict its usage for magnetic moments (integrating the spin densities) or valence charges densities (be careful with the 3D mesh).  \n\nTABLE II. Bader charges $Q_{t,\\mathrm{crystal}}^{\\mathsf{B a d e r}}$ and $Q_{t,\\mathtt{s u p e r}}^{\\mathtt{B a d e r}}$ using the SCF and the free-atom superposed density, respectively, and renormalized atomic charges $Q_{t}^{\\mathsf{r e n}}$ of TiCoSb.   \n\n\n<html><body><table><tr><td>Atom</td><td></td><td>Qeadper</td><td>Qren</td></tr><tr><td>Co</td><td>-0.89</td><td>-0.18</td><td>-1.25</td></tr><tr><td>Ti</td><td>+1.28</td><td>+0.82</td><td>+0.85</td></tr><tr><td>Sb</td><td>-0.39</td><td>-0.64</td><td>+0.40</td></tr></table></body></html>  \n\n# B. Photoelectron spectroscopy  \n\n# 1. Valence-band photoelectron spectroscopy  \n\nExperimental valence band photoelectron spectra (PES) are often just compared to the total DOS. Such a comparison, however, can at best reproduce certain peak positions, but usually not the experimental intensities. This is even more true with modern synchrotron-based hard X-ray PES (HAXPES), where the spectra differ considerably depending on the excitation energy. This is because the cross sections of different atomic orbitals change dramatically as a function of excitation energy and this effect should be taken into account. The pes module179 of WIEN2k uses the partial DOS $\\mathrm{(PDOS}_{t\\ell.}$ ) and multiplies it with the corresponding energydependent atomic orbital cross sections $\\sigma_{t\\ell}$ , taken from various tables,180,181  \n\n$$\nI=\\sum_{t,\\ell}{\\mathrm{PDOS}_{t\\ell}\\sigma_{t\\ell}}.\n$$  \n\nIn addition, pes can use the renormalized PDOS (see Sec. III A), so that the contributions from the less localized orbitals (whose wave functions are mainly in the interstitial region) are also properly taken into account. This module allows us to specify the X-ray energy and can handle unpolarized and linearly polarized light as well as linear dichroism in angular distribution (LDAD). It was successfully applied for various examples179 $\\mathrm{SiO}_{2}$ , $\\mathrm{Pb}{\\mathrm{O}}_{2}$ , $\\mathrm{CeVO_{4}}$ , $\\mathrm{In}_{2}\\mathrm{O}_{3}$ , and $\\mathrm{ZnO}_{\\cdot}^{\\cdot}$ ).  \n\nHere, we compare in Fig. 10 the experimental HAXPES spec$\\mathrm{{trum}^{177}}$ of TiCoSb at $6\\ensuremath{\\mathrm{\\keV}}$ with the theoretical calculation. The theoretical spectrum reproduces the experimental intensities very well, but the bandwidth is too small so that the Sb-s peak has about $1~\\mathrm{eV}$ less binding energy. This is a well-known DFT problem and concerns all states at lower energy. The decomposition of the total spectrum allows us to analyze the contributions to the different peaks in the spectrum. The low energy feature is almost exclusively from Sb-s states, the double peak at $-5\\mathrm{\\eV}$ is from $S{\\mathsf{b}}{-}P$ states, and the double peak at $-2~\\mathrm{eV}$ is from Co- $\\cdot d$ states. However, Co-s also contributes significantly to the feature at $-6~\\mathrm{eV}$ , and $S{\\mathsf{b}}{-}P$ contributions are even larger than $C o{-}3d$ for the lowest binding energy peak at $-1.5\\mathrm{eV}$ . All Ti contributions are very small and thus not shown in Fig. 10.  \n\n![](images/5998269f6b029754011da32e12e314ccad3f23d5912bd79453db34bd4d212a27.jpg)  \nFIG. 10. Experimental177 and theoretical PES of TiCoSb at $6~\\mathsf{k e V}.$ . The theoretical spectrum is further decomposed into its main contributions Sb-s, $p$ and Co-s, p, d.  \n\n# 2. Core-level photoelectron spectroscopy  \n\nX-ray photoelectron spectroscopy (XPS) determines the binding energy (BE) of core states. These BEs are specific to certain atoms, but the possible small changes of BEs (core level shifts) provide important additional information about the chemical environment and, in particular, the oxidation state of that element. Since WIEN2k is an all-electron method, it has the self-consistent core eigenvalues available and one could calculate their BE as energy difference with respect to the Fermi level. However, according to Janak’s theorem,182 DFT eigenvalues represent the partial derivative of the total energy with respect to the orbital occupancy and are therefore not necessarily good approximations of experimental excitation energies. Such BEs are typically $10\\%{-}20\\%$ too small. In fact, even BE differences (core level shifts) from ground-state calculations might not be reliable because screening effects of the final state are not included. Much better approximations to experimental BEs can be obtained according to Slater’s transition state theory,183 where half of a core electron is removed.173,184,185 The corresponding SCF eigenvalue, which represents the slope of the total energy vs occupation at half occupation, is a much better approximation to the actual energy difference for $n$ and $n\\mathrm{~-~}1$ occupation, and typical BE errors are reduced to a few percent. For solids, such calculations should employ large supercells where only one of the atoms gets excited. This method also allows for some possible screening due to the valence electrons but suffers from the fact that, in solids, a neutral unit cell is required. The missing half electron can be compensated by adding a negative background charge, by increasing the number of valence electrons by one half, or by playing slightly with the nuclear charge186 according to a virtual crystal approximation. However, it is not always clear which of these methods should be preferred. Successful applications include, for instance, the N-1s shifts of h-BN covered Pt, Rh, and $\\mathrm{Ru}(111)$ surfaces (with a unit cell of the “nanomesh” containing more than 1000 atoms)173 or to ${\\mathrm{Pb}}{-}5d$ and Ta- $4f$ shifts in the misfit layer compound $(\\mathrm{Pb}\\mathsf{S})_{1.14}\\mathrm{TaS}_{2}$ .185  \n\nThe work function can also be obtained from surface slab calculations as the difference of the Fermi level and the Coulomb potential in the middle of the vacuum region. An example can be found for free and h-BN covered $\\mathrm{\\DeltaNi}$ and Rh(111) surfaces in Refs. 187 and 188. One has to carefully check the convergence of the work function with respect to the size of the vacuum region.  \n\n# C. X-ray absorption/emission spectroscopy and electron energy loss spectroscopy  \n\nExperimental techniques such as $\\mathrm{\\DeltaX}$ -ray emission (XES), nearedge X-ray absorption (XAS, NEXAFS, and XANES), and electron energy loss spectroscopy (EELS) represent an electronic transition between a core state and a corresponding valence/conduction band state, which leads to the measurement of emitted/absorbed X-rays or the energy loss of transmitted electrons. The intensity of such a spectrum is given by Fermi’s golden rule according to dipole transitions between an initial $(\\Psi_{I})$ and a final $(\\Psi_{F})$ state,  \n\n$$\nI(E)\\propto\\langle\\Psi_{I}|\\varepsilon{\\bf r}|\\Psi_{F}\\rangle^{2}\\delta(\\varepsilon_{F}-\\varepsilon_{I}-E).\n$$  \n\nThe dipole selection rule is valid when the $\\mathrm{\\DeltaX}$ -ray energy is not too large and limits transitions between a core state on atom $X$ and angular momentum $\\ell$ into/from a conduction/valence band state with $\\Delta\\ell\\pm1$ on the same atom. In essence, the spectrum is calculated from± the corresponding partial DOS times the squared radial matrix element. In the case of polarized light and oriented samples, the orientation-dependent spectra can be obtained by substituting the $\\ell$ -like partial DOS by an appropriate $\\ell m$ -like DOS, e.g., for a K-spectrum of a tetragonal/hexagonal system by replacing the total $\\boldsymbol{p}$ -DOS by $p_{z}$ and $\\b{p_{x}}+\\b{p_{y}}$ -DOS.189  \n\nSuch a scheme leads to very good results for XES spectra, where the final state has a filled core-hole and the valence-hole is usually well screened. For XANES and EELS spectra, however, the final state190 determines the spectrum. The final state has a core hole and an excited electron in the conduction band, and they will interact with each other leading to strong excitonic effects. In order to describe this effect in a DFT-based band-structure code, one has to create a supercell (as large as possible, but depending on the specific system and the hardware resources) of about 32–256 atoms and remove a core electron from one of the atoms. This electron should be added either to the valence electrons (if there are proper states in the conduction band, e.g., in B–K edges of BN) or to the constant background charge (if the lowest conduction band states are of completely wrong character, e.g., for the $_{\\mathrm{O-K}}$ edge of a TM-oxide) to keep the system neutral. In the following SCF cycle, the valence states on the atom with the core hole will get a lower energy and localize, but the surrounding electrons are allowed to partially contribute to the screening.  \n\nAs an example, the ${\\mathrm{Cs}}{\\mathrm{-}}{\\mathrm{L}}_{3}$ spectrum of $\\mathrm{CsK}_{2}\\mathrm{Sb}$ is shown in Fig. 11(a) and compared with state-of-the-art calculations191 using the Bethe–Salpeter equation (BSE) (for further details on BSE, see Sec. III I 3). Obviously, the spectrum calculated with the ground state electronic structure is very different from core hole supercell or BSE calculations, where the spectral weight is redistributed into the first (excitonic) peak. In particular, the $\\cos-5d$ states come down in energy and partially screen the core hole [Fig. 11(b)].  \n\n![](images/216f2fcf5cb603d409f8dc49d8ddcd659f88ce3f4afb11b40939c5684365ccca.jpg)  \nFIG. 11. (a) $C s\\mathrm{-}L_{3}$ spectrum of $\\mathsf{C s K}_{2}\\mathsf{S b}$ calculated using the ground state or a core hole in 32 or 128 atom supercells. These calculations are compared with the BSE results from Ref. 191, which are aligned at the first main peak as it is not possible to calculate such spectra on an absolute energy scale. The spectra are broadened with a Gaussian of $0.5\\mathsf{e V}$ and a Lorentzian of $0.2\\mathsf{e V}.$ (b) Partial Cs-d DOS for a Cs atom with and without core hole.  \n\nThe Cs-6s contributions (not shown) are 1–2 orders of magnitude smaller because both their dipole matrix elements and the 6s-PDOS are smaller than the corresponding $\\cos-5d$ quantities. On the other hand, the BSE and core hole supercell calculations agree quite well. Note that the BSE data have a larger broadening. In this example, the size of the supercell is easy to converge, but it should be noted that for details of the spectrum, a rather good $\\mathbf{k}$ -mesh $8\\times8\\times8$ for the 128 atom supercell) is necessary.  \n\nWhile the core hole approach works generally quite well, it also has clear limitations or needs extensions:  \n\n(i) For metals or small bandgap semiconductors, a full core hole is sometimes too much because the static screening in the supercell might not be enough. Better results can be obtained using a “partial” hole, 93 although adjusting the size of the hole until the resulting spectra match the experiment is not fully ab initio anymore.   \n(ii) The used DFT approximation may not be accurate. This can concern the ground state, for instance, the $_{\\mathrm{O-K}}$ edge of NiO could be greatly improved using the TB-mBJ approxima$\\mathbf{tion}^{194}$ instead of PBE or $\\mathrm{PBE}+U.$ . Even more problematic are excited-state effects due to the additional $d$ electron in strongly correlated materials (for instance, $3d$ in TM oxides), where very poor $\\mathbf{L}_{2,3}$ edges are obtained in single particle approaches and sophisticated methods such as dynamical mean field theory (DMFT, see Sec. III I 5) or configuration interactions195 are needed.   \n(iii) For early $(^{\\alpha}d^{0}\")$ TM compounds, the L2,3 edges are influenced strongly by interactions and interference effects between the $2p_{1/2}$ and $2p_{3/2}$ states, which are split only by a few eV. This can be accounted for using fully relativistic BSE calculations (see Sec. III I 3), where both the $2p_{1/2}$ and $2p_{3/2}$ states are taken into account simultaneously.196   \n(iv) The B-K edge of hexagonal BN has been investigated many times in the literature .189,197 While the first strong excitonic peak originating from antibonding $\\mathbf{B}{-}p_{z}$ $(\\pi^{*})$ states is well described by supercell calculations and in full agreement with BSE calculations,197 the experimental double peak at around $7\\mathrm{eV}$ above the $\\pi^{*}$ peak originating from $\\sigma^{*}$ $(\\mathrm{B}{-}p_{x,y})$ states shows up in the calculations only as a single peak. This can only be fixed by taking electron–phonon interactions into account. The approach is based on statistical averages over all vibrational eigenmodes of the system.197 Thus, one calculates first the vibrational modes of h-BN and then, using a supercell of e.g., 128 atoms, all atoms are displaced according to the vibrational eigenmodes with amplitudes determined by the Bose–Einstein occupations. Even at $T=0\\mathrm{K},$ the zero-point motion is enough to split the degenerate $p_{x}$ and $\\b{p_{y}}$ states, since, on average, each B will have one $N$ neighbor at a smaller (larger) distance than the two others. This leads to the desired splitting of the single $\\sigma^{*}$ peak into a double peak.19  \n\nEELS has fairly similar basic principles as XAS but differs slightly because of the finite momentum of the electrons.198 The telnes3 module of WIEN2k calculates the double differential scattering cross section on a grid of energy loss values and impulse transfer vectors. This double differential cross section is integrated over a certain momentum transfer $q$ to yield a differential cross section, which can be compared to the experiment. This formalism allows the calculation of relativistic EELS including transitions of arbitrary order (i.e., non-dipole transitions), and it can take into account the relative orientation between the sample and beam.199 Practical aspects on how to perform EELS calculations have been given by Hébert,200 and some examples can be found in Refs. 201–204.  \n\n# D. Optics  \n\nThe optics module of WIEN2k uses the independent-particle approximation (IPA) and calculates the direct transitions (conserving k) between occupied nk and unoccupied $n^{\\prime}\\mathbf{k}$ states, where for both states, KS eigenvalues are used.205 The joint density of states is modified by transition probabilities given by the square of the momentum matrix elements $M=\\langle n^{\\prime}\\mathbf{k}|A{\\cdot}p|n\\mathbf{k}\\rangle$ between these states, which determine the intensity of o⟨ptical spect⟩ra using dipole selection rules and clearly distinguish between optically allowed and forbidden transitions. From the resulting imaginary part $\\varepsilon_{2}$ of the dielectric function, its real part $\\varepsilon_{1}$ can be obtained by the Kramers– Kronig transformation and then additional optical functions such as conductivity, reflectivity, absorption, or the loss function can also be calculated. In a metallic solid, an additional Drude term accounts for the free-electron intraband contribution. For insulators and semiconductors, where the DFT gap is often too small when compared to the experiment, one can use a “scissor operator.” This sounds complicated but is nothing else than a rigid shift of the unoccupied DFT bands to adjust the (too small) DFT bandgap, either using the experimental gap or, more ab initio, using the gap calculated with TB-mBJ. Note that TB-mBJ usually gives very good bandgaps, but the bandwidth of both the valence and conduction bands are too small, and thus, the optical properties with TB-mBJ might not be very accurate, but still more accurate than standard GGA (see Sec. III B 2 in Ref. 104 for a brief summary of literature results with TB-mBJ). Alternatively, hybrid-DFT functionals66 can be used, which give quite good bandgaps for semiconductors, but one should be aware that the optical properties usually require a quite dense k-mesh, which makes hybrid calculations fairly expensive.  \n\nAs an example we present in Fig. 12 the imaginary part of the dielectric function $\\varepsilon_{2}$ for $\\mathrm{CsK}_{2}\\mathrm{Sb}$ using various approximations. First, we note that very dense $\\mathbf{k}$ -meshes are necessary for converged results, which makes the application of more expensive many-body perturbation methods such as $G W$ even more difficult. As expected, PBE calculations yield the smallest bandgap of $1.06\\mathrm{eV}$ , while hybrid YS-PBE0 gives $1.68\\mathrm{~eV}$ and TB-mBJ $2.08\\mathrm{eV}$ . This can be compared to $G_{0}W_{0}$ results191 of $1.62\\ \\mathrm{eV}$ or early experimental estimates of $1.0\\mathrm{-}1.2\\mathrm{eV}$ , which, however, have been criticized. It should be noted that the GW result191 for $\\varepsilon_{2}$ probably suffers from an under converged k-mesh and even with their large smearing (note the large tail below $1.62\\ \\mathrm{eV}$ in the $G W$ results shown in Fig. 12), a distinct peak structure emerges, which is not present in the $\\mathbf{k}$ -converged results.  \n\nExperimental results for optical conductivity, reflectivity, or absorption as well as the low energy valence electron energy loss spectrum (VEELS) can often be successfully interpreted in the IPA.203,204 However, sometimes (in particular, for wide gap insulators) the frequency-dependent dielectric function ε in the IPA may have little in common with the experimental situation. This is because excitations are two-particle processes and the missing electron–hole interaction (i.e., the excitonic effect already mentioned in Sec. III C) can significantly affect the calculated optical response of a material when they are strong. In order to overcome this problem, one needs to include the electron–hole correlation explicitly by solving the BSE (see Sec. III I 3).  \n\n![](images/c565bc2299b7c2373ff81646b962c7e91410e8c6a387bbf66348bb545725d5be.jpg)  \nFIG. 12. Imaginary part of the dielectric function $\\varepsilon_{2}$ for $\\mathsf{C s K}_{2}\\mathsf{S b}$ with various functionals. The PBE calculations are presented with a $6\\times6\\times6$ and $20\\times20\\times20$ k-mesh, and all other calculations have used the larger×mes×h. The GW r×esul s×are from Ref. 191.  \n\n# E. Magnetism  \n\nWhen magnetism occurs in a solid, it may come from localized electrons (e.g., from $f$ electrons of rare-earth atoms) or itinerant (delocalized) electrons (e.g., in Fe, Co, or Ni). In any case, magnetism comes mainly from exchange splitting causing a partial occupation of states, which differ between the spin-up $(N_{\\uparrow})$ and spin-down $(N_{\\downarrow})$ electrons. The corresponding magnetic moment $M$ is defined as the difference between these occupation numbers $(M=N_{\\uparrow}\\ -\\ N_{\\downarrow})$ . For such cases, one must perform spin-polarized calculations $(r u n s p\\textunderscore l a p w)$ and needs the spin density in addition to the total electron density. The default is collinear magnetic order as found in ferromagnets, for example, in Fe, Co, Ni, or antiferromagnets, for example, in Cr. In addition to collinear magnets mentioned here, one can also handle non-collinear magnetism (for example, systems with canted magnetic moments or spin spirals), as described in Sec. III I 2.  \n\nFor spin-polarized calculations of a specific complex antiferromagnetic structure, in most cases, it is essential to specify proper (antiferromagnetic) atomic spin-moments as an input for the SCF cycle and the WIEN2k tool instgen allows us to set this easily. If one is interested in the orientation of the magnetic moments with respect to the crystal structure (easy or hard axis) or the magneto-crystalline anisotropy,206 the SO interaction must be included. For heavy elements or when orbital moments become important, one needs this full relativistic treatment by including the SO interaction.  \n\nNowadays, one can study complicated systems, for example, ${\\tt B a F e}_{2}{\\tt O}_{5}$ , an oxygen-deficient perovskite-like structure, which shows a Verwey transition. At low temperature, this system has a charge-ordered state (with $\\mathrm{Fe}^{2+}$ and $\\mathrm{Fe}^{3+}$ at different Fe-sites), but above the Verwey transition temperature (at about $309\\mathrm{K}$ ), a valence mixed state with the formal oxidation state $\\mathrm{Fe}^{2.5+}$ appears. DFT calculations made it possible to interpret this complicated situation (for details see Ref. 207 and Sec. 7.4.1 of Ref. 208). Another complex system is $\\mathrm{Pr}\\mathrm{O}_{2}$ , which has a Jahn–Teller-distorted $\\mathrm{CaF}_{2}$ structure.209 It contains $\\operatorname{Pr}\\sp{-4f}$ electrons, which form a localized band (lower Hubbard band) for one $4f$ electron, but the others hybridize with the valence electrons forming a situation between $\\mathrm{Pr}^{\\dot{3}+}$ and $\\mathrm{Pr}^{4+}$ . This compound is an antiferromagnetic insulator that requires a relativistic treatment. $\\mathrm{PBE}+U$ calculations93 provide results that are consistent with all experimental data for the bandgap, magnetic moment, and structural distortion.  \n\nIn the 1980s, a numerical problem occurred in connection with several studies of the $\\mathrm{Fe}_{65}\\mathrm{Ni}_{35}$ INVAR alloy, which has a vanishing thermal expansion around room temperature. This is one of the systems for which the magnetization shows a hysteresis when a magnetic field is applied. The hysteresis causes numerical difficulties, since for a certain magnetic field, there are three solutions (magnetic moments) with very similar total energies causing difficulties in the convergence of a conventional SCF procedure. In order to solve this problem, the fixed spin moment (FSM) method was proposed.210,211 It is a computational trick interchanging dependent and independent variables. Physically speaking, one applies a magnetic field and obtains a moment, but computationally one chooses the moment (as input) and calculates the field afterwards. In a conventional spin-polarized calculation, the Fermi energy must be the same for the spin-up and spin-down electrons. The magnetic moment $M$ is an output. In the FSM scheme, one does several constrained calculations, where the moment $M$ is an input, but allowing different Fermi energies for the two spin states. One can interpret the difference in the Fermi energies as a magnetic field. Although one needs to perform several calculations (instead of a single conventional one), they converge rather rapidly. The FSM method allows expanding the usual total-energy vs volume curve to an energy surface $E_{\\mathrm{tot}}(V,M)$ as a function of volume $V$ and moment $M$ , which also provides new insights.  \n\n# F. Hyperfine fields and electric field gradients  \n\nAll aspects of nucleus–electron interactions, which go beyond the electric point-charge model for a nucleus, define the hyperfine interactions. Nuclei with a nuclear quantum number $I\\geqslant1$ have an electric quadrupole moment $Q.$ . The nuclear quadrupole interaction (NQI) stems from the interaction of such a moment and the EFG, the second derivative of the Coulomb potential at the corresponding nuclear site. One can measure the EFG with Mössbauer spectroscopy, NMR, nuclear quadrupole resonance (NQR), or perturbed angular correlation (PAC). The NQI determines the product of Q and the EFG, a traceless tensor. The latter has a principal component and an asymmetry parameter $\\eta$ . The EFG is a ground-state property that can be determined experimentally (measuring NQI), provided the nuclear quadrupole moment is known. In early studies, the EFG was interpreted as a simple point charge model with additional corrections (Sternheimer factor212). However, later it was shown that the EFG can be calculated with DFT, as was illustrated213 for $\\mathrm{LiN}_{3}$ . The EFG is sensitive to the asymmetric charge distribution around a given nucleus and thus is a local probe, which often helps in clarifying the local atomic arrangement. The reader can find a short description of several EFG calculations for selected examples in Chap. 6.4 of Ref. 8, and here, we describe some important aspects and examples below.  \n\nComputationally, it is important to treat both valence and semicore states very accurately because, due to the $1/r^{3}$ factor in the EFG expression, even small asphericities near the nucleus lead to important contributions. This was demonstrated for the first time for $\\mathrm{TiO}_{2}$ in the rutile structure, where the radial functions of the fully occupied Ti- $3p$ semi-core orbitals are slightly different for $p_{x},p_{y}$ , and $p_{z}$ states and thus contribute significantly to the EFG.28  \n\nThe mapping of the two $\\mathtt{C u}$ -EFGs in the high-temperature superconductor $\\mathrm{YBa}_{2}\\mathrm{Cu}_{3}\\mathrm{O}_{7}$ to the “plane” and “chain” Cu sites provides insight into which Cu atom is responsible for superconductivity, and the analysis of the EFG on all other sites helped us to interpret chemical bonding.214 It should be mentioned that the EFG at the (superconducting) Cu-plane site comes out quite inaccurate and ${\\mathrm{GGA}}+U$ calculations are necessary, leading to a redistribution of $0.07~\\mathrm{e}^{-}$ from a $3d_{x^{2}-y^{2}}$ orbital to a $3d_{z^{2}}$ orbital and an EFG in agreement with the experiment.92  \n\nNext, we briefly discuss the study on 16 fluoroaluminates from Ref. 215, for which experimental NMR data were compared to DFT results for the 27Al EFG. In all of these compounds, the aluminum atoms occur in $\\mathrm{AlF}_{6}^{3-}$ octahedra, which have a wide diversity of connectivity and distortions. One of these structures is shown in the inset of Fig. 13 for $\\mathrm{Ba}_{3}\\mathrm{Al}_{2}\\mathrm{F}_{12}$ . These fluoroaluminates illustrate how sensitive the EFG is to the exact position of neighboring atoms. A perfect octahedral symmetry would have a vanishing EFG, but small distortions cause an EFG. The calculations were first done using the less accurate powder diffraction data for the atomic positions [Fig. 13(a)], and the correlation between experiment and theory is not very good. Then, a DFT structure optimization was done leading to an almost perfect correlation between experimental and theoretical EFGs [Fig. 13(b)]. This structure optimization has an even more pronounced effect on the asymmetry parameter, as shown in Ref. 215. We should mention that there can also be a sensitivity to DFT functionals for EFG calculations as described in Refs. 66 and 80.  \n\nLast but not least, we demonstrate how one can determine the nuclear quadrupole moment Q from a combination of theoretical EFG calculations and experimental measurements of the quadrupole splitting, which is proportional to the product of EFG and Q. From the slope of a linear regression for the EFG of several Fe compounds, we could deduce the nuclear quadrupole moment $Q$ of $^{57}\\mathrm{Fe}_{\\mathrm{:}}$ , the most important Mössbauer isotope. It was found to be about twice as large $Q=0.16\\mathrm{~b~}$ ) as the previous literature value $\\cdot Q=0.082{\\mathrm{~b}}^{\\cdot}$ ), suggesting to revise this nuclear property using electronic structure calculations.216  \n\nThe magnetic hyperfine field (HFF) at a nucleus originates from a Zeeman interaction between the magnetic moment $I$ of this nucleus and the magnetic field at this site produced by the spinpolarized electrons in a ferromagnet. The HFF has contributions from the Fermi-contact term (the spin density at the nucleus), an orbital, and a spin dipolar contribution. Here, we skip the details but mention that an all-electron treatment is crucial, especially for the Fermi-contact term, since one needs accurate values of the spindensity close to the nucleus, for which the basis set used in WIEN2k is extremely useful. We recalculate the densities of all electrons, including the core, in each cycle of the SCF scheme, in contrast to the frozen-core approximation. The resulting core polarization can often be the main contribution to the HFF. The high quality of such calculations was demonstrated for the double perovskite ${\\tt B a F e}_{2}{\\tt O}_{5}$ , for which the DFT calculations of EFG and HFF provided new insights (for details, see Ref. 207).  \n\n![](images/664aa9d0e8e2e1e6176599691929c936fc21e251a5720e8a37e38792bcea9091.jpg)  \nFIG. 13. The calculated EFG (principal component) vs the experimental $^{27}\\mathsf{A l}$ quadrupole frequency for 16 fluoroaluminates is shown for two cases based on (a) the experimental structure and (b) the DFT optimized structure. For further details, see Ref. 215. The inset shows as one example of the fluoroaluminates, the $\\mathsf{A l F}_{6}^{3-}$ octahedra in ${\\mathsf{B a}}_{3}{\\mathsf{A l}}_{2}{\\mathsf{F}}_{12}$ (Ba: blue, Al: red, and F: green).  \n\n# G. NMR chemical and Knight shifts  \n\nThe NMR shielding $\\overleftrightarrow{\\boldsymbol{\\sigma}}$ tensor is defined as a constant between an induced magnetic field $\\mathbf{B}_{\\mathrm{ind}}$ at the nucleus at site $\\mathbf{R}$ and the external uniform field $\\mathbf{B}_{\\mathrm{ext}}$ ,  \n\n$$\n{\\displaystyle{\\bf B}_{\\mathrm{ind}}({\\bf R})=-\\overleftrightarrow{\\boldsymbol{\\sigma}}({\\bf R}){\\bf B}_{\\mathrm{ext}}.}\n$$  \n\nIts value is usually in the range of ppm (part per million). Since the magnetic field cannot be controlled with such a precision, the tensor is measured only with respect to some reference,  \n\n$$\n\\delta({\\bf R})=\\sigma_{\\mathrm{ref}}-\\sigma({\\bf R}),\n$$  \n\nand often only its isotropic part $\\sigma({\\bf R})=\\mathrm{tr}[\\overleftrightarrow{\\boldsymbol{\\sigma}}\\left({\\bf R}\\right)]$ is known.  \n\nThe external magnetic field is a relatively weak perturbation compared to the typical energy scale of the electronic structure; therefore, its effect on the spin and orbit of an electron can be separated in the theoretical calculations. Here, we only outline the main features that are specific to the $\\mathrm{\\APW+lo}$ method and the WIEN2k code and are vital for achieving high accuracy of the computed NMR tensor. A more in-depth discussion can be found in the original publications. The formalism for computing the orbital part of the shielding (chemical shift) has been described in Refs. 217 and 218, and the spin part (Knight shift) of the response has been described in Ref. 219. The formalism has been applied for computing shielding in various insulating78,220,221 and metallic systems.222–224  \n\nAs will be explained below, this approach can reach the basis set limit for NMR calculations, and benchmark calculations for small molecular systems have proven that standard quantum chemistry methods can only reach this precision with very large uncontracted quintuple-zeta basis sets and only for light atoms.225  \n\n# 1. Orbital component  \n\nThe orbital part of the shielding, i.e., the orbital component of the induced field $\\mathbf{B}_{\\mathrm{ind}}$ , is obtained directly from the Biot–Savart law (in atomic units, with $c$ as speed of light),  \n\n$$\n\\mathbf{B}_{\\mathrm{ind}}(\\mathbf{R})={\\frac{1}{c}}\\int\\mathbf{j}(\\mathbf{r})\\times{\\frac{\\mathbf{r}-\\mathbf{R}}{\\left|\\mathbf{r}-\\mathbf{R}\\right|^{3}}}d^{3}r,\n$$  \n\nwhere $\\mathbf{j}(\\mathbf{r})$ is the induced orbital current, evaluated as expectation value of the current operator  \n\n$$\n\\mathbf{J}(\\mathbf{r})=-{\\frac{\\mathbf{p}|\\mathbf{r}\\rangle\\langle\\mathbf{r}|+|\\mathbf{r}\\rangle\\langle\\mathbf{r}|\\mathbf{p}}{2}}-{\\frac{\\mathbf{B}_{\\mathrm{ext}}\\times\\mathbf{r}}{2c}}|\\mathbf{r}\\rangle\\langle\\mathbf{r}|.\n$$  \n\nWIEN2k separates the calculation of valence and core states. The core state contribution to the induced current is computed using the spherically symmetric core density only,  \n\n$$\n\\mathbf{j}_{\\mathrm{core}}(\\mathbf{r})=-\\frac{1}{2c}\\rho_{\\mathrm{core}}(\\mathbf{r})\\mathbf{B}_{\\mathrm{ext}}\\times\\mathbf{r}.\n$$  \n\nThe method for computing the valence contribution to $\\mathbf{j}(\\mathbf{r})$ is based on a linear response approach226–228 originally developed by Mauri, Pfrommer, and Louie.226 The expression for the induced current involves only the first-order terms with respect to the external field Bext,  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\bf j}({\\bf r})=\\sum_{o}\\Bigl[\\langle\\psi_{o}^{(1)}|{\\bf J}^{(0)}({\\bf r})|\\psi_{o}^{(0)}\\rangle+\\langle\\psi_{o}^{(0)}|{\\bf J}^{(0)}({\\bf r})|\\psi_{o}^{(1)}\\rangle}}\\\\ {{\\displaystyle~+~\\langle\\psi_{o}^{(0)}|{\\bf J}^{(1)}({\\bf r})|\\psi_{o}^{(0)}\\rangle\\Bigr]},}\\end{array}\n$$  \n\nwhere ψo(0) is an unperturbed KS occupied orbital, J0(r) is the paramagnetic part of the current operator [the first term in Eq. (33)], and $J^{1}(\\mathbf{\\check{r}})$ is the diamagnetic component of the current operator [the second term in Eq. (33)]. $\\psi_{o}^{(1)}$ is the first-order perturbation of $\\psi_{o}^{(0)}$ , given by the standard formula involving a Green’s function,  \n\n$$\n\\big|\\psi_{o}^{(1)}\\big\\rangle=\\sum_{e}\\big|\\psi_{e}^{(0)}\\big\\rangle\\frac{\\big\\langle\\psi_{e}^{(0)}\\big|H^{(1)}\\big|\\psi_{o}^{(0)}\\big\\rangle}{\\varepsilon_{o}-\\varepsilon_{e}}+\\sum_{\\mathrm{core}}\\big|\\psi_{\\mathrm{core}}^{(0)}\\big\\rangle\\frac{\\big\\langle\\psi_{\\mathrm{core}}^{(0)}\\big|H^{(1)}\\big|\\psi_{o}^{(0)}\\big\\rangle}{\\varepsilon_{o}-\\varepsilon_{\\mathrm{core}}},\n$$  \n\nand involves a sum over empty states (first term) as well as a sum over core states (second term) because the core states have been calculated before [Eq. (34)] and Eq. (36) is only correct if all states of a system are included. Note that core wave functions appear in Eq. (36) as if they were unoccupied states. $\\boldsymbol{H}^{(1)}$ is the perturbation due to the external magnetic field in symmetric gauge,  \n\n$$\n\\boldsymbol{H}^{(1)}=\\frac{1}{2c}\\mathbf{r}\\times\\mathbf{p}\\cdot\\mathbf{B}_{\\mathrm{ext}}.\n$$  \n\nAs discussed in Sec. II A, the basis functions of the $\\mathrm{\\APW+lo}$ method are highly tuned to describe the occupied Bloch states everywhere in the unit cell (in particular, also close to the nucleus), but they are not a complete basis set. Therefore, depending on the perturbation of the Hamiltonian, they may not be well suited to expand the perturbations of wave functions. In fact, magnetic fields and NMR are such cases where the perturbed wave functions, in particular, near the nucleus, are very different, which means that the sum in Eq. (36) cannot be converged with the available set of orbitals. In order to remedy this issue, we had to enhance the original WIEN2k basis set. First, the standard set of local orbitals is extended significantly, both in the number of LOs per $\\ell$ (typically to $5{-}8\\ \\mathrm{LOs})$ and also in $\\ell$ (typically to $\\ell+1\\$ ), where $\\ell$ refers to the maximal “chemical” $\\ell$ of this atom.217 These extra local orbitals are referred to as NMRLOs, and the energy at which the radial functions of those NMR-LOs are computed is chosen such that each of the NMR-LO radial functions has zero value at the sphere boundary and the number of nodes inside the sphere of subsequent LOs increases by one corresponding to the next principal quantum number.217 However, these NMRLOs alone cannot completely improve the variational flexibility close to the nuclei.  \n\nThe perturbation of the Hamiltonian due to the external magnetic field is proportional to a product of position and momentum operators. As a result, the perturbation of the radial wave function $\\boldsymbol{u}_{t\\ell}$ contains components proportional to $u_{t\\ell\\pm1}$ and their radial derivative $\\begin{array}{r}{r\\frac{\\partial}{\\partial r}u_{t\\ell\\pm1}}\\end{array}$ . A direct introduction of basi±s functions based on $r\\frac{\\partial}{\\partial r}u$ is not co±nvenient within the APW formalism because such functions are not eigenstates of the radial Schrödinger equation. Therefore, we have proposed to add the desired term directly to the Green’s function present in the formula for the first-order perturbation of the valence state wave function.218 It is referred to as the $\\mathit{\\Omega}^{\\alpha}\\frac{\\partial\\mathit{\\Omega}}{\\partial\\mathit{r}}u$ correction” (DUC).  \n\nThe convergence test with respect to the number of NMR-LO and DUC corrections is presented in Fig. 14. The induced current and shielding calculated within our linear response formalism is compared to the exact value for an isolated Ar atom. The current and shielding for a spherically symmetric atom can be computed exactly using only its density and the same formula [Eq. (34)] as for the core states (diamagnetic current).  \n\nBoth DUC and several additional NMR-LOs are needed to reproduce the shape of the exact diamagnetic current in a region within 0.5 bohr from the nucleus. It appears in Fig. 14(a) that any error in the representation of the current in this region results in substantial errors of the computed shielding values [Fig. 14(b)]. This method can therefore reach the basis set limit.  \n\nThe all-electron nature and the modular concept of WIEN2k makes it very easy to perform NMR calculations with wave functions including SO interactions for heavy nuclei or using different  \n\n![](images/845fd2dba3bf7fae1e317d28fcb313524ec91a1dfd7018343a5a6aeec6919179.jpg)  \nFIG. 14. (a) Comparison of the induced current computed for an Ar atom (valence states 3s and $3p$ ) with and without DUC corrections and for 2 and 7 NMR-LOs. $j_{\\rho}$ is the (exact) diamagnetic current [see Eq. (34)]. (b) Convergence of NMR absolute shielding $\\sigma$ with respect to the number of NMR-LOs in the APW basis and with/without DUC corrections. The vertical blue line represents the exact value of the absolute shielding $\\sigma_{\\mathrm{Ar}}=1245.7$ ppm computed with $j_{\\rho}$ .  \n\nDFT approximations, including $\\mathrm{DFT}+U$ (Sec. II B 3), but, in particular, also hybrid functionals (Sec. II B 2). Thus, we can compare the theoretical shielding with the experimental chemical shifts for several different compounds, and from the correlation and slope of the linear regression curve, the quality of a particular approximation to the XC effects can be evaluated. Ideally, the slope of this linear regression line should be $^{-1}$ , but typically for ionic compounds,78 the slope with PBE is too large $(-1.2)$ , while with the hybrid functional YS-PBE0,66 it is too small $\\left(-0.8\\right)$ . This is quite in contrast to organic molecules, where hybrid f−unctionals perform much better than the GGAs. Surprisingly, the BJ potential106 performs quite well for ionic oxides or halides and yields slopes close to $^{-1}$ .  \n\nMost importantly, theory should not only reproduce measured experimental values but also provide insights. WIEN2k allows us to analyze and identify the contributions to the NMR shielding, and for instance, for the F-shielding in the alkali-fluoride series (Fig. 15), the following observations can be made:221 (i) Basically, all contributions to the F-shielding come from a region inside the F atomic sphere (the current in the rest of the unit cell contributes negligibly). (ii) The large (diamagnetic) shielding comes from the constant F-1s core contribution. (iii) The contributions from the $\\mathrm{F}{-}2s$ bands are still diamagnetic, but much smaller and again constant within the series. (iv) The diamagnetic metal- $\\cdot p$ semi-core contributions $(\\mathrm{Na}-2p$ to $\\mathsf{C s}{-}5p$ ) increase within the series. This can be explained by the fact that for heavier elements the metal $\\boldsymbol{p}$ -states increase in energy and come closer to the $\\mathrm{F}{-}2p$ band. This leads to an increased bonding (in the metal- $d$ band) and anti-bonding (in the $\\mathrm{F}{\\cdot}p$ band) metal- $\\cdot-\\mathrm{F}-P$ interaction giving slightly different (non-canceling) diamagnetic and paramagnetic contributions. (v) The trend of $\\sigma$ within the series comes mainly from the $\\mathrm{F}{-}2p$ valence band. The most important ingredient, which determines the size of the (mostly paramagnetic) $\\mathrm{F}{-}2p$ contribution, is the position of the unoccupied metal- $d$ band. The perturbation due to the magnetic field couples the occupied F- $\\cdot p$ states with $\\Delta\\ell\\pm1$ to unoccupied $d$ states, and due to the energy denominator in E±q. (36), the Cs- $d$ states give the largest contribution because they are the closest in energy to the valence bands. We can even artificially apply a (large) $U$ value to the empty $\\cos-5d$ states shifting them further up. In this way, we do not alter the occupied states but still can increase the F shielding in CsF to reproduce the LiF or NaF shifts.  \n\n![](images/1a9de80f3255d7f90d1fd9f036a43e923693935a272b89b71ef4bec98fda7cf5.jpg)  \nFIG. 15. $^{19}F$ NMR shielding $\\sigma$ (in ppm) in alkali fluorides (data taken from Ref. 221). Black lines: Total $\\sigma$ (full line) and contribution from within the F-sphere (dashed line). Further decomposition into core F-1s (green), F-2s band (blue), metal- $\\cdot p$ band (red), and $F-2p$ band (brown) is also shown.  \n\nBy a similar analysis, we could explain why the $^{33}\\mathrm{S}$ magnetic shielding decreases with the metal nuclear charge $Z$ in the ionic alkali/alkali-earth sulfides but increases in TM sulfides.220  \n\n# 2. Spin component  \n\nIn order to compute the induced spin density and spin part of the NMR shielding tensor, we use a direct approach,219 instead of applying the linear response formalism proposed, for instance, in Ref. 229. This is possible because the interaction of the spin with the external magnetic field does not break the periodicity. Therefore, we perform self-consistent spin polarized calculations with a finite external magnetic field $\\mathbf{B}_{\\mathrm{ext}}$ acting on the electron spin only. The interaction with $\\mathbf{B}_{\\mathrm{ext}}$ is cast into a spin-dependent potential leading to a spin splitting of eigenstates and a finite spin magnetization. The induced magnetic field at a given nucleus is computed using an expression for the magnetic hyperfine field,230  \n\n$$\n\\mathbf{B}_{\\mathrm{hf}}={\\frac{8\\pi}{3}}\\mu_{\\mathrm{B}}\\mathbf{m}_{\\mathrm{av}}+\\left\\langle\\Phi_{1}\\left\\vert{\\frac{S(r)}{r^{3}}}[3(\\mu\\cdot{\\hat{\\mathbf{r}}}){\\hat{\\mathbf{r}}}-\\mu]\\right\\vert\\Phi_{1}\\right\\rangle.\n$$  \n\nThe first term $\\left(\\mathbf{B}_{\\mathrm{c}}\\right)$ is the Fermi contact term, where $\\mathbf{m}_{\\mathrm{av}}$ is the average of the spin density in a region near the nucleus with a diameter equal to the Thomson radius. The second term $(\\mathbf{B}_{\\mathrm{sd}})$ captures the spin-dipolar contribution to the hyperfine field, where $\\Phi_{1}$ is the large component of the wave function, S is the reciprocal relativistic mass enhancement, and $\\mu$ is the magnetic moment operator of the electron. $\\mathbf{B}_{\\mathrm{sd}}$ comes almost entirely from within the atomic sphere, which simplifies its calculation. The spin contribution $(\\sigma_{s})$ , i.e., the Knight shift to the shielding, is therefore given by two terms,  \n\n$$\n\\mathbf{B}_{\\mathrm{hf}}=-{\\overleftrightarrow{\\boldsymbol{\\sigma}_{s}}}\\mathbf{B}_{\\mathrm{ext}}=-\\Big({\\overleftrightarrow{\\boldsymbol{\\sigma}_{c}}}+\\overleftrightarrow{\\boldsymbol{\\sigma}_{s d}}\\Big)\\mathbf{B}_{\\mathrm{ext}}.\n$$  \n\nIn order to obtain a sizable response and evaluate the NMR shielding with a numerical precision at the level of $1\\ \\mathrm{ppm}$ , we apply in our calculations an external magnetic field of $100\\mathrm{T}$ , which induces a spin-splitting of approximately $1\\mathrm{mRy}$ . These small changes require an extremely fine $\\mathbf{k}$ -point sampling (for fcc Al, $10^{6}\\ \\mathbf{k}.$ -points are needed), and this must always be carefully checked.  \n\nFurther details and results of our approach can be found in Refs. 219, 222, and 223, but, here, we summarize the main findings: (i) The previously accepted point of view, namely, that Knight shifts are proportional to the partial $s$ -DOS at the Fermi energy and the orbital contribution $\\sigma_{\\mathrm{orb}}$ is identical to that in the (ionic) reference compound, is only true for simple $\\boldsymbol{s p}$ metals. (ii) In TM $d$ -elements or metallic compounds, the orbital part $\\sigma_{\\mathrm{orb}}$ can be as important as the spin part $\\sigma_{s}$ . (iii) The $s$ -DOS at $E_{F}$ is always important, but an induced TM- $d$ magnetic moment (proportional to the partial $d$ -DOS at $E_{F}$ ) polarizes the core states in the opposite direction so that the valence and core polarizations can partly cancel. (iv) The dipolar contribution $\\sigma_{\\mathrm{sd}}$ is usually small, but, in anisotropic materials, a large dominance of one particular orbital at $E_{F}$ can eventually lead to a very large contribution. We have found this in $\\mathrm{BaGa}_{2}$ , where the $p_{z}{\\mathrm{-DOS}}$ has a large and sharp peak at $E_{F}$ leading to an aspherical magnetization density and a large dipolar contribution.222  \n\n# H. Wannier functions and Berry phases  \n\nA single particle state of a periodic system is conventionally represented as a Bloch state $\\psi_{n\\bf{k}}({\\bf{r}})$ , which is labeled by a band index $n$ and a vector $\\mathbf{k}$ inside the first BZ. It satisfies Bloch’s theorem  \n\n$$\n\\psi_{n\\mathbf{k}}(\\mathbf{r})=u_{n\\mathbf{k}}(\\mathbf{r})e^{i\\mathbf{k}\\cdot\\mathbf{r}},\n$$  \n\nwhere $u_{n\\mathbf{k}}(\\mathbf{r})=u_{n\\mathbf{k}}(\\mathbf{r+R})$ is a lattice periodic function and $\\mathbf{R}$ is a Bravais lattice vector. Alternatively, one can define Wannier functions (WFs) wnR(r) in unit cell R for a set of J bands as231,232  \n\n$$\nw_{n\\mathbf{R}}(\\mathbf{r})={\\frac{V}{(2\\pi)^{3}}}\\int_{\\mathrm{BZ}}e^{-i\\mathbf{k}\\cdot\\mathbf{R}}\\sum_{m=1}^{J}U_{m n}^{\\mathbf{k}}\\psi_{m\\mathbf{k}}(\\mathbf{r})d^{3}k,\n$$  \n\nwhere $V$ is the unit cell volume and $U^{\\mathbf{k}}$ are unitary transformation matrices that mix Bloch states at a given k. Because of the arbitrary phase of $\\psi_{n\\bf{k}}(\\bf{r})$ , the resulting WFs are usually not localized. Maximally localized WFs can be obtained by choosing the $U^{\\mathbf{k}}$ such that the spread $\\Omega$ is minimized,  \n\n$$\n\\Omega=\\sum_{m=1}^{J}\\left[\\left\\langle w_{m\\mathbf{R}}\\middle|r^{2}\\middle|w_{m\\mathbf{R}}\\right\\rangle-\\left\\langle w_{m\\mathbf{R}}\\middle|r\\middle|w_{m\\mathbf{R}}\\right\\rangle^{2}\\right].\n$$  \n\nThis involves overlap integrals $M_{m n}^{\\mathbf{k},\\mathbf{b}}=\\langle u_{m\\mathbf{k}}|u_{n\\mathbf{k}+\\mathbf{b}}\\rangle$ between the periodic part of the wave functions on  uniform grid in the BZ.  \n\nThe maximally localized WFs are calculated by Wannier90,231 and the transformation matrices $U^{\\mathbf{k}}$ are provided by the wien2wannier232 module of WIEN2k. The resulting WF can be used for various tasks. They can be visualized and are useful for the interpretation of chemical bonding to generate tight-binding models or interpolations to very fine k-meshes for properties that require fine $\\mathbf{k}$ -meshes. Such properties can be transport, anomalous Hall conductivity, linear and non-linear optics, Berry curvatures and topology, or electron–phonon interactions. In particular, for the improved description of electronic correlations via the DMFT approximation,233,234 WFs provide a realistic starting point.  \n\nIn connection with wien2wannier, there is also the $B e r r y P i^{235}$ module in WIEN2k, which calculates the polarization of solids using the Berry phase approach.236 BerryPi can calculate the change of polarization $\\Delta P$ in response to an external perturbation to study ferroelectricity, the Born effective charges, pyroelectric coefficients, or the piezoelectric tensor. In addition, one can define Wilson loops and calculate Chern numbers to study topological properties.237  \n\n# I. External programs  \n\n# 1. Thermoelectric transport coefficients  \n\nWIEN2k is interfaced with the BoltzTraP2 program165 for calculating transport coefficients within the relaxation time approximation. The calculation is based on evaluating the transport distribution function,  \n\n$$\n\\sigma(\\varepsilon,T)=\\int\\sum_{n}\\mathbf{v}_{n\\mathbf{k}}\\otimes\\mathbf{v}_{n\\mathbf{k}}\\tau_{n\\mathbf{k}}\\delta{\\bigl(}\\varepsilon-\\varepsilon_{n\\mathbf{k}}{\\bigr)}{\\frac{d^{3}k}{8\\pi^{3}}},\n$$  \n\nusing a fine mesh in $\\mathbf{k}$ -space. To obtain the group velocities, $\\mathbf{v}_{n\\mathbf{k}}$ , and also quasi-particle energies on a fine mesh or effective masses, BoltzTrap2 relies on interpolating the eigenvalues, $\\varepsilon_{n\\mathbf{k}}$ , and possibly also the relaxation times, $\\tau_{n\\mathbf{k}}$ , using Fourier sums.  \n\nThe interpolation is performed so that the calculated eigenvalue energies are reproduced exactly. Within KS theory, the multiplicative potential [Eq. (10)] means that it is often computationally very efficient to calculate a fine mesh of eigenvalues, which can then be interpolated further to evaluate Eq. (43). This argument no longer holds when hybrid functionals (Sec. II B 2), or the $G W$ method (Sec. III I 4), are used to obtain the band structure. Therefore, BoltzTraP2 can also include the $\\mathbf{k}$ -space derivatives in the interpolation scheme, as can be obtained from the momentum matrix elements introduced in Sec. III D. This allows a more efficient interpolation and the use of a coarser $\\mathbf{k}$ -mesh in the actual DFT calculation.  \n\nOnce the transport distribution has been obtained, the temperature and chemical potential dependent transport coefficients  \n\n$$\n\\mathcal{L}^{(\\alpha)}(\\mu;T)=q^{2}\\int\\sigma(\\varepsilon,T)(\\varepsilon-\\mu)^{\\alpha}\\Bigg(-\\frac{\\partial f^{(0)}(\\varepsilon;\\mu,T)}{\\partial\\varepsilon}\\Bigg)\\mathrm{d}\\varepsilon\n$$  \n\ncan be obtained by a simple numerical integration. Figure 16 shows the highest valence bands of TiCoSb calculated using the interpolation scheme of BoltzTraP2. Compared to Fig. 7, a very good agreement with the band structure obtained using DFT is found.  \n\n![](images/dd960b9ff0aa9c005cc69eedec2d8aac2aba5f60eb4355490cd9adbd4c6d7948.jpg)  \nFIG. 16. Highest valence bands of TiCoSb obtained with the BoltzTraP2 interpolation together with the calculated thermoelectric power factor at $300~\\mathsf{K}$ using a constant relaxation time of $\\tau=10^{-14}\\ s$ . The inset shows the constant energy surface at the energy marked by the dotted line.  \n\nSince the Fourier interpolation is done bandwise (marked by color in Fig. 16), the band crossing along the $\\Gamma-X$ direction is not reproduced. However, the error is hardly visible−by eye and, in accordance with the intention of the original algorithm,238,239 no Fourier ripples are seen, which means that errors in the derivatives are isolated to the points where the crossing occurs. The calculated thermoelectric power factor using a constant relaxation time is also shown. The power factor peaks at a high value close to the band edge, where a steep transport distribution can be expected. The high power factor can be attributed to the complex constant energy surface shown in the inset, which is typical for half-Heusler compounds with favorable p-type thermoelectric performance.240  \n\nBoltzTraP2 is written mainly in PYTHON3 and can be used as PYTHON library. The interpolation is handled by a single PYTHON call, fitde3D. Once the Fourier coefficients have been obtained, the interpolation of the bands onto the direction needed for plotting the band structure (getBands), or the fine mesh needed for obtaining the transport distribution (getBTPbands), can be performed. The use of BoltzTraP2 as a library gives a reproducible and flexible work flow. The analysis associated with Fig. 16 can thus be performed with a single PYTHON script, which is included in the most recent distribution.  \n\n# 2. Non-collinear magnetism  \n\nWIEN2k can only compute the electronic structure of magnetic systems with a collinear spin arrangement. For performance benefits, WIEN2k assumes that the spin density matrix along some direction is diagonal for each eigenstate. When SO interactions are taken into account, this condition is sometimes not satisfied. In such cases, the off-diagonal terms of the spin density matrix are simply ignored during the SCF procedure and only the $z$ component of the spin density is properly converged. Generally, this is not a big issue for cases with collinear spin arrangement and is, in fact, common practice. However, such an approximation cannot be applied for systems with a non-collinear spin arrangement for which the full spin density matrix has to be considered. For that purpose, we have written a non-collinear spin version of WIEN2k, referred to as WIENNCM.  \n\nOur implementation is based on a mixed spinor basis set approach.241,242 In the interstitial region, the basis functions are pure spinors given in a global $(g)$ spin coordinate frame,  \n\n$$\n\\varphi_{\\bf K+k,\\sigma}=e^{i({\\bf K+k})\\cdot{\\bf r}}\\chi_{\\sigma}^{g},\n$$  \n\nwher $\\chi_{\\uparrow}^{g}={\\binom{1}{0}}$ $\\chi_{\\downarrow}^{g}={\\binom{0}{1}}$ Inside the atomic spheres, the basis functions are a combination of both up and down spinors, which are set in a local spin coordinate frame with a quantization axis pointing along the direction of the average magnetization of the given atomic sphere. This direction does not have to be the same for each sphere, and the basis functions are [for a LAPW basis set, where we drop the $(r,E)$ -dependency in the radial $u$ functions]  \n\n$$\n\\varphi_{\\mathbf{K}+\\mathbf{k},\\sigma}^{\\mathrm{LAPW}}=\\sum_{\\ell,m,\\sigma^{t}}\\left(A_{t\\ell m}^{\\mathbf{K}+\\mathbf{k},\\sigma\\sigma^{t}}u_{t\\ell}^{\\sigma^{t}}+B_{t\\ell m}^{\\mathbf{K}+\\mathbf{k},\\sigma\\sigma^{t}}\\dot{u}_{t\\ell}^{\\sigma^{t}}\\right)Y_{\\ell m}\\chi_{\\sigma^{t}},\n$$  \n\nwhere $\\chi_{\\sigma^{t}}$ is a spinor given in a local coordinate frame. This choice of the spin coordinate frame allows us to use spin-polarized radial functions with the quantization axis along the direction of the average magnetization. The matching of the $\\varphi_{\\mathbf{K}+\\mathbf{k},\\sigma}^{\\mathrm{LAPW}}$ basis to the plane waves at $r=R_{\\mathrm{MT}}$ is done for up and down plane waves in a global spin coordinate frame,  \n\n$$\ne^{i(\\mathbf{K}+\\mathbf{k})\\cdot\\mathbf{r}}\\chi_{\\sigma}^{g}=\\sum_{\\ell,m,\\sigma^{t}}\\left(A_{t\\ell m}^{\\mathbf{K}+\\mathbf{k},\\sigma\\sigma^{t}}u_{t\\ell}^{\\sigma^{t}}+B_{t\\ell m}^{\\mathbf{K}+\\mathbf{k},\\sigma\\sigma^{t}}\\dot{u}_{t\\ell}^{\\sigma^{t}}\\right)Y_{\\ell m}\\chi_{\\sigma^{t}}^{g}.\n$$  \n\nThus, the $A_{t\\ell m}$ and $B_{t\\ell m}$ depend on global $\\sigma$ and local $\\boldsymbol{\\sigma}^{t}$ spin indices. Multiplying both sides of Eq. (47) by $\\left(\\chi_{\\sigma^{t}}^{g}\\right)^{*}$ , integrating over the spin variable, and comparing to the standard collinear expression, $A_{t\\ell m}^{\\mathbf{K}+\\mathbf{k},\\sigma\\sigma^{t}}$ and $B_{t\\ell m}^{\\mathbf{K}+\\mathbf{k},\\sigma\\sigma^{t}}$ are given by  \n\n$$\n\\begin{array}{r l}&{{\\cal A}_{t\\ell m}^{\\bf K+k,\\sigma\\sigma^{t}}=\\left(\\chi_{\\sigma^{t}}^{g}\\right)^{*}\\chi_{\\sigma}^{g}{\\cal A}_{t\\ell m}^{\\bf K+k,\\sigma^{t}},}\\\\ &{}\\\\ &{{\\cal B}_{t\\ell m}^{\\bf K+k,\\sigma\\sigma^{t}}=\\left(\\chi_{\\sigma^{t}}^{g}\\right)^{*}\\chi_{\\sigma}^{g}{\\cal B}_{t\\ell m}^{\\bf K+k,\\sigma^{t}},}\\end{array}\n$$  \n\nwhere AK+k,σ and BK+k,σt are the collinear matching coefficients calculated for “local” spins. We have extended the original formal$\\mathrm{ism}^{241,243}$ beyond the atomic moment approximation, and the code processes the spin density matrices without any approximations also inside the atomic spheres. Note that the adaptation of the above equations for LOs or the APW+lo basis set is trivial.  \n\nA few more details of the implementation are the following: In WIENNCM, the default scalar relativistic Hamiltonian is extended with SO interactions and this doubles the size of the Hamiltonian. It is also possible to use the $\\mathrm{DFT}+U$ method for correlated systems. The setup and execution of WIENNCM is as in WIEN2k; however, the atomic structure has to be augmented with definitions of the magnetic structure, which requires to define the orientation of the average magnetic moment for each atom. This is to some extent automatized in such a way that the user only needs to provide the orientation for the “magnetic atoms” (e.g., only $\\mathrm{~U~}$ atoms in $\\mathrm{UO}_{2}^{\\cdot}$ ). The orientation for “non-magnetic” atoms (O in $\\mathrm{UO}_{2}$ ) is generated automatically. WIENNCM makes use of the spin symmetry, which simplifies the calculations.  \n\nIf one wants to calculate spin spirals, one can either handle this by (big) supercells or, more efficiently, by using the generalized  \n\nBloch theorem244 (neglecting SO interactions) so that these calculations can be done in the small crystallographic cell.  \n\nExamples of application of our implementation can be found in Refs. 91 and 245.  \n\n# 3. Electron–hole interactions  \n\nThe state-of-the-art method to include electron–hole interactions is based on the solution of the equation of motion of the twoparticle Green’s function, known as the BSE.246–248 The WIENBSE code allows the calculation of the optical response taking into account excitonic effects. The BSE is solved in an approximate manner by representing them in the form of an effective eigenvalue problem with the so-called BSE Hamiltonian,249,250  \n\n$$\n\\sum_{\\nu^{\\prime},c^{\\prime},\\mathbf{k}^{\\prime}}H_{\\nu c\\mathbf{k},\\nu^{\\prime}c^{\\prime}\\mathbf{k}^{\\prime}}^{e}A_{\\nu^{\\prime}c^{\\prime}\\mathbf{k}^{\\prime}}^{\\lambda}=E^{\\lambda}A_{\\nu c\\mathbf{k}}^{\\lambda},\n$$  \n\nwhere the sum runs over occupied (they form the hole upon excitation) valence (v) and unoccupied (they become occupied upon excitation) conduction (c) bands and $\\mathbf{k}$ points (supplied by a DFT calculation performed with WIEN2k) and the electron–hole Hamiltonian consists of three terms, $H^{e}=H^{\\mathrm{diag}}+H^{\\mathrm{dir}}+H^{x}$ , which are given by $[\\mathbf{x}=(\\mathbf{r},\\sigma)]$ ,  \n\n$$\nH_{\\nu c\\mathbf k,\\nu^{\\prime}c^{\\prime}\\mathbf k^{\\prime}}^{\\mathrm{diag}}=\\big(\\varepsilon_{\\nu\\mathbf k}-\\varepsilon_{c\\mathbf k}+\\Delta\\big)\\delta_{\\nu\\nu^{\\prime}}\\delta_{c c^{\\prime}}\\delta_{\\mathbf k\\mathbf k^{\\prime}},\n$$  \n\n$$\nH_{\\nu c\\mathbf k,\\nu^{\\prime}c^{\\prime}\\mathbf k^{\\prime}}^{\\mathrm{dir}}=-\\int\\psi_{\\nu\\mathbf k}(\\mathbf x)\\psi_{c\\mathbf k}^{*}(\\mathbf x^{\\prime})W(\\mathbf r,\\mathbf r^{\\prime})\\psi_{\\nu^{\\prime}\\mathbf k^{\\prime}}^{*}(\\mathbf x)\\psi_{c^{\\prime}\\mathbf k^{\\prime}}(\\mathbf x^{\\prime})d^{3}x d^{3}x^{\\prime},\n$$  \n\n$$\nH_{\\nu\\mathrm{ck},\\nu^{\\prime}\\epsilon^{\\prime}{\\bf k}^{\\prime}}^{x}=\\int\\psi_{\\nu\\bf k}({\\bf x})\\psi_{\\mathrm{ck}}^{*}({\\bf x})\\bar{\\nu}({\\bf r},{\\bf r}^{\\prime})\\psi_{\\nu^{\\prime}{\\bf k}^{\\prime}}^{*}({\\bf x}^{\\prime})\\psi_{c^{\\prime}{\\bf k}^{\\prime}}({\\bf x}^{\\prime})d^{3}x d^{3}x^{\\prime}.\n$$  \n\nThe $\\boldsymbol{H}^{\\mathrm{diag}}$ term depends only on the eigenvalues and accounts for the response in the non-interacting limit. The exchange $H^{x}$ and the direct $H^{\\mathrm{dir}}$ Coulomb terms couple the electron–hole pairs.251 The direct term, in principle, depends on the dynamically screened Coulomb electron–hole interaction and on the excitation energy $(\\boldsymbol{E}^{\\lambda})$ , but here we apply the usual approximation and only account for non-local but static screening. The coupling coefficients $\\boldsymbol{A}_{\\nu c\\mathbf{k}}^{\\lambda}$ define the electron–hole correlation function and enter the expression for the imaginary part of the dielectric function,  \n\n$$\n\\varepsilon_{2}{\\big(}\\omega{\\big)}={\\frac{8\\pi^{2}}{\\Omega}}\\sum_{\\lambda}\\left\\vert\\sum_{\\nu,c,\\mathbf{k}}A_{\\nu c\\mathbf{k}}^{\\lambda}{\\frac{\\langle\\nu\\mathbf{k}|-i\\nabla_{x}|c\\mathbf{k}\\rangle}{\\varepsilon_{\\nu\\mathbf{k}}-\\varepsilon_{c\\mathbf{k}}}}\\right\\vert^{2}\\delta{\\big(}E^{\\lambda}-\\omega{\\big)}.\n$$  \n\nThe BSE approach is very successful in dealing with excitons and the response in the optical regime. The excitonic effects are sometimes small (e.g., in small gap semiconductors), but still important, with binding energies of some tens of meV, sometimes large (in particular, in insulators) with binding energies of a couple of eV such that the resulting optical functions have hardly any resemblance with independent-particle results. The WIENBSE implementation has been used in several works for various semiconductors,252–255 but together with supercell calculations and GW- or TB-mBJ-based single particle states, it can also successfully describe F centers in wide bandgap alkali halides.76,256 It has also been extended to include a fully relativistic treatment of core states to study XANES at $\\mathbf{L}_{2,3}$ edges of 3d TM compounds.196 In particular, for formally $3d^{0}$ compounds such as $\\mathrm{TiO}_{2}$ , the correct $\\mathrm{L}_{2}/\\mathrm{L}_{3}$ branching ratio can be obtained due to interference effects of the $2p_{1/2}$ and $2p_{3/2}$ core states. Even fine differences in line shape between the rutile and anatase modifications are in agreement with the experiment demonstrating the power of this approach.  \n\n# 4. GW approximation for quasi-particle calculations  \n\nThe GW method102,103 is considered as the state-of-the-art method for a first-principles description of the electronic quasiparticle band structure in solids. The name of this many-body perturbation theory based method comes from the interacting Green’s function $G(\\boldsymbol{\\mathbf{r}},\\ \\boldsymbol{\\mathbf{r}}^{\\prime},\\ \\omega)$ , whose poles in the complex frequency plane determine the single-particle excitation energies and $W$ , the dynamically screened Coulomb potential, which is obtained using the polarizability in the random-phase approximation. The central quantity, namely, the self-energy $\\Sigma(\\boldsymbol{\\mathbf{r}},\\ \\boldsymbol{\\mathbf{\\bar{r}}}^{\\prime},\\ \\omega)$ from which the first-order correction to the KS eigenvalues can be calculated, is obtained from  \n\n$$\n\\Sigma(\\mathbf{r},\\mathbf{r}^{\\prime},\\omega)=\\frac{i}{2\\pi}\\int G(\\mathbf{r},\\mathbf{r}^{\\prime},\\omega+\\omega^{\\prime})W(\\mathbf{r}^{\\prime},\\mathbf{r},\\omega^{\\prime})e^{i\\omega^{\\prime}\\eta}d\\omega^{\\prime},\n$$  \n\nwhere $\\eta$ is an infinitesimal positive number.  \n\nThe GAP2 code257,258 is the second version of an all-electron GW implementation based on the WIEN2k code. As WIEN2k, this highly parallelized code can use an arbitrary number of HDLOs,259,260 which ensures that one can obtain fully converged $G W$ results even in difficult cases such as $\\mathrm{znO}$ . Since it is based on an all-electron method and can use orbitals from $\\mathrm{DFT}+U_{\\mathrm{:}}$ , a major advantage of this code is the possibility to explore $d.$ - and $f$ -electron systems in a meaningful way.260 We note that for materials that are traditionally categorized as strongly correlated (e.g., the oxides), the standard semilocal functionals usually fail (see Sec. II B 3). For such systems, using the $\\mathrm{DFT}+U$ (or hybrid) orbitals as input for one-shot, $G_{0}W_{0}$ performs much better.261,262 In addition, the code allows for partially self-consistent $G W_{0}$ calculations by updating $G$ with the modified eigenvalues.  \n\nAnother important feature is the possibility for a first-principles determination of the Hubbard $U$ using the constrained random phase approximation263 and maximally localized WFs with an interface to Wannie $90^{231}$ by wien2wannier.232  \n\n# 5. Dynamical mean field theory  \n\nMany systems have valence electrons in orbitals, which are quite extended in space and overlap strongly with their neighbors. This leads usually to strong bonding-antibonding effects, large bandwidth $W_{;}$ , and dominant contributions from the kinetic energy. If, in addition, the bare Coulomb interaction $U$ between two electrons on the same site is strongly screened (maybe due to metallic character), such systems are usually quite well described by standard DFT approximations. However, as already mentioned in Sec. II B 3, $3d$ or $4f$ electrons may be more localized so that they participate much less in bonding and have a more atomic-like character. In these cases, the bandwidth W (metal) and Coulomb interaction $U$ (Mott insulator) compete as does the crystal-field splitting (low-spin state) and Hund’s rule coupling $J$ (high-spin state). We talk about “correlated electrons” and standard semilocal DFT approximations may fail badly in certain cases. The $\\mathrm{DFT}+U$ and hybrid methods discussed in Secs. II B 3 and $\\mathrm{~I~I~B~}2$ , respectively, can be much more accurate depending on the investigated property. However, the state-of-the-art approach for these correlated electron systems is DMFT,233,234 which is based on the Hubbard model on a lattice, described by the following Hamiltonian:  \n\n$$\nH=\\sum_{<i j>,\\sigma}~t_{i j}c_{i\\sigma}^{\\dag}c_{j\\sigma}+U\\sum_{i}n_{i\\uparrow}n_{i\\downarrow}.\n$$  \n\nThe first term (kinetic energy) describes the hopping $t_{i j}$ of an electron with spin $\\sigma$ from lattice site $j$ to lattice site $i_{:}$ , while the second term (potential energy) accounts for the strong Coulomb repulsion $U$ between two electrons at the same lattice site $i,$ which is responsible for the correlations in the system. Within DMFT, the complicated lattice problem is replaced by a single-site impurity model, which hybridizes with a self-consistently determined non-interacting bath.  \n\nThe basic idea of $\\mathrm{DFT+DMFT}$ is to divide the electrons in the system into two groups: weakly correlated electrons (i.e., electrons in $s-$ and $\\boldsymbol{p}$ -orbitals) that are well described by an approximate DFT functional and strongly correlated electrons (i.e., $d-$ and $f$ -electrons) well described using DMFT. The model Hamiltonian for DFT $^+$ DMFT is then constructed for the correlated subset with a suitable basis usually defined by Wannier functions as discussed in Sec. III H. The full-orbital KS Hamiltonian $H^{\\mathrm{KS}}$ is then projected onto the correlated subspace of the partially filled orbitals and manybody terms $\\boldsymbol{H}^{U}$ as well as a double counting correction $H^{\\mathrm{DC}}$ are added.  \n\nSeveral such DFT $+$ DMFT codes264–269 use WIEN2k as basis, and numerous applications have proven the power of the combined DFT $^+$ DMFT approach. The DMFT approach is often applied to explain optical, XAS, or ARPES spectra (see, e.g., Ref. 270 for $\\mathrm{V}_{2}\\mathrm{O}_{3}\\mathrm{,}$ ) and can also estimate the intensities of the spectral features due to lifetime broadening. Recently, free energies and forces have also been made available,269,271 which allows us to study structural (e.g., α-γ Ce), magnetic (e.g., bcc-fcc Fe), or metal–insulator (e.g., ${\\mathrm{NdNiO}}_{3}$ ) phase transitions with temperature.272  \n\n# 6. Phonons  \n\nWIEN2k does not have its own program to calculate phonon spectra, but it is interfaced with at least three different external phonon programs: phonopy,159,160 Phonon,161,162 and PHON.163,164 They all employ the finite-displacement method162 and a harmonic approximation. First, the crystal structure must be very well relaxed so that residual forces on all atoms are very small. For this structure, the phonon codes suggest a systematic set of displacements (depending on symmetry) in a chosen supercell and WIEN2k calculates the forces for these displacements. These forces are then used by the phonon codes to calculate harmonic force constants and setup and diagonalize the dynamical matrices at the desired $\\mathbf{k}$ -points, which yields the phonon frequencies and their eigenmodes. For Γ-phonons (infrared or Raman spectroscopy), a supercell is not required; otherwise, the supercell should be large enough (typically more than 50 atoms/cell) such that the force constants between atoms separated by more than the supercell size become negligible. In ionic solids, the frequency splitting of the optical vibrational modes parallel and perpendicular to the electric field (the so-called LO-TO splitting) in the small wave-vector limit can be obtained when additionally the Born effective charges (see Sec. III H) are supplied to the phonon programs. An alternative approach for phonon calculations, namely, density functional perturbation theory,273 is not implemented.  \n\nPhonon calculations can be used to investigate various properties of materials. Frequencies at Γ are analyzed according to their symmetry and can be compared to IR and Raman spectra (see, e.g., Ref. 274 for application on ${\\mathrm{PbFBr}}_{1-x}{\\mathrm{I}}_{x}.$ ). The full phonon band structure and the corresponding phonon-DOS can be calculated and integrated, yielding thermodynamic quantities such as the mean square thermal displacements, the specific heat, entropy, or free energy, which together with the quasi-harmonic approximation can be used to determine thermal expansion. Imaginary frequencies at certain $\\mathbf{k}$ -points indicate an instability of this phase (at $0\\mathrm{~K~},$ ) and occur, for instance, in all cubic perovskites.275 Freezing in one of the corresponding eigenmodes with a certain amplitude and subsequent structure relaxation yields a more stable phase in a particular space group of lower symmetry and can be used to detect and analyze second-order phase transitions in various materials.276–278  \n\n# 7. Band structure unfolding  \n\nThe standard way to model defects, vacancies, alloys (disorder), or surfaces is by means of a supercell approach. While sufficiently large supercells can handle the energetics of these problems quite well, it is fairly difficult to describe the effect of the perturbation on the bulk electronic structure. A band structure from a supercell calculation usually looks like a bunch of spaghetti and is very difficult to interpret. It is therefore highly desirable to display the band structure in the original BZ of the bulk material and indicate the original Bloch character as much as possible. This unfolding can be done conveniently by the fold2Bloch utility.166,279  \n\nIn an $h\\times k\\times l$ supercell band structure, each k-point transforms into $h\\times k\\times l$ ×k-points of the bulk BZ. fold2Bloch calculates the corresp×ondi×ng spectral weights $w_{n}(\\mathbf{k})$ , which amounts to the Bloch character $\\mathbf{k}$ of the nth eigenvalue $\\scriptstyle{\\varepsilon_{n}}$ , subject to the normalization that $\\begin{array}{r}{\\sum_{\\mathbf{k}}w_{n}(\\mathbf{k})=1}\\end{array}$ , and displays $w_{n}(\\mathbf{k})$ in the unfolded band structure s∑o that one can distinguish between regular bulk and defect states.  \n\n# 8. de Haas–van Alphen effect  \n\nThe knowledge of the Fermi surface (FS) of a metallic compound is important to understand its electronic and transport properties. de Haas–van Alphen (dHvA) measurements of the quantum oscillatory magnetization contain detailed information about the FS and report frequencies that are proportional to extremal FS cross sections perpendicular to the magnetic field direction.280 However, it is not so easy to reconstruct from the measured data the actual multi-band FS.  \n\nOn the other hand, FS calculations in WIEN2k are rather trivial and can be well presented using XCrysDen.152 For a quantitative comparison with experiment, it is highly desirable to calculate the corresponding dHvA frequencies as well as the corresponding effective masses. This can be done conveniently using the SKEAF (Supercell $\\mathbf{k}$ -space Extremal Area Finder) tool.167,280 An application can be found for the determination of transport properties in the HoBi281 compound.  \n\n# IV. DISCUSSION AND SUMMARY  \n\nIn this paper, we have reviewed the widely used WIEN2k code, which is based on the APW+lo method to solve the KS equations of DFT. Particular emphasis was placed on the various types of basis functions that are available. One of the strengths of the WIEN2k code is the possibility to use an arbitrary number of local orbitals, which allows an accurate calculation of all states, from the low-lying occupied semi-core to the high-lying unoccupied states. For the latter, the use of local orbitals is crucial in order to get converged results for a property that is calculated using perturbation theory such as the NMR chemical shift.  \n\nVarious types of approximations for the treatment of XC effects are mentioned, and the large number of functionals that are available constitutes another strength of the WIEN2k code. They range from the semilocal approximations (all the existing ones can be used via the Libxc library) to the more sophisticated approximations such as $\\mathrm{DFT}+U_{:}$ , the hybrids, or functionals specifically developed for van der Waals interactions. In particular, the popular Tran–Blaha mBJ potential is implemented in WIEN2k, which is a cheap but accurate method to calculate bandgaps in solids. Since the WIEN2k code is a full-potential all-electron code, it is, in principle, able to provide the exact result within a chosen XC approximation. Thus, WIEN2k is ideally suited for the testing of XC functionals.  \n\nThe structure of the WIEN2k code, as well as the workflow of programs in a SCF calculation, has been described. WIEN2k has also a user-friendly interface that is especially useful for beginners. In principle, an APW-based method needs many specific input parameters (various PW and LM cutoffs and case specific LM expansions and specific $E$ -parameters for each atom and angular momentum), but one of the great strengths of our implementation is that for all these parameters very good defaults are provided automatically to the user so that WIEN2k can also be mastered by non-experts. Of course, an all-electron code cannot be as fast as PW pseudopotential codes, where the extensive use of FFTs speeds up the calculations. Despite this, an APW-based method can be fairly efficient when large atomic spheres can be used because of the relatively fast PW convergence in such cases. It also has a fast and robust method to solve the SCF problem, including a simultaneous optimization of the atomic positions. WIEN2k is a very efficient implementation of the APW $+\\mathrm{lo}$ method from the computational point of view. The code is highly optimized and uses whenever possible efficient numerical libraries (BLAS, LAPACK, and ELPA). It has three different parallelization schemes, which allows us to run the code efficiently on a laptop as well as on a huge high performance computing cluster.  \n\nWIEN2k can calculate a large number of different properties. Besides the basic quantities such as the optimized atomic structure, cohesive energy, electronic band structure, or magnetism, numerous more specialized properties are available and can be readily calculated. Among them, those whose corresponding programs or modules are part of the WIEN2k code are, for instance, the optical properties, electric polarization, electric-field gradients, NMR chemical and Knight shifts, or magnetic hyperfine fields. In particular, for the latter two quantities, an all-electron method is mandatory. We also described programs that are not part of the WIEN2k distribution but are compatible with it. This includes, for instance, WIENNCM (non-collinear magnetism), WIENBSE (electron–hole interactions), BoltzTraP2 (thermoelectric transport coefficients), GAP2 (GW), or the various programs that calculate phonons.  \n\nThe main advantage of working with an all-electron code such as WIEN2k is the possibility to implement methods for calculating properties exactly. However, implementations within the $\\mathrm{\\APW+lo}$ method are not always straightforward, since the dual basis-set representation, atomic-like functions inside the atomic spheres, and plane waves between the atoms may lead to complicated equations. In addition, the discontinuity of the derivatives of the basis functions at the sphere boundary may require a careful treatment. DFT codes using a basis set consisting only of plane waves or only of localized basis functions (e.g., Gaussian) lead, in principle, to easier implementations. 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    "Target": null
  },
  {
    "id": "10.1088_1361-648X_ab51ff",
    "DOI": "10.1088/1361-648X/ab51ff",
    "DOI Link": "http://dx.doi.org/10.1088/1361-648X/ab51ff",
    "Relative Dir Path": "mds/10.1088_1361-648X_ab51ff",
    "Article Title": "Wannier90 as a community code: new features and applications",
    "Authors": "Pizzi, G; Vitale, V; Arita, R; Blügel, S; Freimuth, F; Géranton, G; Gibertini, M; Gresch, D; Johnson, C; Koretsune, T; Ibañez-Azpiroz, J; Lee, H; Lihm, JM; Marchand, D; Marrazzo, A; Mokrousov, Y; Mustafa, JI; Nohara, Y; Nomura, Y; Paulatto, L; Poncé, S; Ponweiser, T; Qiao, JF; Thöle, F; Tsirkin, SS; Wierzbowska, M; Marzari, N; Vanderbilt, D; Souza, I; Mostofi, AA; Yates, JR",
    "Source Title": "JOURNAL OF PHYSICS-CONDENSED MATTER",
    "Abstract": "Wannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronic-structure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectively-localised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with high-throughput automation frameworks), as well as the implementation of modern software engineering practices (unit testing, continuous integration, and automatic source-code documentation). These new features, capabilities, and code development model aim to further sustain and expand the community uptake and range of applicability, that nowadays spans complex and accurate dielectric, electronic, magnetic, optical, topological and transport properties of materials.",
    "Times Cited, WoS Core": 1281,
    "Times Cited, All Databases": 1341,
    "Publication Year": 2020,
    "Research Areas": "Physics",
    "UT (Unique WOS ID)": "WOS:000520450700001",
    "Markdown": "# ACCEPTED MANUSCRIPT $\\cdot$ OPEN ACCESS  \n\n# Wannier90 as a community code: new features and applications  \n\nTo cite this article before publication: Giovanni Pizzi et al 2019 J. Phys.: Condens. Matter in press https://doi.org/10.1088/1361-648X/ab51ff  \n\n# Manuscript version: Accepted Manuscript  \n\nAccepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by IOP Publishing of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by IOP Publishing and/or its licensors”  \n\nThis Accepted Manuscript is $\\circledcirc$ 2019 IOP Publishing Ltd.  \n\nAs the Version of Record of this article is going to be / has been published on a gold open access basis under a CC BY 3.0 licence, this Accepted Manuscript is available for reuse under a CC BY 3.0 licence immediately.  \n\nEveryone is permitted to use all or part of the original content in this article, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by/3.0  \n\nAlthough reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions may be required. All third party content is fully copyright protected and is not published on a gold open access basis under a CC BY licence, unless that is specifically stated in the figure caption in the Version of Record.  \n\nView the article online for updates and enhancements.  \n\n# Wannier90 as a community code: new features and applications  \n\nGiovanni Pizzi,1, ∗ Valerio Vitale, ${}^{2,3}$ , ∗ Ryotaro Arita, $^{4,5}$ Stefan Bl¨ugel, $^6$ Frank Freimuth, $^6$ Guillaume G´eranton, $^6$ Marco Gibertini, $^{1,7}$ Dominik Gresch,8 Charles Johnson, $^{9}$ Takashi Koretsune, $^{10,11}$ Julen Iba˜nez-Azpiroz, $^{12}$ Hyungjun Lee, Jae-Mo Lihm,15 Daniel Marchand, $^{16}$ Antimo Marrazzo,1 Yuriy Mokrousov,6, 17 Jamal I. Mustafa, $^{18}$ Yoshiro Nohara, $^{19}$ Yusuke Nomura, $^4$ Lorenzo Paulatto,20 Samuel Ponc´e, $^{21}$ Thomas Ponweiser,22 Junfeng Qiao,23 Florian Th¨ole,24 Stepan S. Tsirkin, $^{12,25}$ Ma gorzata Wierzbowska, $^{26}$ Nicola Marzari,1, ∗ David Vanderbilt, $^{27}$ , ∗ Ivo Souza,12, 28, ∗ Arash A. Mostofi,3, ∗ and Jonathan R. Yates $^{21}$ , $*$ $^{1}$ Theory and Simulation of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL),   \nE´cole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland $^2$ Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, UK $^3$ Departments of Materials and Physics, and the Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, London SW7 2AZ, UK $^4$ RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan $^{5}$ Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan $^6$ Peter Gru¨nberg Institut and Institute for Advanced Simulation, Forschungszentrum Ju¨lich and JARA, 52425 Ju¨lich, Germany Department of Quantum Matter Physics, University of Geneva, Geneva, Switzerland 8ETH Zurich, Zurich, Switzerland $^{9}$ Departments of Materials and Physics, Imperial College London, London SW7 2AZ, UK 10Department of Physics, Tohoku University, Sendai, Japan 11JST PRESTO, Kawaguchi, Saitama, Japan   \nCentro de F´ısica de Materiales, Universidad del Paı´s Vasco, E-20018 San Sebasti´an, Spa   \n$^{13}$ Institute of Physics, E´cole Polytechnique F´ed´erale de   \nLausanne (EPFL), CH-1015 Lausanne, Switzerland   \n$^{14}$ Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea   \n$^{15}$ Department of Physics and Center for Theoretical Physics,   \nSeoul National University, Seoul 08826, Korea   \n$^{16}$ Laboratory for Multiscale Mechanics Modeling (LAMMM),   \nE´cole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland   \n$^{17}$ Institute of Physics, Jonannes-Gutenberg University of Mainz, 55099 Mainz, Germany   \n$^{18}$ Department of Physics, University of California   \nat Berkeley, Berkeley, California 94720, USA   \n$^{19}$ ASMS Co., Ltd., 1-7-11 Higashi-Gotanda,   \nShinagawa-ku, Tokyo 141-0022, Japan   \n$^{\\mathrm{20}}$ Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie (IMPMC),   \nSorbonne Universit´e, CNRS UMR 7590, Case 115,   \n4 place Jussieu, 75252 Paris Cedex 05, France   \n$^{21}$ Department of Materials, University of Oxford,   \nParks Road, Oxford OX1 3PH, UK   \n$^{22}$ Research Institute for Symbolic Computation (RISC),   \nJohannes Kepler University, Altenberger Straße 69, 4040 Linz, Austria   \n$^{23}$ Fert Beijing Institute, School of Microelectronics,   \nBDBC, Beihang University, Beijing, China   \n24Materials Theory, ETH Zu¨rich, Wolfgang-Pauli-Strasse 27, CH-8093 Z¨urich, Switzerland   \n25Department of Physics, University of Zurich,   \nWintherthurerstrasse 190, CH-8057 Zurich, Switzerland   \n6Institute of High Pressure Physics, Polish Academy of Sciences,   \nSoko owska street 29/37, 01-142 Warsaw, Poland   \n27Department of Physics and Astronomy, Rutgers University,   \nPiscataway, New Jersey 08854-8019, USA   \n$^{28}$ Ikerbasque Foundation, E-48013 Bilbao, Spain   \n(Dated: October 10, 2019)  \n\n# Abstract  \n\nWannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronicstructure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectively-localised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with high-throughput automation frameworks), as well as the implementation of modern software engineering practices (unit testing, continuous integration, and automatic source-code documentation). These new features, capabilities, and code development model aim to further sustain and expand the community uptake and range of applicability, that nowadays spans complex and accurate dielectric, electronic, magnetic, optical, topological and transport properties of materials.  \n\n# I. INTRODUCTION  \n\nWannier90 is an open-source code for generating Wannier functions (WFs), in particular maximally-localised Wannier functions (MLWFs), and using them to compute advanced materials properties with high efficiency and accuracy. Wannier90 is a paradigmatic example of interoperable software, achieved by ensuring that all the quantities required as input are entirely independent of the underlying electronic-structure code from which they are obtained. Most of the major and widely used electronic-structure codes have an interface to Wannier90, including Quantum ESPRESSO $^{1}$ , ABINIT2, VASP3–5, Siesta $^6$ , Wien2k7, Fleur $^{8}$ , Octopus $^{\\mathrm{~9~}}$ and ELK $^{10}$ . As a consequence, once a property is implemented within Wannier90, it can be immediately available to users of all codes that interface to it.  \n\nOver the last few years, Wannier90 has undergone a transition from a code developed by a small group of developers to a community code with a much wider developers’ base. This has been achieved in two principal ways: (i) hosting the source code and associated development efforts on a public GitHub repository11; and (ii) building a community of Wannier90 developers and facilitating personal interactions between individuals through community workshops, the most recent in 2016. In response, the code has grown significantly, gaining many novel features contributed by this community, as well as numerous fixes.  \n\nIn this paper, we describe the most important novel contributions to the Wannier90 code, as embodied in its 3.0 release. The paper is structured as follows: In Sec. II we first summarise the background theory for the computation of MLWFs (additional details can be found in Ref. $\\mathrm{12}$ ), and introduce the notation that will be used throughout the paper. In Sec. III we describe the novel features of Wannier90 that are related to the core wannierisation and disentanglement algorithms; these include symmetry-adapted WFs, selective localisation of WFs, and parallelisation using the message-passing interface (MPI). In Sec. IV we describe new functionality enhancements, including the ability to handle spinor-valued WFs and calculations with non-collinear spin that use ultrasoft pseudopotentials (within Quantum ESPRESSO); improved interpolation of the $k$ -space Hamiltonian; a more flexible approach for handling and using initial projections; and the ability to plot WFs in Gaussian cube format on WF-centred grids with non-orthogonal translation vectors. In Sec. V we describe new functionalities associated with using MLWFs for computing advanced electronic-structure properties, including the calculation of shift currents, gyrotropic effects and spin Hall conductivities, as well as parallelisation improvements and the interpolation of bands originating from calculations performed with many-body perturbation theory (GW). In Sec. VI we describe the selected-columns-of-the-density-matrix (SCDM) method, which enables computation of WFs without the need for explicitly defining initial projections. In Sec. VII we describe new post-processing tools and codes, and the integration of Wannier90 with high-throughput automation and workflow management tools (specifically, the AiiDA materials’ informatics infrastructure $^{13}$ ). In Sec. VIII we describe the modern software engineering practices now adopted in Wannier90, that have made it possible to improve the development lifecycle and transform Wannier90 into a communitydriven code. Finally, our conclusions and outlook are presented in Sec. IX.  \n\n# II. BACKGROUND  \n\nWFs form a possible basis set for the electronic states of materials. As we are going to describe in the following, WFs are not unique and they can be optimised to obtain MLWFs. These, thanks to their localisation in real space, are particularly useful in a number of electronic-structure applications. For instance, they allow for efficient interpolation of operator matrix elements on dense grids in the Brillouin Zone (BZ), which is a key step to compute many materials properties. The interpolation is obtained starting from the value of these matrix elements and other properties of the wavefunctions (described below) computed on a coarser grid, usually with an accurate but slower ab initio code. MLWFs play in materials a role analogue to molecular orbitals in molecules and a typical shape of MLWFs (for instance, in the case of the valence bands of GaAs) can be seen in Fig. 1(a) and Fig. 1(e).  \n\nFormally, MLWFs can be introduced as follows in the independent-particle approximation. The electronic structure of a periodic system is conventionally represented in terms of one-electron Bloch states $\\psi_{n\\mathbf{k}}(\\mathbf{r})$ , which are labelled by a band index $n$ and a crystal momentum k inside the first BZ, and which satisfy Bloch’s theorem:  \n\n$$\n\\psi_{n\\mathbf{k}}(\\mathbf{r})=u_{n\\mathbf{k}}(\\mathbf{r})e^{i\\mathbf{k}\\cdot\\mathbf{r}},\n$$  \n\nwhere $u_{n\\mathbf{k}}(\\mathbf{r})=u_{n\\mathbf{k}}(\\mathbf{r+R})$ is a periodic function with the same periodicity of the singleparticle Hamiltonian, and $\\mathbf{R}$ is a Bravais lattice vector. (For the moment we ignore the spin degrees of freedom and work with spinless wave functions; spinor wave functions will be treated in Sec. IV A.) Such a formalism is also commonly applied, via the supercell approximation, to non-periodic systems, typically used to treat point, line and planar defects in crystals, surfaces, amorphous solids, liquids and molecules.  \n\n# A. Isolated bands  \n\nA group of bands is said to be isolated if it is separated by energy gaps from all the other lower and higher bands throughout the BZ (this isolated group of bands may still show arbitrary crossing degeneracies and hybridisations within itself). For such isolated set of $J$ bands, the electronic states can be equivalently represented by a set of $J$ WFs per cell, that are related to the Bloch states via two unitary transformations (one continuous, one discrete)14:  \n\n$$\n|w_{n\\mathbf{R}}\\rangle=V\\int_{\\mathrm{BZ}}\\frac{\\mathrm{d}\\mathbf{k}}{(2\\pi)^{3}}e^{-i\\mathbf{k}\\cdot\\mathbf{R}}\\sum_{m=1}^{J}|\\psi_{m\\mathbf{k}}\\rangle U_{m n\\mathbf{k}},\n$$  \n\nwhere $w_{n\\mathbf{R}}(\\mathbf{r})=w_{n\\mathbf{0}}(\\mathbf{r}-\\mathbf{R})$ is a periodic (but not necessarily localised) WF labelled by the quantum number $\\mathbf{R}$ (the counterpart of the quasi-momentum $\\mathbf{k}$ in the Bloch representation), $V$ is the cell volume and $U_{\\mathbf{k}}$ are unitary matrices that mix Bloch states at a given $\\mathbf{k}$ and represent the gauge freedom that exists in the definition of the Bloch states and that is inherited by the WFs.  \n\nMLWFs are obtained by choosing $U_{\\mathbf{k}}$ matrices that minimise the sum of the quadratic spreads of the WFs about their centres for a reference $\\mathbf{R}$ (say, $\\mathbf R=\\mathbf0$ ). This sum is given by the spread functional  \n\n$$\n\\Omega=\\sum_{n=1}^{J}\\left[\\langle w_{n\\mathbf{0}}|\\mathbf{r}\\cdot\\mathbf{r}|w_{n\\mathbf{0}}\\rangle-|\\langle w_{n\\mathbf{0}}|\\mathbf{r}|w_{n\\mathbf{0}}\\rangle|^{2}\\right].\n$$  \n\nΩ may be decomposed into two positive-definite parts15,  \n\n$$\n\\Omega=\\Omega_{\\mathrm{I}}+\\widetilde{\\Omega},\n$$  \n\n$$\n\\Omega_{\\mathrm{I}}=\\sum_{n}\\left[\\langle w_{n0}|\\mathbf{r}\\cdot\\mathbf{r}|w_{n0}\\rangle-\\sum_{m\\mathbf{R}}{|\\langle w_{m\\mathbf{R}}|\\mathbf{r}|w_{n0}\\rangle|^{2}}\\right]\n$$  \n\nis gauge invariant (i.e., invariant under the action of any unitary $U_{\\mathbf{k}}$ on the Bloch states), and  \n\n$$\n\\widetilde{\\Omega}=\\sum_{n}\\sum_{m\\mathbf{R}\\neq n\\mathbf{0}}|\\langle w_{m\\mathbf{R}}|\\mathbf{r}|w_{n\\mathbf{0}}\\rangle|^{2}\n$$  \n\nis gauge dependent. Therefore, the “wannierisation” of an isolated manifold of bands, i.e., the transformation of Bloch states into MLWFs, amounts to minimising the gauge-dependent part $\\widetilde\\Omega$ of the spread functional.  \n\nCrucially, the matrix elements of the position operator between WFs can be expressed in reciprocal space. Under the assumption that the BZ is sampled on a uniform Monkhorst– Pack mesh of $k$ -points composed of $N$ points $\\begin{array}{r}{\\big(V\\int_{\\mathrm{BZ}}\\frac{\\mathrm{d}\\mathbf{k}}{(2\\pi)^{3}}\\underset{\\mathcal{\\vec{N}}}{\\longrightarrow}\\frac{1}{N}\\sum_{\\mathbf{k}}\\big),}\\end{array}$ , the gauge-independent and gauge-dependent parts of the spread may be expressed, respectively, as15  \n\n$$\n\\Omega_{\\mathrm{I}}={\\frac{1}{N}}\\sum_{\\mathbf{k},\\mathbf{b}}w_{b}\\left[J-\\sum_{m n}\\left|{\\cal M}_{m n}^{(\\mathbf{k},\\mathbf{b})}\\right|^{2}\\right]\n$$  \n\nand  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\widetilde\\Omega}=\\frac{1}{N}\\sum_{{\\bf k},{\\bf b}}w_{b}\\sum_{m\\neq n}\\big\\vert M_{m n}^{({\\bf k},{\\bf b})}\\big\\vert^{2}}\\ ~}\\\\ {{\\displaystyle~+\\frac{1}{N}\\sum_{{\\bf k},{\\bf b}}w_{b}\\sum_{n}(-\\mathrm{Im}\\ln M_{n n}^{({\\bf k},{\\bf b})}-{\\bf b}\\cdot{\\bar{\\bf r}}_{n})^{2}},}\\end{array}\n$$  \n\nwhere $\\mathbf{b}$ are the vectors connecting a $k$ -point to its neighbours, $w_{b}$ are weights associated with the finite-difference representation of $\\nabla_{\\mathbf{k}}$ for a given geometry, the matrix of overlaps ${\\cal M}^{(\\mathbf{k},\\mathbf{b})}$ is defined by  \n\n$$\nM_{m n}^{(\\mathbf{k},\\mathbf{b})}=\\left<u_{m\\mathbf{k}}\\right|u_{n,\\mathbf{k}+\\mathbf{b}}\\right>,\n$$  \n\nand the centres of the WFs are given by  \n\n$$\n\\bar{\\bf r}_{n}\\equiv\\langle w_{n{\\bf0}}|{\\bf r}|w_{n{\\bf0}}\\rangle=-\\frac{1}{N}\\sum_{{\\bf k},{\\bf b}}w_{b}{\\bf b}\\mathrm{Im}\\ln M_{n n}^{({\\bf k},{\\bf b})}.\n$$  \n\nMinimisation of the spread functional is achieved by considering infinitesimal gauge transformations $U_{m n\\mathbf{k}}\\equiv\\delta_{m n}+\\mathrm{d}W_{m n\\mathbf{k}}$ , where dW is anti-Hermitian ( $\\mathrm{d}W^{\\dagger}=-\\mathrm{d}W$ ). The gradient of the spread functional with respect to such variations is given by  \n\n$$\n\\mathcal{G}_{\\mathbf{k}}\\equiv\\frac{\\mathrm{d}\\Omega}{\\mathrm{d}W_{m n\\mathbf{k}}}=4\\sum_{\\mathbf{b}}w_{b}\\left(\\pmb{A}[R_{m n}^{(\\mathbf{k},\\mathbf{b})}]-\\pmb{S}[T_{m n}^{(\\mathbf{k},\\mathbf{b})}]\\right),\n$$  \n\nwhere $\\mathcal{A}$ and $\\boldsymbol{S}$ are the super-operators $\\boldsymbol{\\mathcal{A}}[B]=(B-B^{\\dagger})/2$ and ${\\cal S}[B]=(B+B^{\\dagger})/2i$ ,  \n\nrespectively, and  \n\n$$\n\\begin{array}{l}{{\\displaystyle R_{m n}^{(\\mathbf{k},\\mathbf{b})}=M_{m n}^{(\\mathbf{k},\\mathbf{b})}M_{n n}^{(\\mathbf{k},\\mathbf{b})*},}}\\\\ {{\\displaystyle T_{m n}^{(\\mathbf{k},\\mathbf{b})}=\\frac{M_{m n}^{(\\mathbf{k},\\mathbf{b})}}{M_{n n}^{(\\mathbf{k},\\mathbf{b})}}q_{n}^{(\\mathbf{k},\\mathbf{b})},}}\\\\ {{\\displaystyle q_{n}^{(\\mathbf{k},\\mathbf{b})}=\\operatorname{Im}\\ln M_{n n}^{(\\mathbf{k},\\mathbf{b})}+\\mathbf{b}\\cdot\\bar{\\mathbf{r}}_{n}}.}\\end{array}\n$$  \n\nFor the full derivation of Eq. (11) we refer to Ref. 15. This gradient is then used to generate a search direction $\\mathcal{D}_{\\mathbf{k}}$ for an iterative steepest-descent or conjugate-gradient minimisation of the spread16: at each iteration the unitary matrices are updated according to  \n\n$$\nU_{\\mathbf{k}}\\rightarrow U_{\\mathbf{k}}\\exp[\\alpha\\mathcal{D}_{\\mathbf{k}}],\n$$  \n\nwhere $\\alpha$ is a coefficient that can either be set to a fixed value or determined at each iteration via a simple polynomial line-search, and the matrix exponential is computed in the diagonal representation of $\\mathcal{D}_{\\mathbf{k}}$ and then transformed back in the original representation. Once the unitary matrices have been updated, the updated set of $M^{(\\mathbf{k},\\mathbf{b})}$ matrices is calculated according to  \n\n$$\nM^{(\\mathbf{k},\\mathbf{b})}=U_{\\mathbf{k}}^{\\dagger}M^{(0)(\\mathbf{k},\\mathbf{b})}U_{\\mathbf{k+b}},\n$$  \n\nwhere  \n\nis the set of initial $M^{(\\mathbf{k},\\mathbf{b})}$ matrices, computed once and for all, at the start of the calculation, from the original set of reference Bloch orbitals $|u_{n\\mathbf{k}}^{(0)}\\rangle$  \n\n# B. Entangled bands  \n\nIt is often the case that the bands of interest are not separated from other bands in the Brillouin zone by energy gaps and are overlapping and hybridising with other bands that extend beyond the energy range of interest. In such cases, we refer to the bands as being entangled.  \n\nThe difficulty in constructing MLWFs for entangled bands arises from the fact that, within a given energy window, the number of bands $\\mathcal{T}_{\\mathbf{k}}$ at each $k$ -point $\\mathbf{k}$ in the BZ is not a constant and is, in general, different from the target number $J$ of WFs: $\\mathcal{T}_{\\mathbf{k}}\\geq J$ . Even making the energy window $k$ -dependent would see discontinuous inclusion and exclusion of bands as the BZ is traversed. The treatment of entangled bands requires thus a more complex approach that is typically a two-step process. In the first step, a $J$ -dimensional manifold of Bloch states is selected at each $k$ -point, chosen to be as smooth as possible as a function of $\\mathbf{k}$ . In the second step, the gauge freedom associated with the selected manifold is used to obtain MLWFs, just as described in Sec. II A for the case of an isolated set of bands.  \n\nFocusing on the first step, an orthonormal basis for the $J$ -dimensional subspace $S_{\\mathbf{k}}$ at each $\\mathbf{k}$ can be obtained by performing a semi-unitary transformation on the $\\mathcal{I}_{\\mathbf{k}}$ states at $\\mathbf{k}$ , where $V_{\\mathbf{k}}$ is a rectangular matrix of dimension $\\mathcal{I}_{\\mathbf{k}}\\times\\bar{\\mathcal{I}}$ that is semi-unitary in the sense that $V_{\\mathbf{k}}^{\\dagger}V_{\\mathbf{k}}=\\mathbf{1}$ .  \n\nTo select the smoothest possible manifold, a measure of the intrinsic smoothness of the chosen subspace is needed. It turns out that such a measure is given precisely by what was the gauge-invariant part $\\Omega_{\\mathrm{I}}$ of the spread functional for isolated bands.17 Indeed, Eq. (7) can be expressed as  \n\n$$\n\\mathbb{A}_{\\sf R}=\\frac{1}{N}\\sum_{\\mathbf{k},\\mathbf{b}}w_{b}\\mathrm{Tr}[P_{\\mathbf{k}}Q_{\\mathbf{k}+\\mathbf{b}}],\n$$  \n\nwhere $\\begin{array}{r}{P_{\\mathbf{k}}=\\sum_{n=1}^{J}\\left|\\widetilde{u}_{n\\mathbf{k}}\\right\\rangle\\left\\langle\\widetilde{u}_{n\\mathbf{k}}\\right|}\\end{array}$ is the projection operator onto $S_{\\mathbf{k}}$ , $Q_{\\mathbf{k}}=\\mathbf{1}-P_{\\mathbf{k}}$ is its Hilbertspace complement, eand “eTr” represents the trace over the entire Hilbert space. $\\operatorname{Tr}[P_{\\mathbf{k}}Q_{\\mathbf{k}+\\mathbf{b}}]$ measures the mismatch between the subspaces $S_{\\mathbf{k}}$ and $S_{\\mathbf{k}+\\mathbf{b}}$ , vanishing if they overlap identically. Hence $\\Omega_{\\mathrm{I}}$ measures the average mismatch of the local subspace $S_{\\mathbf{k}}$ across the BZ, so that an optimally-smooth subspace can be selected by minimising $\\Omega_{\\mathrm{I}}$ . Doing this with orthonormality constraints on the Bloch-like states is equivalent to solving self-consistently the set of coupled eigenvalue equations17  \n\n$$\n\\left[\\sum_{\\mathbf{b}}w_{b}P_{\\mathbf k+\\mathbf{b}}\\right]|\\widetilde u_{n\\mathbf k}\\rangle=\\lambda_{n\\mathbf k}|\\widetilde u_{n\\mathbf k}\\rangle.\n$$  \n\nThe solution can be achieved via an iterative procedure, whereby at the $i^{\\mathrm{th}}$ iteration the algorithm traverses the entire set of $k$ -points, selecting at each one the $J$ -dimensional subspace $S_{\\mathbf{k}}^{(i)}$ that has the smallest mismatch with the subspaces $S_{\\mathbf{k}+\\mathbf{b}}^{(i-1)}$ at the neighbouring  \n\n$k$ -points obtained in the previous iteration. This amounts to solving  \n\n$$\n\\left[\\sum_{\\mathbf{b}}w_{b}P_{\\mathbf{k+b}}^{(i-1)}\\right]|\\widetilde{u}_{n\\mathbf{k}}^{(i)}\\rangle=\\lambda_{n\\mathbf{k}}^{(i)}|\\widetilde{u}_{n\\mathbf{k}}^{(i)}\\rangle,\n$$  \n\nand selecting the $J$ eigenvectors with the largest eigenvalues17. Self-consistency is reached when $S_{\\mathbf{k}}^{(i)}=S_{\\mathbf{k}}^{(i-1)}$ (to within a user-defined threshold dis conv tol) at all the $k$ -points. To make the algorithm more robust, the projector appearing on the left-hand-side of Eq. (21) is replaced with $[P_{{\\bf k}+{\\bf b}}^{(i)}]_{\\mathrm{in}}$ , given by  \n\n$$\n[P_{\\mathbf{k}+\\mathbf{b}}^{(i)}]_{\\mathrm{in}}=\\beta P_{\\mathbf{k}+\\mathbf{b}}^{(i-1)}+(1-\\beta)[P_{\\mathbf{k}+\\mathbf{b}}^{(i-1)}]_{\\mathrm{in}},\n$$  \n\nwhich is a linear mixture of the projector that was used as input for the previous iteration and the projector defined by the output of the previous iteration. The parameter $0<\\beta\\leq1$ determines the degree of mixing, and is typically set to $\\beta^{\\gamma}=0.5$ ; setting $\\beta=1$ reverts precisely to Eq. (21), while smaller and smaller values of $\\beta$ make convergence smoother (and thus more robust) but also slower.  \n\nIn practice, Eq. (21) is solved by diagonalising the Hermitian operator appearing on the left-hand-side in the basis of the original $\\mathcal{I}_{\\mathbf{k}}$ Bloch states:  \n\n$$\nZ_{m n\\mathbf{k}}^{(i)}=\\langle u_{m\\mathbf{k}}^{(0)}|\\sum_{\\mathbf{b}}w_{b}[P_{\\mathbf{k+b}}^{(i)}]_{\\mathrm{in}}|u_{n\\mathbf{k}}^{(0)}\\rangle.\n$$  \n\nOnce the optimal subspace has been selected, the wannierisation procedure described in Sec. II A is carried out to minimise the gauge-dependent part $\\widetilde\\Omega$ of the spread functional within that optimal subspace.  \n\n# C. Initial projections  \n\nIn principle, the overlap matrix elements $M_{m n}^{\\left(\\mathbf{k},\\mathbf{b}\\right)}$ are the only quantities required to compute and minimise the spread functional, and generate MLWFs for either isolated or entangled bands. In practice, this is generally true when dealing with an isolated set of bands, but in the case of entangled bands a good initial guess for the subspaces $S_{\\mathbf{k}}$ alleviates problems associated with falling into local minima of $\\Omega_{\\mathrm{I}}$ , and/or obtaining MLWFs that cannot be chosen to be real-valued (in the case of spinless WFs). Even in the case of an isolated set of bands, a good initial guess for the WFs, whilst not usually critical, often results in faster convergence of the spread to the global minimum. (It is important to note that both for isolated and for entangled bands multiple solutions to the wannierisation or disentanglement can exist, as discussed later.)  \n\nA simple and effective procedure for selecting an initial gauge (in the case of isolated bands) or an initial subspace and initial gauge (in the case of entangled bands) is to project a set of $J$ trial orbitals $g_{n}(\\mathbf{r})$ localised in real space onto the space spanned by the set of original Bloch states at each $\\mathbf{k}$ :  \n\n$$\n\\left|\\phi_{n\\mathbf{k}}\\right\\rangle=\\sum_{m=1}^{J{\\mathrm{or}}\\mathcal{T}_{\\mathbf{k}}}\\left|\\psi_{m\\mathbf{k}}\\right\\rangle\\left\\langle\\psi_{m\\mathbf{k}}\\mid g_{n}\\right\\rangle,\n$$  \n\nwhere the sum runs up to either $J$ or $\\mathcal{I}_{\\mathbf{k}}$ , depending on whether the bands are isolated or entangled, respectively, and the inner product $A_{m n\\bf{k}}=\\left<{\\psi_{m\\bf{k}}}\\left|{g_{n}}\\right>$ is over all the Born–von Karman supercell. (In practice, the fact that the gn are localised greatly simplifies this calculation.) The matrices $A_{\\mathbf{k}}$ are square $(J\\times J)$ ) or rectangular $\\left(\\mathcal{I}_{k}\\times J\\right)$ in the case of isolated or entangled bands, respectively. The resulting orbitals are then orthonormalised via a Lo¨wdin transformation18:  \n\n$$\n\\begin{array}{l}{\\displaystyle|\\widetilde\\psi_{n\\mathbf{k}}\\rangle=\\sum_{m=1}^{J}|\\phi_{m\\mathbf{k}}\\rangle S_{m n\\mathbf{k}}^{-1/2}}\\\\ {=\\sum_{m=1}^{J\\mathrm{or}\\mathcal{I}_{\\mathbf{k}}}|\\psi_{m\\mathbf{k}}^{\\prime}\\rangle(A_{\\mathbf{k}}S_{\\mathbf{k}}^{-1/2})_{m n},}\\end{array}\n$$  \n\nwhere $S_{m n\\mathbf{k}}=\\langle\\phi_{m\\mathbf{k}}\\mid\\phi_{n\\mathbf{k}}\\rangle=(A_{\\mathbf{k}}^{\\dagger}A_{\\mathbf{k}})_{m n},$ and $A_{\\mathbf{k}}S_{\\mathbf{k}}^{-1/2}$ is a unitary or semi-unitary matrix. In the case of entangled bands, once an optimally-smooth subspace has been obtained as described in Sec. II B, the same trial orbitals $g_{n}(\\mathbf{r})$ can be used to initialise the wannierisation procedure of Sec. II A. In practice, the matrices $A_{\\mathbf{k}}$ are computed once and for all at the start of the calculation, together with the overlap matrices $M^{(\\mathbf{k},\\mathbf{b})}$ . These two operations need to be performed within the context of the electronic-structure code and basis set adopted; afterwards, all the operations of Wannier90 rely only on $A_{\\mathbf{k}}$ and ${\\cal M}^{(\\mathbf{k},\\mathbf{b})}$ and not on the specific representation of $\\bar{\\psi}_{m\\mathbf{k}}$ (e.g., plane waves, linearised augmented plane waves, localised basis sets, real-space grids, . . . ).  \n\n# III. NEW FEATURES FOR WANNIERISATION AND DISENTANGLEMENT  \n\nIn this section we provide an overview of the new features associated with the core wannierisation and disentanglement algorithms in Wannier90, namely the ability to generate  \n\nWFs of specific symmetry; selectively localise a subset of the WFs and/or constrain their centres to specific sites; and perform wannierisation and disentanglement more efficiently through parallelisation.  \n\n# A. Symmetry-adapted Wannier functions  \n\nIn periodic systems, atoms are usually found at sites q whose site-symmetry group $G_{q}$ is a subgroup of the full point group $F^{'}$ of the crystal $^{19}$ (the symmetry operations in the group $G_{q}$ are those that leave q fixed). The set of points $\\left\\{\\mathbf{q}_{a}\\right\\}$ that are symmetry-equivalent sites to $\\mathbf{q}$ is called an $o r b i t^{20}$ . These are all the points in the unit cell that can be generated from $\\mathbf{q}$ by applying the symmetry operations in the full space group $G$ that do not leave q fixed. If ${\\bf q}_{a}$ is a high-symmetry site then its Wyckoff position has a single orbit20; for low-symmetry sites different orbits correspond to the same Wyckoff position. The number of points in the orbit(s) is the multiplicity $n_{q_{a}}$ of the Wyckoff position. MLWFs, however, are not bound to reside on such high-symmetry sites, and they do not necessarily possess the site symmetries of the crystal $^{17,21,22}$ . When using MLWFs as a local orbital basis set in methods such as first-principles tight binding, DFT+U and DFT plus dynamical-mean-field theory (DMFT), which deal with beyond-DFT correlations in a local subspace such as that spanned by $d$ orbitals (e.g., for systems containing transition metals atoms) or $f$ orbitals (e.g., for systems containing rare-earth or actinide series atoms), it is often desirable to ensure that the WFs basis possesses the local site symmetries.  \n\nSakuma $^{21}$ has shown that such symmetry-adapted Wannier functions (SAWFs) can be constructed by introducing additional constraints on the unitary matrices $U_{\\mathbf{k}}$ of Eq. (2) during the minimisation of the spread. SAWFs, therefore, can be fully integrated within the original maximal-localisation procedure. The SAWF approach gives the user a certain degree of control over the symmetry and centres of the Wannier functions at the expense of some localisation since the final total spread of the resulting SAWFs can only be equal to, or most often larger than, that of the corresponding MLWFs with no constraints (note that in principle some SAWFs can have a smaller individual spread than any MLWFs).  \n\nFor a given point ${\\bf q}_{a}$ in the home unit cell $\\mathbf{R}=\\mathbf{0}$ , the SAWFs centred at that point are denoted by  \n\n$$\n\\{w_{i a}^{(\\varrho)}(\\mathbf{r})\\equiv w_{i}^{(\\varrho)}(\\mathbf{r}-\\mathbf{q}_{a}),\\quad i=1,\\ldots,n_{\\varrho}\\},\n$$  \n\nwhere $\\varrho$ is the character of the irreducible representation (irrep) of the corresponding sitesymmetry group $G_{a}$ with dimension $n_{\\varrho}$ . For instance, in a simple fcc crystal such as copper (Cu), the site-symmetry group associated with the Cu site is $O_{h}$ ; one of its irreps $^{20}$ is, e.g., 3-dimensional $\\operatorname{\\mathrm{~\\AE~}}^{\\prime}2g$ and, assuming the Cu atom is located at the origin $\\mathbf{r}=\\mathbf{0}$ of the unit cell, three associated SAWFs are denoted $w_{10}^{12g}(\\mathbf{r}),w_{20}^{12g}(\\mathbf{r})$ and $w_{30}^{22g}(\\mathbf{r})$ .  \n\nTo find these SAWFs, one needs to specify appropriate unitary transformations $U_{m i a\\mathbf{k}}^{(\\varrho)}$ of the Bloch states, defined by  \n\n$$\n\\begin{array}{r l r}{\\displaystyle}&{}&{\\displaystyle w_{i a}^{(\\varrho)}({\\bf r-R})=\\frac{1}{N}\\sum_{\\bf k}e^{-i{\\bf k\\cdot R}}\\sum_{m=1}^{J}\\psi_{m{\\bf k}}({\\bf r})U_{m i a{\\bf k}}^{(\\varrho)}}\\\\ &{}&{\\displaystyle=\\frac{1}{N}\\sum_{\\bf k}e^{-i{\\bf k\\cdot R}}\\psi_{i a{\\bf k}}^{(\\varrho)}({\\bf r}),}\\end{array}\n$$  \n\nwhere $\\{\\psi_{i a\\mathbf{k}}^{(\\varrho)}(\\mathbf{r})\\}$ are basis functions of the irrep $\\varrho$ and are formed from linear combinations of the $J$ eigenstates $\\{\\psi_{n\\mathbf{k}}(\\mathbf{r})\\}$ of the Hamiltonian $H$ . Since $H_{\\l}$ is invariant under the full spacegroup $G$ , the representation of a given symmetry operation $g=({\\mathcal{R}}|\\mathbf{t})\\in G$ (where $\\mathcal{R}$ and $\\mathbf{t}$ are the rotation and fractional-translation parts of the symmetry operation, respectively) in the basis $\\{\\psi_{n\\mathbf{k}}(\\mathbf{r})\\}$ must be a $J\\times J\\ {\\mathrm{unitary~matrix^{19}~}}\\widetilde{d}_{\\bf k}(g)$ , i.e., $\\widetilde{d}_{\\bf k}(g)$ represents how the $J$ Bloch states are transformed by the symmetry operation $g$ :  \n\n$$\ng\\psi_{n\\mathbf{k}}(\\mathbf{r})=\\sum_{m=1}^{J}\\psi_{m\\mathcal{R}\\mathbf{k}}(\\mathbf{r})\\widetilde{d}_{m n\\mathbf{k}}(g),\\quad g\\in G,\n$$  \n\nwhere the matrix elements $\\bar{d_{\\bf k}}(y)_{\\mathrm{\\bf0\\mathrm{re}}}$ given by  \n\n$$\n\\widetilde{d}_{m n\\mathbf{k}}(g)=\\int\\mathrm{d}\\mathbf{r}\\psi_{m\\mathcal{R}\\mathbf{k}}^{*}(\\mathbf{r})\\psi_{n\\mathbf{k}}\\left(g^{-1}\\mathbf{r}\\right).\n$$  \n\nOn the other hand, the Bloch functions $\\{\\psi_{i a\\mathbf{k}}^{(\\varrho)}(\\mathbf{r})\\}$ , defined in Eq. (28), transform under the action of $g\\in G$ as  \n\n$$\ng\\psi_{i a\\mathbf{k}}^{(\\varrho)}(\\mathbf{r})=\\sum_{i^{\\prime}a^{\\prime}\\varrho^{\\prime}}\\psi_{i^{\\prime}a^{\\prime}\\mathcal{R}\\mathbf{k}}^{(\\varrho^{\\prime})}(\\mathbf{r})D_{i^{\\prime}a^{\\prime},i a\\mathbf{k}}^{(\\varrho^{\\prime},\\varrho)}(g),\n$$  \n\nwhere $D_{\\mathbf{k}}(g)$ is the matrix representation of the symmetry operation $g$ in the basis of $\\{\\psi_{i a\\bf{k}}^{(\\varrho)}({\\bf{r}})\\}$ ; the reader is referred to Refs. 19 and 21 for details.  \n\nFrom Eqs. (28), (29) and (31) it can shown $^{21}$ that, for a symmetry operation $g_{\\mathbf{k}}$ that leaves a given $\\mathbf{k}$ unchanged, the following relationship holds:  \n\n$$\nU_{{\\bf k}}D_{{\\bf k}}(g_{\\bf k})=\\widetilde{d}_{\\bf k}(g_{\\bf k})U_{\\bf k},~g_{\\bf k}\\in G_{\\bf k},\n$$  \n\nand, to obtain SAWFs, the initial unitary matrix $U_{\\mathbf{k}}$ ( $\\mathbf{k}\\in\\mathrm{IBZ}$ ) must satisfy this constraint. This can be achieved iteratively, starting with the initial projection onto localised orbitals as described in Sec. II C, and with knowledge of $\\widetilde{d}_{\\bf k}(g)$ [Eq. (29)] and $D_{\\mathbf{k}}(g)$ [Eq. (31)], as discussed in detail Ref. 21. The matrices $\\widetilde{d}_{\\bf k}(g)$ , which are independent of the underlying basis-set used to represent the Bloch states and are computed only once at the start of the calculation, can be calculated directly from the Bloch states via Eq. (30). The matrices $D_{\\mathbf{k}}(g)$ are calculated by specifying the centre ${\\bf q}_{a}$ and the desired symmetry of the Wannier functions (e.g., $s$ , $p$ , $d$ etc.) and, for each symmetry operation ga in the site-symmetry group $G_{a}$ , calculating the matrix representation of the rotational part.  \n\nFor an isolated set of bands, the minimisation of $\\widetilde\\Omega$ with the constraints defined in Eq. (32) requires the gradient $\\mathcal{G}_{\\bf k}^{\\mathrm{sym}}$ of the total spread $\\Omega$ w th respect to a symmetry-adapted gauge variation, which is then used to generate a search $\\mathrm{direction}\\mathcal{P}_{\\mathbf{k}}^{\\mathrm{sym}}$ . The symmetry-adapted gradient is given by  \n\n$$\n\\mathcal{G}_{\\mathbf{k}}^{\\mathrm{sym}}=\\frac{1}{n_{\\mathbf{k}}}\\sum_{g=(\\mathcal{R}|\\mathbf{t})\\in G}D_{\\mathbf{k}}(g)\\mathcal{G}_{\\mathcal{R}\\mathbf{k}}D_{\\mathbf{k}}^{\\dagger}(g),\n$$  \n\nwhere $\\mathcal{G}_{\\mathbf{k}}$ is the original gradient given in Eq. (11), and $n_{\\mathbf{k}}$ is the number of symmetry operations in $G$ that leave $\\mathbf{k}$ fixed. It is worth noting that there is no guarantee that Eq. (32) can be satisfied for any irrep, for example, when one is considering a target energy window with a limited number of Bloch states whose symmetry might not be compatible with the irrep.  \n\nIn the case of entangled bands, a similar two-step approach is taken as in the case of MLWFs (Sec. II B): first ΩI is minimised by selecting an optimal subspace of Bloch states that are required to transform according to Eq. (31), followed by minimisation of $\\widetilde\\Omega$ with respect to gauge variations that respect the site symmetries within this subspace, as described for the case of isolated bands above, but with the difference that the constraint of Eq. (32) is modified to  \n\n$$\nU_{{\\bf k}}D_{{\\bf k}}(g_{{\\bf k}})=D_{{\\bf k}}(g_{{\\bf k}})U_{{\\bf k}},\\quad g_{{\\bf k}}\\in G_{{\\bf k}},\n$$  \n\nsince the states of the optimal subspace transform according to Eq. (31), rather than Eq. (29). An implementation of the SAWF algorithm for both isolated and entangled bands can be found in pw2wannier90, the interface code between Quantum ESPRESSO and Wannier90. A typical calculation consists of the following steps: (a) Define the symmetry operations of the site-symmetry group. These are either calculated by pw2wannier90.x, if the site-symmetry group is equivalent to the full space group of the crystal, or they can be provided in the .sym file (eg, if the site-symmetry group contains fewer symmetry operations than the full space group); (b) Specify the site location and orbital symmetry of the SAWFs. These are defined in the projection block of the Wannier90 input file .win file. (c) Run a preprocessing Wannier90 calculation to write this info into an intermediate file (with extension .nnkp) which is then read by pw2wannier90.x; (d) Run pw2wannier90.x to calculate the $\\mathbf{D}$ matrix in Eq. (31). pw2wannier90.x computes also the ˜d matrix in Eq. (30) from the Kohn–Sham states of the DFT calculation; (e) These matrixes are then written to a .dmn file which is read by Wannier90 at the start of the optimisation.  \n\n# B. Selectively-localised Wannier functions and constrained Wannier centres  \n\nWang et al. have proposed an alternative method23 to the symmetry-adapted Wannier functions described in Section III A. Their method permits the selective localisation of a subset of the Wannier functions, which may optionally be constrained to have specified centres. Whilst this method does not enforce or guarantee symmetry constraints, it has been observed in the cases that have been studied $^{23}$ that Wannier functions whose centres are constrained to a specific site typically possess the corresponding site symmetries.  \n\nFor an isolated set of $J$ bands, selective localisation of a subset of $J^{\\prime}\\ \\leq\\ J$ Wannier functions is accomplished by minimising the total spread $\\Omega$ with respect to only $J^{\\prime}\\times J^{\\prime}$ degrees of freedom in the unitary matrix $U_{\\mathbf{k}}$ . The spread functional to minimise is then given by  \n\n$$\n\\Omega^{\\prime}=\\sum_{n=1}^{J^{\\prime}\\leq J}\\left[\\langle w_{n\\mathbf{0}}|r^{2}|w_{n\\mathbf{0}}\\rangle-|\\langle w_{n\\mathbf{0}}|\\mathbf{r}|w_{n\\mathbf{0}}\\rangle|^{2}\\right],\n$$  \n\nwhich reduces to the original spread functional $\\Omega$ of Eq. (3) for $J^{\\prime}=J$ . When $J^{\\prime}<J$ , it is no longer possible to cast the functional $\\Omega^{\\prime}$ as a sum of a gauge-independent term $\\Omega_{\\mathrm{I}}$ and gauge-dependent one $\\widetilde\\Omega$ , as done in Eq. (4) for $\\Omega$ . Nevertheless, the minimisation can be carried out with methods very similar to those described in Section II. In fact, for $J^{\\prime}<J$ , $\\Omega^{\\prime}$ can be written as the sum of two gauge-dependent terms, $\\Omega^{\\prime}=\\Omega_{\\mathrm{IOD}}+\\Omega_{\\mathrm{D}}$ , where $\\Omega_{\\mathrm{IOD}}$ is formally given by the sum of $\\Omega_{\\mathrm{I}}$ and the off-diagonal term $(m\\neq n),\\quad m,n\\leq J^{\\prime}<J$ of $\\widetilde\\Omega$ , and $\\Omega_{\\mathrm{{D}}}$ by the diagonal term ( $m=n$ ) of $\\widetilde\\Omega$ . If one adopts the usual discrete representation on a uniform Monkhorst–Pack grid of $k$ -points, $\\Omega_{\\mathrm{IOD}}$ and $\\Omega_{\\mathrm{{D}}}$ are given by23  \n\n$$\n\\Omega_{\\mathrm{IOD}}=\\frac{1}{N}\\sum_{\\mathbf{k},\\mathbf{b}}w_{b}\\left[J^{\\prime}-\\sum_{n}^{J^{\\prime}<J}\\left|M_{n n}^{(\\mathbf{k},\\mathbf{b})}\\right|^{2}\\right]\n$$  \n\nand  \n\n$$\n\\Omega_{\\mathrm{{D}}}=\\frac{1}{N}\\sum_{n=1}^{J^{\\prime}<J}\\sum_{{\\bf{b}},{\\bf{k}}}w_{b}\\left(\\mathrm{Im}\\ln M_{n n}^{({\\bf{k}},{\\bf{b}})}+{\\bf{b}}\\cdot\\bar{\\bf{r}}_{n}\\right)^{2}.\n$$  \n\nWith this new spread functional, we can mimic the procedure used to obtain a set of MLWFs, and derive the gradient $\\mathcal{G}_{\\bf k}^{\\prime}$ of $\\Omega^{\\prime}$ which gives the search direction to be used in the minimisation. The matrix elements of $\\mathcal{G}_{\\bf k}^{\\prime}$ read  \n\n$$\n\\mathcal{G}_{m n\\mathbf{k}}^{\\prime}=\\left\\{\\begin{array}{l l}{\\mathcal{G}_{m n\\mathbf{k}}}&{\\mathrm{~,~}}\\\\ {-2\\sum_{\\mathbf{b}}w_{b}\\left[R_{n m}^{(\\mathbf{k},\\mathbf{b})*}-i T_{n m}^{(\\mathbf{k},\\mathbf{b})*}\\right]}&{\\mathrm{~}m\\mathcal{G}^{\\mathcal{N},\\mathcal{I}^{\\prime}}<n\\leq J,}\\\\ {2\\sum_{\\mathbf{b}}w_{b}\\left[R_{m n}^{(\\mathbf{k},\\mathbf{b})}+i T_{n m}^{(\\mathbf{k},\\mathbf{b})}\\right]}&{\\mathrm{~}j^{\\prime}<m\\leq J,n\\leq J^{\\prime},}\\\\ {0}&{\\mathrm{~}j^{\\prime}<m\\leq J,p^{\\prime}<n\\leq J}\\end{array}\\right.\n$$  \n\nwhere $\\mathcal{G}_{m n\\mathbf{k}}$ are the matrix elements of the original gradient in Eq. (11) (see also Ref. 15), and $R_{m n}^{(\\mathbf{k},\\mathbf{b})}$ and $T_{m n}^{(\\mathbf{k},\\mathbf{b})}$ are given by Eq. (12) and Eq. (13), respectively. As a result of the minimisation, we obtain a set of $J^{\\prime}$ maximally-localised Wannier functions, known as selectively-localised Wannier functions (SLWFs), whose spreads are in general smaller than the corresponding MLWFs. Naturally, the remaining $J{-}J^{\\prime}$ functions will be more delocalised than their MLWF counterparts, as they are not optimised, and the overall sum of spreads will be larger (or in the best case scenario equal).  \n\nThe centres of the SLWFs may be constrained by adding a quadratic penalty function to the spread functional $\\Omega^{\\prime}$ , defining a new functional given by  \n\n$$\n\\begin{array}{c}{{\\displaystyle\\Omega_{\\chi}^{\\prime}=\\sum_{n=1}^{J^{\\prime}\\leq J}[\\langle w_{n\\mathbf{0}}|r^{2}|w_{n\\mathbf{0}}\\rangle-|\\langle w_{n\\mathbf{0}}|\\mathbf{r}|w_{n\\mathbf{0}}\\rangle]^{2}}}\\\\ {{+\\lambda(\\bar{\\mathbf{r}}_{n}-\\mathbf{x}_{n})^{2}],}}\\end{array}\n$$  \n\nwhere λ is a Lagrange multiplier and $\\mathbf{x}_{n}$ is the desired centre for the $n^{\\mathrm{th}}$ WF. The procedure outlined above for minimising $\\Omega^{\\prime}$ can be also adapted to deal with $\\Omega_{\\lambda}^{\\prime}$ (see Ref. 23 for details), and minimising $\\Omega_{\\lambda}^{\\prime}$ results in selectively-localised Wannier functions subject to the constraint of fixed centres (SLWF $^+$ C). As noted above, it is observed that WFs derived using the SLWF+C approach naturally possess site symmetries, and their individual spreads are usually smaller than the corresponding spreads of MLWFs, although the total spread, combination of the $J^{\\prime}$ selectively optimised WFs and the $J-J^{\\prime}$ unoptimised functions, is larger than the total spread of the MLWFs (see, for instance last column in Tab. 1).  \n\nIn the case of entangled bands, the SLWF(+C) method implicitly assumes that a subspace selection has been performed, i.e., that a smooth $J$ -dimensional manifold exists. Since for the $\\Omega^{\\prime}$ and $\\Omega_{\\lambda}^{\\prime}$ functionals it is not possible to define an $\\Omega_{I}$ that measures the intrinsic smoothness of the underlying manifold, the additional constraints in Eq. (35) and Eq. (39) can only be imposed during the wannierisation step. This means that SLWF(+C) can be seamlessly coupled with the disentanglement procedure, with no further additions to the original procedure of Sec. II B.  \n\n# C. SAWF and SLWF+C in GaAs  \n\nAs an example of the capabilities of the SAWF and SLWF+C approaches, we show how to construct atom-centred WFs that possess the local site symmetries in gallium arsenide (GaAs). In particular, we discuss how to obtain one WF from the four valence bands of GaAs that is centred on the As atom and that transforms like the identity under the symmetry operations in $T_{d}$ , the site-symmetry group of the As site (for completeness, we also show one MLWF and one SLWF without constraints). Since we only deal with the four valence bands of GaAs—an isolated manifold—no prior subspace selection is required for the wannierisation. All calculations were carried out with the plane-wave DFT code Quantum ESPRESSO1, employing PAW pseudopotentials $^{24,25}$ from the pslibrary $\\left(\\mathrm{v}1.0\\right)^{26}$ . For the exchange-correlation functional we use the Perdew–Burke–Ernzerhof approximation27. The energy cut-off for the plane-waves basis is set to 35.0 Ry, and a $4\\times4\\times4$ uniform grid is used to sample the Brillouin zone. The lattice parameter is set to the experimental value (5.65 ˚A). The overlap matrices $\\overline{{\\mathscr{M}}}_{m n}^{(\\mathbf{k},\\mathbf{b})}$ in Eq. (9), the projection matrices $A_{m n\\bf{k}}$ in Eq. (26) and both $\\widetilde{d}_{\\mathbf{k}}(\\widetilde{g})$ in Eq. (30) and $D_{\\mathbf{k}}(g)$ in Eq. (31) have been computed with the pw2wannier90.x interface.  \n\nGaAs is a III-V semiconductor that crystallises in the fcc cubic structure, with a twoatom basis: the Ga cation and the As anion (space group $F{-}43m$ ); in our example the Ga atom is placed at the origin of the unit cell, whose Wyckoff letter is $a$ and site-symmetry group is $-43m$ , also known as $T_{d}$ . The As atom is placed at (1/4,1/4,1/4), whose Wyckoff letter is $c$ and site-symmetry group is also $T_{d}$ .  \n\nMarzari and Vanderbilt $^{15}$ have shown that the MLWFs for the 4-dimensional valence manifold are centred on the four As-Ga bonds, have $s p^{3}$ character and can be found by specifying four $s$ -like orbitals on each covalent bond as initial guess (two representatives are shown in Fig. 1(a),(e)). These bond-centred functions correspond to the irreducible representation $A_{1}$ of the site-symmetry group ${C}_{3v}$ of the Wyckoff position $e$ . Hence, the MLWFs can also be obtained with the SAWF approach by specifying the centres and the shapes of the initial projections, e.g. four $s$ -like orbitals centred on the four As–Ga bonds, and the symmetry operations in the point group ${C}_{3v}$ .  \n\nUsing the SAWF method we can enforce the WFs to have the local site symmetries. In particular, since $T_{d}$ has 5 irreps of dimension 1, 1, 2, 3 and 3 respectively, one can form an 1+3–dimensional representation for the four SAWFs. Thus, a set of initial projections compatible with the symmetries of the valence bands is: one $\\mathcal{S}$ -like orbital (1-dimensional irrep whose character is $A_{1}$ ) and three $p$ -like orbitals (3-dimensional irrep whose character is $T_{2}$ ) centred on As. Fig. 1(b) shows the SAWF which corresponds to the $A_{1}$ representation and transforms like the identity under $\\mathcal{I}_{d}$ and in Fig. 1(f) one of the three SAWF corresponding to the 3-dimensional irrep with $p$ character is shown.  \n\nThe same SAWF corresponding to the $A_{1}$ representation can be obtained with the SLWF+C method by selectively localising one function $J^{\\prime}=1$ ( $J\\ =\\ 4$ ) and constraining its centre to sit on the As site $\\left(1/4,1/4,1/4\\right)$ . In the case of GaAs the SLWF+C method turns out to be very robust, to the point that four $s$ -like orbitals randomly centred in the unit cell can be used as initial guess without affecting the result of the optimised function. Fig. 1(c) shows the resulting function using the SLWF method without constraints, while Fig. 1(d) shows the result using SLWF+C, which is identical to the SAWF in Fig. 1(b). Finally, in the second row of Fig. 1 ((e), (f), (g) and (h)) one of the other three Wannier function is shown for all four minimisation scheme. In the case of SAWF, Fig. 1(f), this WF is centred on the As atom, has a larger spread than the corresponding MLWF (Fig. 1(e)) and shows a $p$ -like character as expected. However, in the case of SLWF and SLWF+C (Fig. 1(g) and (h)), these WFs are not optimised and therefore they show a larger spread than the corresponding MLWF and are somewhat less symmetric (see Tab. 1).  \n\nIt is worth to note that for this particular system, it is possible to achieve the result of a $s$ -like and three $p$ -like WFs also with the maximal localisation procedure if one carefully selects the initial projections, i.e., one $s$ -like and three $p$ -like orbitals on the As atom. The resulting WFs will possess the local site symmetries but will not correspond to the global minimum of the spread functional $\\Omega$ . More precisely, they will correspond to a saddle point of $\\Omega$ (unstable against small perturbations of the initial projections).  \n\n# D. Parallelisation  \n\nIn Wannier90 v3.0 we have implemented an efficient parallelisation scheme for the calculation of MLWFs using the message passing interface (MPI).  \n\nCalculation of the spread and distribution of large matrices. The time-consuming part in the evaluation of the spread $\\Omega$ is updating the $M^{(\\mathbf{k},\\mathbf{b})}$ matrices according to Eq. (16), since this requires computing overlap matrix elements between all pairs of bands, and between all $k$ -points $\\mathbf{k}$ and their neighbours $\\mathbf{k}+\\mathbf{b}$ . Therefore, an efficient speed up for the evaluation of the spread can be achieved by distributing over several processes the calculation of the $M^{(\\mathbf{k},\\mathbf{b})}$ matrices for different $k$ -points. In order to compute the ${\\cal M}^{(\\mathbf{k},\\mathbf{b})}$ according to Eq. (16), the $U_{\\mathbf{k}+\\mathbf{b}}$ matrices are sent from process to process prior to the calculation of the overlap matrices. We stress the fact that the $U_{\\mathbf{k}+\\mathbf{b}}$ matrices are the only large arrays that have to be shared between processes, which limits the time spent in communication. The relatively large $M^{(\\mathbf{k},\\mathbf{b})}$ matrices are not sent between processes for the evaluation of Eqs. (7) and (8). Instead, it is enough to collect the contributions to the spread from the different $k$ -points, i.e., a set of scalars, and then sum them up for evaluation of the total spread. This parallelisation scheme is illustrated in Fig. 2 for a $3\\times3$ mesh of $k$ -points with 9 MPI processes.  \n\nMoreover, we emphasise that our parallelisation scheme relies on the evaluation of relevant matrices over $k$ -points on each process (or core, since the only parallelisation scheme currently implemented is MPI and typically each process is assigned to a different CPU core). For systems with large number of $k$ -points and bands, it is also desirable to distribute these matrices across the available processes to reduce the memory requirements. For example, in the case of isolated bands, instead of storing all the ${\\cal M}^{(\\mathbf{k},\\mathbf{b})}$ matrices on all cores (requiring an allocation per core of dimension $J\\times J\\times N\\times N_{b}$ , where $N_{b}$ is the number neighbours of each of the $N$ $k$ -points of the mesh) we distribute the matrices across the $N_{\\mathrm{c}}$ cores. In particular, only the root process stores the full matrices (for I/O purposes) while all other processes just store the ${\\cal M}^{(\\mathbf{k},\\mathbf{b})}$ matrices for the $k$ -points associated with the given process.  \n\n<html><body><table><tr><td>(a)MLWF (b) s-like SAWF</td><td></td><td>(c) opt. SLWF</td><td>(d) opt. SLWF+C</td></tr><tr><td>(e) MLWF</td><td>(f) p-like SAWF</td><td>(g)unopt. SLWF</td><td>(h) unopt.SLWF+C (g)， (h)</td></tr><tr><td colspan=\"4\">(a), (b), (c), (d) Method r</td></tr><tr><td>MLWF</td><td colspan=\"2\">[A] [A2] (-0.857,0.857,0.857) 1.78</td><td>[A2] [A2] (0.857,-0.857,0.857) 1.78 7.12</td></tr><tr><td>SAWF</td><td>(-1.413,1.413,1.413) 1.64</td><td>(-1.413,1.413,1.413) 2.83</td><td>10.14</td></tr><tr><td>SLWF</td><td>(-0.89, 0.89, 0.89)</td><td>1.42 (0.89,-0.89,0.92) 2.14</td><td>9.8</td></tr><tr><td>SLWF+C</td><td>(-1.413,1.413,1.413) 1.63</td><td>(1.23,-1.23,1.08) 2.72</td><td>7.87</td></tr></table></body></html>  \n\nFIG. 1: Top (figure): comparison of two Wannier functions representatives resulting from different minimisation schemes for gallium arsenide (larger pink spheres: Ga cation atoms, smaller yellow spheres:  \n\nAs anions): (a), (e) MLWF; (b), (f) SAWF; (c), (g) SLWF; (d), (h) SLWF $+$ C. For MLWF, SLWF and   \nSLWF $^+$ C, four $s$ -type orbitals centred at the midpoints of the four Ga–As bonds   \n((1/8,1/8,1/8),(1/8,1/8,-3/8),(-3/8,1/8,1/8),(1/8,-3/8,1/8)) were used as initial guess. In the case of SLWF and   \nSLWF+C, we optimise the first WF (and also constrain its centre to sit at (1/4,1/4, $^1/4$ ), i.e. on the As   \natom, for SLWF+C), while all the other WFs are left unoptimised. For SAWF, one $s$ -type and three   \n$p$ -type orbitals centred on the As atom are used as initial guess. Specifically, the first row shows one   \nMLWF (a), one SAWF with s character centred on As (b), one WF obtained with the selective localisation   \nscheme (c) and one WF obtained obtained with the selective localisation scheme with additional   \nconstraints on its centre (d). The second row shows one of the other three WFs for all four methods. In   \nparticular: (e) MLWF, (f) SAWF with $p$ character, (g) unoptimised SAWF and (f) unoptimised SAWF+C.   \nFor all plots we choose an isosurface level of $\\pm\\ 0.5\\mathrm{\\AA}^{-3/2}$ (blue for $^+$ values and red for $-$ values) using the   \nVesta visualisation program28. Bottom (table): Cartesian coordinates of the centres $\\bar{\\mathbf{r}}$ and minimised   \nindividual spreads $\\left\\langle r^{2}\\right\\rangle-\\bar{r}^{2}$ for the two representative Wannier functions of each of the four different   \nminimisation schemes and initial guesses described above. We also report the total spread $\\Omega$ of all four   \nvalence WFs2f0or each method.  \n\n![](images/3cb264d668c72cb0abc1e910350d43788ca8bae2a474ae81d14e54402de5fa87.jpg)  \nFIG. 2: Illustration of the parallelisation scheme for a $3^{\\prime}\\mathrm{\\times3}$ mesh of $k$ -points (black dots) and one MPI process per $k$ -point. The calculation of the M (k,b), $Z_{\\mathbf{k}}$ , $\\Delta W_{\\mathbf{k}}$ and $U_{\\mathbf{k}}$ matrices are distributed over processes by $k$ -point. The $U_{\\mathbf{k}\\pm\\mathbf{b}}$ matrices for the neighbouring $k$ -points are sent from process to process (orange arrows) for the calculation of the ${\\cal M}^{(\\mathbf{k},\\mathbf{b})}$ and $Z_{\\mathbf{k}}$ matrices.  \n\nIn such a way, the memory requirement per core (for the ${\\cal M}^{(\\mathbf{k},\\mathbf{b})}$ matrices) decreases by a factor of approximately $N_{\\mathrm{c}}$ .  \n\nMinimisation of the spread. The minimisation of the spread functional is based on an iterative steepest-descent or conjugate-gradient algorithm. In each iteration, the unitary matrices $U_{\\mathbf{k}}$ are updated according to $U_{\\mathbf{k}}=U_{\\mathbf{k}}\\exp{(\\Delta W_{\\mathbf{k}})^{15}}$ , where $\\Delta W_{\\mathbf{k}}=\\alpha\\mathcal{D}_{\\mathbf{k}}$ , see Eq. (15). Updating the $U_{\\mathbf{k}}$ matrices according to this equation is by far the most time-consuming part in the iterative minimisation algorithm, as it requires a diagonalisation of the $\\Delta W_{\\mathbf{k}}$ matrices. A significant speed-up can be obtained, however, by distributing the diagonalisation of the different $\\Delta W_{\\mathbf{k}}$ matrices over several processes, and performing the calculations fully in parallel. The evaluation of $\\Delta W_{\\mathbf{k}}$ essentially requires the calculation of the overlap matrices M (k,b), as discussed above.  \n\nDisentanglement. The disentanglement procedure is concerned with finding the optimal subspace $\\dot{S}_{\\bf k}$ . As the functional $\\Omega_{\\mathrm{I}}$ measures the global subspace dispersion across the Brillouin zone, at first sight it is not obvious that the task of minimising the spread $\\Omega_{\\mathrm{I}}$ can be parallelised with respect to the $k$ -points. In the iterative algorithm of Eq. (21), the systematic reduction of the spread functional at the $i^{\\mathrm{th}}$ iteration is achieved by minimising the spillage of the subspace $S_{\\mathbf{k}}^{(i)}$ over the neighbouring subspaces from the previous iteration $S_{\\mathbf{k}+\\mathbf{b}}^{(i-1)}$ . This problem reduces to the diagonalisation of $N$ independent matrices ( $N$ is the total number of $k$ -points of the mesh), where an efficient speed-up of the disentanglement procedure can be achieved by distributing the diagonalisation of the $Z_{\\mathbf{k}}^{(i)}$ matrices of Eq. (23) over several processes, which can be done fully in parallel. Since the construction of $Z_{\\mathbf{k}}^{(i)}$ only requires the knowledge of the $U_{\\mathbf{k}+\\mathbf{b}}^{(i-1)}$ matrices, these must be communicated between processes, as shown in Fig. 2. This results in a similar time spent in communication for the disentanglement part of the code as for the wannierisation part.  \n\nPerformance. We have tested the performance of this parallelisation scheme for the calculation of the MLWFs in a FeP $\\mathrm{t}(5)$ /P $\\mathrm{t}$ (18) thin film. Computational details were given in Ref. 29. The benchmarks have been performed on the JURECA supercomputer of the Ju¨lich Supercomputing Center. We have extracted an optimal subspace of dimension $J=414$ from a set of 580 Bloch states per $k$ -point. The upper limit of the inner window was set to 5 eV above the Fermi energy, and 414 MLWFs were constructed by minimising the spread $\\Omega$ . The performance benchmark was based on the average wall-clock time for a single iteration of the minimisation procedure (several thousand iterations are usually needed for convergence). We first analyse the weak scaling of our implementation, i.e., how the computation time varies with the number of cores $N_{\\mathrm{c}}$ for a fixed number of $k$ -points per process. We show in Fig. 3(a) the time per iteration for the disentanglement and wannierisation parts of the minimisation, always using one $k$ -point per process. As we vary the number of $k$ -points $N$ from 4 to 144, the computation time increases only by a factor of 1.3 and 1.8 for disentanglement and wannierisation, respectively. We then demonstrate the strong scaling of our parallelisation scheme in Fig. 3(b), i.e., how the computation time varies with the number of cores $N_{\\mathrm{c}}$ for a fixed number $N=$ 64 of $k$ -points. When varying the number of cores from 4 to 64, we observe a decrease of the computation time per iteration by a factor of 12.6 and 9.5 for disentanglement and wannierisation, respectively. The deviation from ideal scaling is mostly explained by the time spent in inter-core communication of the $U_{\\mathbf{k}+\\mathbf{b}}$ matrices.  \n\n# IV. ENHANCEMENTS IN FUNCTIONALITY  \n\nIn this section we describe a number of enhancements to the functionality of the core Wannier90 code, namely: the ability to compute and visualise spinor-valued WFs, in  \n\n![](images/83a7fdd83a2c8173fc12de9f4010b9eebb2840db6203bb85cf313d0132107e3f.jpg)  \n\nFIG. 3: Plots of the time per single minimisation iteration as a function of the number of cores $N_{\\mathrm{c}}$ . (a) Weak scaling of the implementation, where the number of $k$ -points per process is fixed to one, i.e., $N_{\\mathrm{c}}=N$ . The time only increases by a factor 1.3 (1.8) for the disentanglement (wannierisation) parts of the code, when going from $N_{\\mathrm{c}}=4$ to $N_{\\mathrm{c}}=144$ . (b) Strong scaling of the algorithm for a fixed number of $k$ -points $N=64$ . The time per iteration with one single CPU (serial) is reported in the figure.  \n\ncluding developments to the interface with the Quantum ESPRESSO package to cover also the case of non-collinear spin calculations performed with ultrasoft pseudopotentials (previously not implemented); an improvement to the method for interpolating the $k$ -space Hamiltonian; the ability to select a subset from a larger set of projections of localised trial orbitals onto the Bloch states for initialising the WFs; and new functionality for plotting WFs in Gaussian cube format on WF-centred grids with non-orthogonal translation vectors.  \n\n# A. Spinor-valued Wannier functions with ultrasoft and projector-augmented-wave pseudopotentials  \n\nThe calculation of the overlap matrix in Eq. (17) within the ultrasoft-pseudopotential formalism proceeds via the inclusion of so-called augmentation functions,30  \n\n$$\n\\begin{array}{r l}{\\lefteqn{M_{m n}^{(\\mathbf{k},\\mathbf{b})}=\\left<u_{m\\mathbf{k}}\\ |\\ u_{n,\\mathbf{k}+\\mathbf{b}}\\right>}}\\\\ &{+\\sum_{I i j}Q_{i j}^{I}(\\mathbf{b})\\left<\\psi_{m\\mathbf{k}}^{\\mathrm{ps}}|B_{I i j}^{(\\mathbf{k},\\mathbf{b})}|\\psi_{n,\\mathbf{k}+\\mathbf{b}}^{\\mathrm{ps}}\\right>,}\\end{array}\n$$  \n\nwhere $|\\psi_{m\\mathbf{k}}^{\\mathrm{ps}}\\rangle$ is the pseudo-wavefunction,  \n\n$$\nQ_{i j}^{I}({\\bf b})=\\int\\mathrm{d}{\\bf r}\\ Q_{i j}^{I}({\\bf r})e^{-i{\\bf b}\\cdot{\\bf r}}\n$$  \n\nis the Fourier transform of the augmentation charge, and $B_{I i j}^{(\\mathbf{k},\\mathbf{b})}=|\\beta_{I i}^{\\mathbf{k}}\\rangle\\langle\\beta_{I j}^{\\mathbf{k}+\\mathbf{b}}|$ , where |βIki⟩ denotes the $i^{\\mathrm{th}}$ projector of the pseudopotential on the $I^{\\mathrm{th}}$ atom in the unit cell. We refer to Appendix B of Ref. 30 for detailed expressions.  \n\nWhen spin-orbit coupling is included, the Bloch functions become two-component spinors $(\\psi_{n\\mathbf{k}}^{\\uparrow}(\\mathbf{r}),\\psi_{n\\mathbf{k}}^{\\downarrow}(\\mathbf{r}))^{\\mathrm{T}}$ , where $\\psi_{n\\bf{k}}^{\\sigma}({\\bf{r}})$ is the spin-up (for $\\sigma=\\uparrow$ ) or spin-down $\\mathrm{for}\\delta\\varepsilon=\\downarrow$ ) component with respect to the chosen spin quantisation axis. Accordingly, $Q_{i j}^{I}(\\mathbf{b})$ becomes $Q_{i j}^{I\\sigma\\sigma^{\\prime}}({\\bf b})$ (see Eq. (18) in Ref. 31) and Eq. (40) becomes  \n\n$$\n\\begin{array}{l}{{\\displaystyle M_{m n}^{\\mathrm{({\\bfk},b)}}=\\left\\langle u_{m\\mathbf{k}}\\middle\\vert u_{n,{\\bf k}+{\\bf b}}\\right\\rangle}\\ ~}\\\\ {{\\displaystyle~+\\sum_{I i j\\sigma\\sigma^{\\prime}}Q_{i j}^{I\\sigma\\sigma^{\\prime}}({\\bf b})\\left\\langle\\psi_{m\\mathbf{k}}^{\\mathrm{ps},\\sigma}\\middle\\vert B_{I i j}^{\\mathrm{({\\bfk},b)}}\\middle\\vert\\psi_{n,{\\bf k}+{\\bf b}}^{\\mathrm{ps},\\sigma^{\\prime}}\\right\\rangle}.}}\\end{array}\n$$  \n\nThe above expressions, together with the corresponding ones for the matrix elements of the spin operator, have been implemented in the pw2wannier90.x interface between Quantum ESPRESSO and Wannier90.  \n\nThe plotting routines of Wannier90 have also been adapted to work with the complexvalued spinor WFs obtained from calculations with spin-orbit coupling. It then becomes necessary to decide how to represent graphically the information contained in the two spinor components.  \n\nOne option is to only plot the norm $|\\psi_{n\\mathbf k}(\\mathbf{r})|=\\sqrt{|\\psi_{n\\mathbf k}^{\\uparrow}(\\mathbf r)|^{2}+|\\psi_{n\\mathbf k}^{\\downarrow}(\\mathbf r)|^{2}}$ of spinor WFs, which is reminiscent of the total charge density in the case of a 2 $\\times$ 2 density matrix in non-collinear DFT. Another possibility is to plot independently the up- and downspin components of the spinor WF. Since each of them is in general complex-valued, two options are provided in the code: (i) to plot only the magnitudes $|\\psi_{n\\bf{k}}^{\\uparrow}({\\bf{r}})|$ and $|\\psi_{n\\bf{k}}^{\\downarrow}({\\bf{r}})|$ of the two components; or (ii) to encode the phase information by outputting $|\\psi_{n\\mathbf{k}}^{\\uparrow}(\\mathbf{r})|\\mathrm{sgn}(\\mathrm{Re}\\{\\psi_{n\\mathbf{k}}^{\\uparrow}(\\mathbf{r})\\})$ and $|\\psi_{n\\mathbf{k}}^{\\downarrow}(\\mathbf{r})|\\mathrm{sgn}(\\mathrm{Re}\\{\\psi_{n\\mathbf{k}}^{\\downarrow}(\\mathbf{r})\\})$ , where sgn is the sign function. Which of these various options is adopted by the Wannier90 code is controlled by two input parameters, wannier plot spinor mode and wannier plot spinor phase.  \n\nFinally we note that, for WFs constructed from ultrasoft pseudopotentials or within the projector-augmented-wave (PAW) method, only pseudo-wavefunctions represented on the soft FFT grid are considered in plotting WFs within the present scheme, that is, the WFs are not normalised. We emphasise that this affects only plotting of the WFs in real-space and not the calculation of the MLWFs (the overlap matrices being correctly computed by the interface codes).  \n\n# B. Improved Wannier interpolation by minimal-distance replica selection  \n\nThe interpolation of band structures (and many other quantities) based on Wannier functions is an extremely powerful tool $32\\substack{-34}$ . In many respects it resembles Fourier interpolation, which uses discrete Fourier transforms to reconstruct faithfully continuous signals from a discrete sampling, provided that the signal has a finite bandwidth and that the sampling rate is at least twice the bandwidth (the so-called Nyquist–Shannon condition).  \n\nIn the context of Wannier interpolation, the “sampled signal” is the set of matrix elements  \n\n$$\nH_{m n\\mathbf k_{j}}=\\langle\\chi_{m\\mathbf k_{j}}|H|\\chi_{n\\mathbf k_{j}}\\rangle\n$$  \n\nof a lattice-periodic operator such as the Hamiltonian, defined on the same uniform grid $\\{\\mathbf{k}_{j}\\}$ that was used to minimise the Wannier spread functional (see Sec. II A). The states $|\\chi_{n\\mathbf{k}_{j}}\\rangle$ are the Bloch sums of the WFs, related to ab initio Bloch eigenstates by $|\\chi_{n\\mathbf{k}_{j}}\\rangle=$ $\\begin{array}{r}{\\sum_{m}|\\psi_{m\\mathbf{k}_{j}}\\rangle U_{m n\\mathbf{k}_{j}}}\\end{array}$ .  \n\nTo reconstruct the “continuous signal” $H_{n m\\bf{k}}$ at arbitrary $\\mathbf{k}$ , the matrix elements of Eq. (43) are first mapped onto real space using the discrete Fourier transform  \n\n$$\n\\widetilde{H}_{m n\\mathbf{R}}=\\langle w_{m\\mathbf{0}}|H|w_{n\\mathbf{R}}\\rangle=\\frac{1}{N}\\sum_{j=1}^{N}e^{-i\\mathbf{k}_{j}\\cdot\\mathbf{R}}H_{m n\\mathbf{k}_{j}},\n$$  \n\nwhere $N=N_{1}{\\times}N_{2}{\\times}N_{3}$ is the grid size (which is also the number of $k$ -points in Wannier90). The matrices $H_{m n\\mathbf{k}_{j}}$ are then interpolated onto an arbitrary $\\mathbf{k}$ using an inverse discrete Fourier transform,  \n\n$$\nH_{m n\\mathbf{k}}=\\sum_{\\mathbf{R^{\\prime}}}e^{i\\mathbf{k}\\cdot\\mathbf{R^{\\prime}}}\\widetilde{H}_{m n\\mathbf{R^{\\prime}}},\n$$  \n\nwhere the sum is over $N$ lattice vectors $\\mathbf{R^{\\prime}}$ , and the interpolated energy eigenvalues are obtained by diagonalising $H_{\\mathbf{k}}$ . In the limit of an infinitely dense grid of $k$ -points the procedure is exact and the sum in Eq. (45) becomes an infinite series. Owing to the real-space localisation of the Wannier functions, the matrix elements ${\\widetilde{H}}_{m n\\mathbf{R}}$ become vanishingly small when the distance between the Wannier centres exceeds a critical value $L$ (the “bandwidth” of the Wannier Hamiltonian), so that actually only a finite number of terms contributes significantly to the sum in Eq. (45). This means that, even with a finite $N_{1}\\times N_{2}\\times N_{3}$ grid, the interpolation is still accurate provided that – by analogy with the Nyquist–Shannon condition – the “sampling rate” $N_{i}$ along each cell vector $\\mathbf{a}_{i}$ is sufficiently large to ensure that $N_{i}|\\mathbf{a}_{i}|>2L$ .  \n\nStill, the result of the interpolation crucially depends on the choice of the $N$ lattice vectors to be summed over in Eq. (45). Indeed, when using a finite grid, there is a considerable freedom in choosing the set $\\{{\\bf{R}}^{\\prime}\\}$ as ${\\widetilde{H}}_{m n\\mathbf{R}}$ is invariant under ${\\mathbf{R}}\\rightarrow{\\mathbf{R}}+{\\mathbf{T}}$ for any vector T of the Born–von Karman superlattice generated by $\\left\\{\\mathbf{A}_{i}=N_{i}\\mathbf{a}_{i}\\right\\}$ . The phase factor in Eq. (45) is also invariant when $\\mathbf{k}\\in\\{\\mathbf{k}_{j}\\}$ , but not for arbitrary $\\mathbf{k}$ . Hence we need to choose, among the infinite set of “replicas” ${\\bf R}^{\\prime}={\\bf R}+{\\bf T}$ of $\\mathbf{R}$ , which one to include in Eq. (45). We take the original vectors $\\mathbf{R}$ to lie within the Wigner–Seitz supercell centred at the origin. If some of them fall on its boundary then their total number exceeds $\\varLambda$ and weight factors must be introduced in Eq. (45). For each combination of m, n and R, the optimal choice of $\\mathbf{\\vec{x}}$ is the one that minimises the distance  \n\nbetween the two Wannier centres. With this choice, the spurious effects arising from the artificial supercell periodicity are minimised.  \n\nEarlier versions of Wannier90 implemented a simplified procedure whereby the vectors $\\mathbf{R^{\\prime}}$ in Eq. (45) were chosen to coincide with the unshifted vectors $\\mathbf{R}$ that are closer to the origin than to any other point $\\mathbf{T}$ on the superlattice, irrespective of the WF pair $(m,n)$ . As illustrated in Fig. 4, this procedure does not always lead to the shortest distance between the pair of WFs, especially when some of the $N_{i}$ are small and the Wannier centres are far from the origin of the cell.  \n\nWannier90 now implements an improved algorithm that enforces the minimal-distance condition of Eq. (46), yielding a more accurate Fourier interpolation. The algorithm is the following:  \n\n(a) For each term in Eq. (45) pick, among all the replicas ${\\bf R}^{\\prime}={\\bf R}+{\\bf T}$ of $\\mathbf{R}$ , the one that minimises the distance between Wannier centres (Eq. (46)).   \n(b) If there are $\\mathcal{N}_{m n\\mathbf{R}}$ different vectors $\\mathbf{T}$ for which the distance of Eq. (46) is minimal, then include all of them in Eq. (45) with a weight factor $1/\\mathscr{N}_{m n\\mathbf{R}}$ .  \n\nAn equivalent way to describe these steps is that (a) we choose $\\mathbf{T}$ such that ${\\bf r}_{n}+{\\bf R}+{\\bf T}$ falls inside the Wigner–Seitz supercell centred at $\\mathbf{r}_{m}$ (see Fig. 4), and that (b) if it falls on a face, edge or vertex of the Wigner–Seitz supercell, we keep all the equivalent replicas with an appropriate weight factor. In practice the condition in step (b) is enforced within a certain  \n\n![](images/2b28354059a4b3a9b92d754935c051312bfe3a5bc37d94fc1d826da19c8b9d2e.jpg)  \n\nFIG. 4: Owing to the periodicity of the Wannier functions over the Born-von Karman supercell (with size $2\\times2$ here), the matrix element $\\widetilde{H}_{m n}$ describes the interaction between the $m^{\\mathrm{th}}$ WF $w_{m\\mathbf{0}}$ (shown in orange) with ce tre $\\mathbf{r}_{m}$ inside the home unit cell $\\mathbf R=\\mathbf0$ (green shaded area) and the $n^{\\mathrm{th}}$ WF $w_{n\\mathbf{R}}$ (shown in blue) centred inside the unit cell $\\mathbf{R}$ , or any of its supercell-periodic replicas displaced by a superlattice vector $\\mathbf{\\vec{\\nabla}}^{r}\\mathbf{\\vec{r}}$ . When performing Wannier interpolation, we now impose a minimal-distance condition by choosing the replica $w_{n,\\mathbf{R}+\\mathbf{T}}$ of $w_{n\\mathbf{R}}$ whose centre lies within the Wigner–Seitz supercell centred at $\\mathbf{r}_{m}$ (thick orange line).  \n\ntolerance, to account for the numerical imprecision in the values of the Wannier centres and in the definition of the unit cell vectors. Although step (b) is much less important than (a) for obtaining a good Fourier interpolation, it helps ensuring that the interpolated bands respect the symmetries of the system; if step (b) is skipped, small artificial band splittings may occur at high-symmetry points, lines, or planes in the BZ.  \n\nThe procedure outlined above amounts to replacing Eq. (45) with  \n\n$$\nH_{m n\\mathbf{k}}=\\sum_{\\mathbf{R}}\\frac{1}{\\mathcal{N}_{m n\\mathbf{R}}}\\sum_{j=1}^{\\mathcal{N}_{m n\\mathbf{R}}}e^{i\\mathbf{k}\\cdot(\\mathbf{R}+\\mathbf{T}_{m n\\mathbf{R}}^{(j)})}\\widetilde{H}_{m n\\mathbf{R}},\n$$  \n\nwhere {T(mj)nR} are the NmnR vectors T that minimise the distance of Eq. (46) for a given combination of $\\mathcal{W}$ , $n$ and $\\mathbf{R}$ ; $\\mathbf{R}$ lies within the Wigner–Seitz supercell centred on the origin.  \n\nThe benefits of this modified interpolation scheme are most evident when considering a large unit cell sampled at the $\\Gamma$ point only. In this case $N=1$ so that Eq. (45) with $\\{{\\bf R}^{\\prime}\\}=\\{{\\bf R}\\}=\\{{\\bf0}\\}$ would reduce to $H_{m n\\mathbf{k}}=\\tilde{H}_{m n\\mathbf{0}}$ , yielding interpolated bands that do not disperse with $\\mathbf{k}$ . This is nonetheless an artefact of the choice $\\{{\\bf R}^{\\prime}\\}=\\{{\\bf0}\\}$ (of earlier versions of Wannier90) and not an intrinsic limitation of Wannier interpolation, as first  \n\n![](images/f0cc3e39e1a67dd96b8a509c208d9ba668c46da79a2a108c5ffbc31a48bf8076.jpg)  \n\nFIG. 5: Comparison between the bands obtained using the earlier interpolation procedure (blue lines), those obtained using the (current) modified approach of Eq. (47) (orange   \nlines), and the ab initio bands (black crosses). (a) Linear chain of carbon atoms, with 12 atoms per unit cell (separated by a distance of 1.3 ˚A along the z direction) and $\\Gamma$ -point   \nsampling. 36 Wannier functions have been computed starting from projections over $p_{x}$ and   \n$p_{y}$ orbitals on carbon atoms and $s$ -orbitals midbond between them. A frozen window up to   \nthe Fermi energy (set to zero in the plot) has been considered, while the disentanglement   \nwindow included all states up to 14 eV above the Fermi level. (b) Bulk silicon, with the BZ sampled on an unconverged $3\\times3\\times3$ grid of $k$ -points.  \n\ndemonstrated in Ref. 32 for one-dimensional systems. Indeed, equation (47), which in a sense extends Ref. 32 to any spatial dimension, becomes in this case  \n\n$$\nH_{m n\\mathbf{k}}=\\frac{\\widetilde{H}_{m n\\mathbf{0}}}{\\mathcal{N}_{m n\\mathbf{0}}}\\sum_{j=1}^{\\mathcal{N}_{m n\\mathbf{0}}}e^{i\\mathbf{k}\\cdot\\mathbf{T}_{m n\\mathbf{0}}^{(j)}},\n$$  \n\nwhich can produce dispersive bands. This is illustrated in Fig. 5(a) for the case of a onedimensional chain of carbon atoms: the interpolated bands obtained from Eq. (45) with $\\{{\\bf R}^{\\prime}\\}=\\{{\\bf R}\\}=\\{{\\bf0}\\}$ (earlier version of Wannier90) are flat, while those obtained from Eq. (47) (new versions of Wannier90) are in much better agreement with the dispersive ab initio bands up to a few eV above the Fermi energy.  \n\nClear improvements in the interpolated bands are also obtained for bulk solids, as shown in Fig. 5(b) for the case of silicon. The earlier implementation breaks the two-fold degeneracy along the X W line, with one of the two bands becoming flat. The new procedure recovers the correct degeneracies, and reproduces more closely the ab initio band structure (the remaining small deviations are due to the use of a coarse $k$ -point mesh that does not satisfy the Nyquist–Shannon condition, and would disappear for denser $k$ -grids together with the differences between the two interpolation procedures).  \n\n# C. Selection of projections  \n\nIn many cases, and particularly for entangled bands, it is necessary to have a good initial guess for the MLWFs in order to properly converge the spread to the global minimum. Determining a good initial guess often involves a trial and error approach, using different combinations of orbital types, orientations and positions. While for small systems performing many computations of the projection matrices is relatively cheap, for large systems there is a cost associated with storing and reading the wavefunctions to compute new projection matrices for each new attempt at a better initial guess. Previously, the number of projections that could be specified had to be equal to the number J of WFs to be constructed. The latest version of the code lifts this restriction, making it possible to define in the pre-processing step a larger number $J_{+}>J$ of projection functions to consider as initial guesses. In this way, the computationally expensive and potentially I/O-heavy construction of the projection matrices $A_{\\mathbf{k}}$ is performed only once for all possible projections that a user would like to consider.  \n\nOnce the $A_{\\mathbf{k}}$ matrices (of dimension $J\\times J_{+}$ at each $\\mathbf{k}$ ) have been obtained, one proceeds with constructing the MLWFs by simply selecting, via a new input parameter (select projections) of the Wannier90 code, which $J$ columns to use among the $J_{+}$ that were computed by the interface code. Experimenting with different trial orbitals can thus be achieved by simply selecting a different set of projections within the Wannier90 input file, without the need to perform the pre-processing step again.  \n\nSimilarly, another use case for this new option is the construction of WFs for the same material but for different groups of bands. Typically one would have to modify the Wannier90 input file and run the interface code multiple times, while now the interface code may compute $A_{\\mathbf{k}}$ for a superset of trial orbitals just once, and then different subsets may be chosen by simple modification of a single input parameter. As a demonstration, we have adapted example11 of the Wannier90 distribution (silicon band structure), that considers two band groups: (a) the valence bands only, described by four bond-centred $s$ orbitals, and (b) the four valence and the four lowest-lying conduction bands together, described by atom-centred $\\boldsymbol{s p}^{3}$ orbitals. In the example, projections onto all 12 trial orbitals are provided, and the different cases are covered by specifying in the Wannier90 input file which subset of projections is required.  \n\n# D. Plotting cube files with non-orthogonal vectors  \n\nIn Wannier90 v3.0 it is possible to plot the MLWFs in real-space in Gaussian cube format, including the case of non-orthogonal cell lattice vectors. Many modern visualisation programs such as Vesta $^{\\mathrm{28}}$ are capable of handling non-orthogonal cube files and the cube file format can be read by many computational chemistry programs. Wannier90’s representation of MLWFs in cube format can be significantly more compact than using the alternative xsf format. With the latter, MLWFs are calculated (albeit with a coarse sampling) on a supercell of the computational cell that can be potentially large (the extent of the supercell is controlled by an input parameter wannier plot supercell). Whereas, with the cube format, each Wannier function is represented on a grid that is centred on the Wannier function itself and has a user-defined extent, which is the smallest parallelepiped (whose sides are aligned with the cell vectors) that can enclose a sphere with a user-defined radius wannier plot radius. Because MLWFs are strongly localised in real space, relatively small cut-offs are all that is required, significantly smaller than the length-scale over which the MLWFs themselves are periodic. As a result, the cube format is particularly useful when a more memory-efficient representation is needed. The cube format can be activated by setting the input parameter wannier plot mode to cube, and the code can handle both isolated molecular systems (treated within the supercell approximation) as well as periodic crystals by setting wannier plot mode to either molecule or crystal, respectively.  \n\n# V. NEW POST-PROCESSING FEATURES  \n\nOnce the electronic bands of interest have been disentangled and wannierised to obtain well-localised WFs, the Wannier90 software package includes a number of modules and utilities that use these WFs to calculate various electronic-structure properties. Much of this functionality exists within postw90.x, an MPI-parallel code that forms an integral part of the Wannier90 package. In v2.x of Wannier90, postw90.x included functionality for computing densities of states and partial densities of states, energy bands and Berry curvature along specified lines and planes in $k$ -space, anomalous Hall conductivity, orbital magnetisation and optical conductivity, Boltzmann transport coefficients within the relaxation time approximation, and band energies and derivatives on a generic user-defined list of $k$ -points. Some further functionality exists in a set of utilities that are provided as part of the Wannier90 package, including a code (w90pov.F90) to plot WFs rendered using the Persistence of Vision Raytracer (POV-Ray) $^{35}$ code and to compute van der Waals interactions with WFs (w90vdw.F90).  \n\nIn addition, there are a number of external packages for computing advanced properties based on WFs and which interface to Wannier90. These include codes to generate tight-binding models such as pythTB $^{36}$ and tbmodels37, quantum transport codes such as sisl $^{38}$ , gollum $^{39}$ , omen $^{40}$ and nanoTCAD-ViDES41, the EPW $^{42}$ code for calculating properties related to electron-phonon interactions and WannierTools $^{43}$ for the investigation of novel topological materials.  \n\nBelow we describe some of the new post-processing features of Wannier90 that have been introduced in the latest version of the code, v3.0.  \n\n# A. postw90.x: Shift Current  \n\nThe photogalvanic effect (PGE) is a nonlinear optical response that consists in the generation of a direct current (DC) when light is absorbed. $^{44-46}$ It can be divided phenomenologically into linear (LPGE) and circular (CPGE) effects, which have different symmetry requirements within the acentric crystal classes. The CPGE requires elliptically-polarised light, and occurs in gyrotropic crystals (see next subsection). The LPGE occurs with linearly or unpolarised light as well; it is present in piezoelectric crystals and is given by  \n\n$$\nJ_{a}(0)=2\\sigma_{a b c}(0;\\omega,-\\omega)E_{b}(\\omega)E_{c}(-\\omega),\n$$  \n\nwhere $\\mathbf{J}(0)$ is the induced DC photocurrent density, $\\mathbf{E}(\\omega)=\\mathbf{E}^{*}(-\\omega)$ is the amplitude of the optical electric field, and $\\sigma_{a b c}=\\sigma_{a c b}=\\sigma_{a b c}^{*}$ is a nonlinear photoconductivity tensor.  \n\nThe shift current is the part of the LPGE photocurrent generated by interband light absorption.47 Intuitively, it arises from a coordinate shift accompanying the photoexcitation of electrons from one band to another. Like the intrinsic anomalous Hall effect48, the shift current involves off-diagonal velocity matrix elements between occupied and empty bands, depending not only on their magnitudes but also on their phases49–52.  \n\nThe shift current along direction $a$ induced by light that is linearly polarised along $b$ is described by the following photoconductivity tensor:52,53  \n\n$$\n\\begin{array}{l}{{\\displaystyle\\sigma_{a b b}^{\\mathrm{shift}}(0;\\omega,-\\omega)=-\\frac{\\pi|e|^{3}}{\\hbar^{2}}\\int_{\\mathrm{BZ}}\\frac{\\mathrm{d}\\mathbf{k}}{(2\\pi)^{3}}\\sum_{n,m}f_{n m\\mathbf{k}}R_{n m\\mathbf{k}}^{a b}}}\\\\ {{\\displaystyle~\\times~\\left|r_{n m\\mathbf{k}}^{b}\\right|^{2}\\delta(\\omega_{m n\\mathbf{k}}-\\omega).}}\\end{array}\\overset{\\longleftrightarrow}{\\underbrace{(\\mathbf{e}_{\\mathbf{\\Gamma}}\\mathbf{e}_{\\mathbf{\\Gamma}})}}~,\n$$  \n\nHere, $f_{n m\\mathbf{k}}=f_{n\\mathbf{k}}-f_{m\\mathbf{k}}$ is the difference between occupation factors, $\\hbar\\omega_{m n\\bf{k}}=\\epsilon_{m\\bf{k}}-\\epsilon_{n\\bf{k}}$ is the difference between energy eigenvalues of the Bloch bands, rbnmk is the bth Cartesian component of the interband dipole matrix (the off-diagonal part of the Berry connection matrix $\\mathbf{A}_{n m\\mathbf{k}}=i\\langle u_{n\\mathbf{k}}|\\partial_{\\mathbf{k}}u_{m\\mathbf{k}}\\rangle,$ ), and  \n\n$$\nR_{n m\\mathbf k}^{a b}=\\partial_{k_{a}}\\arg\\left(r_{n m\\mathbf k}^{b}\\right)-A_{n n\\mathbf k}^{a}+A_{m m\\mathbf k}^{a}\n$$  \n\nis the shift vector (not to be confused with the lattice vector $\\mathbf{R}$ , or with the matrix $R^{(\\mathbf{k},\\mathbf{b})}$ defined in Eq. (12)). The shift vector has units of length, and it describes the real-space shift of wavepackets under photoexcitation.  \n\nThe numerical evaluation of Eq. (51) is tricky because the individual terms therein are gauge-dependent, and only their sum is unique. Different strategies were discussed in the early literature in the context of model calculations $^{51,54}$ and more recently for ab initio calculations. The ab initio implementation of Young and Rappe $^{55}$ employed a gauge-invariant $k$ -space discretisation of Eq. (51), inspired by the discretised Berry-phase formula for electric polarisation.56  \n\nThe implementation in Wannier90 is based instead on the formulation of Sipe and coworkers.52,57 In this formulation, the shift (interband) contribution to the LPGE tensor in Eq. (49) is expressed as  \n\n$$\n\\begin{array}{l}{\\displaystyle{\\sigma_{a b c}^{\\mathrm{shift}}(0;\\omega,-\\omega)=\\frac{i\\pi|e|^{3}}{4\\hbar^{2}}\\int_{\\mathrm{BZ}}\\frac{\\mathrm{d}\\mathbf{k}}{(2\\pi)^{3}}\\sum_{n,m}f_{n m\\mathbf{k}}}}\\\\ {\\displaystyle{\\qquad\\times\\left(r_{m n\\mathbf{k}}^{b}r_{n m\\mathbf{k}}^{c;a}+r_{m n\\mathbf{k}}^{c}r_{n m\\mathbf{k}}^{b;a}\\right)}}\\\\ {\\displaystyle{\\qquad\\times\\left[\\delta(\\omega_{m n\\mathbf{k}}-\\omega)+\\delta(\\omega_{n m\\mathbf{k}}-\\omega)\\right],}}\\end{array}\n$$  \n\n$$\nr_{n m\\mathbf{k}}^{b;a}=\\partial_{k_{a}}r_{n m\\mathbf{k}}^{b}-i\\left(A_{n n\\mathbf{k}}^{a}-A_{m m\\mathbf{k}}^{a}\\right)r_{n m\\mathbf{k}}^{b}\n$$  \n\nis the generalised derivative of the interband dipole. When $b=c$ , Eq. (52) becomes equivalent to Eq. (50).52  \n\nThe generalised derivative $r_{n m\\mathbf{k}}^{b;a}$ is a well-behaved (covariant) quantity under gauge transformation but – as in the case of the shift vector – this is not the case for the individual terms in Eq. (53), leading to numerical instabilities. To circumvent this problem, Sipe and co-workers used $\\mathbf{k}\\cdot\\mathbf{p}$ perturbation theory to recast Eq. (53) as a summation over intermediate virtual states where the individual terms are gauge covariant.52,57 That strategy has been successfully employed to evaluate the shift-current spectrum from first principles.58,59  \n\nAs it is well known, similar “sum-over-states” expressions can be written for other quantities involving k derivatives, such as the inverse effective-mass tensor and the Berry curvature. When evaluating those expressions, a sufficient number of virtual states should be included to achieve convergence. Alternatively, one can work with a basis spanning a finite number of bands, such as a tight-binding or Wannier basis, and carefully reformulate $\\mathbf{k}\\cdot\\mathbf{p}$ perturbation theory within that incomplete basis to avoid truncation errors. This reformulation was carried out in Ref. 60 for the inverse effective-mass tensor, and in Ref. 33 for the Berry curvature; the formalism of Ref. 33 is at the core of the berry.F90 module of postw90, where Berry curvatures and related quantities are computed by Wannier interpolation. The same interpolation strategy was used in Refs. 61 and 62 to evaluate Eq. (52), and the approach of Ref. 62 is now implemented in the berry.F90 module.  \n\n# B. postw90.x: Gyrotropic module  \n\nIn the previous subsection we considered the shift current, an effect that occurs in piezoelectric crystals. Here we turn to a host of effects that occur in a different group of acentric crystals: those belonging to the gyrotropic crystal classes, which include the chiral, polar, and optically-active crystal classes.44  \n\nTo motivate the gyrotropic effects considered below, let us start from the more familiar magneto-optical effects. To review, the spontaneous magnetisation of ferromagnets endows their conductivity tensor $\\sigma_{a b}(\\omega)$ with an antisymmetric part. In the DC limit this antisymmetric conductivity describes the anomalous Hall effect (AHE), and at finite frequencies it describes magneto-optical effects such as Faraday rotation in transmission and magnetic circular dichroism in absorption. In paramagnets, those effects appear under applied magnetic  \n\nfields.  \n\nAs first pointed out in Refs. 63 and 64, an antisymmetric conductivity can be induced in certain nonmagnetic (semi)conductors by purely electrical means: by passing a current through the sample. Symmetry arguments indicate that this is allowed in the gyrotropic crystal classes, and the first experimental demonstration consisted in the measurement of a current-induced change in the rotatory power of $p$ -doped trigonal tellurium.65,66 When linearly polarised light of frequency $\\omega$ propagates along the trigonal ˆz axis in the presence of a current density $\\mathbf{j}=j_{z}\\hat{\\mathbf{z}}$ , the change in rotatory power is proportional to $\\widetilde{D}_{z z}(\\omega)j_{z}$ , where  \n\n$$\n\\widetilde{D}_{a b}(\\omega)=\\int_{\\mathrm{BZ}}\\frac{\\mathrm{d}\\mathbf{k}}{(2\\pi)^{3}}\\sum_{n}f_{0}(\\epsilon_{n\\mathbf{k}})\\partial_{k_{a}}\\widetilde{\\Omega}_{n\\mathbf{k}}^{b}(\\omega).\n$$  \n\nIn this expression $f_{0}$ is the equilibrium occupation factor, and  \n\n$$\n\\widetilde{\\Omega}_{n\\mathbf{k}}(\\omega)=-\\sum_{m}\\frac{\\omega_{m n\\mathbf{k}}^{2}}{\\omega_{m n\\mathbf{k}}^{2}-\\omega^{2}}\\mathrm{Im}[\\mathbf{A}_{n m\\mathbf{k}}\\times\\mathbf{A}_{m n\\mathbf{k}}],\n$$  \n\nwhere $\\mathbf{A}_{n m\\mathbf{k}}$ is the Berry connection matrix introduced in Sec. V A. At zero frequency, $\\widetilde{\\Omega}_{n\\mathbf{k}}(\\omega)$ reduces to the Berry curvature $\\Omega_{n\\mathbf k}=\\pmb{\\nabla}_{\\mathbf k}\\times\\mathbf k_{n n\\mathbf k}$ .  \n\nThe DC or transport limit of this current-induced Faraday effect is the current-induced AHE, or nonlinear $A H E^{67-71}$ . Like the linear (spontaneous) AHE in ferromagnetic metals, the nonlinear (current-induced) AHE in gyrotropic conductors has an intrinsic contribution associated with the Berry curvature. It is given by $j_{a}\\propto\\tau\\varepsilon_{a d c}D_{b d}E_{b}E_{c}$ , where $\\mathbf{E}$ is the electric field, $\\tau$ is the relaxation time of the conduction electrons, $\\varepsilon_{a b c}$ is the alternating tensor, and $D_{a b}=\\widetilde{D}_{a b}(\\omega=0)$ is the “Berry-curvature dipole.”67 After performing an integration by parts in Eq. (54), the quantities $D_{a b}$ and $\\widetilde{D}_{a b}(\\omega)$ can be easily evaluated with the help of the berry.F90 module.  \n\nAlong with nonlinear magneto-optical and anomalous Hall effects, the flow of electrical current in a gyrotropic conducting medium also generates a net magnetisation. This kinetic magnetoelectric effect was originally proposed for bulk chiral conductors,64,72 and later for two-dimensional (2D) inversion layers with an out-of-plane polar axis $^{73,74}$ , where it has been studied intensively75. The kinetic magnetoelectric effect in 2D – also known as the Edelstein effect – is a purely spin effect, whereas in bulk crystals an orbital contribution is also present.72 The orbital kinetic magnetoelectric effect is given by $M_{a}\\propto\\tau K_{b a}E_{b}$ , where the tensor $\\vec{K}_{a b}$ is obtained from $D_{a b}$ by replacing the Berry curvature with the intrinsic magnetic moment of the Bloch states, $^{76-78}$ a quantity that is also provided by the berry.F90 module.79  \n\nAnother phenomenon characteristic of gyrotropic crystals is the circular photogalvanic effect (CPGE) that was mentioned briefly in Sec. V A. This nonlinear optical effect consists in the generation of a photocurrent that reverses sign with the helicity of light44–46,64,80, and it occurs when light is absorbed via interband or intraband scattering processes. The intraband contribution to the CPGE can be expressed in terms of the Berry curvature dipole as $\\begin{array}{r}{j_{a}\\propto\\frac{\\omega\\tau^{2}D_{a b}}{1+\\omega^{2}\\tau^{2}}\\mathrm{Im}\\left[\\mathbf{E}(\\omega)\\times\\mathbf{E}^{*}(\\omega)\\right]_{b}}\\end{array}$ .67,81,82  \n\nThe above effects are being very actively investigated in connection with novel materials ranging from topological semimetals68,83,84 to monolayer and bilayer transition-metal dichalcogenides $69–71$ . The sensitivity of both the Berry curvature and the intrinsic orbital moment to the details of the electronic structure, together with the need to sample them on a dense mesh of $k$ -points, calls for the development of accurate and efficient ab initio methodologies, and the Wannier interpolation technique is ideally suited for this task.  \n\nThe Wannier interpolation methodology for gyrotropic effects was presented in Ref. 78, where it was applied to $p$ -doped trigonal tellurium, and the resulting computer code has been incorporated in postw90 as the gyrotropic.F90 module. The reader is referred to Ref. 78 for more details such as the prefactors in the expressions above, as well as the formulas for natural optical activity, which has also been implemented in the same module.  \n\n# C. postw90.x: Spin Hall conductivity  \n\nThe spin Hall effect (SHE) is a phenomenon in which a spin current is generated by applying an electric field. The current is often transverse to the field (Hall-like), but this is not always the case.85 The SHE is characterised by the spin Hall conductivity (SHC) tensor $\\sigma_{a b}^{\\mathrm{spin},c}$ as follows:  \n\n$$\nJ_{a}^{\\mathrm{spin},c}(\\omega)=\\sigma_{a b}^{\\mathrm{spin},c}(\\omega)E_{b}(\\omega),\n$$  \n\nwhere ${\\cal J}_{a}^{\\mathrm{spin},c}$ is the spin-current density along direction $a$ with its spin pointing along $c$ , and $E_{b}$ is the external electric field of frequency $\\omega$ applied along $b$ . In non-magnetic materials the equal number of up- and down-spin electrons forces the AHE to vanish, resulting in a pure spin current.  \n\nLike the AHC, the SHC contains both intrinsic and extrinsic contributions. $^{86}$ The intrinsic contribution to the SHC can be calculated from the following Kubo formula,87  \n\n![](images/38fd8ded5dbec1381433317428efa194299fcb49f1c6a171056185aaae6fa564.jpg)  \n\n$$\n\\begin{array}{l}{\\sigma_{a b}^{\\mathrm{spin},c}(\\omega)=-\\displaystyle\\frac{e^{2}}{\\hbar}\\frac{1}{V N}\\sum_{\\mathbf{k}}\\sum_{n}f_{n\\mathbf{k}}\\Omega_{n\\mathbf{k},a b}^{\\mathrm{spin},c}(\\omega),\\qquad}\\\\ {\\Omega_{n\\mathbf{k},a b}^{\\mathrm{spin},c}(\\omega)=\\hbar^{2}\\sum_{m\\neq n}\\frac{-2\\mathrm{Im}[\\langle\\psi_{n\\mathbf{k}}|\\frac{2}{\\hbar}j_{a}^{\\mathrm{spin},c}|\\psi_{m\\mathbf{k}}\\rangle\\langle\\psi_{m\\mathbf{k}}|v_{b}|\\psi_{n\\mathbf{k}}\\rangle]}{(\\epsilon_{n\\mathbf{k}}-\\epsilon_{m\\mathbf{k}})^{2}-(\\hbar\\omega+i\\eta)^{2}},\\qquad}\\end{array}\n$$  \n\nwhere $s_{c}$ , $v_{a}$ and $\\begin{array}{r}{j_{a}^{\\mathrm{spin,}c}=\\frac{1}{2}\\{s_{c},v_{a}\\}}\\end{array}$ are the spin, velocity and spin current operators, respectively; $V$ is the cell volume, and $N$ is the total number of $k$ -points used to sample the BZ. Equations (57) are very similar to the Kubo formula for the AHC, except for the replacement of a velocity matrix element by a spin-current matrix element. As mentioned in the previous two subsections, Wannier-interpolation techniques are very efficient at calculating such quantities.  \n\nA Wannier-interpolation method scheme for evaluating the intrinsic SHC was developed in Ref. 87 (see also Ref. 88 for a related but independent work). The required quantities from the underlying ab initio calculation are the spin matrix elements S(m0)nk,a $S_{m n\\mathbf k,a}^{(0)}=\\langle\\psi_{m\\mathbf k}^{(0)}|s_{a}|\\psi_{n\\mathbf k}^{(0)}\\rangle$ , the Hamiltonian matrix elements $H_{m n\\mathbf{k}}^{(0)}=\\langle\\psi_{m\\mathbf{k}}^{(0)}|H|\\psi_{n\\mathbf{k}}^{(0)}\\rangle=\\epsilon_{m\\mathbf{k}}^{(0)}\\delta_{m n}$ , and the overlap matrix elements of Eq. (17). Since the calculation of all these quantities has been previously implemented in pw2wannier90.x (the interface code between pwscf and Wannier90), this advantageous interpolation scheme can be readily used while keeping to a minimum the interaction between the ab initio code and Wannier90.  \n\nThe application of the method to fcc Pt is illustrated in Fig. 6. Panel (a) shows the calculated SHC as a function of the Fermi-level position, and panel (b) depicts the “spin Berry curvature” of Eq. (57b) that gives the contribution from each band state to the SHC. The aforementioned functionalities have been incorporated in the berry.F90, kpath.F90 and kslice.F90 modules of postw90.x.  \n\n# D. postw90.x: Parallelisation improvements  \n\nThe original implementation of the berry.F90 module in postw90.x (for computing Berry-phase properties such as orbital magnetisation and anomalous Hall conductivity $^{79}$ ), introduced in Wannier90 v2.0, was written with code readability in mind and had not been optimised for computational speed. In Wannier90 v3.0, all parts of the berry.F90 module have been parallelised while keeping the code readable; moreover, its scalability has been improved, accelerating its performance by several orders of magnitude.89  \n\n![](images/727a7b6b3d2529ded471b32bc7c0a9bd15bcd1f46e24d37d47c6f4b5b4a7da6b.jpg)  \n\nFIG. 6: (a) Intrinsic spin Hall conductivity $\\sigma_{x y}^{\\mathrm{spin,}z}$ of fcc Pt, plotted as a function of the shift in Fermi energy relative to its self-consistent value. (b) Band structure of fcc $\\mathrm{Pt}$ , colour-coded by a dimensionless function $r(\\Omega_{n\\mathbf{k},x y}^{\\mathrm{spun},z})$ of the spin Berry curvature [Eq. (57b)]. The function $r(x)$ is equal to $x/10$ when $|x|<10,\\mathrm{andto}\\log_{10}(|x|)\\mathrm{sgn}(x)$ when $|x|\\geq10$ .  \n\nTo illustrate the improvements in performance we present calculations on a 128-atom supercell of GaAs interstitially doped with Mn (we emphasise that here we are not interested in the results of the calculation but simply on its performance testing, and that the choice of the system does not affect the scaling results that we report). We use a lattice constant of the elementary cell of 5.65 ˚A. We use norm-conserving relativistic pseudopotentials with the PBE exchange-correlation functional. The energy cut-off for the plane waves is set to 40 Ry, and the Brillouin-zone sampling of the supercell is $3\\times3\\times3$ . We use a Gaussian metallic smearing with a broadening of 0.015 Ry. For the non-self-consistent step of the calculation, 600 bands are computed and used to construct 517 Wannier functions. The initial projections are chosen as a set of $s p^{3}$ orbitals centred on each Ga and As atom, and a set of $d$ orbitals on Mn. The calculations were performed on the Prometheus supercomputer of PL-GRID (in Poland). The code was compiled with the Intel ifort compiler (v15.0.2), using the OpenMPI libraries (v1.8.4) and BLAS/LAPACK routines from Intel MKL (v11.3.1).  \n\nThe Berry-phase calculations can be performed in three distinct ways: (i) 3D quantities in $k$ -space (routine berry main), (ii) the same quantities resolved on 2D planes (routine kslice.F90), and (iii) 1D paths (routine kpath.F90) in the Brillouin zone. In the benchmarks, we will refer to these three cases as “Berry 3D”, “Berry 2D”, and “Berry 1D”,  \n\nrespectively.  \n\nThe first optimisation target was the function utility rotate in the module utility.F90, which calculates a matrix product of the form $B=R^{\\dagger}A R$ using Fortran’s built-in matmul function. The new routine utility rotate new uses instead BLAS and performs about 5.7 times better than the original one, giving a total speedup for berry main of about $55\\%$ .  \n\nA second performance-critical section of code was identified in the routine get imfgh k list which took more than $50\\%$ of the total run-time of berry main. This routine computes three quantities: $F_{\\alpha\\beta}$ , $G_{\\alpha\\beta}$ and $H_{\\alpha\\beta}$ , which are defined in Eqs. (51), (66) and (56) of Ref. 79. By some algebraic transformations, it was possible to reduce 25 calls to matmul, carried out in the innermost runtime-critical loop, to only 5 calls. After replacement of matmul with the Basic Linear Algebra Subprogram (BLAS), the speed up of this routine exceeds a factor of 11, and the total time spent in berry main is 2.5 times shorter (including the speed-up from the first optimisation).  \n\nIn the third step, a bottleneck was eliminated in the initialisation phase, where mpi bcast was waiting more than two minutes for the master rank to broadcast the parameters. The majority of this time was spent in loops computing matrix products of the form $S=(V_{1})^{\\dagger}S_{0}V_{2}$ . Again, we replaced this with two calls to the BLAS gemm routine. This resulted in a speed-up of a factor of 610 for the calculation of this matrix product in our test case, and the total initialisation time dropped to less than 15 seconds. In total, the berry main routine runs about 5 times faster than it did originally.  \n\nFinally, the routines kslice.F90 and kpath.F90 were parallelised. The scalability results of berry main, kslice.F90 and kpath.F90 are presented in Fig. 7, and a comparison with the scalability of the previous version of berry main is also given. Absolute times for some of the calculations are reported in Table I.  \n\n# E. GW bands interpolation  \n\nWhile density-functional theory (DFT) is the method of choice for most applications in materials modelling, it is well known that DFT is not meant to provide spectral properties such as band structures, band gaps and optical spectra. Green’s function formulation of many-body perturbation theory (MBPT) $^{90}$ overcomes this limitation, and allows the excitation spectrum to be obtained from the knowledge of the Green’s function. Within  \n\nMBPT the interacting electronic Green’s function $G(\\mathbf{r},\\mathbf{r}^{\\prime},\\omega)$ may be expressed in terms of the non-interacting Green’s function $G^{0}({\\bf r},{\\bf r}^{\\prime},\\omega)$ and the so-called self-energy $\\Sigma(\\mathbf{r},\\mathbf{r}^{\\prime},\\omega)$ , where several accurate approximations for $\\Sigma$ have been developed and implemented into first-principles codes91. While maximally-localised Wannier functions for self-consistent GW quasiparticles have been discussed in Ref. 92, here we focus on the protocol to perform bands interpolation at the one-shot ${\\mathrm{{G}}_{0}}{\\mathrm{{W}}_{0}}$ level. For solids, the ${\\mathrm{{G}}_{0}}{\\mathrm{{W}}_{0}}$ approximation has proven to be an excellent compromise between accuracy and computational cost and it has become the most popular MBPT technique in computational materials science $^{93}$ . In the standard one-shot $\\mathrm{{{G}_{0}}\\mathrm{{W}_{0}}}$ approach, $\\Sigma$ is written in terms of the Kohn–Sham (KS) Green’s function and the RPA dielectric matrix, both obtained from the knowledge of DFT-KS orbitals and eigenenergies. Quasi-particle (QP) energies are obtained from:  \n\n$$\n\\epsilon_{n\\mathbf{k}}^{\\mathrm{QP}}=\\epsilon_{n\\mathbf{k}}+Z_{n\\mathbf{k}}\\langle\\psi_{n\\mathbf{k}}|\\Sigma(\\epsilon_{n\\mathbf{k}})-V_{\\mathrm{xc}}|\\psi_{n\\mathbf{k}}\\rangle,\n$$  \n\nwhere $\\psi_{n\\mathbf{k}}$ and $\\epsilon_{n\\mathbf{k}}$ are the KS orbitals and eigenenergies, $Z_{n\\mathbf{k}}$ is the so-called renormalisation factor and $V_{\\mathrm{xc}}$ is the DFT exchange-correlation potential. In addition, in the standard $\\mathrm{{{G}_{0}}\\mathrm{{W}_{0}}}$  \n\n<html><body><table><tr><td colspan=\"2\">Mode k-grid Nc Time</td></tr><tr><td>version 3.0</td><td>S</td></tr><tr><td>30×30×30</td><td>24 6903</td></tr><tr><td>30×30×30</td><td>48 3527</td></tr><tr><td>Berry 3D 30×30×30</td><td>480 441</td></tr><tr><td>100×100×100</td><td>480 13041</td></tr><tr><td>100×100×100 7680</td><td>957</td></tr><tr><td>Berry 2D 100×100</td><td>24 1389</td></tr><tr><td>Berry 1D 10000</td><td>24 12639</td></tr><tr><td>version 2.0</td><td></td></tr><tr><td>30×30×30 Berry 3D</td><td>24 56497</td></tr><tr><td>30×30×30 48</td><td>40279</td></tr></table></body></html>  \n\nTABLE I: Wall-time for some of the runs performed with the Berry module, before (Wannier90 v2.0) and after (Wannier90 v3.0) the optimisations, for the test system described in the main text. $N_{\\mathrm{c}}$ indicates the number of cores used in the calculation.  \n\n![](images/82c4a4d4f69ab4ff9ab99a88141adee95f177905bfeb3f5996637a3616a2096d.jpg)  \n\nFIG. 7: (Top) Speedup of the new Wannier90 v3.0 with respect to v2.0, for a run of the berry module (mode “Berry 3D”) on the test system described in the text, demonstrating the improvements implemented in the new version of the code. (Bottom) Total CPU time (defined as total walltime times number of CPUs) for the three cases “Berry 3D”, “Berry 2D” and “Berry 1D” (whose meaning is described in the main text), normalised with respect to the same case run with $N_{\\mathrm{cpu}}/=24$ , for the Wannier90 v3.0 code. The “Berry 1D” and “Berry 2D” tests scan a 1D or 2D grid of points in the BZ, respectively; for these tests, the total number of grid points is 10000, therefore they can scale only up to a few hundreds of cores, above which the communication cost overweights the advantage coming from parallelisation. Instead, we emphasise that calculations with $N_{\\mathrm{cpu}}\\geq480$ for “Berry 3D” were run on a denser grid ( $100\\times100\\times100$ rather than $30\\times30\\times30$ ) and values have been rescaled using the time measured for both grids at $N_{\\mathrm{cpu}}=480$ to show the scalability of the code on thousands of CPUs.  \n\napproximation the QP orbitals are approximated by the KS orbitals. At variance with DFT, QP corrections for a given $k$ -point require knowledge of the KS orbitals and eigenenergies at all points $(\\mathbf{k}+\\mathbf{q})$ in reciprocal space. In practice, codes such as Yambo $^{94}$ compute QP corrections on a regular grid and rely on interpolation schemes to obtain the full band structure along high-symmetry lines. Wannier90 supports the use of G0W $_0$ QP corrections through the general interface gw2wannier90.py distributed with Wannier90, while dedicated tools for Quantum ESPRESSO and Yambo allow for an efficient use of symmetries. Thanks to the software interface, QP corrections can be computed in the irreducible BZ (IBZ) and later unfolded to the full BZ to comply with Wannier90 requirements. In addition, the interface facilitates the use of a denser $k$ -point grid to converge the self-energy and of a coarser grid to obtain MLWFs, as long as the two grids are commensurate. This is particularly efficient in the case of two-dimensional materials, where the $k$ -point convergence of the self-energy is typically very slow while Wannier interpolation is already accurate with much coarser $k$ -point grids. Finally, the interface takes care of correcting and possibly reordering in energy both the KS eigenvalues and the corresponding input matrices (like $M_{m n}^{(\\mathbf{k},\\mathbf{b})}$ , $A_{m n\\bf{k}}$ ). After reading these eigenvalues and matrices, Wannier90 can proceed as usual and all functionalities are available (band-structure interpolation and beyond) at the level of ${\\mathrm{{G}}_{0}}{\\mathrm{{W}}_{0}}$ calculations.  \n\n# VI. AUTOMATIC WANNIER FUNCTIONS: THE SCDM METHOD  \n\nAn alternative method for generating localised Wannier functions, known as the selected columns of the density matrix (SCDM) algorithm, has been proposed by Damle, Lin and Ying $^{95,96}$ . At its core the scheme exploits the information stored in the real-space representation of the single-particle density matrix, a gauge-invariant quantity. Localisation of the resulting functions is a direct consequence of the well-known nearsightedness principle $^{97,98}$ of electronic structure in extended systems with a gapped Hamiltonian, i.e., insulators and semiconductors. In these cases, the density matrix is exponentially localised along the off-diagonal direction in its real-space representation $\\rho({\\bf r},{\\bf r^{\\prime}})$ and it is generally accepted that Wannier functions with an exponential decay also exist; numerical studies have confirmed this claim for a number of materials, and there exist formal proofs for multiband time-reversal-invariant insulators99–101. Since the SCDM method does not minimise a given gauge-dependent localisation measure via a minimisation procedure, it is free from any issue regarding the dependence on initial conditions, i.e., it does not require a good initial guess of localised orbitals. It also avoids other problems associated with a minimisation procedure, such as getting stuck in local minima. More generally, the localised Wannier functions provided by the SCDM method can be used as starting points for the MLWF minimisation procedure, by using them to generate the $A_{\\mathbf{k}}$ projection matrices needed by Wannier90.  \n\nFor extended insulating systems, the density matrix is given by  \n\n$$\n\\rho=\\sum_{\\mathbf{k}}P_{\\mathbf{k}}=\\sum_{n=1}^{J}\\sum_{\\mathbf{k}}\\left|\\psi_{n\\mathbf{k}}\\right\\rangle\\left\\langle\\psi_{n\\mathbf{k}}\\right|.\n$$  \n\nAs shown in Sec. II, the $P_{\\mathbf{k}}$ are the spectral projectors associated with the crystal Hamiltonian operator $H_{\\mathbf{k}}$ onto the valence space $S_{\\mathbf{k}}$ , hence their rank is $\\ensuremath{N_{\\mathrm{e}}}$ (number of valence electrons). Moreover, they are analytic functions of $\\mathbf{k}$ and also manifestly gauge invariant102,103. As mentioned above, the nearsightedness principle98 guarantees that the columns of the kernels $P_{\\mathbf{k}}(\\mathbf{r},\\mathbf{r}^{\\prime})=\\langle\\mathbf{r}|P_{\\mathbf{k}}|\\mathbf{r}^{\\prime}\\rangle$ are localised along the off-diagonal direction and therefore they may be used to construct a localised basis. If we consider a discretisation of the $J$ Bloch states at each $\\mathbf{k}$ on a real-space grid of $N_{\\mathrm{g}}$ points, we can arrange the wavefunctions into the columns of a unitary $N_{\\mathrm{g}}\\times J$ $k$ -dependent matrix Ψk  \n\n$$\n\\Psi_{\\mathbf{k}}=\\left(\\begin{array}{c c c}{\\psi_{1\\mathbf{k}}(\\mathbf{r}_{1})}&{\\hdots}&{\\psi_{J\\mathbf{k}}(\\mathbf{r}_{1})}\\\\ {\\vdots}&{\\ddots}&{\\vdots}\\\\ {\\psi_{1\\mathbf{k}}(\\mathbf{r}_{N_{\\mathrm{g}}})}&{\\hdots}&{\\psi_{J\\mathbf{k}}(\\mathbf{r}_{N_{\\mathrm{g}}})}\\end{array}\\right),\n$$  \n\nsuch that $P_{\\mathbf{k},i j}=\\left(\\Psi_{\\mathbf{k}}\\Psi_{\\mathbf{k}}^{\\dagger}\\right){}_{i j}$ is $\\mathrm{a}\\ N_{\\mathrm{g}}\\times N_{\\mathrm{g}}$ matrix. In this representation, it is straightforward to see that the columns of $P_{\\mathbf{k}}(\\mathbf{r}_{i},\\mathbf{r}_{j})$ are projections of extremely localised functions (i.e., Dirac-delta functions localised on the grid points) onto the valence eigenspace. As a result, selecting any linearly-independent subset of $J$ of them will yield a localised basis for the span of $P(\\mathbf{r},\\mathbf{r}^{\\prime})$ . However, randomly selecting $J$ columns does not guarantee that a wellconditioned basis will be obtained. For instance, there could be too much overlap between the selected columns. Conceptually, the most well conditioned columns may be found via a QR factorisation with column pivoting (QRCP) applied to $P(\\mathbf{r},\\mathbf{r}^{\\prime})$ , in the form $P\\Pi=Q R$ , with $\\Pi$ being a matrix permuting the columns of $P$ , $Q$ a unitary matrix and $R$ an uppertriangular matrix (not to be confused with the lattice vector $\\mathbf{R}$ , or with the matrix $R^{(\\mathbf{k},\\mathbf{b})}$ defined in Eq. (12), or with the shift vector of Eq. (51)), and where $\\Pi$ is chosen so that $|R_{11}|\\geq|R_{22}|\\geq\\cdots\\geq|R_{n n}|$ . Then the $J$ columns forming a localised basis set are chosen to be the first $J$ of the matrix with permuted columns $P\\Pi$ .  \n\nThe SCDM- $k^{96}$ method suggests that it is sufficient to apply the QRCP factorisation at k = 0 ( $\\Gamma$ point) only, and use the same selection of columns at all $k$ -points. However, this is still often impractical since $\\boldsymbol{P_{\\mathbf{\\tilde{I}}}}$ is prohibitively expensive to construct and store in memory. Therefore an alternative procedure is proposed, for which the columns can be computed via the QRCP of the (smaller) matrix $\\Psi_{\\mathbf{r}}^{\\dagger}$ instead:  \n\n$$\n\\Psi_{\\bf{r}}^{\\dagger}\\Pi=Q^{\\prime}R^{\\prime},\n$$  \n\ni.e., the same $\\Pi$ matrix is obtained by computing a QRCP on $\\Psi^{\\dagger}$ only. Once the set of columns has been obtained, we need to impose the orthonormality constraint on the chosen columns without destroying their locality in real space. This can be achieved by a Lo¨wdin orthogonalisation, similarly to Eq. (26). In particular, the selection of columns of $\\Psi_{\\mathbf{T}}$ can be used to select the columns of all $\\Psi_{\\mathbf{k}}$ , which in turn define the $A_{m n\\bf{k}}$ matrices needed as input by Wannier90 to start the MLWF minimisation procedure, by defining $A_{m n\\mathbf{k}}=\\psi_{m\\mathbf{k}}^{*}(\\mathbf{r}_{\\Pi(n)})$ , where the $\\Pi(n)$ is the index of the $n^{\\mathrm{th}}$ column of $P\\vert$ after permutation with $\\Pi$ . In fact, we can write the $n^{\\mathrm{th}}$ column of $P$ after permutation, $\\widetilde{\\mathcal{P}_{\\mathbf{k}}}(\\mathbf{r},\\mathbf{r}_{\\Pi(n)})$ , as  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\cal P}_{{\\bf k}}({\\bf r},{\\bf r}_{\\Pi(n)})=\\sum_{m=1}^{J}\\psi_{m{\\bf k}}({\\bf r})\\psi_{m{\\bf k}}^{*}({\\bf r}_{\\Pi(n)})}\\ ~}\\\\ {{\\displaystyle~=\\phi_{\\Pi(n),{\\bf k}}\\equiv\\sum_{m=1}^{J}\\psi_{m{\\bf k}}({\\bf r})A_{m n{\\bf k}}}.}\\end{array}\n$$  \n\nThe unitary matrix $U_{\\mathbf{k}}$ sought for is then constructed via Lo¨wdin orthogonalisation  \n\n$$\nU_{\\mathbf{k}}=A_{\\mathbf{k}}(A_{\\mathbf{k}}^{\\dagger}A_{\\mathbf{k}})^{-1/2}=A_{\\mathbf{k}}S_{\\mathbf{k}}^{-1/2}.\n$$  \n\nWe can also extend the SCDM- $k$ method to the case where the Bloch states are represented as two-component spinor wavefunctions $\\psi_{n\\mathbf{k}}(\\mathbf{r},\\alpha)$ , e.g., when including spin-orbit interaction in the Hamiltonian. Here, $\\alpha=\\uparrow,\\downarrow$ is the spinor index. In this case, we include the spin index as well as the position index to perform QRCP. First, we define the $2N_{g}\\times J$ matrix $\\Psi_{\\mathbf{k}}$  \n\n$$\n\\Psi_{\\mathbf{k}}=\\left(\\begin{array}{c c c}{\\psi_{1\\mathbf{k}}(\\mathbf{r}_{1},\\uparrow)}&{\\ldots}&{\\psi_{J\\mathbf{k}}(\\mathbf{r}_{1},\\uparrow)}\\\\ {\\psi_{1\\mathbf{k}}(\\mathbf{r}_{1},\\downarrow)}&{\\ldots}&{\\psi_{J\\mathbf{k}}(\\mathbf{r}_{1},\\downarrow)}\\\\ {\\vdots}&{\\ddots}&{\\vdots}\\\\ {\\psi_{1\\mathbf{k}}(\\mathbf{r}_{N_{g}},\\uparrow)}&{\\ldots}&{\\psi_{J\\mathbf{k}}(\\mathbf{r}_{N_{g}},\\uparrow)}\\\\ {\\psi_{1\\mathbf{k}}(\\mathbf{r}_{N_{g}},\\downarrow)}&{\\ldots}&{\\psi_{J\\mathbf{k}}(\\mathbf{r}_{N_{g}},\\downarrow)}\\end{array}\\right).\n$$  \n\nNext, as in the spinless case, the QRCP of $\\Psi_{\\mathbf{r}}^{\\dagger}$ is computed, and the first $J$ columns of the Π matrix are selected. Now, $\\Pi(n)$ , the index of the $n^{\\mathrm{th}}$ column of $P$ after permutation with $\\Pi$ , determines both the position index $\\mathbf{r}_{\\Pi(n)}$ and the spin index $\\alpha_{\\Pi(n)}$ . We define $\\dot{\\bf\\cal A}_{m n\\bf\\bf\\bf\\bf\\phi}=$ $\\psi_{m\\mathbf{k}}^{*}\\big(\\mathbf{r}_{\\Pi(n)},\\alpha_{\\Pi(n)}\\big)$ and perform Lo¨wdin orthogonalisation to obtain the unitary matrix $U_{\\mathbf{k}}$ . In the case of entangled bands, we need to introduce a so-called quasi-density matrix defined as  \n\n$$\nP_{{\\bf k}}=\\sum_{n}\\left|\\psi_{n{\\bf k}}\\right\\rangle f(\\epsilon_{n{\\bf k}})\\left\\langle\\psi_{n{\\bf k}}\\right|,\n$$  \n\nwhere $f(\\epsilon_{n\\mathbf{k}})~\\in~[0,1]$ is a generalisation of the Fermi-Dirac probability for the occupied states. Also in this case we only use the information at $\\Gamma$ to generate the permutation matrix. Depending on what kind of entangled manifold one is interested in, $f(\\epsilon)$ can be modelled with various functional forms. In particular, the authors of Ref. 96 suggest the following three forms:  \n\n1. Isolated manifold, e.g., the valence bands of an insulator or a semiconductor: $f(\\epsilon)$ is a step function, with the step inside the energy gap $\\Delta\\epsilon_{\\mathrm{g}}=\\epsilon_{\\mathrm{c}}-\\epsilon_{\\mathrm{v}}$ , where $\\epsilon_{\\mathrm{c(v)}}$ represents the minimum (maximum) of the conduction (valence) band:  \n\n$$\nf(\\epsilon)=\\theta(\\epsilon_{\\mathrm{v}}+\\Delta\\epsilon_{\\mathrm{g}}/2-\\epsilon).\n$$  \n\nBoth $\\Delta\\epsilon_{\\mathrm{g}}$ and $\\epsilon_{\\mathrm{v}}$ are not free parameters, as they may be obtained directly from the ab initio calculation.  \n\n2. Entangled manifold (case I), e.g., the valence bands and low-lying conduction bands in a semiconductor: $\\mathcal{f}(\\boldsymbol{\\epsilon})$ is a complementary error function:  \n\n$$\nf(\\epsilon)=\\frac{1}{2}\\mathrm{erfc}\\left(\\frac{\\epsilon-\\mu}{\\sigma}\\right),\n$$  \n\nwhere $\\mu$ is used to shift the mid-value of the complementary error function, so that states with energy equal to $\\mu$ have a weight of $f(\\mu)=1/2$ . The parameter $\\sigma$ is used to gauge the “broadness” of the distribution function.  \n\n3. Entangled manifold (case II), e.g., the $d$ bands in a transition metal: $f(\\epsilon)$ is a Gaussian function  \n\n$$\nf(\\epsilon)=\\exp\\left(-\\frac{(\\epsilon-\\mu)^{2}}{\\sigma^{2}}\\right).\n$$  \n\nThe procedure then follows as in the previous case, by computing a QRCP factorisation on the quasi-density matrix. It is worth to note that in the case of an entangled manifold, the  \n\nSCDM method requires the selection of two real numbers: $\\mu$ and $\\sigma$ , as well as the number of Wannier functions to disentangle $J$ . These parameters play a crucial role in the selection of the columns of the density matrix. While the selection of these parameters requires some care, as a rule of thumb (e.g., in entangled case I) $\\sigma$ is of the order 2 − 5 eV (which is the energy range of a typical bandwidth), while $\\mu$ can often be set around the Fermi energy (but the exact value depends on various factors, including the number $J$ of bands chosen and the specific properties of the bands of interest). It is worth to mention that since the SCDM- $k$ method is employed as an alternative way of specifying a set of initial projections and hence to compute the $A_{\\mathbf{k}}$ matrices in Eq. (26), the disentanglement procedure can be used in exactly the same way as described in Sec. II B. However, in the case of entangled bands the column selection is done on a quasi-density matrix, which implicitly defines a working subspace larger than the target subspace of dimension $J$ . We find that for wellknown systems SCDM- $k$ is typically already capable of selecting a smooth manifold and no further subspace selection is needed.  \n\nThis method is now implemented as part of the pw2wannier90.x interface code to Quantum ESPRESSO. We have decided to implement the algorithm in the interface code(s) rather than in Wannier90 itself, because the SCDM method requires knowledge of the wavefunctions $\\psi_{n\\mathbf{k}}$ , which are only available in the ab initio code.  \n\nIn Wannier90 only a single new input parameter auto projections is required. This disables the check on the number of projections specified in the input file (as we rely on SCDM to provide us with the initial guesses) and adds a new entry to the <seedname>.nnkp file (which is read by pw2wannier90.x in order to compute the quantities required by Wannier90) that specifies the number of Wannier functions required. The remaining control parameters for the SCDM method are specified in the input file for the pw2wannier90.x code, including whether to use the SCDM method, the functional form of the $f(\\epsilon)$ function in Eq. (66) and, optionally, the values of $\\mu$ and $\\sigma$ in the definition of $f(\\epsilon)$ .  \n\n# VII. AUTOMATION AND WORKFLOWS: AIIDA-WANNIER90 PLUGIN  \n\nAiiDA $^{13}$ (Automated Interactive Infrastructure and Database for Computational Science) is an informatics infrastructure that helps researchers in managing, automating, storing and sharing their computations and results. AiiDA automatically tracks the entire provenance of every calculation to ensure full reproducibility, which is also stored in a tailored database for efficient querying of previous results. Moreover, it provides a workflow engine, allowing researchers to implement high-level workflows to automate sequences of tedious or complex calculation steps. AiiDA supports simulation codes via a plugin interface, and over 30 different plugins are available to date104.  \n\nAmong these, the AiiDA-Wannier90 plugin provides support for the Wannier90 code. Users interact with the code (to submit calculations and retrieve the results) via the highlevel python interface provided by AiiDA rather than directly creating the Wannier90 input files. AiiDA will then handle automatically the various steps involved in submitting calculations to a cluster computer, retrieving and storing the results, and parsing them into a database. Furthermore, using the AiiDA workflow system users can chain pre-processing and post-processing calculations automatically (e.g., the preliminary electronic structure calculation with an ab initio code). These scientific workflows, moreover, can encode in a reproducible form the scientific knowledge of expert computational researchers in the field on how to run the simulations, choose the numerical parameters and recover from potential errors. In turn, their availability reduces the training time of new researchers, eliminates sources of error and enables large-scale high-throughput simulations.  \n\nThe AiiDA-Wannier90 plugin expects that each calculation takes a few well-defined input parameters. Among the most important ones, a Wannier90 calculation run via AiiDA requires that the following input nodes are provided: an input crystal structure, a node of parameters with a dictionary of input flags for Wannier90, a node with the list of kpoints, a node representing the atomic projections, and a local input folder or remote input folder node containing the necessary input files (.amn, .mmn, .nnkp, .eig, .dmn) for the Wannier90 calculation as generated by an ab initio code.  \n\nAll of these parameters, with the exception of projections, are generic to AiiDA to facilitate their reuse with different simulation codes. More detailed information on all inputs can be found in the AiiDA-Wannier90 package documentation105.  \n\nAfter the Wannier90 execution is completed, the AiiDA-Wannier90 plugin provides parsers that are able to detect whether the convergence was successful and retrieve key parameters including the centres of the Wannier functions and their spread, as well as the different components of the spread ( $\\Omega_{\\mathrm{I}}$ , ΩD, ΩOD and $\\Omega$ ), and (if computed) the maximum imaginary/real ratio of the Wannier functions and the interpolated band structure.  \n\nThe whole simulation is stored in the form of a graph, representing explicitly the provenance of the data generated including all inputs and outputs of the codes used in the workflow. An example of a provenance graph, automatically generated by AiiDA when running a Quantum ESPRESSO calculation followed by a Wannier90 calculation, is shown in Fig. 8.  \n\nTo demonstrate the usefulness of this approach, we refer to Ref. 106 that reports the implementation and verification results of a fully-automated workflow (implemented within AiiDA, using the AiiDA-Wannier90 plugin described in this section) to compute Wannier functions of any material without any user input (besides its crystal structure). In addition, a virtual machine containing the codes (AiiDA with its plugins, Quantum ESPRESSO and Wannier90 including the SCDM implementation described in Sec. VI, and the automation workflows) is distributed. This virtual machine allows any researcher to reproduce the results of the paper and, even more, to perform simulations on new materials using the same protocol, without the need of installing and configuring all codes.  \n\nWe emphasise that the availability of a platform to run Wannier90 in a fully-automated high-throughput way via the AiiDA-Wannier90 plugin has already proved to be beneficial for the Wannier90 code itself. Indeed, it has pushed the development of additional features or improvements now part of Wannier90 v3.0, including additional output files to facilitate output parsing and improvements in some of the algorithms and their default parameters to increase robustness.  \n\n# VIII. MODERN SOFTWARE ENGINEERING PRACTICES  \n\nIn this section, we describe a number of modern software engineering practices that are now part of the development cycle of the Wannier90 code. In particular, Wannier90 includes a number of tests that are run at every commit via a continuous integration approach, as well as nightly in a dedicated test farm. Version control is handled using git and the code is hosted on the GitHub platform $^{107}$ . We follow the fork and pull-request model, in which users can duplicate (fork) the project into their own private repository, make their own changes, and make a pull request (i.e., request that their changes be incorporated back into the main repository). When a pull request is made, a series of tests are automatically performed: the test suite is run both in serial and parallel using the Travis continuous in  \n\n![](images/3b86a13be5543cc7107e47b49125e127b70511c4664c2f9eaf19eb1516dfae67.jpg)  \n\nFIG. 8: The provenance graph automatically generated by AiiDA when running a   \nWannier90 calculation for a diamond crystal using Quantum ESPRESSO as the DFT   \ncode. Rectangles represent executions of calculations, ellipses represent data nodes, and diamonds are code executables. Graph edges connect calculations to their inputs and   \noutputs. In particular, the following calculations are visible: Quantum ESPRESSO pw.x   \nSCF (dark blue) and NSCF (green), Quantum ESPRESSO pw2wannier90.x (brown), and Wannier90 pre-processing (yellow) and minimisation run (purple). The initial   \ndiamond structure (light blue) and the final interpolated band structure (dark grey) are also highlighted.  \n\ntegration platform108, and code coverage is checked using codecov $^{\\cdot109}$ . If these tests are successful then the changes are reviewed by members of the Wannier90 developers group and, if the code meets the published coding guidelines, it can be merged into the development branch.  \n\nIn addition, while interaction with end users happens via a mailing-list forum, discussion among developers is now tracked using GitHub issues. This facilitates the maintenance of independent conversation threads for each different code issue, new feature proposal or bug.  \n\nThese can easily reference code lines as well as be referenced in code commit messages. Moreover, for every new bug report a new issue is opened, and pull requests that close the issue clearly refer to it. This approach facilitates tracking back the reasoning behind the changes in case a similar problem resurfaces.  \n\nIn the remainder of this section we describe more in detail some of these modern software engineering practices.  \n\n# A. Code documentation (FORD)  \n\nThe initial release of Wannier90 came with extensive documentation in the form of a User Guide describing the methodology, input flags to the program and format of the input and output files. This document was aimed at the end users running the software. Documentation of the code itself was done via standard code comments. In order to foster not only a community of users but also of code contributors to Wannier90, we have now created an additional documentation of the internal structure of the code. This makes the code more approachable, particularly for new contributors. To create this code documentation in a fully automated fashion, we use the FORD (FORtran Documenter) $^{110}$ documentation generator. We have chosen this over other existing documentation solutions because of FORD’s specific support for Fortran. This tool parses the Fortran source, and generates a hyperlinked (HTML) index of source files, modules, procedures, types and programs defined in the code. Furthermore, it constructs graphs showing the dependencies between different modules and subroutines. Additional information can be provided in the form of special in-code comments (marked with double exclamation marks) describing in more detail variables, modules or subroutines. By tightly coupling the code to its documentation using in-code comments, the documentation maintenance efforts are greatly reduced, decreasing the risk of having outdated documentation. The compiled version of the documentation for the most recent code version is made available on the Wannier90 website $^{111}$ .  \n\n# B. Testing infrastructure and continuous integration  \n\nWith the recent opening to the community of the Wannier90 development, it has become crucial to create a non-regression test suite to ensure that new developments do not break existing functionalities of the code. Its availability facilitates the maintenance of the code and ensures its long-term stability.  \n\nThe Wannier90 test suite relies on a modified version of James Spencer’s python testcode.py $^{112}$ . This provides the functionality to run tests and compare selected quantities parsed from the output files against benchmarked values.  \n\nAt present, the Wannier90 test suite includes over 50 tests which are run both in serial and parallel and cover over $60\\%$ of the source code (with many modules exceeding $80\\%$ coverage). The code coverage is measured with the codecov software109. Developers are now required to add tests when adding new features to the code to ensure that their additions work as expected. This also ensures that future changes to the code will never break that functionality. Two different test approaches are implemented, serving different purposes.  \n\nFirst, the Wannier90 repository is now linked with the Travis continuous integration platform $^{108}$ to prevent introducing errors and bugs into the main code branch. Upon any commit to the GitHub repository, the test suite is run both in serial and in parallel. Any test failure is reported back to the GitHub webpage. Additionally, for tests run against pull requests, any failed test results in the pull request being blocked and not permitted to merge. Contributors will first need to change their code to fix the problems highlighted in the tests; pull requests are able to be merged only after all tests pass successfully.  \n\nSecond, nightly automatic tests are run on a Buildbot test-farm. The test-farm compiles and runs the code with a combination of compilers and libraries (current compilers include GFortran v6.4.0 and v7.3.0, Intel Fortran Compiler v17 and v18, and PGI compiler v18.05; current MPI libraries include Open MPI v1.10.7 and v3.1.3, Intel MPI v17 and MVAPICH v2.3b). This ensures that the code runs correctly on various high-performance computer (HPC) architectures. More information on the test-farm can be found on the Wannier90 GitHub wiki website113.  \n\nIn addition to these tests, we have implemented git pre-commit hooks to help keep the same code style in all source files. The current pre-commit hooks run Patrick Seewald’s Fortran source code formatter fprettify $^{\\cdot114}$ to remove trailing whitespaces at the end of a line and to enforce a consistent indentation style. These precommit hooks, besides validating the code, can reformat it automatically. Developers may simply run the formatter code to convert the source to a valid format. If a developer installs the pre-commit hooks, these will be run automatically before every commit. Even if this is not the case, these tests are also run on Travis; therefore, a pull request that does not conform to the standard code style cannot be merged before the style is fixed.  \n\n# C. Command-line interface and dry-run  \n\nThe command-line interface of the code has been improved. Just running wannier90.x without parameters shows a short explanation of the available command line options. In addition, a -v flag has been added to print the version of the code, as well as a new -d dry-run mode, that just parses the input file to perform all needed checks of the inputs without running the actual calculation. The latter functionality is particularly useful to be used in input validators for Wannier90 or to precalculate quantities computed by the code at the beginning of the simulation (such as nearest-neighbour shells, $b$ -vectors or expected memory usage) and use this information to validate the run or optimise it (e.g., to decide the parallelisation strategy within automated AiiDA workflows).  \n\n# D. Library mode  \n\nWannier90 also comes with a library mode, where the core code functionality can be compiled into a library that can then be linked by external programs. This library mode is used as the default interaction protocol by some interface codes. The library mode provides only support for a subset of the full functionality, in particular at the moment it only supports serial execution. We have now added and improved support for the use of excluded bands also within the library mode. Moreover, beside supporting the generation of a staticallylinked library, we now also support the generation of dynamically-linked versions. Finally, we have added a minimal test code, run together with all other tests in the test suite, that serves both to verify that the library functionality works as expected, and as an example of the interface of the library mode.  \n\n# IX. CONCLUSIONS AND OUTLOOK  \n\nWannier90 v2.0 was released in October 2013 with a small update for v2.1 in January 2017. The results and developments of the past years, presented in this work, were released in Wannier90 v3.0 in February 2019. Thanks to the transition of Wannier90 to a community code, Wannier90 includes now a large number of new functionalities and improvements that make it very robust, efficient and rich with features. These include the implementation of new methods for the calculation of WFs and for the generation of the initial projections; parallelisation and optimisations; interfaces with new codes, methods and infrastructures; new user functionality; improved documentation; and various bug fixes. The effect of enlarging the community of developers is not only visible in the large number of contributions to the code, but also in the modern software engineering practices that we have put in place, that help improve the robustness and reliability of the code and facilitate its maintenance by the core Wannier90 developers group and its long-term sustainability.  \n\nThe next major improvement that we are planning is the implementation of a more robust and general library mode. The features that we envision are: (1) the possibility to call the code from C or Fortran codes without the need to store files but by passing all variables from memory; (2) a more general library interface that is easily extensible in the future when new functionality is added; and (3) the possibility to run Wannier90 from a parallel MPI code, both by running each instance in parallel and by allowing massively-parallel codes to call, in parallel, various instances of Wannier90 on various structures or with different parameters. This improvement will demand a significant restructuring of most of the codebase and requires a good design of the new interface. Currently we are drafting the new library interface, by collecting feedback and use cases from the various contributors and users of the code, to ensure that the new library mode can be beneficial to all different possible use cases.  \n\n# ACKNOWLEDGMENTS  \n\nWe acknowledge code contributions by Daniel Aberg (w90pov code), Lampros Andrinopoulos (w90vdw code), Pablo Aguado Puente (gyrotropic.F90 module), Raffaello Bianco ( $k$ -slice plotting), Marco Buongiorno Nardelli (dosqc v1.0 subroutines upon which some of transport.F90 is based), Stefano de Gironcoli (pw2wannier90.x interface to Quantum ESPRESSO), Pablo Garcia Fernandez (matrix elements of the position operator), Nicholas Hine (w90vdw code), Young-Su Lee (specialised $\\Gamma$ -point routines and transport), Antoine Levitt (preconditioning), Graham Lopez (extension of pw2wannier90.x to add terms needed for orbital magnetisation), Radu Miron (constrained centres), Nicolas Poilvert (transport routines), Michel Posternak (original plotting routines), Rei Sakuma (symmetry-adapted Wannier functions), Gabriele Sclauzero (disentanglement in spheres in $k$ -space), Matthew Shelley (transport routines), Christian Stieger (routine to print the U matrices), David Strubbe (various bug fixes and improvements), Timo Thonhauser (extension of pw2wannier90.x to add terms needed for orbital magnetisation), as well as the participants of the first Wannier90 developers meeting in San Sebasti´an (Spain) in 2016 for useful discussions (Daniel Fritsch, Victor Garcia Suarez, Pablo Garcia Fernandez, Jan-Philipp Hanke, Ji Hoon Ryoo, Ju¨rg Hutter, Javier Junquera, Liang Liang, Michael Obermeyer, Gianluca Prandini, Christian Stieger, Paolo Umari). The WDG acknowledges financial support from the NCCR MARVEL of the Swiss National Science Foundation, the European Union’s Centre of Excellence E-CAM (grant no. 676531), and the Thomas Young Centre for Theory and Simulation of Materials (grant no. TYC-101).  \n\n1 P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. K¨u¸c¨ukbenli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V. Nguyen, A. O. de-la Roza, L. Paulatto, S. Ponc´e, D. Rocca, R. Sabatini, B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thonhauser, P. Umari, N. Vast, X. Wu and S. Baroni, Advanced capabilities for materials modelling with Quantum ESPRESSO, J. Phys. Cond. 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    "Target": null
  },
  {
    "id": "10.1088_1361-648X_ab4007",
    "DOI": "10.1088/1361-648X/ab4007",
    "DOI Link": "http://dx.doi.org/10.1088/1361-648X/ab4007",
    "Relative Dir Path": "mds/10.1088_1361-648X_ab4007",
    "Article Title": "QuantumATK: an integrated platform of electronic and atomic-scale modelling tools",
    "Authors": "Smidstrup, S; Markussen, T; Vancraeyveld, P; Wellendorff, J; Schneider, J; Gunst, T; Verstichel, B; Stradi, D; Khomyakov, PA; Vej-Hansen, UG; Lee, ME; Chill, ST; Rasmussen, F; Penazzi, G; Corsetti, F; Ojanperä, A; Jensen, K; Palsgaard, MLN; Martinez, U; Blom, A; Brandbyge, M; Stokbro, K",
    "Source Title": "JOURNAL OF PHYSICS-CONDENSED MATTER",
    "Abstract": "QuantumATK is an integrated set of atomic-scale modelling tools developed since 2003 by professional software engineers in collaboration with academic researchers. While different aspects and individual modules of the platform have been previously presented, the purpose of this paper is to give a general overview of the platform. The QuantumATK simulation engines enable electronic-structure calculations using density functional theory or tight-binding model Hamiltonians, and also offers bonded or reactive empirical force fields in many different parametrizations. Density functional theory is implemented using either a plane-wave basis or expansion of electronic states in a linear combination of atomic orbitals. The platform includes a long list of advanced modules, including Green?s-function methods for electron transport simulations and surface calculations, first-principles electron-phonon and electron-photon couplings, simulation of atomic-scale heat transport, ion dynamics, spintronics, optical properties of materials, static polarization, and more. Seamless integration of the different simulation engines into a common platform allows for easy combination of different simulation methods into complex workflows. Besides giving a general overview and presenting a number of implementation details not previously published, we also present four different application examples. These are calculations of the phonon-limited mobility of Cu, Ag and Au, electron transport in a gated 2D device, multi-model simulation of lithium ion drift through a battery cathode in an external electric field, and electronic-structure calculations of the composition-dependent band gap of SiGe alloys.",
    "Times Cited, WoS Core": 1409,
    "Times Cited, All Databases": 1439,
    "Publication Year": 2020,
    "Research Areas": "Physics",
    "UT (Unique WOS ID)": "WOS:000499348200001",
    "Markdown": "PAPER  \n\n# QuantumATK: an integrated platform of electronic and atomic-scale modelling tools  \n\nTo cite this article: Søren Smidstrup et al 2020 J. Phys.: Condens. Matter 32 015901  \n\nView the article online for updates and enhancements.  \n\n# Recent citations  \n\n![](images/ce30ad4319607b20f8886af0b4b8c62546836f885dd6bc5a7512ed9d247b9483.jpg)  \n\nThis content was downloaded from IP address 132.174.255.215 on 14/11/2019 at 16:14  \n\n# QuantumATK: an integrated platform of electronic and atomic-scale modelling tools  \n\nSøren Smidstrup1 , Troels Markussen1, Pieter Vancraeyveld1, Jess Wellendorff $^1\\textcircled{\\circ}$ , Julian Schneider1, Tue Gunst1,2, Brecht Verstichel1, Daniele Stradi1, Petr A Khomyakov1, Ulrik G Vej-Hansen $1\\textcircled{\\circ}$ , Maeng-Eun Lee1, Samuel T Chill1, Filip Rasmussen1, Gabriele Penazzi1, Fabiano Corsetti $^1\\textcircled{\\circ}$ , Ari Ojanperä1, Kristian Jensen1, Mattias L N Palsgaard1,2, Umberto Martinez $^1\\textcircled{\\circ}$ , Anders Blom $1_{\\textcircled{1}}$ , Mads Brandbyge2 and Kurt Stokbro1  \n\n1  Synopsys Denmark, Fruebjergvej 3, Postbox 4, DK-2100 Copenhagen, Denmark 2  DTU Physics, Center for Nanostructured Graphene (CNG), Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark  \n\nE-mail: soren.smidstrup@synopsys.com  \n\nReceived 6 May 2019, revised 9 August 2019   \nAccepted for publication 30 August 2019   \nPublished 10 October 2019  \n\n![](images/a18f6dbae5447d77598d293f6d35f1060d4eeefb301a14f34382348842c67650.jpg)  \n\n# Abstract  \n\nQuantumATK is an integrated set of atomic-scale modelling tools developed since 2003 by professional software engineers in collaboration with academic researchers. While different aspects and individual modules of the platform have been previously presented, the purpose of this paper is to give a general overview of the platform. The QuantumATK simulation engines enable electronic-structure calculations using density functional theory or tightbinding model Hamiltonians, and also offers bonded or reactive empirical force fields in many different parametrizations. Density functional theory is implemented using either a plane-wave basis or expansion of electronic states in a linear combination of atomic orbitals. The platform includes a long list of advanced modules, including Green’s-function methods for electron transport simulations and surface calculations, first-principles electron-phonon and electron-photon couplings, simulation of atomic-scale heat transport, ion dynamics, spintronics, optical properties of materials, static polarization, and more. Seamless integration of the different simulation engines into a common platform allows for easy combination of different simulation methods into complex workflows. Besides giving a general overview and presenting a number of implementation details not previously published, we also present four different application examples. These are calculations of the phonon-limited mobility of Cu, Ag and Au, electron transport in a gated 2D device, multi-model simulation of lithium ion drift through a battery cathode in an external electric field, and electronic-structure calculations of the composition-dependent band gap of SiGe alloys.  \n\nKeywords: atomic-scale modelling, density functional theory, semi-empirical methods, tight-binding, force fields, first-principles simulations, non-equilibrium Green’s function (Some figures may appear in colour only in the online journal)  \n\n# Contents  \n\n# 1.  Introduction  \n\n1.  Introduction\b 2.  Overview\b   \n3.  Atomistic configurations\b   \n4.  DFT simulation engines\b 4.1.  LCAO representation 4.2.  PW representation 4.3.  Pseudopotentials and LCAO basis sets 4.4.  Exchange-correlation methods 4.5.  Boundary conditions and Poisson solvers   \n5.  Semi-empirical models\b   \n6.  Empirical force fields\b 7.  Ion dynamics\b 7.1.  Local structural optimization 7.2.  Global structural optimization 7.3.  Reaction pathways and transition states 7.4.  Molecular dynamics 7.5.  Adaptive kinetic Monte Carlo   \n8.  Phonons\b 8.1.  Calculating the dynamical matrix 8.2.  Wigner–Seitz method 8.3.  Phonon band structure and density of states 8.4.  Electron-phonon coupling 8.5.  Transport coefficients   \n9.  Polarization and Berry phase\b   \n10.  Magnetic anisotropy energy\b   \n11.  Quantum transport\b 11.1.  NEGF method 11.2.  Retarded Green’s function 11.3.  Complex contour integration 11.4.  Bound states 11.5.  Spill-in terms 11.6.  Device total energy and forces 11.7.  Transmission coefficient and current 11.8.  Inelastic transmission and inelastic current\b 11.9.  Thermoelectric transport 11.10.  Photocurrent   \n12.  QuantumATK parallelization 12.1.  Bulk DFT and semi-empirical simulations 12.2.  DFT-NEGF device simulations 12.3.  FF simulations   \n13.  NanoLab simulation environment 13.1.  Python scripting 13.2.  NanoLab graphical user interface 13.3.  Documentation   \n14.  QuantumATK applications\b 14.1.  \u0007Large-scale simulations of 2D field-effect transistors\b 14.2.  Phonon-limited mobility of metals 14.3.  \u0007Multi-model dynamics with an applied electric field 14.4.  Electronic structure of binary alloys\b   \n15.  Summary\b   \nAcknowledgments   \nAppendix.  Computational details  \n\nReferences  \n\n2 Atomic-scale modelling is increasingly important for indus  \n4 trial and academic research and development in a wide range   \n5 of technology areas, including semiconductors [1, 2], batteries   \n5 [3], catalysis [4], renewable energy [5], advanced materials   \n5 [6], next-generation pharmaceuticals [7], and many others.   \n6 Surveys indicate that the return on investment of atomic-scale   \n6 modelling is typically around 5:1 [8]. With development of   \n8 increasingly advanced simulation algorithms and more pow  \n9 erful computers, we expect that the economic benefits of   \n1 atomic-scale modelling will only increase.  \n\nThe current main application of atomic-scale modelling is in early-stage research into new materials and technology designs, see [9, 10] for examples. The early research stage often has a very large design space, and experimental trial and error is a linear process that will explore only a small part of this space. Atomic-scale simulations make it possible to guide experimental investigations towards the most promising part of the technology design space. Such insights are typically achieved by simulating the underlying atomic-scale processes behind failed or successful experiments, to understand the physical or (bio-)chemical origins. Such insight can often rule out or focus research to certain designs or mat­erial systems [8]. Recently, materials screening has also shown great promise. In this approach, atomic-scale calculations are used to obtain important properties of a large pool of materials, and the most promising candidates are then selected for exper­imental verification and/or further theoretical refinement [6, 11, 12].  \n\nThe scientific field of atomic-scale modelling covers everything from near-exact quantum chemical calculations to approximate simulations using empirical force fields. Quantum chemical methods (based on wave-function theory) attempt to fully solve the many-body Schrödinger equation for all electrons in the system, and can provide remarkably accurate descriptions of molecules [13]. However, the computational cost is high: in practice, one is usually limited to calcul­ations involving far below 100 atoms in total. Such methods are cur­ rently not generally useful for industrial research into advanced materials and next-generation electronic devices.  \n\nOn the contrary, force-field (FF) methods are empirical but computationally efficient: all inter-atomic interactions are described by analytic functions with pre-adjusted parameters. It is thereby possible in practice to simulate systems with millions of atoms. Unfortunately, this often also hampers the applicability of a force field for system types not included when fitting the FF parameters.  \n\nAs an attractive intermediate methodology, density func  \n27 tional theory (DFT) [14–17] provides an approximate but   \n29 computationally tractable solution to the electronic many  \nbody problem. This allows for good predictive power with   \n30 respect to experiments with minimal use of empirical param­   \n32 eters at a reduced computational cost. Standard DFT simu  \n33 lations may routinely be applied to systems containing more   \n33 than one thousand atoms, and DFT is today the preferred   \n33 framework for industrial applications of ab initio electronic  \n33 structure theory.  \n\nTable 1.  Simulation engines in the QuantumATK platform, with examples of other simulation platforms using the same underlying methodology. LCAO and PW means linear combination of atomic orbitals and plane wave, respectively.   \n\n\n<html><body><table><tr><td>Engine</td><td>Description</td><td>First release</td><td>Related platforms</td></tr><tr><td>ATK-LCAO</td><td>Pseudopotential DFT using LCAO basis [19]</td><td>2003</td><td>SIESTA [20],OpenMX [21]</td></tr><tr><td>ATK-PlaneWave</td><td>Pseudopotential DFT using PW basis</td><td>2016</td><td>VASP [22], Quantum ESPRESSO [23]</td></tr><tr><td>ATK-SE</td><td>Semi-empirical TB methods [24]</td><td>2010</td><td>DFTB + [25], NEMO [26],OMEN [27]</td></tr><tr><td>ATK-ForceField</td><td>All types of empirical force fields [28]</td><td>2014</td><td>LAMMPS [29], GULP [30]</td></tr></table></body></html>  \n\nSemi-empirical (SE) electronic-structure methods based on tight-binding (TB) model Hamiltonians are more approximate, but have a long tradition in semiconductor research [18]. Whereas DFT ultimately aims to approximate the true many-body electronic Hamiltonian in an efficient but parameter-free fashion, a TB model relies on parameters that are adjusted to very accurately describe the properties of a number of reference systems. This leads to highly specialized electronic-structure models that typically reduce the computational expense by an order of magnitude compared to DFT methods. Such SE methods may be convenient for large-scale electronic-structure calculations, for example in simulations of electron transport in semiconductor devices.  \n\nThe QuantumATK platform offers simulation engines covering the entire range of atomic-scale simulation methods relevant to the semiconductor industry and materials science in general. This includes force fields, SE methods, and several flavors of DFT. These are summarized in table  1, including examples of other platforms that offer similar methodology.  \n\nTo give a bird’s-eye view of the computational cost of the different atomic-scale simulation methods mentioned above, we compare in figure  1 the computational speed of the methods when simulating increasingly larger structures of amorphous $\\mathrm{{Al}}_{2}\\mathrm{{O}}_{3}$ . The measure of speed is here the number of molecular dynamics steps that are feasible within $24\\mathrm{h}$ when run in parallel on 16 computing cores. Although the parallel computing techniques used may differ between some of the methods, we find that figure 1 gives a good overview of the scaling between the different methods.  \n\nIt is important to realize that the simulation methods listed in table  1 should ideally complement each other: for successful use of atomic-scale modelling, it is essential to have easy access to all the methods, in order to use them in combination. The vast majority of atomic-scale simulation tools are developed by academic groups, and most of them focus on a single method. Using the tool typically requires a large effort for compilation, installation, learning the input/output syntax, etc. The tool is often not fully compatible with any other tool, so learning an additional tool within a new modelling class requires yet another large effort. Even within one modelling class, for example DFT, a single simulation tool may not have all the required functionality for a given application, so several different tools within each modelling class may be needed to solve a given problem, and a significant effort must be invested to master each of them. As a commercially developed platform, QuantumATK aims to circumvent these issues.  \n\nAcademic development of atomic-scale simulation platforms, often made available through open-source licenses, is essential for further technical progress of the field. However, the importance of commercial platforms in progressing the industrial uptake of the technology is often underestimated. Commercial software relies on payment from end users. This results in a strong focus on satisfying end-user requirements in terms of usability, functionality, efficiency, reliability, and support. The revenue enables the commercial software provider to establish a stable team of developers and thereby provide a software solution that will be maintained, extended, and supported for decades.  \n\n![](images/8f206934e73d1384f2dd92834da8e6e5fcc7cf2529e7a498c1576cac951fabc5.jpg)  \nFigure 1.  Comparison of the simulation methods available in QuantumATK, showing the total number of molecular dynamics steps performed in $24\\mathrm{{h}}$ ( $\\#$ MD steps) against system size ( $\\#$ atoms) for amorphous $\\mathrm{Al}_{2}\\mathrm{O}_{3}$ with constant density. Each step includes evaluation of the total energy and atomic forces. The simulations were run on a 16-core central processing unit (CPU) of the type Inte $\\mathsf{\\Omega}_{\\mathsf{X}}\\mathsf{e o n}^{\\circledast}\\mathsf{E}5\\mathsf{-}2670$ . The FF simulations (section 6) were performed using threading only, whereas full MPI parallelization was used for the TB (section 5) and DFT (section 4) simulations. For the latter, we have considered either semi-local exchange-correlation functionals using linear-combination-ofatomic-orbitals (DFT-LCAO) and plane-wave (DFT-PW) basis sets, or a hybrid exchange-correlation functional using a PW basis set (Hybrid DFT). Further details of the calculations are given in appendix.  \n\nThe ambition of the QuantumATK platform is to provide a state-of-the-art and easy-to-use integrated toolbox with all important atomic-scale modelling methodologies for a growing number of application areas. The methods are made available through a modern graphical user interface (GUI) and a Python scripting-based frontend for expert users. Our current focus is semiconductor devices, polymers, glasses, catalysis, batteries, and materials science in general. In this context, semiconductor devices is a broad area, ranging from silicon-based electronic logics and memory elements [31, 32], to solar cells composed of novel materials [33] and nextgeneration electronic devices based on spintronic phenomona [34]. One key strength of a unified framework for a large selection of simulation engines and modelling tools is within multiphysics and multiscale problems. Such problems often arise in physical modelling of semiconductor devices, and the QuantumATK platform is widely used for coupling technology computer-aided design (TCAD) tools with atomic-scale detail, for instance to provide first-principles simulations of defect migration paths and subsequently the temperature-dependent diffusion constant for continuum-level simulation of semiconductor processes [2]. Furthermore, QuantumATK provides a highly flexible and efficient framework for coupling advanced electrostatic setups with state-of-the-art transport simulations including electron–phonon coupling and light-matter interaction. This has enabled predictions of gate-induced phonon scattering in graphene gate stacks [35], atomistic description of ferroelectricity driven by edge-absorbed polar molecules in gated graphene [36], and new 2D material science such as prediction of the room-temperature photocurrent in emerging layered Janus materials with a large dipole across the plane [37]. The flexibility of the QuantumATK framework supports the imagination of researchers, and at the same time enables solutions to both real-world and cutting-edge semiconductor device and material science problems.  \n\n![](images/67fae50bcba7df7713f3ac431e953da8b0411c4f81ceaf7df0ea5eaabdcb8490.jpg)  \nFigure 2.  Supported atomistic configurations in QuantumATK. (a) Molecule configuration of a pentane molecule. (b) Bulk configuration of a gold crystal. (c) Device configuration of a gold-silver interface. The structure consists of a left electrode (transparant yellow), central region (orange box), and a right electrode (transparent yellow). Both electrodes are semi-infinite in the left-right transport direction. The device is in this case periodic in the two directions perpendicular to the transport direction, but would be nonperiodic in one or both perpendicular directions in case of a nanosheet or nanotube device, respectively. (d) Surface configuration of a gold surface. The structure consists of a left electrode (transparant yellow) and a central region (orange box). We note that an electric field can be applied to the surface by choice of boundary condition on the right-hand face of the central region.  \n\nThe purpose of this paper is to give a general overview of the QuantumATK platform with appropriate references to more thorough descriptions of several aspects of the platform. We also provide application examples that illustrate how the different simulation engines can complement each other. The paper is organized as follows: In section 2 we give a general overview of the QuantumATK platform, while section 3 introduces the types of system geometries handled by the platform. The next three sections  4–6 describe the DFT, SE, and FF simulation engines, respectively. We then introduce a number of simulation modules that work with the different engines. These modules include ion dynamics (section 7.1), phonon properties (section 8), polarization (section 9), magn­etic aniso­tropy energy (section 10), and quantum transport (section 11). We next describe the parallel computing strategies of the different engines, and present parallel scaling plots in section 12. We then in section 13 describe the scripting and GUI simulation environment in the QuantumATK platform. This is followed by four application examples in section 14, and the paper is summarized in section 15.  \n\n# 2.  Overview  \n\nThe core of QuantumATK is implemented in $^{C++}$ modules with Python bindings, such that all $^{C++}$ modules are accessible from ATK-Python, a customized version of Python built into the software. The combination of a $^{C++}$ backend and a Python-based frontend offers both high computational performance and a powerful but user-friendly scripting platform for setting up, running, and analyzing atomic-scale simulations. All simulation engines listed in table  1 are invoked using ATK-Python scripting. More details are given in section 13.1. QuantumATK also relies on a number of open-source packages, including high-performance numerical solvers.  \n\nAll computationally demanding simulation modules may be run in parallel on many processors at once, using message passing between processes and/or shared-memory threading, and often in a multi-level approach. More details are given in section 12.  \n\nThe full QuantumATK package is installed on Windows or Linux/Unix operating systems using a binary installer obtained from the Synopsys SolvNet website, https://solvnet. synopsys.com. All required external software libraries are precompiled and shipped with the installer. Licensing is handled using the Synopsys common licensing (SCL) system.  \n\n# 3.  Atomistic configurations  \n\nThe real-space physical system to be simulated is defined as an ATK-Python configuration object, including lattice vectors, element types and positions, etc. QuantumATK currently offers four main types of such configurations: molecule, bulk, device, and surface. Examples of these are given in figure 2.  \n\nThe simplest configuration is the molecule configuration shown in figure 2(a). It is used for isolated (non-periodic) systems, and is defined by a list of elements and their positions in Cartesian coordinates.  \n\nThe bulk configuration, shown in figure  2(b), defines an atomic-scale system that repeats itself in one or more directions, for example a fully periodic crystal (periodic in 3D), a 2D nanosheet (or a slab), or a 1D nanowire. The bulk system is defined by the Bravais lattice and the position of the atomic elements inside the primitive cell.  \n\nThe two-probe device configuration is used for quantum transport simulations. As shown in figure 2(c), the device consists of a central region connected to two semi-infinite bulk electrodes. The central region, where scattering of electrons travelling from one electrode to the other may take place, can be periodic in zero (1D wire), one (2D sheet), or two (3D bulk) directions, but is bounded by the electrodes along the third dimension. The device configuration is used to simulate electron and/or phonon transport via the non-equilibrium Green’s function (NEGF) method [38].  \n\nFinally, for physically correct simulations of a surface, QuantumATK provides the one-probe surface configuration. This is basically a device configuration with only one electrode, as illustrated in figure  2(d). By construction, the surface configuration realistically describes the electronic structure of a semi-infinite crystal beyond the approximate slab model [19].  \n\nThe remainder of this paper is devoted to describing the computational methods available for calculating the properties of such configurations using QuantumATK.  \n\n# 4.  DFT simulation engines  \n\nDensity functional theory is implemented in the Kohn–Sham (KS) formulation [14–17] within the framework of the linear combination of atomic orbitals (LCAO) and plane-wave (PW) basis set approaches, combined with the pseudopotential method. The electronic system is seen as a non-interacting electron gas of density $n$ in the effective potential $V^{\\mathrm{eff}}[n]$ ,  \n\n$$\n\\begin{array}{r}{V^{\\mathrm{eff}}[n]=V^{\\mathrm{H}}[n]+V^{\\mathrm{xc}}[n]+V^{\\mathrm{ext}}[n],}\\end{array}\n$$  \n\nwhere $V^{\\mathrm{H}}$ is the Hartree potential describing the classical electrostatic interaction between the electrons, $V^{\\mathrm{xc}}$ is the exchange-correlation (XC) potential, which in practise needs to be approximated, and $V^{\\mathrm{ext}}$ is the sum of the electrostatic potential energy of the electrons in the external potential of ions and other electrostatic field sources. The total external potential is in QuantumATK given by  \n\n$$\nV^{\\mathrm{ext}}=\\sum_{a}V_{a}^{\\mathrm{pseudo}}+V^{\\mathrm{gate}},\n$$  \n\nwhere $V_{a}^{\\mathrm{pseudo}}$ includes the local $(V_{a}^{\\mathrm{loc}})$ and nonlocal $(V_{a}^{\\mathrm{nl}})$ contrib­utions to the pseudopotential of the ath atom. The term $V^{\\mathrm{gate}}$ is a potential that may originate from other external sources of electrostatic fields, for example metallic gates.  \n\nThe KS Hamiltonian consists of the single-electron kinetic energy and the effective potential,  \n\n$$\n\\hat{H}^{\\mathrm{KS}}=-\\frac{\\hbar^{2}}{2m}\\nabla^{2}+V^{\\mathrm{eff}},\n$$  \n\nand the single-electron energies $(\\epsilon_{\\alpha})$ and wave functions $(\\psi_{\\alpha})$ are solutions to eigenvalue problem  \n\n$$\n\\hat{H}^{\\mathrm{KS}}\\psi_{\\alpha}=\\epsilon_{\\alpha}\\psi_{\\alpha}.\n$$  \n\nThe electronic ground state is found by iteratively minimizing the KS total-energy density functional, $E[n]$ , with respect to the electron density,  \n\n$$\n\\boldsymbol E[n]=\\boldsymbol T+\\boldsymbol E^{\\mathrm{H}}[n]+\\boldsymbol E^{\\mathrm{xc}}[n]+\\boldsymbol E^{\\mathrm{ext}}[n],\n$$  \n\nwhere $T$ is the kinetic energy. The forces (acting on the atoms) and stress tensor of the electronic system may then be computed as derivatives of the ground-state total energy with respect to the atomic coordinates and the strain tensor, respectively.  \n\n# 4.1.  LCAO representation  \n\nThe DFT-LCAO method uses a LCAO numerical representation of the KS equations, closely resembling the SIESTA formalism [20]. This allows for a localized matrix representation of the KS Hamiltonian in (3), and therefore an efficient implementation of KS-DFT for molecules, bulk materials, interface structures, and nanoscaled devices.  \n\nIn the DFT-LCAO method, the single-electron KS eigenfunctions, $\\psi_{\\alpha}$ , are expanded in a set of finite-range atomic-like basis functions $\\phi_{i},$  \n\n$$\n\\psi_{\\alpha}(\\mathbf{r})=\\sum_{i}c_{\\alpha i}\\phi_{i}(\\mathbf{r}).\n$$  \n\nThe KS equation can then be represented as a matrix equation for determining the expansion coefficients $c_{\\alpha i}$ ,  \n\n$$\n\\sum_{j}H_{i j}^{\\mathrm{KS}}c_{\\alpha j}=\\varepsilon_{\\alpha}\\sum_{j}S_{i j}c_{\\alpha j},\n$$  \n\nwhere the Hamiltonian matrix $H_{i j}^{\\mathrm{KS}}=\\langle\\phi_{i}|\\hat{H}^{\\mathrm{KS}}|\\phi_{j}\\rangle$ and overlap matrix $S_{i j}=\\langle\\phi_{i}|\\phi_{j}\\rangle$ are given by integrals with respect to the electron coordinates. Two-center integrals are computed using 1D radial integration schemes employing a Fourier transform technique, while multiple-center integrals are computed on a real-space grid [20].  \n\nFor molecules and bulk systems, diagonalization of the Hamiltonian matrix yields the density matrix $D_{i j}$ ,  \n\n![](images/8cd273210efa84f2c18c8b98e77af03fbbee40dc9782cb083f775106b8831f96.jpg)  \nFigure 3.  Time per 10 selfconsistent field (SCF) iterations for different sized gold melts at $900\\mathrm{K}$ . For each system, we use a single $\\mathbf{k}$ -point and the simulation runs on a 16-core CPU. The timings of the DFT-PW method are compared to those of the DFT-LCAO method using the Ultra (LCAO-U), High (LCAO-H), and Medium (LCAO-M) basis sets.  \n\n$$\nD_{i j}=\\sum_{\\alpha}c_{\\alpha i}^{*}c_{\\alpha j}f\\left(\\frac{\\varepsilon_{\\alpha}-\\varepsilon_{\\mathrm{{F}}}}{k_{\\mathrm{{B}}}T}\\right),\n$$  \n\nwhere $f$ is the Fermi–Dirac distribution of electrons over energy states, $\\varepsilon_{\\mathrm{{F}}}$ the Fermi energy, $T$ the electron temperature, and $k_{\\mathrm{B}}$ the Boltzmann constant. For device and surface configurations, the density matrix is calculated using the NEGF method, as described in section 11.  \n\nThe electron density is computed from the density matrix,  \n\n$$\nn(\\mathbf{r})=\\sum_{i j}D_{i j}\\phi_{i}(\\mathbf{r})\\phi_{j}(\\mathbf{r}),\n$$  \n\nand is represented on a regular real-space grid, which is the same grid as used for the effective potential in (1).  \n\n# 4.2.  PW representation  \n\nA PW representation of the KS equations was recently implemented in QuantumATK. It is complimentary to the LCAO representation discussed above. The ATK-PlaneWave engine is intended mainly for simulating bulk configuratins with periodic boundary conditions. The KS eigenfunctions are expanded in terms of PW basis functions,  \n\n$$\n\\psi_{\\alpha}(\\mathbf{r})=\\sum_{|\\mathbf{g}|<g_{\\mathrm{max}}}c_{\\alpha,\\mathbf{g}}\\mathrm{e}^{\\mathrm{i}\\mathbf{g}\\cdot\\mathbf{r}},\n$$  \n\nwhere $\\alpha$ denotes both the wave vector $\\mathbf{k}$ and the band index $n$ , and $\\mathbf{g}$ are reciprocal lattice vectors. The upper threshold for the reciprocal lattice-vector lengths included in the PW expansion $(g_{\\mathrm{max}})$ is determined by a kinetic-energy (wavefunction) cutoff energy $E_{\\mathrm{cut}}$ ,  \n\n$$\n\\frac{\\hbar^{2}g_{\\mathrm{max}}^{2}}{2m}<E_{\\mathrm{cut}}.\n$$  \n\nThe DFT-PW method has its distinct advantages and disadvantages compared to the DFT-LCAO approach. In particular, the PW expansion is computationally efficient for relatively small bulk systems, and the obtained physical quantities can be systematically converged with respect to the PW basis-set size by increasing $E_{\\mathrm{cut}}$ . However, the PW representation is computationally inefficient for lowdimensional systems with large vacuum regions. It is also incompatible with the DFT-NEGF methodology for electron transport calculations in nanoscaled devices, unlike the LCAO representation, which is ideally suited for dealing with open boundary conditions, and is also more efficient for large systems.  \n\nTable 2.  Summary of QuantumATK $\\Delta$ -tests for elemental solids [48], and RMS errors of the lattice constant (a) and bulk modulus $(B)$ of rock-salt and perovskite test sets [49], using SG15 and PseudoDojo PPs, and the ATK-PlaneWave and ATK-LCAO engines, the latter with different basis sets. All errors are relative to allelectron calculations.   \n\n\n<html><body><table><tr><td>Medium</td><td>High</td><td>Ultra</td><td>PW</td></tr><tr><td colspan=\"4\">Elemental solids: Delta tests</td></tr><tr><td>SG15 (meV)</td><td>1.88</td><td>2.03</td><td>1.32</td></tr><tr><td>3.45 PseudoDojo  (meV) 4.53</td><td>1.52</td><td>1.40</td><td>1.04</td></tr><tr><td colspan=\"4\">Rock salts: RMS of a and B</td></tr><tr><td>SG15 (%)</td><td>0.24</td><td>0.23</td><td>0.16</td></tr><tr><td>PseudoDojo (%)</td><td>0.18</td><td>0.15</td><td>0.09</td></tr><tr><td colspan=\"4\">Perovskites: RMS of a and B</td></tr><tr><td>SG15 (%)</td><td></td><td>0.18</td><td>0.13</td></tr><tr><td>PseudoDojo (%)</td><td></td><td>0.13</td><td>0.06</td></tr></table></body></html>  \n\nThe ATK-PlaneWave engine was implemented on the same infrastructure as used by the ATK-LCAO engine, though a number of routines were modified to reach state-of-the-art PW efficiency. For example, we have adopted iterative algorithms for solving the KS equations [39], and fast Fourier transform (FFT) techniques for applying the Hamiltonian operator and evaluating the electron density [40, 41].  \n\nIn figure 3 we compare the CPU times of DFT-PW vesrus DFT-LCAO calculations for different LCAO basis sets. The figure shows the CPU time for the different methods as function of the system size. The PW approach is computationally efficient for smaller systems, while the LCAO approach can be more than an order of magnitude faster for systems with more than 100 atoms.  \n\n# 4.3.  Pseudopotentials and LCAO basis sets  \n\nQuantumATK uses pseudopotentials (PPs) to avoid explicit DFT calculations of core electrons, and currently supports both scalar-relativistic and fully relativistic normconserving PPs [42]. Projector augmented-wave (PAW) potentials [43] is cur­rently available for the ATK-PlaneWave simulation engine only.  \n\nThe QuantumATK platform is shipped with built-in databases of well-tested PPs, covering all elements up to $Z=83$ (Bi), excluding lanthanides. The current default PPs are those of the published SG15 [44] and PseudoDojo [45] sets. These are two modern normconserving PP types with multiple projectors for each angular momentum, to ensure high accuracy. Both sets contain scalar-relativistic and fully relativistic PPs for each element. The fully relativistic PPs are generated by solving the Dirac equation for the atom, which naturally includes spin–orbit coupling, and then mapping the solution onto the scalar-relativistic formalism [42, 46].  \n\nTable 3.  Fundamental band gaps (in units of eV) for a range of semiconductors and simple oxides, calculated using different XC methods, and compared to experimental values. The ATK-LCAO simulation engine was used for PBE, TB09, and PBE-1/2 calculations, while the ATK-PlaneWave engine was used for simulations using HSE06. PseudoDojo PPs were used, combined with Ultra basis sets for DFTLCAO, except for TB09 calculations, which were done using FHI-DZP. Default cutoff energies were used, and a $\\mathbf{k}$ -point grid density of $7\\mathring{\\mathrm{A}}$ . For bulk silicon, this corresponds to a $15\\times15\\times15{\\bf k}$ -point grid. Experimental band gaps are from [53] unless otherwise noted. The bottom row lists the RMS deviation between theory and experiments.   \n\n\n<html><body><table><tr><td>Material</td><td>Experiment</td><td>PBE</td><td>TB09</td><td>PBE-1/2</td><td>HSE06</td></tr><tr><td>C</td><td>5.48</td><td>4.19</td><td>5.11</td><td>5.59</td><td>5.33</td></tr><tr><td>Si</td><td>1.17</td><td>0.57</td><td>1.20</td><td>1.16</td><td>1.17</td></tr><tr><td>Ge</td><td>0.74</td><td>0.00</td><td>1.11</td><td>0.81</td><td>0.55a</td></tr><tr><td>SiC</td><td>2.42</td><td>1.36</td><td>2.31</td><td>2.66</td><td>2.27</td></tr><tr><td>BP</td><td>2.40</td><td>1.24</td><td>1.79</td><td>1.63</td><td>2.01</td></tr><tr><td>BAs</td><td>1.46</td><td>1.25</td><td>1.94</td><td>1.58</td><td>2.05</td></tr><tr><td>AIN</td><td>6.13</td><td>4.16</td><td>6.97</td><td>5.83</td><td>5.54</td></tr><tr><td>AIP</td><td>2.51</td><td>1.55</td><td>2.36</td><td>2.46</td><td>2.30</td></tr><tr><td>AlAs</td><td>2.23</td><td>1.45</td><td>2.45</td><td>2.38</td><td>2.27</td></tr><tr><td>AlSb</td><td>1.68</td><td>1.22</td><td>1.82</td><td>1.92</td><td>1.76</td></tr><tr><td>GaN</td><td>3.50</td><td>1.89</td><td>4.10</td><td>3.27</td><td>2.87</td></tr><tr><td>GaP</td><td>2.35</td><td>1.59</td><td>2.38</td><td>2.22</td><td>2.26</td></tr><tr><td>GaAs</td><td>1.52</td><td>0.63</td><td>1.81</td><td>1.23</td><td>1.11</td></tr><tr><td>GaSb</td><td>0.73</td><td>0.11</td><td>0.76</td><td>0.52</td><td>0.64</td></tr><tr><td>InN</td><td>0.69</td><td>0.00</td><td>1.74</td><td>1.20</td><td>0.49</td></tr><tr><td>InP</td><td>1.42</td><td>0.69</td><td>2.17</td><td>1.30</td><td>1.26</td></tr><tr><td>InAs</td><td>0.41</td><td>0.00</td><td>1.08</td><td>0.51</td><td>0.23</td></tr><tr><td>InSb</td><td>0.23</td><td>0.00</td><td>0.49</td><td>0.32</td><td>0.27</td></tr><tr><td>TiO2</td><td>3.0b</td><td>1.91</td><td>3.11</td><td>3.00</td><td>3.37</td></tr><tr><td>SiO2</td><td>8.9c</td><td>6.07</td><td>11.31</td><td>8.16</td><td>7.83</td></tr><tr><td>ZrO2</td><td>5.5℃</td><td>3.65</td><td>4.96</td><td>5.26</td><td>5.16</td></tr><tr><td>HfO2</td><td>5.7℃</td><td>4.17</td><td>5.54</td><td>5.87</td><td>5.76</td></tr><tr><td>ZnO</td><td>3.44d</td><td>0.95</td><td>3.24</td><td>2.78</td><td>2.47</td></tr><tr><td>MgO</td><td>7.22</td><td>4.79</td><td>8.51</td><td>6.75</td><td>6.49</td></tr><tr><td>RMS error</td><td></td><td>1.34</td><td>0.71</td><td>0.33</td><td>0.43</td></tr></table></body></html>\n\na Direct band gap $\\Gamma\\rightarrow\\Gamma$ ), different in size from the $0.72\\mathrm{eV}$ reported in [67], but similar to the $0.56\\mathrm{eV}$ reported in [53], both using theoretical lattice constants rather than experimental ones. b [68]. c [69]. d [70].  \n\nFor each PP, we have generated an optimized LCAO basis set, consisting of orbitals $\\phi_{\\mathrm{nlm}}$ ,  \n\n$$\n\\phi_{\\mathrm{nlm}}(\\mathbf{r})=\\chi_{n l}(r)Y_{l m}(\\hat{\\mathbf{r}}),\n$$  \n\nwhere $Y_{l m}$ are spherical harmonics, and $\\chi_{n l}$ are radial functions with compact support, being exactly zero outside a confinement radius. The basis orbitals are obtained by solving the radial Schrödinger equation for the atom in a confinement potential [20]. For the shape of the confinement potential, we follow [47].  \n\nTo construct high-accuracy LCAO basis sets for the SG15 and PseudoDojo PPs, we have adopted a large set of pseudoatomic orbitals that are similar to the ‘tight tier $2^{\\circ}$ basis sets used in the FHI-aims package [47]. These basis sets typically have 5 orbitals per PP valence electron, and a range of $\\Dot{5}\\Dot{\\mathrm{A}}$ for all orbitals, and include angular momenta up to $l=5$ . From this large set, we have constructed three different series of reduced DFT-LCAO basis sets implemented in QuantumATK:  \n\n1.\t\u0007Ultra: generated by reducing the range of the original pseudo-atomic orbitals, requiring that the overlap of each contracted orbital with the corresponding original orbital can change by no more than $0.1\\%$ . Also denoted ‘LCAO-U’. 2.\t\u0007High: generated by reducing the number of basis orbitals in the Ultra basis set, requiring that the DFT-obtained total energy of suitably chosen test systems change by no more than 1 meV/atom. Also denoted ‘LCAO- $\\mathbf{\\nabla}\\cdot\\mathbf{H}^{\\prime}$ . 3.\t\u0007Medium: generated by further reduction of the number of orbitals in the High basis set, requiring that the subsequent change of the DFT-obtained total energies do not exceed 4 meV/atom. Also denoted ‘LCAO-M’. The number of pseudo-atomic orbitals in a Medium basis set is typically comparable to that of a double-zeta polarized (DZP) basis set.  \n\nTable 4.  Silicon and germanium equilibrium lattice constants and fundamental band gaps, both calculated using the PPS-PBE XC method, and compared to experiments at $300\\mathrm{K}$ . The SG15- High combination of PPs and LCAO basis sets was used, and a $15\\times15\\times15{\\bf k}$ -point grid. The lattice constants were determined by minimizing the first-principles stress on the primitive unit cells, using a maximum stress criterion of 0.1 GPa $(0.6~\\mathrm{meV}\\mathring{\\mathbf{A}}^{-3},$ ).   \n\n\n<html><body><table><tr><td>Material</td><td>Property</td><td>PPS-PBE</td><td>Experiment</td></tr><tr><td>Silicon</td><td>Lattice constant</td><td>5.439 A</td><td>5.431A</td></tr><tr><td></td><td>Band gap</td><td>1.14 eV</td><td>1.12eV</td></tr><tr><td>Germanium</td><td>Lattice constant</td><td>5.736A</td><td>5.658 A</td></tr><tr><td></td><td>Band gap</td><td>0.65eV</td><td>0.67eV</td></tr></table></body></html>  \n\nTo validate the PPs and basis-sets, we have used the $\\Delta$ -test [48, 50] to check the accuracy of the equation  of state for elemental, rock-salt, and perovskite solids against all-electron reference calculations, as shown in table  2. For each bulk crystal, the equation of state was calculated at fixed internal ion coordinates, and the equilibrium lattice constant and bulk modulus were computed. In table  2, the $\\Delta$ -value is defined as the root-mean-square (RMS) energy difference between the equations  of state obtained with QuantumATK and the all-electron reference, averaged over all crystals in a purely elemental benchmark set.  \n\nTable 2 suggests a general trend that the PseudoDojo PPs are slightly more accurate than the SG15 ones. Since the PseudoDojo PPs are in general also softer, requiring a lower real-space density mesh cutoff energy, these are the default PPs in QuantumATK.  \n\nTable 2 also shows that the accuracy of the DFT-LCAO calculations done with High or Ultra basis sets is rather close to that of the PW calculations. The Medium basis sets give on average a larger deviation from the PW results. However, we also find that LCAO-M provides sufficient accuracy for many applications, and it is therefore the default ATK-LCAO basis set in QuantumATK. We note that in typical applications, using Medium instead of the High (Ultra) basis sets decreases the computational cost by a factor of 2–4 (10–20), as seen in figure 3.  \n\nMore details on the construction and validation of the LCAO basis sets can be found in [19].  \n\n# 4.4.  Exchange-correlation methods  \n\nThe XC functional in (5) is the only formal approximation in KS-DFT, since the exact functional is unknown [15–17]. QuantumATK supports a large range of approximate XC functionals, including the local density approximation (LDA), generalized gradient approximations (GGAs), and meta-GGA functionals, all supplied through the Libxc library [51]. The ATK-PlaneWave engine also allows for calculations using the HSE06 screened hybrid functional [52–54]. The ATK-LCAO and ATK-PlaneWave engines both support van der Waals dispersion methods using the two-body and three-body dispersion corrections by Grimme [55]. Both DFT engines support different spin variants for each XC functional: spin-unpolarized and spin-polarized (both collinear and noncollinear). Spin-polarized noncollinear calculations may include spin– orbit interaction through the use of fully relativistic PPs.  \n\n4.4.1.  Semilocal functionals.  During the past 20 years, the semilocal (GGA) XC approximations have been widely used, owing to a good balance between accuracy and efficiency for DFT calculations. QuantumATK implements many of the popular GGAs, including the general-purpose PBE [56], the PBEsol (designed for solids) [57], and the revPBE/RPBE functionals (designed for chemistry applications) [58]. Recently, the meta-GGA SCAN functional [59] was also included in QuantumATK, often providing improved accuracy of DFT calculations as compared to PBE.  \n\n4.4.2.  Hybrid functionals.  Hybrid XC approximations mix local and/or semilocal functionals with some amount of exact exchange in order to provide higher accuracy for electronicstructure calculations [52, 60]. However, the computational cost is usually much higher than for semilocal approx­ imations. New methodological developments based on the adaptively compressed exchange operator (ACE) method [61] allow reducing the computational burden of hybrid functionals. The ACE algorithm was recently implemented in Quant­ umATK for HSE06 calculations, which gives a systematically good description of the band gap of most semiconductors and insulators, see table 3.  \n\n4.4.3.  Semiempirical methods.  Using hybrid functionals is computationally demanding for simulating large systems, often even prohibitive. QuantumATK offers a number of semiempirical XC methods that allow for computationally efficient simulations while giving rather accurate semiconductor band gaps. These include the DFT-1/2 method [62, 63], the TB09 XC potential [64], and the pseudopotential projectorshift approach of [19].  \n\nThe selfconsistent DFT-1/2 methods, including LDA1/2 and GGA-1/2, do contain empirical parameters. In QuantumATK, these parameters are chosen by fitting the calculated band gaps to measured ones for bulk crystals. Table  3 suggests that the DFT-1/2 method, as implemented in QuantumATK, allows for significantly improved band gaps at almost no extra computational cost. We note that a recent study has shown certain limitations of the DFT-1/2 method, in particular for anti-ferromagnetic transition metal oxides [65]. Furthermore, this method does not provide reliable force and stress calculations. It is also important to note that not all species in the system necessarily require the DFT-1/2 correction. In general, it is advisable to apply this correction to the anionic species only, keeping the cationic species as normal [62, 63].  \n\nThe Tran–Blaha meta-GGA XC functional (TB09) [64] introduces a parameter, $c$ , which can be calculated selfconsistently according to an empirical formula given in [64]. Table  3 includes band gaps computed using this approach. The $c$ -parameter may also be adjusted to obtain a particular band gap for a given material, and QuantumATK allows for setting different TB09 $\\boldsymbol{c}$ -parameters on different regions in the simulation cell. This may be useful for studying electronic effects at interfaces between dissimilar materials, for example in oxide-semiconductor junctions, where the appropriate (and material-dependent) $c$ -parameter may be significantly different in the oxide and in the semiconductor.  \n\nQuantumATK also offers a pseudopotential projector-shift (PPS) method, that introduces empirical shifts of the nonlocal projectors in the PPs, in spirit of the empirical PPs proposed by Zunger and co-workers [66]. The PPS method is usually combined with ordinary PBE calculations [19]. The two main advantages of this PPS-PBE approach are that (1) for each semiconductor, the projector shifts can be fitted such that the DFT-predicted fundamental band gap and lattice parameters are both fairly accurate compared to measured ones, and (2) the PPS method does yield first-principles forces and stress, and therefore can be used for geometry optimization, unlike the DFT- $1/2$ and TB09 methods. Table  4 shows that the PPS-PBE predicted equilibrium lattice parameters are only slightly overestimated, and the PPS-PBE band gaps are fairly close to experiments. We note that the PPS-PBE parameters are currently available in QuantumATK for the elements silicon and germanium only.  \n\n4.4.4. $D F T+U$ methods.  QuantumATK supports the mean-field Hubbard-U correction by Dudarev et  al [71] and Cococcioni et  al [72], denoted $\\mathrm{DFT+U}$ , $\\mathrm{LDA}+\\mathrm{U}$ , $\\mathrm{GGA}+\\mathrm{U}$ , or $\\mathrm{XC}+\\mathrm{U}$ . This method aims to include the strong on-site Coulomb interaction of localized electrons (often localized $d$ and $f$ electrons), which are not correctly described by LDA or GGA. A Hubbard-like term is added to the XC functional,  \n\n$$\nE_{U}=\\frac{1}{2}\\sum_{l}U_{l}(n_{l}-n_{l}^{2}),\n$$  \n\nwhere $n_{l}$ is the projection onto an atomic shell $l$ , and $U_{l}$ is the Hubbard $\\mathrm{~U~}$ for that shell. The energy term $E_{U}$ is zero for a fully occupied or unoccupied shell, but positive for a fractionally occupied shell. This favors localization of electrons in the shell $l$ , typically increasing the band gap of semiconductors.  \n\n# 4.5.  Boundary conditions and Poisson solvers  \n\nAs already mentioned in section  4.1, the electron density, $n(\\mathbf{r})$ in (9), and the effective potential, $V^{\\mathrm{eff}}(\\mathbf{r})$ in (3), are in QuantumATK represented on a real-space regular grid. The corresponding Hartree potential $V^{\\mathrm{H}}({\\bf r})$ is then calculated by solving the Poisson equation  on this grid with appropriate boundary conditions (BCs) imposed on the six facets of the simulation cell,  \n\n$$\n\\nabla^{2}V^{\\mathrm{H}}(\\mathbf{r})=-{\\frac{e^{2}}{4\\pi\\epsilon_{0}}}n(\\mathbf{r}),\n$$  \n\n![](images/9503d9ca903f8d1dfcea6f3f71067a65b3d0af67658b88bf9b7fb584c5120158.jpg)  \nFigure 4.  QuantumATK supports many different BCs. (a) Multipole BCs for a charged molecule in all directions, (b) 3D periodic BCs for a bulk configuration, (c) mixed Dirichlet and Neumann BCs for a slab model, (d) Dirichlet and Neumann BCs are also the natural choice for a surface configuration, (e) Dirichlet BCs at the interfaces between the semi-infinite electrodes and the central region in a device configuration. Note that periodic BCs are imposed in the directions perpendicular to the $C\\mathrm{.}$ -axis in (b)–(e).  \n\nwhere $e$ is the elementary charge, and $\\epsilon_{0}$ is the vacuum permittivity.  \n\nIn QuantumATK, one may also specify metallic or di­electric continuum regions in combination with a microscopic, atomistic structure, as demonstrated for a 2D device in figure 14 in section 14.1. This affects the solution of the Poisson equation (14) in the following way. For a metallic region denoted $\\Omega$ , the electrostatic potential is fixed to a constant potential value $(V_{0})$ within this region, i.e. the Poisson equation  is solved with the constraint  \n\nTable 5.  Classes of TB models currently supported by ATK-SE. The model types are either two-center Slater–Koster (SK) or based on environment-dependent parameters (Env). The model may be orthogonal $(H)$ or non-orthogonal $(H,S)$ . Short-ranged models include nearest-neighbour interactions only (range up to a few A˚), while the long-ranged Hückel models have a typical range of 5–10 $\\mathring\\mathbf{A}$ . As indicated in the right-hand column, not all models support calculation of total energies, forces, and stress, but are used mainly for simulating the electronic structure of materials.   \n\n\n<html><body><table><tr><td>Model</td><td>Ref.</td><td>Type</td><td>Range</td><td>E,F,0</td></tr><tr><td>Hickel</td><td>[79]</td><td>SK, (H,S)</td><td>long</td><td>no</td></tr><tr><td>Empirical TB</td><td>[18]</td><td>SK, (H)</td><td>short</td><td>no</td></tr><tr><td>DFTB</td><td>[78]</td><td>SK, (H,S)</td><td>medium</td><td>yes</td></tr><tr><td>Purdue</td><td>[80]</td><td>Env, (H)</td><td>short</td><td>no</td></tr><tr><td>NRL</td><td>[81]</td><td>Env, (H,S)</td><td>long</td><td>yes</td></tr></table></body></html>  \n\n$$\nV^{\\mathrm{H}}(\\mathbf{r})=V_{0},\\mathbf{r}\\in\\Omega.\n$$  \n\nFor a dielectric region denoted $\\Upsilon$ , the right-hand side of the Poisson equation will be modified as follows:  \n\n$$\n\\begin{array}{l l l}{{\\displaystyle\\nabla^{2}V^{\\mathrm{H}}({\\bf r})=-\\frac{e^{2}}{4\\pi\\epsilon_{0}}n({\\bf r}),{\\bf r}\\notin\\Upsilon,}}\\\\ {{\\displaystyle\\nabla^{2}V^{\\mathrm{H}}({\\bf r})=-\\frac{e^{2}}{4\\pi\\epsilon_{r}\\epsilon_{0}}n({\\bf r}),{\\bf r}\\in\\Upsilon,}}\\end{array}\n$$  \n\nwhere $\\epsilon_{r}$ is the relative dielectric constant, which can be specified as an external parameter in QuantumATK calculations.  \n\n4.5.1.  Boundary conditions.  QuantumATK implements four basic types of BCs; multipole, periodic, Dirichlet and Neumann BCs. It is also possible to impose mixed BCs on the six facets of the simulation cell to simulate a large variety of physical systems at different levels of approximation.  \n\nA multipole BC means that the Hartree potential at the boundary is determined by calculating the monopole, dipole and quadrupole moments of the charge distribution inside the simulation cell, and that these moments are used to extrapolate the value of the potential at the boundary. A Dirichlet BC means that the potential has been fixed to a certain potential $V_{0}(\\mathbf{r})$ at the boundary, such that, for a facet $S$ of the simulation cell,  \n\n$$\nV^{\\mathrm{H}}(\\mathbf{r})=V_{0}(\\mathbf{r}),\\mathbf{r}\\in S.\n$$  \n\nA Neumann BC means that the normal derivative of the potential on a facet has been fixed to a given function $V_{0}^{\\prime}(\\mathbf{r})$ ,  \n\n$$\n\\frac{{\\partial{{V}^{\\mathrm{H}}}}({\\bf{r}})}{{\\partial{\\bf{n}}}}={\\bf{n}}\\cdot\\nabla{{V}^{\\mathrm{H}}}({\\bf{r}})=V_{0}^{\\prime}({\\bf{r}}),{\\bf{r}}\\in S,\n$$  \n\nwhere $\\mathbf{n}$ denotes the normal vector of the facet. Next, we briefly describe applications of the different BCs.  \n\n•\t\u0007Multipole $B C s$ are used for molecule configurations, ensuring the correct asymptotic behavior of the Hartree potential, even for charged systems (ions or charged molecules), as shown in figure 4(a).  \n\nTable 6.  Selected potential models included in ATK-ForceField.   \n\n\n<html><body><table><tr><td>Potential model</td><td>Special properties</td><td>References</td></tr><tr><td>Stillinger-Weber (SW)</td><td>Three-body</td><td>[85]</td></tr><tr><td>Embedded atom model (EAM)</td><td>Many-body</td><td>[86]</td></tr><tr><td>Modified embedded atom model (MEAM)</td><td>Many-body Directional bonding</td><td>[87]</td></tr><tr><td>Tersoff/Brenner</td><td>Bond-order</td><td>[88,89]</td></tr><tr><td>ReaxFF</td><td>Bond-order Dynamical charges</td><td>[90]</td></tr><tr><td>COMB/COMB3</td><td>Bond-order Dynamical charges Induced dipoles</td><td>[91]</td></tr><tr><td>Core-shell</td><td>Dynamical charge fluctuations</td><td>[92]</td></tr><tr><td>Tangney-Scandolo (TS)</td><td>Induced dipoles</td><td>[93]</td></tr><tr><td>Aspherical ion model</td><td>Induced dipoles and quadrupoles Dynamical ion distortion</td><td>[94]</td></tr><tr><td>Biomolecular and valence force fields</td><td>Static bonds</td><td>[95,96]</td></tr></table></body></html>  \n\n•\t\u0007Periodic $B C s$ is the natural choice along all directions for fully periodic bulk materials, as shown in figure 4(b). Periodic BCs are also often used to model heterostructures or interfaces, as well as surfaces using a slab model. •\t\u0007Dirichlet–Neumann BCs for a slab model. In slab calcul­ ations, it can be more advantageous to impose mixed BCs, such as Neumann (fixed potential gradient) and Dirichlet (fixed potential) on the left- and right-hand side of the slab, respectively, combined with periodic BCs in the in-plane-directions, as shown in figure  4(c). These mixed BCs provide a physically sound alternative to the often-used dipole correction for slab calculations [73]. •\t\u0007Dirichlet–Neumann BCs for a surface configuration. For accurate surface simulations, the surface configuration may be used, in combination with mixed BCs: Neumann in the right-hand-side vacuum region, Dirichlet at the left electrode, and periodic BCs in the in-plane directions, see figure 4(d). In this case, one can account, e.g. for the charge transfer from the near-surface region to the semiinfinite electrode, which acts as an electron reservoir [19]. •\t\u0007Dirichlet BCs for a device configuration. Two-probe device simulations are in QuantumATK done using Dirichlet BCs at the left and right boundaries to the electrodes. Periodic BCs may then be applied in the directions perpendicular to the electron transport direction, as shown in figure 4(e). For complex devices, one may need to apply a more mixed set of BCs, as discussed in the following.  \n\n•\t\u0007General mixed BCs. QuantumATK also allows for combining Neumann, Dirichlet and periodic BCs. This can be used to, e.g. model a 2D device in a field-effect transistor setup, such as that in figure 14.  \n\nTable 7.  ATK-ForceField timings as compared to LAMMPS [97]. Absolute timings for molecular dynamics (MD) simulations, in units of microseconds per atom per MD step, using one computing core for all potentials. Potential abbreviations are defined in table 6. In addition, LJ means Lennard–Jones. More details of the benchmark systems can be found in [28].   \n\n\n<html><body><table><tr><td></td><td>LJ</td><td>Tersoff</td><td>SW</td><td>EAM</td><td>ReaxFF</td><td>COMB</td><td>TS</td></tr><tr><td>QuantumATK</td><td>3.8</td><td>6.3</td><td>5.2</td><td>3.8</td><td>180</td><td>320</td><td>360</td></tr><tr><td>LAMMPS</td><td>1.9</td><td>7.8</td><td>5.2</td><td>2.4</td><td>190</td><td>240</td><td>N/A</td></tr></table></body></html>  \n\nWe note that for systems with periodic or Neumann BCs in all directions, the Hartree potential can only be determined up to an additive constant. In this case, in order to obtain a uniquely defined solution, we require the average of the Hartree potential to be zero when solving the Poisson equation.  \n\n4.5.2.  Poisson solvers.  To handle such different BCs, the QuantumATK simulation engines use Poisson solvers based on either FFT methods or real-space finite-difference (FD) methods. The FD methods are implemented using a multigrid solver [74], a parallel conjugate-gradient-based solver [75], and the MUMPS direct solver [76]. The real-space methods also allow for specifying spatial regions with specific di­electric constants or values of the electrostatic potential, as mentioned above.  \n\nFor systems with 2D or 3D periodic BCs, and no dielectric regions or metallic gates, the Poisson equation  (14) is most efficiently solved using the FFT solvers. For a bulk configuration with 3D periodic directions, we use a 3D-FFT method, see figure 4(b). In the case of only 2 periodic directions, for example in slab models, surface configurations, and device configurations, we use a 2D-FFT method combined with a 1D finite-difference method, see figures 4(c)–(d) [77].  \n\n# 5.  Semi-empirical models  \n\nAs a computationally fast alternative to DFT, the ATK-SE engine allows for semi-empirical TB-type simulations [24]. The TB models consist of a non-selfconsistent Hamiltonian that can be extended with a selfconsistent correction for charge fluctuations and spin polarization. These corrections closely follow the density functional tight-binding (DFTB) approach [78]. The main aspects of these TB models have been described in [24] and below we give only a brief description of the models.  \n\nTable 5 summarizes the available models for the non-selfconsistent part of the SE Hamiltonian, $H_{i j}^{0}$ . Most of the models are non-orthogonal, that is, include a parametrization of the overlap matrix $S_{i j}$ . In most of the models, the Hamiltonian matrix elements depend only on two centers, parameterized in terms of Slater–Koster parameters. These models include Hückel models [79, 82], Slater–Koster orthogonal TB models [18, 83], and DFTB models [78]. The ATK-SE engine also supports models that take into account the position of atoms around the two centers. These currently include the environ­ment-dependent TB models from Purdue [80] and those from the U.S. Naval Research Laboratory [81].  \n\nIt is possible to add a selfconsistent correction to the nonselfconsistent TB models [24]. The selfconsistent correction use the change in the onsite Mulliken population of each orbital, relative to a reference system, to assign an orbitaldependent charge to each atom. The charge on the orbital is represented by a Gaussian orbital, and the width of the Gaussian, $\\sigma_{l}$ , can be related to an onsite repulsion, $U_{l},$ where $l$ is the angular momentum of the orbital. The relation is given by [24]  \n\n$$\nU_{l}=\\frac{2e^{2}}{\\sqrt{\\pi}\\sigma_{l}}.\n$$  \n\nThis onsite repulsion can be calculated from the chargedependent onsite energies [78],  \n\n$$\nU_{l}=\\frac{\\mathrm{d}\\varepsilon_{l}}{\\mathrm{d}n_{l}},\n$$  \n\nwhere $\\varepsilon_{l}$ is the orbital energy of the atom and $n_{l}$ the charge in orbital $l.$ QuantumATK comes with a database of $U_{l}$ calculated using DFT all-electron simulations of the atom. In practice, it is more reliable for each element to use a single averaged value [78],  \n\n$$\nU=\\frac{1}{N}\\sum_{l}n_{l}U_{l},\n$$  \n\nwhere the average is determined by the number of valence electrons of each orbital, nl; $N=\\textstyle\\sum_{l}n_{l}$ . The ATK-SE default is to use such a single value.  \n\nIn the ATK-SE selfconsistent loop, the Mulliken population is calculated for each orbital. Based on the change in charge relative to the reference system, a Gaussian charge is added at the orbital position. We note that in the default case, where an atom-averaged $U$ is used on each orbital, only changes in the atomic charge will have an affect. From the atom-centered charge we set up a real-space charge density from which the Hartree potential $V(\\mathbf{r})$ is calculated using the same methods as used for DFT, see section 4.5. It is added to the TB Hamiltonian through  \n\n$$\nH_{i j}=H_{i j}^{0}+\\frac{1}{2}(V(\\mathbf{r}_{i})+V(\\mathbf{r}_{j}))S_{i j},\n$$  \n\nwhere $\\mathbf{r}_{i}$ is the position of orbital $i$ .  \n\nThe ATK-SE engine also supports spin polarization through the term [84]  \n\n$$\nH_{i j}^{\\sigma}=\\pm\\frac{1}{2}S_{i j}\\left(\\mathrm{d}E_{l_{i}}+\\mathrm{d}E_{l_{j}}\\right),\n$$  \n\n![](images/7f007cc206f0d1b737f5168e103c73f60f322a7db668021010a031deba4fe708.jpg)  \nFigure 5.  Flowchart of a typical QuantumATK MD loop.  \n\nwhere the sign depends on the spin. The spin splitting of shell $l,\\mathrm{d}E_{l_{i}}$ , is calculated from the spin-dependent Mulliken populations $m_{l\\uparrow},m_{l\\downarrow}$ of each shell at the local site $\\mu_{l}$ :  \n\n$$\nd E_{l}=\\sum_{l^{\\prime}\\in\\mu_{l}}W_{l l^{\\prime}}\\left(m_{l^{\\prime}\\uparrow}-m_{l^{\\prime}\\downarrow}\\right).\n$$  \n\nThe shell-dependent spin-splitting strength $W_{l l^{\\prime}}$ is calculated from a spin-polarized atomic calculation [84], and ATK-SE provides a database with the parameters.  \n\nThe main advantage of the SE models compared to DFT methods are their computational efficiency. For large systems, the main computational cost of both DFT and TB simulations is related to diagonalization of the Hamiltonian, the speed of which depends strongly on the number of orbitals on each site and their range. This makes TB Hamiltonians very attractive for large systems, provided the SE parametrization is appropriate for the particular simulation. Furthermore, orthogonal Hamiltonians have inherent performance advantages. The Empirical and Purdue environment-dependent models are the most popular TB models for electron transport calcul­ ations. We also note that for many two-probe device systems, it is mainly the band structure and quantum confinement that determine electrical characteristics such as current-voltage curves. TB model Hamiltonians can provide good results for such simple devices. Finally, DFTB models are popular for total-energy calculations, although we find in general that the accuracy should be cross-checked against DFT.  \n\n# 6.  Empirical force fields  \n\nATK-ForceField is a state-of-the-art FF simulation engine that is fully integrated into the Python framework. This has already been described in detail in [28], and we therefore only summarize some of the main features.  \n\nTable 6 lists the empirical potential models supported by ATK-ForceField, which includes all major FF types. The simulation engine also allows for combining models, such that different FFs can be assigned to different sub-systems. The empirical potential for each sub-system, and the interactions between them, can be customized as desired, again using Python scripting. ATK-ForceField currently includes more than 300 predefined literature parameter sets, which can conveniently be invoked from the NanoLab GUI. Additionally, it is also possible to specify custom FF parameters via the Potential Editor tool in NanoLab or in a Python script, or even use builtin Python optimization modules to optimize the parameters against reference data.  \n\nTable 7 compares the computational speed of ATKForceField molecular dynamics simulations to that of the popular LAMMPS package [97]. For most of the FF potential types, the two codes have similar performance.  \n\n# 7.  Ion dynamics  \n\nOne very powerful feature of QuantumATK is that ion dynamics is executed using common modules that are not specific to the chosen simulation engine. This means that modules for calculating energy, forces, and stress may be used with any of the supported engines, including DFT, SE methods, and classical FFs. Options for ion-dynamics simulations are defined using Python scripting, which allows for easy customization, extension, and combination of different simulation methods, without loss of performance. In section  14.3 we illustrate this by combining the DFT and FF engines in a single molecular dynamics simulation. Several methods related to ion dynamics in QuantumATK have been described in detail in [28], so here we only summarize the main features.  \n\n![](images/fd214cb720845607e3c164822652d16925ee02eb18244914c45269894e8d2673.jpg)  \nFigure 6.  Free energy map of a metadynamics simulation of surface vacancy diffusion on a $\\mathrm{Cu}(111)$ surface using QuantumATK. The collective variables CV1 and CV2 refer to the $x-$ and $y$ -position of a surface atom close to the vacancy. The atom positions of the surface layer of the lattice are depicted by the white circles.  \n\n# 7.1.  Local structural optimization  \n\nThe atomic positions in molecules and clusters are optimized by minimizing the forces, while for periodic crystals, the unitcell vectors can also be included in the optimization, possibly under an external pressure that may be anisotropic. The simultaneous optimization of positions and cell vectors is based on [98], where the changes to the system are described as a combined vector of atomic and strain coordinates.  \n\nThe default method for optimization is the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) quasiNewton-type minimization algorithm [99], but QuantumATK also implements the fast inertial relaxation engine (FIRE) method [100].  \n\n# 7.2.  Global structural optimization  \n\nThe previous section considered methods for local geometry optimization, which locate the closest local minimum-energy configuration. However, often the goal is to find the globally most stable configuration, for example, the minimumenergy crystal structure. QuantumATK therefore implements a genetic algorithm for crystal structure prediction. It works by generating an initial set of random configurations and then evolving them using genetic operators, as described in [101]. An alternative approach is to perform simulated annealing using molecular dynamics [102].  \n\n# 7.3.  Reaction pathways and transition states  \n\nThe minimum-energy path (MEP) for changes to the atomic positions from one stable configuration to another may be found using the nudged elastic band (NEB) method [103]. The QuantumATK platform implements state-of-the-art NEB [104], including the climbing-image method [105]. The initial set of images are obtained from linear interpolation between the NEB end points, or by using the image-dependent pair potential (IDPP) method [106]. The IDPP method aims to avoid unphysical starting guesses, and leads in general to an initial NEB path that is closer to the (unknown) MEP. This typically reduces the number of required NEB optimization steps by a factor of 2.  \n\nIn some implementations, the projected NEB forces for each image are optimized independently. However, the L-BFGS algorithm is in that case known to behave poorly [107]. In QuantumATK, the NEB forces for each image are combined into a single vector, $\\mathbf{F}_{\\mathrm{NEB}}\\in\\mathbb{R}^{3m n}$ , where $m$ is the number of images and $n$ the number of atoms. This combined approach is more efficient when used with L-BFGS, and has been referred to as the global L-BFGS method [107].  \n\n# 7.4.  Molecular dynamics  \n\nMolecular dynamics (MD) simulations provide insights into dynamic atomic-scale processes or sample microscopic ensembles. The essential functional blocks in a typical QuantumATK MD loop are depicted in figure  5. Different thermodynamic ensembles can be simulated. The basic NVE ensemble uses the well-known velocity-Verlet algorithm [108]. Additionally, thermostats or barostats can be applied to different parts of the system to simulate NVT or NPT ensembles, using for example the chained Nosé–Hoover thermostat [109], an impulsive version of the Langevin thermostat [110], or the barostat proposed by Martyna et al in [111] for isotropic and anisotropic pressure coupling.  \n\nFigure 5 also shows that one may apply so-called pre-step hooks and post-step hooks during a QuantumATK MD simulation. These hook functions are scripted in ATK-Python, and may vastly increase flexibility with respect to specialized MD simulation techniques and custom on-the-fly analysis. This makes it easy to employ predefined or user-defined custom operations during the MD simulation. The pre-step hook is called before the force calculation, and may modify atomic positions, cell vectors, velocities, etc. This is often used to implement custom constraints on atoms or to apply a nontrivial strain to the simulation cell. The post-step hook is typically used to modify the forces and/or stress. It may, for example, be used to add external forces and stress contrib­ utions, such as a bias potential, to the regular interaction forces.  \n\nQuantumATK is shipped with a number of predefined hook functions, implementing thermal transport via reverse non-equilibrium molecular dynamics (RNEMD) [112], metadynamics, and other methods. For metadynamics, QuantumATK integrates with the PLUMED package [113], so that all methods implemented in PLUMED are available in QuantumATK as well. Figure 6 illustrates the free-energy map of surface vacancy diffusion on $\\mathrm{Cu}(111)$ using the ATKForceField engine with an EAM potential [114].  \n\n# 7.5.  Adaptive kinetic Monte Carlo  \n\nAdaptive kinetic Monte Carlo (AKMC) is an algorithm for modelling the long-timescale kinetics of solid-state mat­erials [115–117]. For a given configuration, AKMC involves 3 steps: (1) locate all kinetically relevant product states; (2) determine the saddle point between the reactant and product states; (3) select a reaction using kinetic Monte Carlo (KMC).  \n\nStep 1 is in QuantumATK performed using high-temper­ ature MD. At regular intervals, the MD simulation is stopped and a geometry optimization is performed to check if the system has left the initial basin. This procedure is repeated until all relevant reactions are found within a user-specified confidence [117, 118].  \n\nIn step 2, the saddle-point geometry for each reaction is determined by performing a NEB optimization for each reaction, and the reaction rates $k$ are determined via harmonic transition-state theory (HTST) [119],  \n\n$$\nk_{\\mathrm{HTST}}=\\frac{\\prod_{i}^{3N}\\nu_{i}^{\\mathrm{min}}}{\\prod_{i}^{3N-1}\\nu_{i}^{\\ddag}}\\exp\\left[-\\left(E^{\\ddag}-E^{\\mathrm{min}}\\right)/k_{\\mathrm{B}}T\\right],\n$$  \n\nwhere $N$ is the number of atoms, $\\nu_{i}^{\\mathrm{{min}}}$ and $\\nu_{i}^{\\ddag}$ are the positive (stable) normal-mode frequencies at the minimum and saddle points, $E^{\\mathrm{min}}$ and $E^{\\ddagger}$ the corresponding energies, $k_{\\mathrm{B}}$ is the Boltzmann constant, and $T$ the temperature. The ratio of the products of the vibrational frequencies in (25) is often called the attempt frequency or the prefactor, and can be computationally expensive to obtain. Instead of calculating the prefactor for each reaction mechanism, a user-given value may therefore be used.  \n\nFinally, in step 3, a reaction is selected using KMC, the system evolves to the corresponding product configuration, and the entire procedure is repeated. More details of the QuantumATK implementation of AKMC may be found in [117].  \n\n# 8.  Phonons  \n\nThe ground-state vibrational motion of atoms is of paramount interest in modern materials science. Within the harmonic approximation, which is valid for small thermal displacements of atoms around their equilibrium position, the vibrational frequencies of a configuration are eigenvalues of the dynamical matrix $D$ ,  \n\n$$\nD_{a,\\alpha;b,\\beta}=\\frac{1}{\\sqrt{m_{a}m_{b}}}\\frac{\\mathrm{d}F_{b,\\beta}}{\\mathrm{d}r_{a,\\alpha}},\n$$  \n\nwhere $m_{a}(m_{b})$ is the atomic mass of atom $a\\left(b\\right)$ and $d F_{b,\\beta}/\\mathrm{d}r_{a,\\alpha}$ is the force constant. Computing and diagonalizing $D$ yields the vibrational modes of the system (molecular or periodic), and is also used to obtain the phonon density of states for a periodic crystal.  \n\n# 8.1.  Calculating the dynamical matrix  \n\nQuantumATK calculates the dynamical matrix using a FD method, where each matrix element in (26) is computed by displacing atom $a$ along Cartesian direction $\\alpha$ , and then calculating the resulting forces on atom $b$ along directions $\\beta$ . This approach is sometimes referred to as the frozen-phonon or supercell method, and applies equally well to isolated (molecular) systems. The method lends itself to heavy computational parallelization over many computing cores, since all displacements may be calculated independently. Crystal symmetries are taken into account in that only symmetrically unique atoms in the unit cell are displaced, and the forces resulting from displacement of the equivalent atoms are obtained using the corresponding symmetry operations [120].  \n\n![](images/9ca8f9981516b2115473f1d2c413015978fa38948fd0a73cff390c561804ba61.jpg)  \nFigure 7.  Phonon dispersions of the three FCC metals Au, $\\mathbf{A}\\mathbf{g}$ and $\\mathrm{Cu}$ , obtained from supercell calculations using the ATK-LCAO engine and the SG15-M LCAO basis set. Supercells were generated from $9\\times9\\times9$ repetitions of the primitive cells.  \n\n# 8.2.  Wigner–Seitz method  \n\nFor crystals with small unit cells, periodic repetition of the cell is usually needed to accurately account for long-range interactions in $D$ . For larger simulation cells, including cells with defects and amorphous structures, this is not always necessary, since the cell may already include the entire interaction range. In order to recover the correct phonon dispersion across periodic boundaries, the Wigner–Seitz method can be employed. Here, a Wigner–Seitz cell is centered around the displaced atom and the forces on each atom in the simulation cell is assigned to its periodic image that is located within the Wigner–Seitz cell [121].  \n\n# 8.3.  Phonon band structure and density of states  \n\nThe phonon band structure (or phonon dispersion) consists of bands with index λ of vibrational frequencies ω = ωλq throughout the Brillouin zone (BZ) of phonon wave vectors q. The phonon density of states (phonon DOS) per unit cell, $g(\\omega)$ , is defined as  \n\n$$\ng(\\omega)=\\frac{1}{N}\\sum_{\\mathbf{q}\\lambda}\\delta(\\omega-\\omega_{\\lambda\\mathbf{q}}),\n$$  \n\nwhere $N$ is the number of $\\mathbf{q}$ -points in the sum. In practice, the phonon DOS is calculated using the tetrahedron method [122]. Additionally, quantities such as vibrational free energy, entropy, and zero-point energy can easily be calculated from the vibrational modes and energies.  \n\nFigure 7 gives an example of phonon simulations for different metals using the ATK-ForceField and ATK-LCAO engines. The ATK-LCAO supercell calculation yields accurate vibrational properties, as exemplified by the excellent agreement between the two methods. The dispersions follow the same trends, which is expected, since the three metals have the same FCC crystal symmetry. We note that the higher phonon frequencies in $\\mathrm{{Cu}}$ can be understood from the similar bond strength as in $\\operatorname{Ag}$ and Au, but significantly lower $\\mathrm{{Cu}}$ atomic mass.  \n\n![](images/fd523521142cbfe953f5d5d3e6f327ca37720a87a994dd502ecb67059a8353e9.jpg)  \nFigure 8.  (a) Illustrative $\\mathbf{k}-$ and $\\mathbf{q}$ -point selections in the Brillouin zone for the case of a two-dimensional semiconductor with two valleys $K$ and $K^{\\prime}$ ). In semiconductors, it is possible to make a clever selection of $\\mathbf{k}$ - and $\\mathbf{q}$ -points to minimize the computational load while including all relevant scattering processes. Typically, a sparse $\\mathbf{k}$ -point sampling is used for the mobility integral, while a denser $\\mathbf{q}$ -point sampling is needed to secure a correct scattering rate at each $\\mathbf{k}$ -point. (b) Fermi-surface of bulk Au. In metals, $\\mathbf{k}$ -points contributing to the neighborhood of the Fermi surface are not located in a small subset of the Brillouin zone. Therefore, $\\mathbf{k}-$ and $\\mathbf{q}$ -points are sampled in the full Brillouin zone, and $\\mathbf{q}$ -space is integrated using the tetrahedron method to minimize the sampling density. (c) Resistivity convergence with respect to the number of $\\mathbf{q}$ -points for bulk Au with either a direct or tetrahedron integration over the full Brillouin zone. Resistivities were calculated with $N_{k}\\times N_{k}\\times N_{k}\\mathbf{k}$ -points for a sequence of $N_{q}\\times N_{q}\\times N_{q}\\bullet$ q-points.  \n\n# 8.4.  Electron-phonon coupling  \n\nThe electron–phonon coupling (EPC) is an imporant quanti­ty in modern electronic-structure theory. It is, for example, used to calculate the transport coefficients in bulk crystals (see section  8.5) and inelastic scattering of electrons in two-probe devices (see section 11.8).  \n\nTo obtain the EPC, we calculate the derivative of the Hamiltonian matrix with respect to the position of atom $a$ , $\\mathbf{r}_{a}$ ,  \n\n$$\n(\\delta\\hat{H}_{\\mathbf{r}_{a}})_{i j}=\\langle\\phi_{i}|\\frac{\\partial\\hat{H}}{\\partial\\mathbf{r}_{a}}|\\phi_{j}\\rangle,\n$$  \n\nwhere $\\partial\\hat{H}/\\partial\\mathbf{r}_{a}$ is calculated using finite differences, similar to the calculation of the dynamical matrix described above. A unit cell is repeated to form a supercell (for a device configuration, only the atoms in the central region are displaced). The terms that contribute to the Hamiltonian derivative is the local and non-local PP terms. The real-space Hamiltonian matrix is expanded in electron eigenstates, $n\\mathbf{k}$ , and Fourier transformed using the phonon polarization vectors, to finally obtain the electron–phonon couplings $g$ ,  \n\n$$\ng_{\\mathbf{k}\\mathbf{k}^{\\prime}\\mathbf{q}}^{\\lambda n n^{\\prime}}=\\langle n^{\\prime}\\mathbf{k}^{\\prime}|\\delta\\hat{H}_{\\lambda\\mathbf{q}}|n\\mathbf{k}\\rangle,\n$$  \n\nwhere $\\mathbf{q}$ is the phonon momentum and $\\lambda$ the phonon branch index.  \n\nFurther details of how QuantumATK calculates the EPC are given in [123].  \n\n# 8.5. Transport coefficients  \n\nThe electron/hole mobility in a semiconductor material is an important quantity in device engineering, and also determines the conductivity of metals. Electronic transport coefficients for bulk materials, including the conductivity, Hall conductivity, and thermoelectric response, may be calculated from the Boltzmann transport equation  (BTE) as linear-response coefficients related to the application of an electric field, magn­etic field, or temperature gradient. In QuantumATK, this is done by expanding the current density $\\mathbf{j}$ to lowest order in the electric field $\\mathcal{E}$ , magnetic field $B$ , and temperature gradient $\\boldsymbol{\\nabla}T$ ,  \n\nTable 8.  Born effective charges $(Z^{*})$ and piezoelectric tensor components ( $\\stackrel{\\cdot}{\\epsilon}_{33}$ and $\\epsilon_{14}$ ) for $\\mathrm{III-V}$ wurtzite nitrides and zincblende GaAs. Reference vales for the nitrides are from [127] and from [126] for GaAs. QuantumATK calculations were performed using the DFT-LCAO engine with the LDA XC functional and a DZP basis set.   \n\n\n<html><body><table><tr><td></td><td colspan=\"2\">Z*</td><td colspan=\"2\">E33</td></tr><tr><td></td><td>Reference</td><td>QuantumATK</td><td>Reference</td><td>QuantumATK</td></tr><tr><td>AIN</td><td>-2.70</td><td>-2.67</td><td>1.46</td><td>1.65</td></tr><tr><td>GaN</td><td>-2.72</td><td>-2.75</td><td>0.73</td><td>0.86</td></tr><tr><td>InN</td><td>-3.02</td><td>-2.98</td><td>0.97</td><td>1.21</td></tr><tr><td>GaAs</td><td>-1.98</td><td>-2.07</td><td></td><td>-0.26</td></tr></table></body></html>  \n\n$$\nj_{\\alpha}=\\sigma_{\\alpha\\beta}\\mathcal{E}_{\\beta}+\\sigma_{\\alpha\\beta\\gamma}\\mathcal{E}_{\\beta}B_{\\gamma}+\\nu_{\\alpha\\beta}\\nabla_{\\beta}T,\n$$  \n\nwhere the indices label Cartesian directions and σαβ, σαβγ and $\\nu_{\\alpha\\beta}$ are the electronic conductivity, Hall conductivity, and thermoelectric response, respectively. Following [124], the band-dependent thermoelectric transport coefficients and Hall coefficients are obtained as  \n\n$$\n\\begin{array}{r l}&{\\sigma_{\\alpha\\beta}(n\\mathbf{k})=e^{2}\\tau_{n\\mathbf{k}}\\mathbf{v}_{\\alpha}(n\\mathbf{k})\\mathbf{v}_{\\beta}(n\\mathbf{k}),}\\\\ &{\\sigma_{\\alpha\\beta\\gamma}(n\\mathbf{k})=e^{3}\\tau_{n\\mathbf{k}}^{2}\\epsilon_{\\gamma u v}\\mathbf{v}_{\\alpha}(n\\mathbf{k})\\mathbf{v}_{v}(n\\mathbf{k})\\mathbf{M}_{\\beta u}^{-1}(n\\mathbf{k}),}\\\\ &{\\nu_{\\alpha\\beta}(n\\mathbf{k})=(\\varepsilon_{n\\mathbf{k}}-\\mu)e/T\\tau_{n\\mathbf{k}}\\mathbf{v}_{\\alpha}(n\\mathbf{k})\\mathbf{v}_{\\beta}(n\\mathbf{k}),}\\end{array}\n$$  \n\nwhere $\\mu$ is the chemical potential and $\\epsilon_{\\gamma u v}$ the Levi–Civita symbol. The band group velocities $\\mathbf{v}(n\\mathbf{k})$ and effective mass tensors $\\mathbf{M}(n\\mathbf{k})$ are obtained from perturbation theory. Importantly, we may in (31) include the full scattering rate $\\tau_{n\\mathbf{k}}$ , and thereby go beyond the constant scattering-rate approx­ imation used in [124]. As we will see in section 14.2, this may not only be important in order to obtain quantitatively correct results; it is also required to reproduce experimental trends in the conductivity of different materials.  \n\nThe scattering rate is given by  \n\n$$\n\\frac{1}{\\tau_{n\\mathbf{k}}}=\\sum_{n^{\\prime}\\lambda\\mathbf{q}}\\left[B_{\\mathbf{k}(\\mathbf{k}+\\mathbf{q})}^{n n^{\\prime}}P_{\\mathbf{k}(\\mathbf{k}+\\mathbf{q})\\mathbf{q}}^{\\lambda n n^{\\prime}}+B_{\\mathbf{k}(\\mathbf{k}-\\mathbf{q})}^{n n^{\\prime}}\\bar{P}_{\\mathbf{k}(\\mathbf{k}-\\mathbf{q})\\mathbf{q}}^{\\lambda n n^{\\prime}}\\right],\n$$  \n\nwhere $B$ is a temperature-dependent scattering weight,  \n\n$$\nB_{\\mathbf{k}\\mathbf{k}^{\\prime}}^{n n^{\\prime}}=\\frac{1-f_{n^{\\prime}\\mathbf{k}^{\\prime}}}{1-f_{n\\mathbf{k}}}\\left[1-\\cos(\\theta_{\\mathbf{k}\\mathbf{k}^{\\prime}})\\right],\n$$  \n\nwhere $f$ is the Fermi function, and the scattering angle is defined by  \n\n$$\n\\cos(\\theta_{\\mathbf{k}\\mathbf{k}^{\\prime}})=\\frac{\\mathbf{v}(n^{\\prime}\\mathbf{k}^{\\prime})\\cdot\\mathbf{v}(n\\mathbf{k})}{|\\mathbf{v}(n^{\\prime}\\mathbf{k}^{\\prime})||\\mathbf{v}(n\\mathbf{k})|}.\n$$  \n\nFurthermore, $P({\\bar{P}})$ are transition rates due to phonon absorption (emission). They are obtained from Fermi’s golden rule,  \n\n$$\n\\begin{array}{l}{{\\displaystyle P_{{\\bf k}{\\bf k}^{\\prime}{\\bf q}}^{\\lambda n n^{\\prime}}=\\frac{2\\pi}{\\hbar}|g_{{\\bf k}{\\bf k}^{\\prime}{\\bf q}}^{\\lambda n n^{\\prime}}|^{2}n_{\\lambda{\\bf q}}\\delta\\left(\\varepsilon_{n^{\\prime}{\\bf k}^{\\prime}}-\\varepsilon_{n{\\bf k}}-\\hbar\\omega_{\\lambda{\\bf q}}\\right),}}\\\\ {{\\displaystyle\\bar{P}_{{\\bf k}{\\bf k}^{\\prime}{\\bf q}}^{\\lambda n n^{\\prime}}=\\frac{2\\pi}{\\hbar}|g_{{\\bf k}{\\bf k}^{\\prime}-{\\bf q}}^{\\lambda n n^{\\prime}}|^{2}\\left(n_{\\lambda-{\\bf q}}+1\\right)\\delta\\left(\\varepsilon_{n^{\\prime}{\\bf k}^{\\prime}}-\\varepsilon_{n{\\bf k}}+\\hbar\\omega_{\\lambda-{\\bf q}}\\right),}}\\end{array}\n$$  \n\nwhere $n_{\\lambda\\mathbf{q}}$ is the phonon occupation operator, and $g_{\\mathbf{k}\\mathbf{k}^{\\prime}\\mathbf{q}}^{\\lambda n n^{\\prime}}$ the EPC constant from (29).  \n\nQuantumATK offers two different methods for performing the $\\mathbf{q}$ -integral in (32). In the first method, the delta functions in (35) are represented by Gaussians with a certain width, and we perform the discrete sum over q. In the second method, we realize that the integral closely resembles the numerical problem of obtaining a density of states, and use the tetrahedron method [122] for the integration. In particular for metals, we find the tetrahedron method to be most efficient. Figure 8(c) shows the convergence of the Au resistivity as the number of $\\mathbf{q}$ -points increases, using both Gaussian and tetrahedron integration. The tetrahedron calculation seems conv­ erged for $N_{q}=20$ , that is, a $20\\times20\\times20~\\mathbf{q}$ -point sampling. The result with a finite Gaussian broadening may converge fast if using a rather large broadening, but the resistivity then appears to converge to a wrong result. In general, we therefore recommend the tetrahedron integration method for calculation of metallic resistivity.  \n\nTo further improve the computational performance when calculating transport coefficients, it is possible to use the energy-dependent isotropic-scattering-rate approximation, introduced in [125]. A two-step procedure is used for the $\\mathbf{k}$ -point sampling, which significantly reduces simulation time without affecting the resulting mobilities for many materials (those that have a fairly isotropic scattering rate in momentum space). In step one, an initial $\\mathbf{k}$ -space with a low sampling density and a well-converged $\\mathbf{q}$ -point sampling are used. The initial $\\mathbf{k}$ -point grid is automatically reduced further by including only $\\mathbf{k}$ -points where the band structure has energies in a specific range around the Fermi level. This limits the simulations to the relevant range of initial states (and relevant carrier densities), which significantly increases simulation speed and reduces memory usage. Typically, the variation of the scattering rates from the different directions in momentum space will be small. Fom the obtained data, we may therefore generate an isotropic scattering rate that only depends on energy,  \n\n$$\n\\frac{1}{\\tau(E)}=\\frac{1}{n(E)}\\sum_{n\\mathbf{k}}\\frac{1}{\\tau_{n\\mathbf{k}}}\\delta(E_{n\\mathbf{k}}-E),\n$$  \n\nwhere we have integrated over bands $n$ and wave vectors $\\mathbf{k}$ , and $n(E)$ is the density of states. In the second step, we then perform a calculation on a fine $\\mathbf{k}$ -point grid, but using the energydependent isotropic scattering rate $\\tau(E)$ . Since the scattering rate often varies slowly on the Fermi surface (for metals), this is a good approximation. The second step therefore requires only an evaluation of band velocities and effective masses on the dense $\\mathbf{k}$ -point grid, while the scattering rate is reused. This two-step procedure, combined with either direct integration for semiconductors and semimetals, or tetrahedron integration for metals, makes QuantumATK an efficient platform for simulating phonon-limited mobilities of materials.  \n\nTable 9.  MAE (in units of meV) for various Fe-based $\\mathrm{L}1_{0}$ phases. Atomic structures and reference results (SIESTA and VASP) are from [129]. The QuantumATK selfconsistent and non-selfconsistent calculations were performed with a $17\\times\\times17\\times14\\mathbf{k}$ -point grid, while the band energies were sampled on a $40\\times40\\times34\\mathbf{k}$ -point grid. PseudoDojo pseudopotentials were used for both LCAO and PW calculations. The High basis set was used for LCAO.   \n\n\n<html><body><table><tr><td></td><td>SIESTA</td><td>VASP</td><td>QuantumATK</td><td>QuantumATK</td></tr><tr><td>Structure</td><td>LCAO</td><td>PW</td><td>LCAO</td><td>PW</td></tr><tr><td>FeCo</td><td>0.45</td><td>0.55</td><td>0.66</td><td>0.66</td></tr><tr><td>FeCu</td><td>0.42</td><td>0.45</td><td>0.45</td><td>0.45</td></tr><tr><td>FePd</td><td>0.20</td><td>0.13</td><td>0.12</td><td>0.15</td></tr><tr><td>FePt</td><td>2.93</td><td>2.78</td><td>2.43</td><td>2.57</td></tr><tr><td>FeAu</td><td>0.36</td><td>0.62</td><td>0.22</td><td>0.56</td></tr></table></body></html>  \n\nIn addition, it is possible to input a predefined scattering rate as a function of energy. This is relevant for adding extra scattering mechanisms, for example impurity scattering, on top of the electron–phonon scattering, or in the case where a scattering-rate expression is known analytically. One special case of the last situation is the limit of a constant relaxation time, which is the basis of the popular Boltztrap code [124]. We note that such constant-relaxation-time calculations are easily performed within the more general QuantumATK framework outlined above. Moreover, since electron velocities are calculated from perturbation theory, accuracy is not lost due to band crossings, which is the case when velocities are obtained from FD methods, as is done in [124]. In some cases, the constant relaxation time approximation can give a good first estimate of thermoelectric parameters for a rough screening of materials, but for quantitative predictions, the more accurate models of the relaxation time outlined above must be used.  \n\n# 9.  Polarization and Berry phase  \n\nElectronic polarization in materials has significant interest, for example in ferroelectrics, where the electric polarization $\\mathbf{P}$ can be controlled by application of an external electric field, or in piezoelectrics, where charge accumulates in response to an applied mechanical stress or strain [126].  \n\nIt is common to divide the polarization into ionic and electronic parts, $\\mathbf{P}=\\mathbf{P}_{\\mathrm{i}}+\\mathbf{P}_{\\mathrm{e}}$ . The ionic part can be treated as a classical electrostatic sum of point charges,  \n\n$$\n\\mathbf{P}_{\\mathrm{i}}={\\frac{|e|}{\\Omega}}\\sum_{a}Z_{a}^{\\mathrm{ion}}\\mathbf{r}_{a},\n$$  \n\nwhere $Z_{a}^{\\mathrm{ion}}$ and $\\mathbf{r}_{a}$ are the valence charge and position vector of atom $a$ , $\\Omega$ is the unit-cell volume, and the sum runs over all ions in the unit cell.  \n\nThe electronic contribution to the polarization in direction $\\alpha$ is obtained as [126]  \n\n$$\n\\mathbf{P}_{\\mathrm{e},\\alpha}=-\\frac{|e|}{\\Omega}\\frac{\\Phi_{\\alpha}}{2\\pi}\\mathbf{R}_{\\alpha},\n$$  \n\nwhere $\\ensuremath{\\mathbf{R}}_{\\alpha}$ is the lattice vector in direction $\\alpha$ , and the Berry phase $\\Phi_{\\alpha}$ is obtained as  \n\n$$\n\\Phi_{\\alpha}=\\frac{1}{N_{\\bot}}\\sum_{\\mathbf{k}_{\\bot}}\\phi_{\\alpha}(\\mathbf{k}_{\\bot}),\n$$  \n\nwhere the sum runs over $N_{\\bot}\\textbf{k}_{\\bot}$ -points in the BZ plane perpend­icular to $\\mathbf{R}_{\\alpha}.$ , and  \n\n$$\n\\phi_{\\alpha}(\\mathbf{k}_{\\perp})=2\\operatorname{Im}\\left[\\ln\\prod_{j=0}^{J-1}\\operatorname*{det}S(\\mathbf{k}_{j},\\mathbf{k}_{j+1})\\right],\n$$  \n\nwith the overlap integrals  \n\n$$\n\\begin{array}{r}{S_{n m}(\\mathbf{k}_{j},\\mathbf{k}_{j+1})=\\langle\\mathbf{u}_{\\mathbf{k}_{j}n}\\big\\vert\\mathbf{u}_{(\\mathbf{k}_{j+1})m}\\rangle,}\\end{array}\n$$  \n\nand with the $J\\textbf{k}$ -points given by $\\mathbf{k}_{j}=\\mathbf{k}_{\\perp}+\\mathbf{k}_{\\parallel,j}$ lying on a line along the ${\\mathbf{R}}_{\\alpha}$ direction.  \n\nThe polarization depends on the coordinate system chosen since it is related to the real-space charge position, and is determined by the Berry phase, which is only defined modulo $2\\pi$ . Consequently, the polarization is a periodic function and constitutes a polarization lattice itself. The polarization lattice in direction $\\alpha$ is written as  \n\n$$\n\\mathbf{P}_{\\alpha}^{(n)}=\\mathbf{P}+n\\mathbf{P}_{\\mathcal{Q},\\alpha},\n$$  \n\nwhere $n$ is an integer labeling a polarization branch, and the polarization quantum in direction α is PQ,α = |eΩ| Rα. All measurable quantities are related to changes in the polarization, which is a uniquely defined variable, provided that the different polarization values are calculated for the same branch in the polarization lattice.  \n\nQuantumATK supports calculation of the polarization itself, as well as the derived quantities piezoelectric tensor,  \n\n$$\n\\epsilon_{i\\alpha}=\\frac{\\partial{\\mathbf{P}_{\\alpha}}}{\\partial\\epsilon_{i}},\n$$  \n\nwhere Voigt notation is used for the strain component, that is, $i\\in(x x,y y,z z,y z,x z,x y)$ , and the Born effective charge tensor  \n\n$$\nZ_{a,\\alpha\\beta}^{*}=\\frac{\\partial\\mathbf{P}_{\\alpha}}{\\partial\\mathbf{r}_{a,\\beta}},\n$$  \n\nwhere the derivative is with respect to the position of atom $a$ in direction $\\beta$ .  \n\nTable 8 shows calculated values of the Born effective charges (only the negative components for each structure) and elements of the piezoelectric tensor for III–V wurtzite nitrides and zincblende GaAs. The calculated Born effective charges and piezoelectric tensor components agree well with the reference calculations.  \n\n![](images/b3f0c1f8889bc17fbc72785d3391dad73139587e1e1dc81dd3c24d998cf2f7e8.jpg)  \nFigure 9.  MAE for a $\\mathrm{Fe/MgO}$ interface, calculated using the QuantumATK implementation of the FT method. The total MAE is $1.59\\mathrm{\\meV},$ in close agreement with previous results obtained with VASP $(1.56\\mathrm{meV})$ [130]. The black circles show the atomprojected MAE for all the atoms, while the colored squares show the projection onto the Fe $d.$ -orbitals, which contribute the most to the total MAE. Positive energies correspond to perpendicular ( ) magnetization, while negative energies correspond to in-plane ( ) magnetization.  \n\n# 10.  Magnetic anisotropy energy  \n\nThe magnetic anisotropy energy (MAE) is an important quanti­ty in spintronic magnetic devices. The MAE is defined as the energy difference between two spin orientations, often referred to as in-plane $(\\parallel)$ and out-of-plane $(\\perp)$ with respect to a crystal plane of atoms, a surface, or an interface between two materials:  \n\n$$\n\\mathrm{MAE}=E_{\\parallel}-E_{\\perp}.\n$$  \n\nThe MAE can be split into two contributions: a classical dipole-dipole interaction resulting in the so-called shape anisotropy, and a quantum mechanical contribution often refered to as the magnetocrystalline anisotropy, which arises as a consequence of spin–orbit coupling (SOC). In this section we will focus on the magnetocrystalline anisotropy and refer to this as the MAE.  \n\nThere are at least three different ways of calculating the MAE: (i) Selfconsistent total-energy calculations including SOC with the noncollinear spins constrained in the in-plane and out-of-plane directions, respectively, (ii) using the force theorem (FT) to perform non-selfconsistent calculations (including SOC) of the band-energy difference induced by rotating the noncollinear spin from the in-plane to the outof-plane direction, and (iii) second-order perturbation theory (2PT) using constant values for the SOC. While it has been demonstrated that methods (i) and (ii) give very similar results [128, 129], the 2PT method can lead to significantly different results [129]. In QuantumATK we have implemented an easyto-use workflow implementing the FT method (ii). Using the FT gives the advantage over method (i) that the calculated MAE can be decomposed into contributions from individual atoms or orbitals, which may give valuable physical and chemical insight.  \n\nThe QuantumATK workflow for calculating the MAE using the FT method is the following:  \n\n1.\t\u0007Perform a selfconsistent spin-polarized calculation.   \n2.\t\u0007For each of the considered spin orientations  \n\n(a)\t \u0007Perform a non-selfconsistent calculation, in a noncollinear spin representation including SOC, using the effective potential and electron density from the polarized calculation but rotated to the specified spin direction.   \n(b)\t \u0007Calculate the band energies $\\epsilon_{n}$ and projection weights wn,p .  \n\n3.\t\u0007Calculate the total MAE as  \n\n$$\n\\mathrm{MAE}=\\sum_{n}f_{n}^{\\parallel}\\epsilon_{n}^{\\parallel}-\\sum_{n}f_{n}^{\\perp}\\epsilon_{n}^{\\perp},\n$$  \n\nwhere $f_{n}^{\\parallel}$ is the occupation factor for band $n$ (including both band and $\\mathbf{k}$ -point index) for the $\\parallel$ spin orientation and $\\epsilon_{n}^{\\parallel}$ is the corresponding band energy, and likewise for the $\\perp$ spin orientation.  \n\nThe contribution to the total MAE for a particular projection $p$ (atom or orbital projection) is  \n\n$$\n\\mathrm{MAE}_{p}=\\sum_{n}f_{n}^{\\parallel}\\epsilon_{n}^{\\parallel}w_{n,p}^{\\parallel}-\\sum_{n}f_{n}^{\\perp}\\epsilon_{n}^{\\perp}w_{n,p}^{\\perp},\n$$  \n\nwhere the projection weight is  \n\n$$\nw_{n,p}=\\langle\\psi_{n}|(\\mathbf{SP}+\\mathbf{PS})/2|\\psi_{n}\\rangle,\n$$  \n\nwith $\\left|\\psi_{n}\\right\\rangle$ being the eigenstate, $\\mathbf{s}$ the overlap matrix, and $\\mathbf{P}$ the projection matrix. $\\mathbf{P}$ is a diagonal, singular matrix with ones in the indices corresponding to the orbitals we wish to project onto and zeros elsewhere.  \n\nTable 9 shows the calculated MAE for a number of Fe-based $\\mathbf{L}1_{0}$ alloys. Atomic structures as well as reference values calculated with SIESTA and VASP using the FT method are from [129]. We first note that the calculated MAEs agree rather well among the four codes, the only exception being FeAu, where the LCAO representations give somewhat smaller values than obtained with PW expansions. In this case it seems that the LCAO basis set has insufficient accuracy, which could be related to the fact that the LCAO basis functions are generated for a scalar-relativistic PP derived from a fully relativistic pseudopotential.  \n\nFigure 9 shows the atom- and orbital-projected MAE for a $\\mathrm{Fe/MgO}$ interface. The structure is similar to the one reported in [130]. We use periodic BCs in the transverse directions. The calculated interfacial anisotropy constant $K_{1}=\\mathbf{MAE}/(2A)$ , where $A$ is the cross-sectional area, is $K_{1}=1.41~\\mathrm{\\mJ}~\\mathrm{m}^{-2}$ , in close agreement with a previous reported value [130] of $K_{1}=1.40\\mathrm{\\mJ\\m}^{-2}$ . From the atomprojected MAE (black circles) it is clear that the interface Fe atoms favor perpendicular MAE (since $\\mathbf{MAE}>0$ ), while the atoms in the center of the Fe slab contribute with much smaller values. From the orbital projections it is evident that the MAE peak at the interface is caused primarily by a transition from negative to positive MAE contributaions from the Fe $d_{x y}$ and $d_{x^{2}-y^{2}}$ orbitals, which hybridize with the nearby oxygen atom.  \n\n![](images/33e2425e0ce42d968a090aa98039c93fff60a8a1f5a23ecc6d1b2e760ea160fa.jpg)  \nFigure 10.  Illustration of the NEGF quantum transport module in QuantumATK. The left and right electrode regions (orange background) have an equilibrium electron distribution with chemical potentials $\\mu_{\\mathrm{L}}$ and $\\mu_{\\mathrm{R}}$ , related through the applied sample bias, $\\mu_{\\mathrm{R}}-\\mu_{\\mathrm{L}}=e V_{\\mathrm{bias}}$ . At $T=0\\mathrm{K}$ , the electrons with energies in the bias window, $\\mu_{\\mathrm{L}}\\leqslant\\varepsilon\\leqslant\\mu_{\\mathrm{R}}$ , give rise to a steady-state electrical current from the right to left electrode. Note that the electron transport direction is from the left to right electrode. For higher temperatures, the electrons above (below) $\\mu_{\\mathrm{L}}\\left(\\mu_{\\mathrm{L}}\\right)$ will also contribute to the current because of the corresponding broadening of the Fermi–Dirac distribution at $T>0\\mathrm{K}$ . The system is modelled selfconsistently at the DFT or TB level using the NEGF method. It is possible to include the effect of gate potentials in the selfconsistent solution. Inelastic effects due to phonon or photon scattering can be included through perturbation theory.  \n\n# 11.  Quantum transport  \n\nThe signature feature of QuantumATK is simulation of device systems. While most DFT device simulation codes are constructed on top of an electronic structure code designed for simulating bulk systems, QuantumATK is designed from scratch to achieve the highest accuracy and performance for both bulk and device systems.  \n\nFigure 10 shows a device (two-probe) geometry. It consists of a left electrode, a central region, and a right electrode. The three regions have the same BCs in the two lateral directions perpendicular to the left-right electron transport direction, as defined in figure 10. The left and right electrodes are assumed to have bulk properties, and the first step of the device simulation is to perform a bulk calculation of each electrode with periodic BCs in the transport direction. Using Bloch’s theorem, we describe the wave functions in terms of transverse $\\mathbf{k}$ -points, and to seamlessly connect the three regions, the same $\\mathbf{k}$ -point sampling is used in the transverse directions for all three regions. In the transport direction, the central-region wave functions are described by using scattering BCs, while the electrode wave functions are described by using periodic BCs. To have a seamless connection, it is important that the electrode wave functions very accurately reproduce the infinite-crystal limit in the transport direction. A very dense electrode $\\mathbf{k}$ -point grid is therefore needed in the transport direction.  \n\nThe left and right electrodes are modelled in their ground states with chemical potentials $\\mu_{\\mathrm{L}}$ and $\\mu_{\\mathrm{R}}$ , respectively. This is only a correct model if the electrodes are not affected by the contact with the central region. The central-region electrostatic potential should therefore be sufficiently screened in the regions interfacing with the electrodes (denoted ‘electrode extensions’), such that the potential in each electrode extension virtually coincides with that in the electrode. Furthermore, the approximation is not valid if the finite-bias current density is high; in this case a non-equilibrium electron occupation is needed to accurately model the electrodes. A device with no electron scattering in the central region can therefore not be modelled reliably at finite bias.  \n\nThe electronic structures of the isolated electrodes are defined with respect to an arbitrary energy reference. When used in a device simulation, they must be properly aligned to a common reference. This is achieved by applying a potential shift to the electronic structure of the right electrode, chosen to fulfill the condition  \n\n$$\n\\mu_{\\mathrm{L}}-\\mu_{\\mathrm{R}}=-e V_{\\mathrm{bias}},\n$$  \n\nwhere $V_{\\mathrm{bias}}$ is the bias applied on the electrodes. It is clear that $\\mu_{\\mathrm{R}}=\\mu_{\\mathrm{L}}$ at zero bias. The electrode electrostatic potentials, including the right-electrode potential shift, sets up the BCs for the central-region electrostatic potential. Thus, the whole system is aligned to a common reference, and device built-in potentials, if any, are properly included.  \n\nThe electrostatic potential enters the KS equation  from which the electron density in the central region is determined. We assume the system is in a steady state, that is, the centralregion electron density does not change with time. The density can then be described in terms of extended electronic states from the left and right electrodes, as well as bound states in the central region,  \n\n$$\nn(\\mathbf{r})=n_{\\mathrm{L}}(\\mathbf{r})+n_{\\mathrm{R}}(\\mathbf{r})+n_{\\mathrm{B}}(\\mathbf{r}).\n$$  \n\nWe now focus on the contribution from the extended states of the left $(n_{\\mathrm{L}})$ and right $(n_{\\mathrm{R}})$ electrodes, and delay the discussion of bound states $(n_{\\mathrm{B}})$ for later. The former may be obtained by calculating the scattering states incoming from the left $(\\psi_{\\alpha}^{\\mathrm{L}})$ and right $(\\psi_{\\alpha}^{\\mathbf{R}})$ electrodes, which can be obtained by first calculating the Bloch states in the electrodes, and subsequently solving the KS equation  for the central region using those Bloch states as matching BCs.  \n\nThe left and right electron densities can then be calculated by summing up the occupied scattering states,  \n\n$$\nn_{\\mathrm{L}}(\\mathbf{r})=\\sum_{\\alpha}|\\psi_{\\alpha}^{\\mathrm{L}}(\\mathbf{r})|^{2}f\\left({\\frac{\\varepsilon_{\\alpha}-\\mu_{\\mathrm{L}}}{k_{\\mathrm{B}}T_{\\mathrm{L}}}}\\right),\n$$  \n\n$$\nn_{\\mathrm{R}}(\\mathbf{r})=\\sum_{\\alpha}|\\psi_{\\alpha}^{\\mathrm{R}}(\\mathbf{r})|^{2}f\\left(\\frac{\\varepsilon_{\\alpha}-\\mu_{\\mathrm{R}}}{k_{\\mathrm{B}}T_{\\mathrm{R}}}\\right),\n$$  \n\nwhere $f(x)=(1+\\mathbf{e}^{x})^{-1}$ is the Fermi–Dirac distribution.  \n\n# 11.1.  NEGF method  \n\nInstead of using the scattering states to calculate the nonequilibrium electron density, QuantumATK uses the NEGF method; the two approaches are formally equivalent and give identical results [38].  \n\nThe electron density is given in terms of the electron density matrix. We split the density matrix into left and right contributions,  \n\n$$\nD=D^{\\mathrm{L}}+D^{\\mathrm{R}}.\n$$  \n\nThe left contribution is calculated using the NEGF method as [38]  \n\n$$\nD^{\\mathrm{L}}=\\int{\\rho^{\\mathrm{L}}(\\varepsilon)f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{L}}}{k_{\\mathrm{B}}T_{\\mathrm{L}}}\\right)\\mathrm{d}\\varepsilon},\n$$  \n\nwhere  \n\n$$\n\\rho^{\\mathrm{L}}(\\varepsilon)\\equiv\\frac{1}{2\\pi}G(\\varepsilon)\\Gamma^{\\mathrm{L}}(\\varepsilon)G^{\\dagger}(\\varepsilon)\n$$  \n\nis the spectral density matrix, expressed in terms of the retarded Green’s function $G$ and the broadening function $\\Gamma^{\\mathrm{L}}$ of the left electrode,  \n\n$$\n\\Gamma^{\\mathrm{L}}=\\frac{1}{\\mathrm{i}}(\\Sigma^{\\mathrm{L}}-(\\Sigma^{\\mathrm{L}})^{\\dagger}),\n$$  \n\nwhich is given by the left electrode self-energy $\\Sigma^{\\mathrm{L}}$ . Note that while there is a non-equilibrium electron distribution in the central region, the electron distribution in the left electrode is described by a Fermi–Dirac distribution $f$ with an electron temperature $T_{\\mathrm{L}}$ .  \n\nSimilar equations exist for the right density matrix contrib­ ution. The next section describes the calculation of $G$ and $\\Sigma$ in more detail.  \n\nWe note that the implemented NEGF method supports spintronic device simulations, using a noncollinear electronic spin representation, and possibly including spin–orbit coupling. This enables, for example, studies of spin-transfer torque driven device physics [131].  \n\n# 11.2.  Retarded Green’s function  \n\nThe NEGF key quantity to calculate is the retarded Green’s function matrix for the central region. It is calculated from the central-region Hamiltonian matrix $H$ and overlap matrix $S$ by adding the electrode self-energies,  \n\n$$\n\\begin{array}{r}{G(\\varepsilon)=\\left[(\\varepsilon+\\mathrm{i}\\delta_{+})S-H-\\Sigma^{\\mathrm{L}}(\\varepsilon)-\\Sigma^{\\mathrm{R}}(\\varepsilon)\\right]^{-1},}\\end{array}\n$$  \n\nwhere $\\delta_{+}$ is an infinitesimal positive number.  \n\nCalculation of $G$ at a specific energy $\\varepsilon$ requires inversion of the central-region Hamiltonian matrix. The latter is stored in a sparse format, and we only need the density matrix for the same sparsity pattern. This is done by block diagonal inversion [132], which is $\\mathcal{O}(N)$ in the number of blocks along the diagonal.  \n\nThe self-energies describe the effect of the electrode states on the electronic structure in the central region, and are calculated from the electrode Hamiltonians. QuantumATK provides a number of different methods [133–136], where our preferred algorithm use the recursion method of cite134, which in our implementation exploits the sparsity pattern of the electrode. This can greatly speed up the NEGF calculation as compared to using dense matrices.  \n\n# 11.3.  Complex contour integration  \n\nThe integral in (54) requires a dense set of energy points due to the rapid variation of the spectral density along the real axis. We therefore follow [38] and divide the integral into an equilibrium part, which can be integrated on a complex contour, and a non-equilibrium part, which needs to be integrated along the real axis, but only for energies within the bias window. We have  \n\n$$\nD=D_{\\mathrm{eq}}^{\\mathrm{L}}+\\Delta_{\\mathrm{neq}}^{\\mathrm{R}},\n$$  \n\nwhere  \n\n$$\nD_{\\mathrm{eq}}^{\\mathrm{L}}=\\int\\mathrm{d}\\varepsilon(\\rho^{\\mathrm{L}}(\\varepsilon)+\\rho^{\\mathrm{R}}(\\varepsilon)+\\rho^{\\mathrm{B}}(\\varepsilon))f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{L}}}{k_{\\mathrm{B}}T_{\\mathrm{L}}}\\right),\n$$  \n\n$$\n\\Delta_{\\mathrm{neq}}^{\\mathrm{R}}=\\int\\mathrm{d}\\varepsilon\\rho^{\\mathrm{R}}(\\varepsilon)\\left[f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{R}}}{k_{\\mathrm{B}}T_{\\mathrm{R}}}\\right)-f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{L}}}{k_{\\mathrm{B}}T_{\\mathrm{L}}}\\right)\\right],\n$$  \n\nwhere $\\rho^{\\mathbf{B}}$ is the density of states of any bound states in the central region. Equivalently, we could write the density matrix as  \n\n$$\nD=D_{\\mathrm{eq}}^{\\mathrm{R}}+\\Delta_{\\mathrm{neq}}^{\\mathrm{L}},\n$$  \n\nwhere $\\mathrm{~L~}$ and $\\mathtt{R}$ are exchanged in (59) and (60).  \n\nDue to the finite accuracy of the integration along the real axis, (58) and (61) are numerically different. We therefore use a double contour [38], where (58) and (61) are weighted such that the main fraction of the integral is obtained from the equilibrium parts, $D_{\\mathrm{eq}}^{\\mathrm{L}}$ and $D_{\\mathrm{eq}}^{\\mathrm{R}}$ , which are usually much more accurate than the non-equilibrium parts, due to the use of high-precision contour integration. We have  \n\n$$\n\\begin{array}{r}{D_{i j}=W_{i j}^{\\mathrm{L}}\\left[D_{\\mathrm{eq}}^{\\mathrm{L}}+\\Delta_{\\mathrm{neq}}^{\\mathrm{R}}\\right]_{i j}+W_{i j}^{\\mathrm{R}}\\left[D_{\\mathrm{eq}}^{\\mathrm{R}}+\\Delta_{\\mathrm{neq}}^{\\mathrm{L}}\\right]_{i j},}\\end{array}\n$$  \n\nwhere $W^{\\mathrm{L}}$ and $W^{\\mathrm{R}}$ are chosen according to [38], i.e. such that at each site, the equilibrium part of the density matrix gives the largest contribution and $\\bar{W}^{\\mathrm{L}}+W^{\\mathrm{R}}=1$ .  \n\n# 11.4.  Bound states  \n\nThe non-equlibrium integrals, $\\Delta_{\\mathrm{neq}}^{\\mathrm{L}}$ and $\\Delta_{\\mathrm{neq}}^{\\mathrm{R}}$ , do not include any density from bound states in the central region. However, the equilibrium part of the density matrix is calculated from a complex contour integral of the retarded Green’s function, and this calculation includes bound states with energies below the chemical potential of the contour.  \n\nAssume $\\mu_{\\mathrm{L}}<\\mu_{\\mathrm{R}}$ , then a bound state with energy $\\varepsilon_{\\mathrm{B}}<\\mu_{\\mathrm{L}}$ will be included in both $D_{\\mathrm{eq}}^{\\mathrm{L}}$ and $D_{\\mathrm{eq}}^{\\mathrm{R}}$ , but a bound state in the bias window, $\\mu_{\\mathrm{L}}<\\varepsilon_{\\mathrm{B}}<\\mu_{\\mathrm{R}}$ , will only be included in $D_{\\mathrm{eq}}^{\\mathrm{R}}$ . Thus, from (62) we see that the state will be included with weight 1 if $\\varepsilon_{\\mathrm{B}}<\\mu_{\\mathrm{L}}$ and only with a fractional weight if $\\mu_{\\mathrm{L}}<\\varepsilon_{\\mathrm{B}}<\\mu_{\\mathrm{R}}$ . The weight will depend on the position of the bound state along the transport direction, that is, if the bound state is in a region that is well connected with the right electrode, the occupation will follow the right electrode and thus be close to 1. If it is in a region that is not well connected with the right electrode, the occupation will follow the left electrode, and thus for the current example the occupation be close to 0.  \n\nThe true occupation of a bound state in the bias window will depend on the physical mechanism responsible for the occupation and de-occupation, for example electron–phonon scattering, defects, etc. However, the matrix element will typically be higher with the electrode that is well connected with the region around the bound state, so we believe that the use of a double contour gives a qualitatively correct description of the occupation of the bound states in the bias window. Furthermore, we find that if we do not use such weighting schemes, bound states in the bias window can cause instabilities in the selfconsistent finite-bias NEGF calculation.  \n\n# 1 .5.  Spill-in terms  \n\nGiven the density matrix $D$ , the electron density is obtained from the LCAO basis functions $\\phi$ :  \n\n$$\nn(\\mathbf{r})=\\sum_{i j}D_{i j}\\phi_{i}(\\mathbf{r})\\phi_{j}(\\mathbf{r}).\n$$  \n\nThe Green’s function of the central region gives the density matrix of the central region, $D^{\\mathrm{CC}}$ . However, to calculate the density correctly close to the central-region boundaries towards the electrodes, the terms involving $D^{\\mathrm{{\\bar{L}L}}},D^{\\mathrm{{LC}}},D^{\\mathrm{{CR}}},$ , and $D^{\\mathrm{RR}}$ are also needed. These are denoted spill-in terms [137].  \n\nQuantumATK implements an accurate scheme for including all the spill-in terms, both for the electron density and for the Hamiltonian integrals [137]. This gives additional stability and well-behaved convergence in device simulations.  \n\n# 11.6.  Device total energy and forces  \n\nA two-probe device is an open system where charge can flow in and out of the central region through the left and right electrode reservoirs. Since the two reservoirs may have different chemical potentials, and the particle number from a reservoir is not conserved, it is necessary to use a grand canonical potential to describe the energetics of the system [138],  \n\n$$\n\\Omega[n]=E_{\\mathrm{KS}}[n]-N_{\\mathrm{L}}\\mu_{\\mathrm{L}}-N_{\\mathrm{R}}\\mu_{\\mathrm{R}},\n$$  \n\nwhere $N_{\\mathrm{L/R}}$ is the number of electrons contributed to the central region from the left/right electrode, and $E_{\\mathrm{KS}}[n]$ is the KS total energy.  \n\nDue to the screening approximation, the central region will be charge neutral, and therefore $N_{\\mathrm{L}}+N_{\\mathrm{R}}=N$ , where $N$ is the ionic charge in the central region. At zero bias $(\\mu_{\\mathrm{L}}=\\mu_{\\mathrm{R}})$ , the particle term is constant, so that $N\\mu_{\\mathrm{L}}=N\\mu_{\\mathrm{R}}$ , and is thus independent of atom displacements in the central region. However, at finite bias ${\\bf\\chi}_{\\mu_{\\mathrm{L}}}\\neq{\\mu_{\\mathrm{R}}})$ , the particle terms in $\\Omega$ will affect the forces.  \n\nIf one neglects current-induced forces [139, 140], as done in QuantumATK simulations, the force acting on atom $a$ at position $\\mathbf{r}_{a}$ in the device central region is given by  \n\n$$\n\\mathbf{F}_{a}=-\\frac{\\partial\\Omega[n]}{\\partial\\mathbf{r}_{a}}.\n$$  \n\nIt can be shown that the calculation of this force is identical to the calculation of the equilibrium (zero-bias) force, but in the non-equilibrium (finite-bias) case the density and energy density matrix must be calculated within the NEGF framework [38, 138, 141].  \n\n# 11.7. Transmission coefficient and current  \n\nWhen the selfconsistent non-equilibrium density matrix has been obtained, it is possible to calculate various transport properties of the system. One of the most notable is the transmission spectrum from which the current and differential conductance are obtained. The transmission coefficient $T$ at electron energy $\\varepsilon$ is obtained from the retarded Green’s function [142],  \n\n$$\nT(\\varepsilon)=\\mathrm{Tr}\\left[G(\\varepsilon)\\Gamma^{\\mathrm{L}}(\\varepsilon)G^{\\dagger}(\\varepsilon)\\Gamma^{\\mathrm{R}}(\\varepsilon)\\right],\n$$  \n\nand the electrical current is given by the Landauer formula,  \n\n$$\nI=\\frac{2e}{h}\\int_{-\\infty}^{\\infty}\\mathrm{d}\\varepsilon T(\\varepsilon)\\left[f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{L}}}{k_{\\mathrm{B}}T_{\\mathrm{L}}}\\right)-f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{R}}}{k_{\\mathrm{B}}T_{\\mathrm{R}}}\\right)\\right].\n$$  \n\n# 11.8.  Inelastic transmission and inelastic current  \n\nQuantumATK implements the lowest-order expansion (LOE) method [143] for calculating the inelastic current due to electron–phonon scattering, which is not included in (66)  \n\nand (67). The LOE method is based on perturbation theory in the first Born approximation, and requires calculation of the dynamical matrix and the Hamiltonian derivative with respect to atomic positions in the central region, $\\nabla H(\\mathbf{r})$ . Calculation of these derivatives are described in section 8.  \n\nFirst-principles calculation of $\\nabla H(\\mathbf{r})$ can be prohibitive for large device systems. However, if the atomic configuration of the central region can be generated by repeating the left electrode along the transport direction, then $\\nabla H(\\mathbf{r})$ can be obtained to a good approximation by using the $\\nabla H(\\mathbf{r})$ of the left electrode only [144].  \n\nFrom $\\nabla H(\\mathbf{r})$ of the central region we get the electron– phonon matrix elements in reciprocal space [123],  \n\n$$\n\\begin{array}{l}{{\\displaystyle M_{\\lambda,{\\bf k},{\\bf q}}^{i j}=\\sum_{m n}\\mathrm{e}^{\\mathrm{i}{\\bf k}\\cdot({\\bf R}_{n}-{\\bf R}_{m})-\\mathrm{i}{\\bf q}\\cdot{\\bf R}_{m}}}\\ ~}\\\\ {{\\displaystyle~\\times~\\langle\\phi_{j}{\\bf R}_{m}|{\\bf v}_{\\lambda,{\\bf q}}\\cdot\\nabla H_{0}({\\bf r})|\\phi_{i}~{\\bf R}_{n}\\rangle},}\\end{array}\n$$  \n\nwhere the $(m n)$ -sum runs over repeated unit cells in the supercell calculation of the Hamiltonian derivatives [123], and the subscript 0 indicates that the derivatives are only calculated for atoms in the unit cell with index 0. Moreover, $\\left|\\phi_{i}\\mathbf{R}_{n}\\right\\rangle$ $(|\\phi_{j}\\mathbf{R}_{m}\\rangle)$ denotes the $i(j)^{:}$ ’th LCAO basis orbital in the unit cell displaced from the reference cell by the lattice vector ${\\bf R}_{n}$ $(\\mathbf{R}_{m})$ , while $\\mathbf{q}$ is the phonon momentum, and $\\mathbf{v}_{\\lambda,\\mathbf{q}}$ is the massscaled mode vector of phonon mode λ with frequency ωλ,q.  \n\nFollowing [143], we obtain the inelastic transmission functions for a finite transfer of momentum. From these we calculate the total electrical current, including inelastic effects [144, 145]. The complete formulas for the QuantumATK implementation can be found in [144].  \n\n11.8.1.  Special thermal displacement method.  In [146] we showed that the average transmission from a thermal distribution of configurations accurately describes the inelastic electron transmission spectrum due to electron–phonon scattering at this temperature. In the special thermal displacement (STD) method, the average is replaced with a single representative configuration, which may drastically reduce the computational cost of inelastic transport simulations [147].  \n\nTo obtain the STD configuration, we first calculate the phonon eigenspectrum using the dynamical matrix of the central region. We consider only $\\mathbf{q}=\\mathbf{0}$ , since only relative displacements between atoms in the cell will be important, and to account for finite $\\mathbf{q}$ -vectors we will have to increase the cell size. The phonon modes are labeled by $\\lambda$ with frequency $\\omega_{\\lambda}$ , eigenmode vector $\\mathbf{e}_{\\lambda}$ , and characteristic length $l_{\\lambda}$ .  \n\nThe STD vector of atomic displacements is given by [147]  \n\n$$\n{\\bf u}_{\\mathrm{STD}}(T)=\\sum_{\\lambda}s_{\\lambda}(-1)^{\\lambda-1}\\sigma_{\\lambda}(T){\\bf e}_{\\lambda},\n$$  \n\nwhere $s_{\\lambda}$ denotes the sign of the first non-zero element in $\\mathbf{e}_{\\lambda}$ , enforcing the same choice of ‘gauge’ for the modes. The Gaussian width $\\sigma$ is related to the mean square displacement $\\begin{array}{r}{\\langle\\mathbf{u}_{\\lambda}^{2}\\rangle=l_{\\lambda}^{2}(2n_{B}(\\frac{\\hbar\\omega_{\\lambda}}{k_{B}T})+1)=\\sigma_{\\lambda}^{2}(T)}\\end{array}$ at temperature $T$ , where $n_{\\mathrm{B}}$ is the Bose–Einstein distribution.  \n\nAn essential feature of the STD method is the use of opposite phases for phonons with similar frequencies; in this way phonon-phonon correlation functions average to zero and the transmission spectrum of the STD configuration becomes similar to a thermal average of single phonon excitations.  \n\nThe final step in the STD method is to calculate the selfconsistent Hamiltonian of the system displaced by uSTD, and use that to calculate the transmission spectrum. Thus, the computational cost of the inelastic transmission calculation is for the STD method similar to that of an ordinary elastic transmission calculation.  \n\nFormally, this method becomes accurate for systems where the central region is a large unit cell generated by the repetition of a basic unit cell.  \n\n# 11.9. Thermoelectric transport  \n\nThe thermoelectric figure  of merit, ZT, quantifies how efficiently a temperature difference (heat) can be converted into a voltage difference in a thermoelectric material,  \n\n$$\n\\mathrm{ZT}=\\frac{G\\mathrm{e}^{\\cal S^{2}T}}{\\kappa},\n$$  \n\nwhere $G_{\\mathrm{e}}$ is the electronic conductance, $S$ the Seebeck coefficient, $T$ the temperature, and $\\kappa=\\kappa_{\\mathrm{e}}+\\kappa_{\\mathrm{ph}}$ the summed electron and phonon heat transport coefficients. Following [148], and given a set of electron and phonon transmission spectra for a device configuration, QuantumATK uses linear-response theory to compute the above-mentioned thermoelectric coefficients and the Peltier coefficient, $\\Pi$ ,  \n\n$$\nG_{\\mathrm{e}}=\\left.\\frac{\\mathrm{d}I}{\\mathrm{d}V_{\\mathrm{bias}}}\\right|_{\\mathrm{d}T=0},\n$$  \n\n$$\nS=-\\left.\\frac{\\mathrm{d}V_{\\mathrm{bias}}}{\\mathrm{d}T}\\right|_{I=0},\n$$  \n\n$$\n\\kappa_{\\mathrm{e}}=\\left.\\frac{\\mathrm{d}I_{Q}}{\\mathrm{d}T}\\right|_{I=0},\n$$  \n\n$$\n\\Pi=\\left.\\frac{I_{Q}}{I}\\right|_{\\mathrm{d}T=0}=S V_{\\mathrm{bias}},\n$$  \n\nwhere $I_{Q}=\\mathrm{d}Q/\\mathrm{d}T$ is the electronic contribution to the heat current. It is calculated in a similar way as the electronic cur­ rent [149],  \n\n$$\n\\begin{array}{l}{{\\displaystyle I_{Q}=\\frac{2e}{h}\\int_{-\\infty}^{\\infty}\\mathrm{d}\\varepsilon T(\\varepsilon)\\left[\\varepsilon-\\mu\\right]}}\\\\ {{\\displaystyle\\qquad\\times\\left[f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{L}}}{k_{\\mathrm{B}}T_{\\mathrm{L}}}\\right)-f\\left(\\frac{\\varepsilon-\\mu_{\\mathrm{R}}}{k_{\\mathrm{B}}T_{\\mathrm{R}}}\\right)\\right],}}\\end{array}\n$$  \n\nwhere $\\mu=(\\mu_{\\mathrm{L}}+\\mu_{\\mathrm{R}})/2$ is the average chemical potential, and the difference to (67) is the inclusion of the factor $(\\varepsilon-\\mu)$ in the integral.  \n\nNote that one may use DFT or a TB model for obtaining the electron transmission and a force field to calculate the  \n\n![](images/923ceeaa98f350f4c56a2d09b5f06c7536e8914bb281763da5513580f95dcde1.jpg)  \nFigure 11.  Scaling performance of QuantumATK DFT simulations for a 64-atom ${\\mathrm{Si}}_{0.5}{\\mathrm{Ge}}_{0.5}$ random-alloy supercell when executed in parallel (using MPI) on 1, 2, 4, and 8 computing nodes (16 cores per node). (a) Total wall-clock times for LCAO and PW selfconsistent total-energy calculations, and b) the corresponding peak memory requirements per core. Grey lines indicate ideal scaling of the wallclock time. PseudoDojo PPs with LCAO-High basis sets were used. Note that the Ge PP contains semicore states. The supercell has 32 irreducible $\\mathbf{k}$ -points, corresponding to two computing nodes for full MPI parallelization over $\\mathbf{k}$ -points. With 4 (8) full nodes, 2 (4) MPI processes are assigned to eack $\\mathbf{k}$ -point.  \n\n![](images/c6b3970192e15f3eb14d05f59f19fbeb81546c0a8840c13ac249ba38ae49952f.jpg)  \nFigure 12.  Scaling performance of equilibrium DFT-NEGF simulations for a $10\\mathrm{nm}$ long silicon $p{-}n$ junction with doping levels of $5\\cdot10^{20}\\ \\mathrm{cm}^{-3}$ . The junction cross section is $1.3\\hat{3}\\ \\mathrm{nm}^{2}$ , corresponding to $684\\mathrm{Si}$ atoms in the device central region. The NEGF calculations were done using a PseudoDojo PP with the LCAO-Low basis set, and 2 irreducible $\\mathbf{k}$ -points in the centralregion 2D Brillouin zone, resulting in 96 generalized contour points. The simulations were run on up to eight 24-core Intel Xeon nodes, using both MPI (purple) and hybrid parallelization schemes. Hybrid parallelization was done using 2 (orange), 4 (green), and 24 (blue) threads per MPI process, with processes distributed evenly over the nodes. Gray dashed line indicates ideal scaling of the wallclock time.  \n\nphonon transmission, constituting a computaionally efficient workflow for investigating thermoelectric materials.  \n\n# 11.10.Photocurrent  \n\nQuantumATK allows for calculating photocurrent using firstorder perturbation theory within the first Born approximation [150–152]. In brief, the electron-light interaction is added to the Hamiltonian,  \n\n$$\n{\\hat{H}}={\\hat{H}}_{0}+{\\frac{e}{m_{0}}}\\mathbf{A}_{\\omega}\\cdot{\\hat{\\mathbf{p}}},\n$$  \n\nwhere $\\hat{H}_{0}$ is the Hamiltonian without the electron-light interaction, $e$ the electron charge, $m_{0}$ the free-electron mass, $\\hat{\\mathbf{p}}$  \n\nthe momentum operator, and $\\mathbf{A}_{\\omega}$ the electromagnetic vector potential from a single-mode monocromatic light source with frequency $\\omega$ .  \n\nThe first-order coupling matrix is  \n\n$$\nM_{i j}=\\frac{e}{m_{0}}\\langle i|\\mathbf{A}_{\\omega}\\cdot\\hat{\\mathbf{p}}|j\\rangle,\n$$  \n\nwhere $\\left|j\\right\\rangle$ is an LCAO basis function.  \n\nThe first Born electron-photon self-energies are  \n\n$$\n\\begin{array}{r}{\\pmb{\\Sigma}_{\\mathtt{p h}}^{\\mathtt{>}}=[N\\mathbf{M}^{\\dagger}\\mathbf{G}_{0}^{\\mathtt{>}}(\\varepsilon^{+})\\mathbf{M}+(N+1)\\mathbf{M}\\mathbf{G}_{0}^{\\mathtt{>}}(\\varepsilon^{-})\\mathbf{M}^{\\dagger}],}\\\\ {\\pmb{\\Sigma}_{\\mathtt{p h}}^{\\mathtt{<}}=[N\\mathbf{M}\\mathbf{G}_{0}^{\\mathtt{<}}(\\varepsilon^{-})\\mathbf{M}^{\\dagger}+(N+1)\\mathbf{M}^{\\dagger}\\mathbf{G}_{0}^{\\mathtt{>}}(\\varepsilon^{-})\\mathbf{M}],}\\end{array}\n$$  \n\n![](images/436d20589fb19a7204e1d1418e246b94f8f645650e8c9d4fcccc2026cfd001a4.jpg)  \nFigure 13.  Scaling performance of ATK-ForceField simulations for a $\\mathrm{SiO}_{2}$ supercell containing one million atoms, using a force field from [154]. The simulation used one MPI process per CPU core for parallelization, and was run on up to six 16-core Intel Xeon nodes. Gray dashed line indicates ideal scaling of the wall-clock time.  \n\nwhere $\\varepsilon^{\\pm}=\\varepsilon\\pm\\hbar\\omega$ , and $N$ is the number of photons. The Green’s function including electron-photon interactions to first order is then  \n\n$$\n\\mathbf{G}^{>/<}=\\mathbf{G}_{0}^{r}\\left(\\pmb{\\Sigma}_{L}^{>/<}+\\pmb{\\Sigma}_{R}^{>/<}+\\pmb{\\Sigma}_{p h}^{>/<}\\right)\\mathbf{G}_{0}^{a},\n$$  \n\nwhere ${\\bf G}_{0}^{r,>,<}$ denote the non-interacting Green’s functions, and $\\pmb{\\Sigma}_{\\mathrm{L,R}}^{>/<}$ are the lesser and greater self-energies due to coupling to the electrodes. The current in electrode $\\alpha$ (left or right) with spin $\\sigma$ is calculated as  \n\n$$\nI_{\\alpha,\\sigma}=\\frac{e}{\\hbar}\\int\\frac{\\mathrm{d}\\varepsilon}{2\\pi}\\sum_{k}T_{\\alpha}(\\varepsilon,k,\\sigma),\n$$  \n\nwhere the effective transmission coefficients are given by [152]  \n\n$$\nT_{\\alpha}(\\varepsilon,k,\\sigma)=\\mathrm{Tr}\\left\\{\\mathrm{i}\\Gamma_{\\alpha}(\\varepsilon,k)[1-f_{\\alpha}]G^{<}+f_{\\alpha}G^{>}\\right\\}_{\\sigma\\sigma}.\n$$  \n\nWe note that it is possible to include also the effect of phonons through the STD method, which is important for a good description of photocurrent in indirect-band-gap materials such as silicon [153].  \n\n# 12.  QuantumATK parallelization  \n\nAtomic-scale simulations for small configurations (systems with only a few atoms) may often be executed in serial on a single CPU core, but most production simulations require execution in parallel on several cores (often many) to increase computational speed and/or to reduce the per-core memory footprint. The QuantumATK platform offers several parallelization techniques depending on the type of computational task.  \n\n# 12.1.  Bulk DFT and semi-empirical simulations  \n\nFor bulk DFT-LCAO calculations, the basic unit of computational work to distribute in parallel is a single $\\mathbf{k}$ -point.  \n\nQuantumATK uses the message passing interface (MPI) proto­col to distribute such work units as individual computing processes on individual, or small groups of, CPU cores, and also allows for assigning multiple processes to each work unit. Moreover, each MPI process may be further distributed in a hybrid parallelization scheme by employing shared-memory threading of each process.  \n\nFigure 11 shows an example of how the total wall-clock time and peak memory requirement for DFT-LCAO and DFT-PW calculations scale with the number of 16-core computing nodes used with MPI parallelization. We considered a 64 atom SiGe random-alloy supercell with $N_{\\mathrm{k}}=32\\mathrm{\\bfk}$ -points. In this case, 2 full nodes, $N_{\\mathrm{n}}=2$ , with 32 cores in total ${N_{\\mathrm{c}}}={N_{\\mathrm{n}}}\\times16=32)$ , yields full MPI parallelization over k-points. The PW calculations were done using a blocked generalized Davidson algorithm [39, 155] to iteratively diagonalize the Hamiltonian matrix, which in the QuantumATK implementation parallelizes the computational work over both $\\mathbf{k}$ -points and plane waves. The LCAO calculations use the LAPACK [156] (when $N_{\\mathrm{c}}/N_{\\mathrm{k}}\\leqslant1)$ or ELPA [157] (when $N_{\\mathrm{c}}/N_{\\mathrm{k}}>1)$ libraries to distribute Hamiltonian diagonalization over MPI processes. It is clear from figure  11 that the LCAO engine is both fast and requires less memory than the PW representation for the 64 atom supercell, although communication overhead causes the LCAO computational speed to start breaking off from ideal scaling when the number of processes (cores) exceeds the number of $\\mathbf{k}$ -points in the DFT calculation (when $N_{\\mathrm{c}}/N_{\\mathrm{k}}>1\\AA,$ . On the contrary, MPI parallelization over both $\\mathbf{k}$ -points and plane waves enables approximately ideal scaling of the PW wall-clock time up to at least 8 nodes (128 cores), corresponding to $4~\\mathrm{MPI}$ processes per $\\mathbf{k}$ -point.  \n\n# 12.2.  DFT-NEGF device simulations  \n\nAs discussed in section 11.3, the NEGF equilibrium density matrix at a single $\\mathbf{k}$ -point is obtained from integrating the spectral density matrix over $M_{\\varepsilon}$ energy points on a complex contour. This integral must be performed at all transverse $\\mathbf{k}$ -points in the 2D Brillouin zone of the device central region, yielding $N_{k}\\times M_{\\varepsilon}$ generalized contour points. Each of these constitute a unit of computational work in equilibrium NEGF calculations, equivalent to $\\mathbf{k}$ -point parallelization in DFT calcul­ations for periodic bulks.  \n\nSince we typically have $M_{\\varepsilon}=48$ contour energies, an equilibrium NEGF simulation may easily require evaluation of hundreds of generalized contour points. MPI parallelization over contour points is therefore a highly efficient strategy. For devices with relatively large transverse cross sections, and therefore relatively few contour points (because of small $N_{k})$ , assignment of several processes to each contour point enables scaling of NEGF computational speed to numbers of computing cores well beyond the number of contour points. This can also be combined with more than one thread per process in a hybrid parallelization scheme, for a smaller speedup, but with a reduced per-core memory footprint.  \n\nFigure 12 shows an example of how the total wall-clock time and peak memory usage for a DFT-NEGF calculation scale with the number of computing nodes used with both MPI and hybrid parallelization schemes. Calculations for this $10\\mathrm{nm}$ long silicon $p$ -n junction require evaluation of 96 generalized contour points, in this case corresponding to 4 nodes for full MPI distribution of computational work. As expected, we find that using only MPI parallelization requires most memory per core, but also results in the smallest wall-clock time for the NEGF calculation, although communication overhead causes a deviation from ideal scaling for more than 1 node, see figure 12(a). We also note that the per-core memory consumption is in this case almost constant in figure 12(b), except for a modest decrease for 8 nodes, where 2 processes (cores) are assigned to each contour point. It is furthermore clear from figure 12 that hybrid parallelization enables significant memory reduction, although at the cost of decreased computational speed. Taking simulation on 4 nodes as an example, hybrid parallelization with 4 threads per process (green lines) requires in this case $50\\%$ more wall-clock time as compared to the MPI-only simulation (purple lines), but at a $70\\%$ smaller memory footprint.  \n\nAlthough NEGF computational efficiency and memory consumption depend significantly on the device length and transverse dimensions, the general trend is that MPI parallelization over contour points yields computational speedup, while threading of processes reduce the NEGF memory footprint at a comparatively smaller computational speedup.  \n\n# 12.3.  FF simulations  \n\nThe ATK-ForceField engine uses shared-memory threading for parallelization of relatively small systems, while additional parallelization by domain decomposition over MPI processes is available for large systems. As explained in detail in [28], the MPI distribution of ATK-ForceField workload is implemented via functionality from the Tremolo-X MD package [158], which is developed by the Fraunhofer Institute for Algorithms and Scientific Calculations (SCAI).  \n\nIn figure  13, we show the wall-clock time per MD step for a simulation of $\\mathrm{SiO}_{2}$ with 1 million atoms, using a force field from Pedone et al [154] This illustrates how the use of domain decomposition over MPI processes results in a significant speedup when parallelizing over a large number of nodes and cores.  \n\n# 13.  NanoLab simulation environment  \n\n# 13.1.  Python scripting  \n\nThe QuantumATK software is programmed in the $^{C++}$ and Python languages. Around $80\\%$ of the code lines are in Python, and only low-level numerically demanding parts are written in $^{C++}$ . The use of Python allows for using a large number of high-level physics and mathematics libraries, and this has greatly helped building the rich functionality of QuantumATK in a relatively short time.  \n\nThe user input file is a Python script and the user has through the script access to the same functionality as a  \n\nQuantumATK developer. This enables the user to transform input files into advanced simulation scripts, which do not only set up advanced workflows and analysis, but may also alter the functionality of the simulation engines, for example by adding new total-energy terms. QuantumATK supplies a public application programming interface (API) with currently more than 350 classes and functions. These all take a number of arguments with detailed checks of the input parameters to ensure correct usage. For example, if the input argument is a physical quantity, the physical units must be supplied. A wide range of units are supported, e.g.for energy, the user may select units of joule, calorie, electron volt, kilojoule per mole, kilocalories per mole, Hartree, or Rydberg. All physical units are automatically converted to the internal units used by QuantumATK. The user also has access to internal quantities such as the Hamiltonian, Green’s function, self-energies, etc through the API.  \n\nThrough Python scripting it is possible to build advanced workflows that automate complex simulations and analysis. However, some simulations may require a large number of time consuming calculation tasks that are combined into a final result, and scripting such workflows can be impractical. For instance, if the computer crashes during a loop in the script, how to restart the script at the right step in a loop in the middle of the script? Or perhaps some additional tasks are needed after a custom simulation has finished; how to combine the already calculated data with the new data?  \n\nTo simplify such simulations, QuantumATK has introduced a framework called a study object. The study object keeps track of complex simulations that rely on execution and combination of a number of basic tasks. It allows for running the basic tasks in parallel and will be able to resume if the calcul­ation is terminated before completion. A study object also allows for subsequently extending the number of tasks, and will only perform tasks that have not already completed. This framework is currently used for a number of complex simulations, for instance for coupling atomic-scale simulations with continuum-level TCAD tools. Examples include simulation of the formation energy and diffusion paths of charged point defects, scans over source-drain and gate bias for two-terminal devices, relaxation of devices, and calcul­ ation of the dynamical matrix and Hamiltonian derivatives by finite differences.  \n\nTo store data we use the cross-platform HDF5 binary format [159], which allows for writing and reading data in parallel to/from a single file. This file can also hold many different objects, so the entire output from a QuantumATK simulation can be stored efficently in a single file.  \n\n# 13.2.  NanoLab graphical user interface  \n\nWhile scripting is very efficient for production runs, it requires knowledge of the scripting language, and it takes time to manually build up scripts for setting up the configuration, simulation, and analysis of interest. The NanoLab GUI eliminates this barrier to productivity by enabling the user to fully set up the Python input script in a professional  \n\n![](images/087b48d3bc3e90fa55245a80928ed58daf4487ac09e167c3f9b9496598d0f272.jpg)  \nFigure 14.  Structure of the $\\mathrm{M_{D}/M o T e_{2}/S n S_{2}/M_{S}}$ device. Mo, Te, Sn and S atoms are shown in cyan, orange, dark green and yellow, respectively. The atoms of the $\\mathbf{M}_{\\mathrm{D}}$ (Au, Al) and $\\mathbf{M}_{\\mathrm{S}}$ (Au) regions are shown in pink and yellow, respectively. The metallic gate regions (top and bottom gates) are shown as light grey rectangles. The dielectric regions are shown as dark purple ( $\\epsilon=6$ ) or light purple ${\\bf\\epsilon}(\\epsilon=25\\$ ) rectangles. The dashed green lines highlight the boundaries of the different device regions indicated in figures 16(a) and (b). Note that the region of $40.2\\mathrm{nm}$ is the 2D device central region without the left and right electrode extensions included, as defined in section 11. A vertical black solid line highlights the boundary between that region and the left (right) electrode extension. The semi-infinite, periodic left (right) electrode is visualized with the corresponding unit cell structure of $\\mathrm{MoTe}_{2}$ $(\\mathrm{{SnS}}_{2})$ , which is highlighted with a dark grey-shaded rectangle adjacent to the left (right) electrode extension region. The Dirichlet BC is imposed on the left (right) boundary plane between the left (right) electrode and its extension. The top (bottom) horizontal black solid line highlights the top (bottom) boundary of the device simulation box. Mixed BCs are imposed on the corresponding boundary planes: Dirichet BCs on the metal gate surfaces, and Neumann BCs on the boundary planes in the vacuum regions (white rectangles). A periodic BC is applied in the lateral direction, which is perpendicular to the transport direction and the $\\mathbf{MoTe}_{2}$ $\\mathrm{\\overline{{SnS}}}_{2}$ ) sheet.  \n\nGUI environ­ment. NanoLab is itself programmed in Python, and each tool in NanoLab can interpret and generate Python scripts, thus, it is possible to seamlessly shift from using the GUI tools in NanoLab to manually editing the Python scripts. It is the ambition that all NanoLab functions are also available as Python commands, such that any GUI workflow can be documented and reproduced in a Python script.  \n\nNanoLab is developed around a plugin concept, which makes it easy to extend it and add new functionality. Plugins can be downloaded and installed from an add-on server, and the majority of the plugins are available as source code, making it easy to modify or extend them with new userdefined functionality.  \n\nNanoLab also provides GUI tools for communicating with online databases (‘Databases’), setting up the atomic-scale geometry of configurations (‘Builder’), writing the Python script (‘Scripter’), submitting the script to a remote or local computing unit (‘Job Manager’), and visualizing and analyzing the results (‘Viewer’). It is possible to connect thirdparty simulation cods with NanoLab by writing plugins that translate the input/output files into the internal NanoLab format. Such plugins are currently available for the VASP [22], Quantum ESPRESSO [23], ORCA [160], GPAW [161], and CASTEP [162] codes.  \n\nThe plugin concept also allows for many specialized functions, for example specialized Builder tools like surface builders, interface builders [163], NEB setups [106], etc. The Job Manager has plugins that provide support for a wide range of job schedulers on remote computing clusters. Moreover, NanoLab has a large selection of graphical analysis tools, which can be used to visualize and analyze simulations with respect to a wide range of properties, all implemented as plugins. For instance, with the ‘MD analyser’ plugin, a MD trajectory can be analyzed with respect to angular and radial distribution functions, or different spatial and time correlation functions. Other examples are interactive band structure analysis with extraction of effective masses, and analysis of transmission in device simulations with on-the-fly inspection of transmission eigenstates at specified points in the transmission spectrum. NanoLab currently ships with more than 100 preinstalled plugins, and additional plugins are available through the add-on server.  \n\n![](images/76c38b7b2e492220f6704d54718019f2427f0e039c8097f3a58568ef69ef440d.jpg)  \nFigure 15.  (a) and (c) $I_{\\mathrm{DS}^{-}}V_{\\mathrm{GS}}$ transconductance curves calculated for the symmetrically gated $\\mathrm{Au/MoTe_{2}/S n S_{2}/A u}$ device at drainsource biases of $-0.2\\mathrm{V}$ (purple circles, solid line) and $-0.4\\mathrm{V}$ (purple circles, dashed line), and for the asymmetrically gated Al/ $\\mathrm{MoTe}_{2}/\\mathrm{SnS}_{2}/\\mathrm{Au}$ device at drain-source biases of $-0.2\\mathrm{V}$ (green squares circles, solid line) and $-0.4\\mathrm{V}$ (green circles, dashed line).  \n\n# 13.3.  Documentation  \n\nKeeping an updated documentation system for the large set of QuantumATK classes and functions pose a challenge. To synchronize the documentation with the source code, we have developed an automated documentation system where the information for the QuantumATK reference manual is extracted directly from the Python source code using the Sphinx documentation generator [164]. The reference manual is available from an online platform [165] together with tutorials, whitepapers, webinars, etc. Through a search engine it is thus easy to find all available information for a given problem.  \n\n![](images/8892b455169cf84f5fb16d22960a83b3815bb74a1830e76a53872a06080c5289.jpg)  \nFigure 16.  (a) Cut-planes of the Hartree difference potential, $\\Delta V^{\\mathrm{H}}$ , along the transport direction of the symmetrically contacted $\\mathrm{Au/MoTe}_{2}/$ $\\mathrm{SnS}_{2}/\\mathrm{Au}$ device. The potential is plotted in the range $-0.2\\ \\mathrm{eV}\\leqslant\\Delta V^{\\mathrm{H}}\\leqslant0.2\\ \\mathrm{eV}.$ , with equipotential lines shown at every $0.025\\mathrm{meV}.$ Regions of negative, zero, and positive potential are shown in blue, white, and red, respectively. The capital letters indicate the sections of the device corresponding to the drain (D) and source (S) electrodes, the overlap region (O), and the exposed region (E). (c) and (e) Projected local density of states along the transport direction for the SC device at $V_{\\mathrm{DS}}=-0.2\\:\\mathrm{V}$ and $V_{\\mathrm{GS}}=0.0~\\mathrm{V}$ (c) and at $V_{\\mathrm{GS}}=0.6\\:\\mathrm{V}$ (e). The red solid lines indicate the position of the left $(\\mu_{\\mathrm{D}})$ and right $(\\mu\\mathrm{{s})}$ chemical potentials. The green dashed lines mark the boundaries of the different device regions. (b),(d) and (f) Same as (a),(c) and (e), but for the asymmetrically contacted $\\mathrm{Al/MoTe_{2}/S n S_{2}/A u}$ device.  \n\n# 14.  QuantumATK applications  \n\n# 14.1.  Large-scale simulations of 2D field-effect transistors  \n\nAs already described in section 11, the combination of DFTLCAO with the NEGF method makes it possible to use QuantumATK to simulate the electronic structure and electrical characteristics of devices at the atomistic level. Fieldeffect transistor (FET) device configurations [166, 167] are simulated by including dielectric regions and electrostatic gates, see section 4.5.  \n\nHere, we show how this framework can be used to study the electrical characteristics of a tunnel FET (TFET) device, where the channel is formed by a heterojunction based on two-dimensional semiconductors [168, 169]. We demonstrate how the characteristics of the device can be tuned by using an asymmetric contact scheme. The latter is similar to that proposed for graphene-based photodetectors [170], where two different metals are used to contact the graphene channel.  \n\nFigure 14 shows the 2D-TFET device considered here. The device comprises a semiconducting channel formed by a $\\mathrm{MoTe}_{2}/\\mathrm{SnS}_{2}$ heterojunction [171]. We consider two different contact schemes by including atomistic metallic contacts: In the symmetrically contacted (SC) $\\mathrm{M_{D}/M o T e_{2}/S n S_{2}/M_{S}}$ device, Au is used for both the source $(\\mathbf{M}_{\\mathrm{S}})$ and drain $\\mathbf{\\Gamma}(\\mathbf{M}_{\\mathrm{D}})$ metallic contacts, whereas in the asymmetrically contacted (ASC) device, we set $\\mathbf{M}_{\\mathrm{D}}=\\mathbf{A}\\mathbf{l}$ and $\\mathbf{M}_{\\mathrm{S}}=\\mathbf{A}\\mathbf{u}$ , in order to have a rather large work function difference $(\\Delta\\Phi)$ between $\\mathbf{M}_{\\mathrm{D}}$ and $\\mathbf{M}_{\\mathrm{S}}$ [172]. In both devices, the metallic contacts to $\\mathbf{MoTe}_{2}$ and $\\mathrm{SnS}_{2}$ are represented by $\\langle110\\rangle$ -oriented 4-layer slabs.  \n\nThe device configurations were constructed from the optim­ized structures of the $\\mathrm{Au}(110)/\\mathrm{MoTe}_{2}$ and $\\mathrm{Au}(110)/\\mathrm{SnS}_{2}$ electrodes, and the interlayer distance in the overlap region was set to $3.1\\mathrm{~\\AA~}$ . Following [171], the devices were encapsulated in a high- $\\cdot\\kappa$ dielectric region $\\mathrm{(HfO}_{2}$ , $\\kappa=25.0\\AA$ ), and a thin low- $\\boldsymbol{\\kappa}$ dielectric region (h-BN, $\\kappa=6.0$ ) was placed above the ‘exposed’ $\\mathbf{MoTe}_{2}$ region that is not contacted or forms part of the overlap region, hereafter denoted $\\mathrm{E}(\\mathrm{MoTe}_{2})$ . Electrostatic top and bottom gates were defined outside the high- $\\boldsymbol{\\kappa}$ di­electric region, covering the overlap and half of the $\\mathrm{E}(\\mathrm{MoTe}_{2})$ and $\\mathrm{E}(\\mathrm{{SnS}}_{2})$ regions. The ASC device was constructed by replacing the Au atoms in the left electrode with Al atoms, with no further structural optimization3. Additional computational details are given in appendix.  \n\n![](images/3a61587de0619bd164461ef5496ef91482d73a7169761622a1429e0eca0b1504.jpg)  \nFigure 17.  Temperature-dependent phonon-limited resistivity of the three metals Au, $\\mathrm{Ag}$ and $\\mathrm{Cu}$ evaluated from first-principles simulations using the ATK-LCAO engine.  \n\nTo study the impact of the contact asymmetry on the device characteristics, the reverse-bias $I_{\\mathrm{DS}^{-}}V_{\\mathrm{GS}}$ curves (the transconductance) were simulated for both devices and for two values of the drain-source voltage, $V_{\\mathrm{DS}}=-0.2\\mathrm{~V~}$ and $V_{\\mathrm{DS}}=-0.4~\\mathrm{V},$ by grounding the top gate and by sweeping the bottom gate. The same physical picture emerges for both values of $V_{\\mathrm{DS}}$ , and we discuss here only the results obtained for $V_{\\mathrm{DS}}=-0.4~\\mathrm{V}.$ The $I_{\\mathrm{DS}}-V_{\\mathrm{GS}}$ curves in figure  15 show that the drain-source current is higher in the SC device than in the ASC device across the entire range of gate-source volt­ ages. However, in the SC device, $I_{\\mathrm{DS}}$ increases only by a factor of ${\\sim}10$ , from $I_{\\mathrm{DS}}(V_{\\mathrm{GS}}=0.05~\\mathrm{V})=4.28\\times10^{-8}~\\mathrm{A}~\\mathrm{cm}^{-1}$ to $I_{\\mathrm{DS}}(V_{\\mathrm{GS}}=0.6~\\mathrm{V})=1.29\\times10^{-6}~\\mathrm{A}~\\mathrm{cm}^{-1}$ . Conversely, in the ASC device, $I_{\\mathrm{DS}}$ increases by about six orders of magnitude in the same $V_{\\mathrm{GS}}$ range, from $I_{\\mathrm{DS}}(V_{\\mathrm{GS}}=0.05~\\mathrm{V})=4.09\\times$ $10^{-15}\\ \\mathrm{A\\cm^{-1}}$ to $I_{\\mathrm{DS}}(V_{\\mathrm{GS}}=0.6~\\mathrm{V})=1.74\\times10^{-9}~\\mathrm{A}~\\mathrm{cm}^{-1}$ .  \n\nUnderstanding these trends requires considering that the asymmetric contact scheme has a two-fold effect on the electronic structure of the device. On the one hand, the use of two metals with different work functions leads to an additional built-in electric field in the channel region, when the chemical potentials of the drain and source electrodes, $\\mu_{\\mathrm{D}}$ and $\\mu_{\\mathrm{{S}}}$ , are aligned on a common energy scale. On the other hand, the interaction between the metallic contact and $\\mathbf{MoTe}_{2}$ is expected to depend also on the chemical nature of the metal.  \n\nThe presence of an additional built-in electric field, and its effect on the device electrostatics, are evident by comparing the Hartree difference potential $(\\Delta V^{\\mathrm{H}})$ in the two devices at $V_{\\mathrm{GS}}=0~\\mathrm{V}$ along the channel, as shown in figures 16(a) and (b). While in the SC device the potential changes smoothly along the channel region, a sudden increase in the potential is observed in the ASC device around the $\\mathrm{E}\\left(\\mathrm{MoTe}_{2}\\right)$ region. Here, the potential lines run parallel to the transport direction, indicating the presence of a left-pointing local electric field. The sign of this field is consistent with that generated by an asymmetric contact scheme with $\\Phi^{M_{\\mathrm{S}}}>\\Phi^{M_{\\mathrm{D}}}$ , that is, the same as that of the ASC device.  \n\nTable 10.  First-principles phonon-limited resistivities at $300\\mathrm{K}$ (in units of $\\mathbf{n}\\Omega\\cdot\\mathbf{m}^{\\cdot}$ ), compared with experimental values from [173]. Au nanowire results from [146].   \n\n\n<html><body><table><tr><td></td><td>DFT</td><td>Experiment</td></tr><tr><td>Au, Bulk</td><td>15.9</td><td>20.5</td></tr><tr><td>Au, NW (d ≈ 1 nm)</td><td>56.0</td><td>一</td></tr><tr><td>Ag,Bulk</td><td>4.9</td><td>14.7</td></tr><tr><td>Ag, NW (d ≈ 1 nm)</td><td>28.7</td><td>一</td></tr><tr><td>Cu, Bulk</td><td>14.2</td><td>15.4</td></tr><tr><td>Cu, NW (d ≈ 1 nm)</td><td>98.3</td><td>一</td></tr></table></body></html>  \n\nThe projected local density of states (PLDOS) along the devices reveal that the different electrostatics also affect their electronic structure. For both contact schemes, the DOS within the bias window, $[\\mu_{\\mathrm{D}}-\\mu_{\\mathrm{S}}]\\pm k_{\\mathrm{B}}T=\\Delta\\mu\\pm k_{\\mathrm{B}}T$ , is strongly inhomogeneous along the channel, as the conduction bands (CBs) of $\\mathbf{MoTe}_{2}$ and $\\mathrm{SnS}_{2}$ are pinned to $\\mu_{\\mathrm{D}}$ and $\\mu_{\\mathrm{{S}}}$ , respectively (see figures 16(c) and (d)). This results in a vanishing DOS in the $\\mathrm{E}(\\mathrm{MoTe}_{2})$ region within the bias window. Here, the DOS is even smaller in the ASC device, due to (i) the weaker pinning of the CBs to $\\mu_{\\mathrm{D}}$ , and (ii) the effect of the local electric field, which bends and depletes even more the CBs, moving them further away from the bias window. In the SC device, the field is much weaker, and the CBs are bent only in the proximity of the overlap region.  \n\nThe transconductance behavior can be understood from the combined analysis of $\\Delta V^{\\mathrm{H}}$ and of the PLDOS. The DOS within $\\Delta\\mu\\pm k_{\\mathrm{{B}}}T$ in the $\\mathrm{E}(\\mathrm{MoTe}_{2})$ region, described in terms of an effective barrier $\\phi_{\\mathrm{MoTe}_{2}}$ , ultimately determines the reverse-bias current in the channel. In the SC device, $\\phi_{\\mathrm{MoTe}_{2}}$ is lower for the case $V_{\\mathrm{GS}}=0\\:\\mathrm{V},$ and depends only weakly on $V_{\\mathrm{GS}},$ , as shown in figure 16(e). This results in a higher absolute value of $I_{\\mathrm{DS}}$ , and in a lower variation of $I_{\\mathrm{DS}}$ with $V_{\\mathrm{GS}}$ . Conversely, in the ASC device, $\\phi_{\\mathrm{MoTe_{2}}}$ is higher at comparable values of $V_{\\mathrm{GS}}$ , and varies appreciably when $V_{\\mathrm{GS}}$ is increased, see figure 16(f). This explains the lower values of the drainsource current, and its higher variation with the gate-source voltage. These trends are consistent with those of the transconductance curves shown in figure 15.  \n\nIn summary, DFT-NEGF simulations for $\\mathbf{M}_{\\mathrm{D}}/\\mathbf{M}\\mathrm{oTe}_{2}/$ $\\mathrm{SnS}_{2}/\\mathrm{M}_{\\mathrm{S}}$ ultra-scaled 2D-TFET devices show that the transconductance can be engineered by an appropriate choice of the metallic electrodes, and highlight the importance of atomistic device simulations for optimization of the electrical characteristics of devices based on non-conventional semiconductors.  \n\n# 14.2.  Phonon-limited mobility of metals  \n\nThe continued downscaling of nanoelectronics makes the metal interconnects an increasingly critical part of transistor designs [174]. Present-day transistors use $\\mathrm{Cu}$ as an interconnect material, and a good understanding of the origin of resistance increase with downscaling of interconnects will be important for the design and performance of future nanoscale devices.  \n\nWe here present first-principles calculations of the phononlimited resistivity of three FCC metals; Cu, Ag, and Au. We solve the Boltzmann transport equation for the mobility, using first-principles EPC constants, as described in section  8.5. Such DFT calculation of the resistivity of metals is computationally demanding, as one needs to integrate the EPC over both electron and phonon wave vectors $\\mathbf{k}-\\mathbf{\\lambda}$ and $\\mathbf{q}$ -space), and we know of only few studies of the EPC in metals that includes a full integration [146, 175, 176]. We here show that the tetrahedron integration method enables computationally efficient mobility calculations. The method may therefore be used for computational screening of materials, and first-principles simulations become accessible for identifying promising replacement materials for future interconnects.  \n\nTo calculate the scattering rate related to EPC, the phonon modes and derivatives of the Hamiltonian with displacements are needed. The supercell method for calculation of phonons and EPC from first principles was described in section 8, and figure  7 showed the phonon band structures of Cu, Ag and Au, calculated using the ATK-LCAO simulation engine. For the integration of the scattering rate in (32) we use a sampling of $20\\times20\\times20\\textbf{q}$ -points and tetrahedron integration. In addition, we apply the two-step procedure, where a k-space isotropic but energy-dependent scattering rate is used to efficiently evaluate the resistivity.  \n\nFigure 17 shows the DFT results for the temperaturedependent phonon-limited resistivity of bulk Cu, Ag, and Au (Debye temperatures of 347, 227 and $162\\mathrm{~K~}$ [177], respectively). The resistivity increases with temperature as the phonon occupation increases, and becomes linearly dependent on temperature above the Debye temperature.  \n\nTable 10 presents the calculated room-temperature bulk resistivities, and compares them to experiments and to calculated values for metal nanowires (NWs) with diameters $d=1{\\mathrm{nm}}$ . In agreement with experiments, we find that Au has the largest resistivity, and that Ag is more conductive than Cu. In addition, the resistivity increases significantly when forming nanowires of the elements. Despite the fact that the phonon dispersions of bulk Au and $\\mathrm{Ag}$ are very similar, the resistivity is quite different. In the minimal free-electron model of metals, the conductivity is given by $\\begin{array}{r}{1/\\rho(T)=\\frac{1}{3}e^{2}v_{\\mathrm{F}}^{2}\\tau(T)n(\\varepsilon_{\\mathrm{F}})}\\end{array}$ . In the three FCC metals considered here, the Fermi velocity, $v_{\\mathrm{F}},$ , and the DOS, $n(\\varepsilon_{\\mathrm{F}}).$ , (and resulting carrier density) are almost identical, and the difference in the resistivity is traced back to the variation in the scattering rate. This shows how full firstprinciples Boltzmann transport simulations of the scattering rate is needed to capture the origin of the resistivities of different metals. While the resistivity of bulk $\\operatorname{Ag}$ is slightly underestimated by the simulations, we find good agreement with experiments for bulk Au and Cu, as well as the correct ranking of the individual metals. This illustrates the predictive power of the method. In general, we find that the resistivity of $d=1{\\mathrm{nm}}$ nanowires is increased by a factor of three for Au and even more for $\\mathrm{Ag}$ and $\\mathrm{{Cu}}$ , as compared to bulk, due to the increased EPC in nanowires.  \n\n![](images/4020fc808db89432da9ee37d9e470f475e6cf5d30990cb37fbe843bd355fb716.jpg)  \nFigure 18.  Average position of the ${\\mathrm{Li}^{+}}$ ions in $\\mathrm{LiFePO_{4}}$ along the $y$ Cartesian direction (along the [0 1 0] channel), as a function of time, calculated at temperatures $300\\mathrm{K}$ (a) and $1000\\mathrm{K}$ (b), and for electric field strengths from $D_{y}=0.0$ (purple lines) to $D_{y}=0.3$ (red lines) VA˚. The data obtained from multi-model and FF simulations are shown as solid and dashed lines, respectively. (c) and (d) Snapshots obtained from the simulations after $40\\mathrm{ps}$ of simulation for $\\bar{3}00\\mathrm{K}$ (c) and $1000\\mathrm{K}$ (d). The lithium and iron ions are shown in pink and orange, respectively, while the magenta tetrahedra represent the phosphate groups. The black arrow indicates the direction of the applied field.  \n\n![](images/9543990520f970312510e3513242eada916469e41d64f9a6d57c7a7377cb5304.jpg)  \nFigure 19.  Conduction band energies ( $\\cdot\\ E_{\\mathrm{X}}$ and $E_{\\mathrm{L}}$ ) of ${\\mathrm{Si}}_{1-x}{\\mathrm{Ge}}_{x}$ alloy as a function of Ge content, $x.$ , calculated using PPS-PBE and HSE06 functionals in combination with the LCAO basis set and the PW basis set, respectively. The $E_{\\mathrm{X}}$ and $E_{\\mathrm{L}}$ energies are defined with respect to the top of the valence band $(E_{\\mathrm{val}})$ at the $\\Gamma$ -point. Details on the definition of band energies at special $\\mathbf{k}$ -points for disordered alloys can be found elsewhere [186]. Reference experimental data (open markers) on the band-gap compositional dependence, $E_{\\mathrm{gap}}(x).$ , are given for low $(4.2\\mathrm{K})$ and room (296 K) temperatures [187] The dashed (solid) lines correspond to linear (quadratic) interpolation of the DFT-calculated band energies, $E_{\\mathrm{L}}$ $(E_{\\mathrm{X}})$ , given with filled markers; the interpolation formulas are given in table 11.  \n\n# 14.3.  Multi-model dynamics with an applied electric field  \n\nThe tight integration of different atomic-scale simulation engines within the same software framework allows for straight-forward combination of multiple atomistic models into one single simulation workflow. This enables elaborate computational workflows and extend the functionality of QuantumATK beyond that of methods based on a single atomistic model. We here show how such a multi-model approach can be used to implement a hybrid method that combines classical FF MD simulations with a DFT description of timedependent fluctuations of the atomic charges as the MD simulation progresses.  \n\nTable 11.  First-principles interpolation formulas for the ${\\mathrm{Si}}_{1-x}{\\mathrm{Ge}}_{x}$ composition-dependent band gap and lattice constant. The variables $b_{\\mathrm{w}},\\bar{b}_{\\mathrm{w}}^{\\prime}.$ and $b_{\\mathrm{w}}^{\\prime\\prime}$ are bowing parameters.   \n\n\n<html><body><table><tr><td></td><td>Band gap interpolation formula (eV)</td></tr><tr><td>PPS-PBE</td><td>Ex =1.116－0.764x+bwx2 bw = 0.526 eV EL= 2.104－1.425x</td></tr><tr><td>HSE06</td><td>Ex =1.204 -0.444x +bx2 b\"=0.228 eV EL=2.032－1.267x</td></tr><tr><td></td><td>Lattice constant interpolation formula (A)</td></tr><tr><td>PPS-PBE</td><td>a(x)= 5.431+0.257x+bwx2 bw=0.034A</td></tr></table></body></html>  \n\nWe study here ${\\mathrm{LiFePO}}_{4}$ , a promising cathode material of the olivine family for Li-ion batteries [141, 178]. In this class of materials, the olivine scaffold provides natural diffusion channels for the ${\\mathrm{Li}^{+}}$ ions, which have been shown to diffuse via a hopping mechanism, preferentially through [0 1 0]-oriented channels [179, 180]. MD simulations aimed at understanding the diffusion process have focused mainly on its temperature $(T)$ dependence. In this case, relatively high temperatures, usually in the range $500{-}2000\\mathrm{K}$ , are required to reach a sufficiently high hopping probability within a reasonable MD simulation time, and allow for calculation of the associated diffusion constants. These simulations have demonstrated that the diffusion increases with T, as a natural consequence of the increased hopping probability favored by Brownian motion.  \n\nHowever, in an electrochemical cell under operating conditions, the motion of the ${\\mathrm{Li}^{+}}$ ions may also have a non-negligible drift component, due to the displacement field resulting from the voltage difference applied between the anode and the cathode. This potentially rather important effect is rarely taken into account in atomistic simulations [181, 182].  \n\nAnother significant issue in the simulation of Li-ion batteries is related to the inclusion of electronic effects. In order to reach reasonably long simulation times, to describe atom diffusion at temperatures close to $300\\mathrm{~K~}$ , most low- $\\mathbf{\\nabla}\\cdot\\boldsymbol{T}_{} $ MD simulations are based on FFs, which by construction neglect any time-dependent fluctuations of the electronic density during the MD run. A number of models have tried to address this issue by either including approximate models to account for the charge fluctuation [183], or by running semi-classical dynamics on precalculated potential-energy surfaces based on DFT [184].  \n\nA QuantumATK multi-model approach can be used to address these issues by including first-principles charge fluctuations in the FF MD. The applied displacement field should add a force term $\\mathbf{F}_{a}^{\\prime}=Q_{a}\\mathbf{D}$ on the ath ion with formal charge $Q_{a}$ and $\\mathbf{D}$ being the field vector. However, in a FF, $Q_{a}$ is a timeindependent parameter, so the field-induced force will also be time-independent. In a multi-model approach, we instead use DFT simulations to determine the instantaneous charge $Q_{a}$ at regular intervals during the MD. Time-dependency in the field-induced force term is then included by use of a MD hook function (see section  7.4) by defining the time-dependent formal charge $Q_{a}^{\\prime}(t)$ as  \n\n![](images/19e4bf9bea0fc500b8c8c2d3de2cbb49c18a2e82fe3591a94ec5eb246aa5445b.jpg)  \nFigure 20.  Band structure of bulk Si (left panel) and Ge (right panel) obtained using the PPS-PBE (solid line) and HSE06 (dashed line) methods. The calculations used a $\\Gamma$ -centered $12\\times12\\times12{\\bf{k}}$ -point grid to sample the Brillouin zone of the 2 atom primitive cells.  \n\n$$\n\\begin{array}{r}{Q_{a}^{\\prime}=\\mathrm{Q}_{a}^{\\mathrm{FF}}+\\Delta Q_{a}(t),}\\end{array}\n$$  \n\nwhere $\\Delta Q_{a}$ describes the time-dependent fluctuation. In principle, $\\Delta Q_{a}$ can be defined arbitrarily, provided that charge neutrality is maintained in the system. In the present case, we chose a simple definition,  \n\n$$\n\\Delta Q_{a}(t)=\\mathrm{Q}_{a}^{\\mathrm{DFT}}(t)-\\mathrm{Q}_{a}^{\\mathrm{DFT,ref}},\n$$  \n\nwhere $Q_{a}^{\\mathrm{DFT}}(t)$ and $\\mathrm{Q}_{a}^{\\mathrm{DFT,ref}}$ are the time-dependent charge of the ith atom obtained from a DFT calculation for the MD configuration at time $t$ , and a time-independent charge obtained for a reference configuration at $T=0\\mathrm{K}$ , respectively. We note that, in the present case, the lack of consistency between the methods used to calculate $\\mathrm{Q}_{a}^{\\mathrm{FF}}$ and $\\Delta Q_{a}(t)$ does not constitute an issue, since the charge fluctuations during the dynamics are of the order $\\Delta Q_{a}\\sim0.1~e^{-}$ .  \n\nWe have applied this multi-model approach to investigate the interplay between Brownian and drift components of the diffusion of ${\\mathrm{Li}^{+}}$ ions along the [0 1 0] channels in $\\mathrm{LiFePO_{4}}$ in the presence of an applied displacement field. The system was described by a $1\\times2\\times1~\\mathrm{LiFePO_{4}}~112$ atom supercell, that is, 2 times the conventional unit cell (16 formula units). For the classical part of the multi-model simulations, we used a FF potential by Pedone et al [154], which has been shown to describe qualitatively correctly the geometry and transport properties of olivine materials [185]. The ATK-LCAO engine was used for the DFT part. MD simulations were performed at temperatures $300~\\mathrm{K}$ and $1000\\mathrm{~K~}$ for a displacement field $\\mathbf{D}=[0,D_{y},0]$ , with $0.0\\mathrm{~V~}\\mathring{\\mathrm{A}}^{-1}\\leqslant D_{y}\\leqslant0.3\\mathrm{~V~}\\mathring{\\mathrm{A}}^{-1}.$ . For each temperature, a 5 ps equilibration run using a NPT ensemble was performed, starting from the structure optimized at $0\\mathrm{\\:K}$ , using a Maxwell–Boltzmann distribution of initial velocities, followed by a 45 ps production run using a NVT ensemble. The MD time step was 1.0 fs, and $\\Delta Q_{a}(t)$ was recalculated every $100~\\mathrm{{MD}}$ steps, see (84), with $Q_{a}^{\\mathrm{DFT}}(t)$ and $\\mathrm{Q}_{a}^{\\mathrm{DFT}}$ ,ref obtained from Mulliken population analysis. Further computational details are given in appendix.  \n\nFigures 18(a) and (c) shows the average displacement $\\langle{y_{\\mathrm{Li^{+}}}}\\rangle$ of the ${\\mathrm{Li}^{+}}$ ions along the $y$ Cartesian direction, that is, along the [0 1 0] channels of the $\\mathrm{FePO_{4}}$ scaffold, calculated for temper­atures 300 and $1000\\mathrm{~K~}$ and for increasingly higher values of the applied field, using either FFs only or the $\\mathrm{FF+DFT}$ multi-model approach. In the absence of an applied field and at $300\\mathrm{~K~}$ , the average $\\mathrm{Li^{+}}$ -ion displacement remains constant at $\\langle y_{\\mathrm{Li^{+}}}\\rangle=4.67\\pm0.11\\stackrel{\\circ}{\\mathrm{A}}$ during the entire simulation, indicating the absence of hopping events. At $1000\\mathrm{K}$ , the situation is rather similar, as $\\left\\langle{y_{\\mathrm{Li}_{\\mathrm{\\Omega}}^{+}}}\\right\\rangle$ increases only slightly from an initial value of $7.12\\pm0.19\\check{\\mathrm{~A~}}$ (obtained from an average of the snapshots collected during the first picosecond of the FF-only MD) to a final value of $9.16\\pm0.{\\overset{\\cdot}{1}}3{\\overset{\\circ}{\\mathrm{A}}}$ (obtained from an average of the snapshots collected during the last picosecond). For the multi-model simulation, we observe instead a small decrease of $\\left\\langle y_{\\mathrm{Li^{+}}}\\right\\rangle$ over time. This indicates that, at both temperatures, ${\\mathrm{Li}^{+}}$ hopping due to Brownian motion is a rare event.  \n\nApplying an increasingly stronger displacement field leads to a progressive increase in the ${\\mathrm{Li}^{+}}$ hopping probability. At $300~\\mathrm{K}$ , the average $\\mathrm{Li^{+}}$ -ion displacement increases steadily from the beginning of the MD run for $D_{y}\\geqslant0.20\\mathrm{~V~mathring{A}}^{-1}$ , indicating that, for these values of $D_{y}$ , ${\\mathrm{Li}^{+}}$ hopping is primarily due to field-induced drift. The ${\\mathrm{Li}}^{+}$ ions accelerate until they reach a constant velocity, as shown by the tendency of the $\\langle{y_{\\mathrm{Li^{+}}}}\\rangle$ versus time curves to continually decrease their slope, corresponding to a straight line on a linear plot.  \n\nIn the absence of an applied field, increasing the temper­ ature should increase the probability of $\\mathrm{Li^{+}}$ ion diffusion due to increased Brownian motion [180, 185]. However, in the present case we find that the $\\mathrm{Li^{+}}$ ions move less at $1000\\mathrm{~K~}$ than at $300\\mathrm{K}$ . For comparable values of $D_{y}$ , the $\\left\\langle{y_{\\mathrm{Li^{+}}}}\\right\\rangle$ versus time curve has a smaller slope at $1000\\mathrm{K}$ than those calculated at $300~\\mathrm{K}$ . The reason is that collision events of the ${\\mathrm{Li}^{+}}$ ions with the $\\mathrm{LiFePO_{4}}$ lattice, where phonons are considerably more excited at higher temperatures than at room temperature, limits the effective velocity of the ${\\mathrm{Li}}^{+}$ ions.  \n\nThis is evident by comparing the ${\\mathrm{LiFePO}}_{4}$ structures at the two temperatures. Figures 18(c) and (d) shows two snapshots extracted at the end of the MD runs at $D_{y}=0.3\\mathrm{V}\\mathring{\\mathrm{~A}}^{-1}$ and at temperatures 300 and $1000~\\mathrm{K}$ , respectively. At $300~\\mathrm{K}$ , the $\\mathrm{LiFePO_{4}}$ structure is relatively unperturbed. Consequently, the ${\\mathrm{Li}^{+}}$ ions are able to travel through the [0 1 0] channels with relatively few scattering events with the ${\\mathrm{LiFePO}}_{4}$ lattice. Conversely, at $1000\\mathrm{K}$ , the ${\\mathrm{LiFePO}}_{4}$ structure is significantlty perturbed, leading to a high probability of collisions between the $\\mathrm{Li^{+}}$ ions and the olivine lattice.  \n\nIn summary, we have studied the diffusion of ${\\mathrm{Li}^{+}}$ in olivine $\\mathrm{LiFePO_{4}}$ , using a multi-model computational approach that combines a classical FF with DFT, the latter to include the effect of the field and of time-dependent charge fluctuations. Our analysis highlights the importance of considering the combined effect of both Brownian and drift contributions to the ${\\mathrm{Li}^{+}}$ hopping to describe the overall process, which strongly depends on not only the temperature itself, but also on the probability of collision events between the diffusing ions and the ${\\mathrm{FePO}}_{4}$ lattice.  \n\n# 14.4.  Electronic structure of binary alloys  \n\nUnderstanding the physical properties of semiconductor alloys, such as silicon-germanium binary compounds, is highly relevant, since such alloys are commonly used in microelectronics as a semiconductor material for, e.g. heterojunction bipolar transistors or as a strained semiconductor layer in CMOS transistors [188]. Moreover, device-level TCAD simulations, frequently used in industrial semiconductor research and development, usually require material-dependent input parameters such as band gap, effective masses, deformation potentials, and many others [189]. Atomic-scale simulations may be used to calculate such parameters from first principles if experimental values are not available, including composition dependence [186]. However, simulating randomly dis­ ordered alloys may be computationally challenging since the traditional approach to random-alloy (RA) simulations use stastical sampling of multiple large supercells with random atomic arrangements (configurational averaging) to take into account the effect of disorder on the physical properties of alloys.  \n\nWe here adopt the special quasi-random structure (SQS) approach [190] for DFT modelling of SiGe random alloys, which significantly reduces the computational cost. Unlike in the RA approach, in the SQS method the configurational averaging of band energies is captured by a single supercell structure. We study 64 atom ${\\mathrm{Si}}_{1-x}{\\mathrm{Ge}}_{x}$ supercells in the full range of compositions, $0\\leqslant x\\leqslant1$ , by calculating composition-dependent lattice constants and band energies. The PPS-PBE method [19], discussed in section  4.4, was used with the ATK-LCAO simulation engine, and we compare the band energies to those obtained with the HSE06 hybrid functional using the ATK-PlaneWave engine. We also compare  \n\nSQS band energies to those calculated using the traditional RA approach, obtained by averaging over 5 randomly generated RA configurations. In both the SQS and RA cases, the band energies were computed by averaging the energies of the conduction (valence) band states split by alloy disorder [186].  \n\nWe used a NanoLab SQS module to generate the SQS configurations. The module uses a genetic algorithm to optim­ize finite-size alloy configurations to reproduce selected correlation functions of the infinite alloy system. The genetic optim­ ization algorithm is very efficient, and systems with many hundred atoms are easily handled. In this case, the SQS structures were generated by fitting all pair, triplet, and quadruplet correlation functions with figure sizes up to 7.0, 5.0 and $4.0\\mathring\\mathrm{A}$ , respectively, such as to match those correlation functions for the truly random alloy, as detailed in [190, 191]. Generation of a single 64 atom SiGe SQS alloy takes about $4\\mathrm{{min}}$ on a modern 4-core processor. The alloy configurations were then relaxed using PPS-PBE followed by band structure analysis. HSE06-level band structures were calculated without further relaxation. More computational details are given in appendix. Figure 19 shows the $\\mathrm{Si}_{1-x}\\mathrm{Ge}_{x}$ composition dependent conduction band minima (CBM), referenced to the valence band maximum, for both the $X\\mathrm{-}$ and $L$ -valley in the SiGe BZ. We first note that SQS band energies are very similar to the those calculated using the more expensive RA approach. It is well known that the $\\mathrm{Si}_{1-x}\\mathrm{Ge}_{x}$ fundamental band gap changes character at $x\\sim0.85$ . The PPS-PBE and HSE06 predictions of the transition point are $x\\sim0.88$ and 0.82, respectively. As expected, the calculated $X$ -valley conduction-band energies exhibit bowing, i.e. nonlinear behavior of these quantities with respect to Ge content, $x.$ The best-fit interpolation formulas, shown as lines in figure 19, are listed in table 11, including the band-gap compositional bowing parameters. The PPS-PBE band gaps are in good agreement with room-temperature experiments (within ${\\sim}50\\mathrm{meV}$ for the entire range of Ge content), while the HSE06 band gaps are in better agreement with low-temperature experiments. Moreover, the HSE06-based approach appears to more accurately describe the band-gap bowing parameter, while PPS-PBE tends to overestimate it. Finally, the calculated SiGe lattice constant also exhibits compositional bowing, as indicated by the interpolation formula in table 11. The bowing parameter of $0.034\\overset{\\circ}{\\mathrm{A}}$ is overestimated by $\\sim26\\%$ as compared to experiments $(0.027\\mathring\\mathbf{A})$ ).   \nTo benchmark the empirical PPS-PBE method against the  \n\nparameter-free HSE06 approach, we also calculated the band structure of bulk Si and Ge using both methods, as shown in figure 20. The PPS-PBE conduction and valence bands around the Fermi energy are in good agreement with the HSE06 band structure. This is consistent with the fact that the PPS-PBE method was fitted to experimental data, and that the HSE06 hybrid functional accurately simulates the band structure of bulk semiconductors.  \n\nIn summary, we find that the SQS approach is well suited to describe the compositional bowing of the band energies in $\\mathrm{Si}_{1-x}\\mathrm{Ge}_{x}$ random alloys, suggesting that SQS provides an accurate and efficient approach to random-alloy simulations. The HSE06 hybrid functional accurately describes the conduction-band energies of SiGe alloys and their compositional bowing, while the PPS-PBE method offers a computationally efficient alternative if only bands around the Fermi level are important.  \n\n# 15.  Summary  \n\nIn this paper we have presented the QuantumATK platform and details of its atomic-scale simulation engines, which are ATK-LCAO, ATK-PlaneWave, ATK-SE, and ATK-ForceField. We have compared the accuracy and performance of the different engines, and illustrated the application range of each. The platform includes a wide range of modules for application of the different simulation engines in solid-state and device physics, including electron transport, phonon scattering, photocurrent, phonon-limited mobility, optical properties, static polarizations, molecular dynamics, etc  \n\nThe simulation engines are complimentary and through the seamless Python integration in the QuantumATK platform, it is easy to shift between different levels of theory or integrate different engines into complex computational workflows. This has been illustrated in several application examples, where we for example showed how ATK-LCAO and ATK-ForceField can be combined to study $\\mathrm{Li^{+}}$ -ion drift in a battery cathode material. We also presented applications of QuantumATK for simulating electron transport in 2D materials, phonon-limited resistivity of metals, and electronic-structure simulations of SiGe random alloys.  \n\nWhile several of the simulation engines and methods have been described independently before [19, 24, 28, 106, 123, 132, 137, 147, 153, 163], we have here provided an overview of the entire platform, including implementation details not previously published. We expect that this paper can become a general reference for documenting the QuantumATK platform, and is a reference to its applications for atomic-scale modelling in semiconductor physics, catalysis, polymer mat­ erials, battery materials, and other fields.  \n\n$0\\mathrm{K}$ to avoid extremely small bond distances. The MD simulations were then performed at $300\\mathrm{K}$ using a random Boltzmann distribution of initial velocities and a Langevin thermostat. The FF simulations used a Pedone potential [154], while the TB simulations used a Slater–Koster parametrization. For the DFT-LCAO and DFT-PW simulations, we used normconserving PseudoDojo pseudopotentials with a Medium basis set and a kinetic-energy cutoff energy of $1360\\mathrm{eV}$ $\\mathrm{50\\Ha)}$ , respectively. For TB, DFT-LCAO, and DFT-PW simulations, the Brillouin zone was sampled using a Monkhorst–Pack [192] (MP) $\\mathbf{k}$ -point density of $3{-}4\\mathring{\\mathrm{A}}$ . For systems with sizes between 240 and 960 atoms, 2 processes/k-point was used, whereas for the 1920 atom system, 16 processes/k-point was used.  \n\nFor the DFT-NEGF device simulations presented in section  14.1, we used the PBE density functional with SG15- Medium (FHI-DZP) combinations of pseudopotentials and basis sets for $\\mathbf{MoTe}_{2}$ and $\\mathrm{SnS}_{2}$ (Au and Al). The real-space cutoff energy was 2721 eV $(100\\mathrm{Ha})$ , and $\\mathbf{MPk}$ -point grids of $12\\times1\\times100$ and $12\\times1$ were used to sample the BZ of the electrode and of the device, respectively.  \n\nIn the study of multi-model dynamics presented in section 14.3, we used the ATK-LCAO engine with a DZP basis set and a real-space cutoff energy of $2180\\mathrm{eV}$ ( $\\mathrm{80\\Ha)}$ . Exchangecorrelation effects were described by the PBE functional, and the $\\mathrm{FePO_{4}B Z}$ was sampled using a $3\\times3\\times2\\mathrm{MP}\\mathbf{k}$ -point grid.  \n\nFor the electronic-structure calculations for SiGe random alloys presented in section  14.4, we used a $3\\times3\\times3$ MP $\\mathbf{k}$ -point grid and an electron temperature of $0.025\\mathrm{eV}$ for the Fermi–Dirac occupation function. SG15 (FHI) pseudopotentials were used for the PSS-PBE (HSE06) simulations. The LCAO mesh density cutoff was 2721 eV ( $\\mathrm{100~Ha)}$ , and the PW kinetic-energy cutoff was $544\\mathrm{eV}$ $20\\ \\mathrm{Ha})$ . The LCAO simulations used Medium (High) bais sets for silicon (germanium). Relaxation of unit-cell volume and ion positions was done using the PPS-PBE method with total energy, forces and stress converged to $10^{-5}\\mathrm{eV},0.01\\mathrm{eV}\\mathring{\\mathrm{A}}^{-1}$ , and $0.05\\mathrm{GPa}$ , respectively.  \n\n# Acknowledgments  \n\nAuthors acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreements No. 713481 (SPICE), No. 766726 (COSMICS), and No. 723867 (EMMC-CSA), as well as funding from the Quantum Innovation Center (QUBIZ) and the Lundbeck Foundation (R95-A10510). CNG is sponsored by the Danish National Research Foundation (DNRF103).  \n\n# Appendix.  Computational details  \n\nIn the simulations presented in figure 1, we have considered noncrystalline $\\mathrm{a-Al}_{2}\\mathrm{O}_{3}$ structures with a constant density of $2.81\\mathrm{g}\\ \\mathrm{cm}^{-3}$ . The system sizes considered were formed by 5, 30, 60, 120, 240, 480, 960, 1920, 3840, 7680, 15360, and 30720 atoms, respectively, each system with the appropriate unit-cell volume. 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  {
    "id": "10.1063_1.5143190",
    "DOI": "10.1063/1.5143190",
    "DOI Link": "http://dx.doi.org/10.1063/1.5143190",
    "Relative Dir Path": "mds/10.1063_1.5143190",
    "Article Title": "DFTB plus , a software package for efficient approximate density functional theory based atomistic simulations",
    "Authors": "Hourahine, B; Aradi, B; Blum, V; Bonafé, F; Buccheri, A; Camacho, C; Cevallos, C; Deshaye, MY; Dumitrica, T; Dominguez, A; Ehlert, S; Elstner, M; van der Heide, T; Hermann, J; Irle, S; Kranz, JJ; Köhler, C; Kowalczyk, T; Kubar, T; Lee, IS; Lutsker, V; Maurer, RJ; Min, SK; Mitchell, I; Negre, C; Niehaus, TA; Niklasson, AMN; Page, AJ; Pecchia, A; Penazzi, G; Persson, MP; Rezác, J; Sánchez, CG; Sternberg, M; Stöhr, M; Stuckenberg, F; Tkatchenko, A; Yu, VWZ; Frauenheim, T",
    "Source Title": "JOURNAL OF CHEMICAL PHYSICS",
    "Abstract": "DFTB+ is a versatile community developed open source software package offering fast and efficient methods for carrying out atomistic quantum mechanical simulations. By implementing various methods approximating density functional theory (DFT), such as the density functional based tight binding (DFTB) and the extended tight binding method, it enables simulations of large systems and long timescales with reasonable accuracy while being considerably faster for typical simulations than the respective ab initio methods. Based on the DFTB framework, it additionally offers approximated versions of various DFT extensions including hybrid functionals, time dependent formalism for treating excited systems, electron transport using non-equilibrium Green's functions, and many more. DFTB+ can be used as a user-friendly standalone application in addition to being embedded into other software packages as a library or acting as a calculation-server accessed by socket communication. We give an overview of the recently developed capabilities of the DFTB+ code, demonstrating with a few use case examples, discuss the strengths and weaknesses of the various features, and also discuss on-going developments and possible future perspectives. (C) 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).",
    "Times Cited, WoS Core": 780,
    "Times Cited, All Databases": 806,
    "Publication Year": 2020,
    "Research Areas": "Chemistry; Physics",
    "UT (Unique WOS ID)": "WOS:000521986100001",
    "Markdown": "#  \n\n# DFTB+, a software package for efficient approximate density functional theory based atomistic simulations  \n\nCite as: J. Chem. Phys. 152, 124101 (2020); https://doi.org/10.1063/1.5143190 Submitted: 20 December 2019 . Accepted: 27 February 2020 . Published Online: 23 March 2020 B. Hourahine , B. Aradi , V. Blum , F. Bonafé , A. Buccheri , C. Camacho , C. Cevallos , M. Y. Deshaye, T. Dumitrică , A. Dominguez, S. Ehlert , M. Elstner, T. van der Heide, J. Hermann , S. Irle $\\oplus_{i}$ , J. J. Kranz, C. Köhler, T. Kowalczyk , T. Kubař , I. S. Lee, V. Lutsker, R. J. Maurer $\\oplus_{i}$ , S. K. Min $\\oplus_{\\cdot}$ I. Mitchell $\\oplus_{\\cdot}$ C. Negre, T. A. Niehaus $\\oplus_{\\cdot}$ A. M. N. Niklasson $\\oplus,$ A. J. Page , A. Pecchia $\\oplus,$ G. Penazzi , M. P. Persson , J. ${\\check{\\mathsf{R e z i c}}}^{\\oplus},$ C. G. Sánchez , M. Sternberg, M. Stöhr , F. Stuckenberg, A. Tkatchenko, V. W.-z. Yu, and T. Frauenheim  \n\n# COLLECTIONS  \n\nPaper published as part of the special topic on Electronic Structure Software Note: This article is part of the JCP Special Topic on Electronic Structure Softwar  \n\n![](images/4b27b96ae92caa8618c66577692555a501be003003584103aae0bacc1c338105.jpg)  \n\nThis paper was selected as Featured  \n\n![](images/045c4fab7d3bd7a241e839342ded0e5b653bb8dff97ebb6db57598dbcb184803.jpg)  \n\n# ARTICLES YOU MAY BE INTERESTED IN  \n\nAdventures in DFT by a wavefunction theorist The Journal of Chemical Physics 151, 160901 (2019); https://doi.org/10.1063/1.5116338  \n\nGeneralized spin mapping for quantum-classical dynamics The Journal of Chemical Physics 152, 084110 (2020); https://doi.org/10.1063/1.5143412  \n\nA consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu The Journal of Chemical Physics 132, 154104 (2010); https://doi.org/10.1063/1.3382344  \n\n![](images/bb3263f3e5931e72d435127f205eaee238964f315d39c4df411b0c59a5abdbcc.jpg)  \n\n# DFTB+, a software package for efficient approximate density functional theory based atomistic simulations  \n\nCite as: J. Chem. Phys. 152, 124101 (2020); doi: 10.1063/1.514319 Submitted: 20 December 2019 $\\cdot$ Accepted: 27 February 2020 • Published Online: 23 March 2020  \n\nB. Hourahine,1 $\\textcircled{1}$ B. Aradi,2,a) $\\textcircled{1}$ V. Blum,3 $\\textcircled{1}$ F. Bonafé,4 $\\textcircled{1}$ A. Buccheri,5 $\\textcircled{1}$ C. Camacho,6 $\\textcircled{1}$ C. Cevallos,6   \nM. Y. Deshaye,7 T. Dumitric˘a,8 A. Dominguez,2,9 S. Ehlert,10 M. Elstner,11 T. van der Heide,2 J. Hermann,12   \nS. Irle,13 J. J. Kranz,11 C. Köhler,2 T. Kowalczyk,7 T. Kubarˇ, $11\\textcircled{\\scriptsize{\\parallel}}$ I. S. Lee,14 V. Lutsker,15 R. J. Maurer,16   \nS. K. Min,14 I. Mitchell,17 C. Negre,18 T. A. Niehaus,19 $\\textcircled{1}$ A. M. N. Niklasson,18 A. J. Page,20 A. Pecchia,21 $\\textcircled{1}$   \nG. Penazzi,2 M. P. Persson,22 J. Rˇ ezáˇc,23 C. G. Sánchez,24 M. Sternberg,25 M. Stöhr,26   \nF. Stuckenberg,2 A. Tkatchenko,26 V. W.-z. Yu,3 and T. Frauenheim2,9  \n\n# AFFILIATIONS  \n\n1 SUPA, Department of Physics, The University of Strathclyde, Glasgow G4 0NG, United Kingdom   \n2 Bremen Center for Computational Materials Science, University of Bremen, Bremen, Germany   \n3 Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA   \n4 Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany   \n5 School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom   \n6 School of Chemistry, University of Costa Rica, San José 11501-2060, Costa Rica   \n7 Department of Chemistry and Advanced Materials Science and Engineering Center, Western Washington University,   \nBellingham, Washington 98225, USA   \n8 Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA   \n9 Computational Science Research Center (CSRC) Beijing and Computational Science Applied Research (CSAR)   \nInstitute Shenzhen, Shenzhen, China   \n10University of Bonn, Bonn, Germany   \n11 Institute of Physical Chemistry, Karlsruhe Institute of Technology, Karlsruhe, Germany   \n12Freie Universität Berlin, Berlin, Germany   \n13Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA   \n14Department of Chemistry, Ulsan National Institute of Science and Technology, Ulsan, South Korea   \n15Institut I – Theoretische Physik, University of Regensburg, Regensburg, Germany   \n16Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom   \n17Center for Multidimensional Carbon Materials, Institute for Basic Science (IBS), Ulsan 44919, South Korea   \n18Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA   \n19Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, F-69622 Villeurbanne, France   \n20School of Environmental and Life Sciences, University of Newcastle, Callaghan, Australia   \n21CNR-ISMN, Via Salaria km 29.300, 00015 Monterotondo Stazione, Rome, Italy   \n22Dassault Systemes, Cambridge, United Kingdom   \n23Institute of Organic Chemistry and Biochemistry AS CR, Prague, Czech Republic   \n24Instituto Interdisciplinario de Ciencias Básicas, Universidad Nacional de Cuyo, CONICET,   \nFacultad de Ciencias Exactas y Naturales, Mendoza, Argentina   \n25Argonne National Laboratory, Lemont, Illinois 60439, USA   \n26Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg City, Luxembourg  \n\nNote: This article is part of the JCP Special Topic on Electronic Structure Software. a)Author to whom correspondence should be addressed: aradi@uni-bremen.de  \n\n# ABSTRACT  \n\n${\\mathrm{DFTB}}+$ is a versatile community developed open source software package offering fast and efficient methods for carrying out atomistic quantum mechanical simulations. By implementing various methods approximating density functional theory (DFT), such as the density functional based tight binding (DFTB) and the extended tight binding method, it enables simulations of large systems and long timescales with reasonable accuracy while being considerably faster for typical simulations than the respective ab initio methods. Based on the DFTB framework, it additionally offers approximated versions of various DFT extensions including hybrid functionals, time dependent formalism for treating excited systems, electron transport using non-equilibrium Green’s functions, and many more. ${\\mathrm{DFTB}}+$ can be used as a userfriendly standalone application in addition to being embedded into other software packages as a library or acting as a calculation-server accessed by socket communication. We give an overview of the recently developed capabilities of the ${\\mathrm{DFTB}}+$ code, demonstrating with a few use case examples, discuss the strengths and weaknesses of the various features, and also discuss on-going developments and possible future perspectives.  \n\n$\\circledcirc$ 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5143190.,  \n\n# I. INTRODUCTION  \n\nDensity Functional Theory $(\\mathrm{DFT})^{1,2}$ dominates the landscape of electronic structure methods, being the usual go-to technique to model large, chemically complex systems at good accuracy. For larger systems and time scales, force-field models instead dominate materials and chemical modeling. Between these is the domain of semi-empirical methods, derived from approximations to Hartree– Fock or DFT based methods. Within this space, density functional based tight binding (DFTB)3–5 effectively offers a reduced complexity DFT method, being derived from a simplification of Kohn–Sham DFT to a tight binding form.6  \n\nThis paper describes the ${\\mathrm{DFTB}}+$ code,7 an open source implementation, which aims at collecting the developments of this family of methods and making them generally available to the chemical, materials, and condensed matter communities. This article describes extensions to this code since its original release in $2007,^{8}$ there being a lack of a more recent overview of its features and underlying theory.  \n\n# II. DFTB $^+$ FEATURES  \n\n# A. The core DFTB-model  \n\nThe basic DFTB-equations are presented below. They can be easily generalized for periodic cases $k$ -points) as well as for other boundary conditions, as implemented in ${\\mathrm{DFTB}}+$ . All equations throughout this paper are given in atomic units with Hartree as the energy unit.  \n\n# 1. Expansion of the total energy  \n\nThe DFTB models are derived from Kohn–Sham (KS) $\\mathrm{DFT}^{2}$ by expansion of the total energy functional. Starting from a properly chosen reference density $\\rho_{0}$ (e.g., superposition of neutral atomic densities), the ground state density is then represented by this reference, as perturbed by density fluctuations: $\\rho({\\bf r})=\\rho_{0}({\\bf r})+\\delta\\rho({\\bf r})$ . The total energy expression then expands the energy functional in a  \n\nTaylor series up to third order,  \n\n$$\n\\begin{array}{r}{\\begin{array}{r}{E^{\\mathrm{DFTB3}}\\big[\\rho_{0}+\\delta\\rho\\big]=E^{0}\\big[\\rho_{0}\\big]+E^{1}\\big[\\rho_{0},\\delta\\rho\\big]+E^{2}\\big[\\rho_{0},\\big(\\delta\\rho\\big)^{2}\\big]}\\\\ {+E^{3}\\big[\\rho_{0},\\big(\\delta\\rho\\big)^{3}\\big]~}\\end{array}}\\end{array}\n$$  \n\nwith  \n\n$$\n\\begin{array}{l}{{\\displaystyle{E}^{l}[\\rho_{\\mathrm{o}}]=\\frac{1}{2}\\sum_{i=1}^{Z_{\\mathrm{o}}\\zeta_{\\mathrm{o}}}\\frac{1}{K_{i k}}-\\frac{1}{2}\\int\\int\\frac{\\rho_{\\mathrm{o}}(\\boldsymbol{\\mathbf{r}})\\rho_{\\mathrm{o}}(\\boldsymbol{\\mathbf{r}}^{\\prime})}{|\\boldsymbol{\\mathbf{r}}-\\boldsymbol{\\mathbf{r}}^{\\prime}|}\\mathrm{d}\\boldsymbol{\\mathbf{r}}\\mathrm{d}^{\\prime}}\\ ~}\\\\ {{\\displaystyle~\\qquad-\\int\\ V^{\\mathrm{x}}[\\rho_{\\mathrm{o}}]\\rho_{\\mathrm{o}}(\\boldsymbol{\\mathbf{r}})\\mathrm{d}\\boldsymbol{\\mathbf{r}}+E^{\\mathrm{x}}[\\rho_{\\mathrm{o}}]},}\\\\ {{\\displaystyle E^{l}[\\rho_{\\mathrm{o}},\\delta\\rho]=\\sum_{i}n\\langle|\\psi|\\hat{H}[\\rho_{\\mathrm{o}}]|\\psi\\rangle_{i}\\rangle,}}\\\\ {{\\displaystyle E^{\\frac{1}{2}}[\\rho_{\\mathrm{o}},(\\hat{\\rho}\\rho)^{2}]=\\frac{1}{2}\\iint\\left(\\frac{1}{|\\boldsymbol{\\mathbf{r}}-\\boldsymbol{\\mathbf{r}}^{\\prime}|}+\\frac{\\delta^{2}E^{\\mathrm{x}}[\\rho_{\\mathrm{o}}]}{\\delta\\rho(\\boldsymbol{\\mathbf{r}})\\delta(\\rho^{\\prime})}\\Big|_{\\rho_{\\mathrm{o}}}\\right)\\delta\\rho(\\boldsymbol{\\mathbf{r}})\\delta\\rho(\\boldsymbol{\\mathbf{r}}^{\\prime})\\mathrm{d}\\boldsymbol{\\mathbf{r}}\\mathrm{d}\\boldsymbol{\\mathbf{r}}^{\\prime},}}\\\\ {{\\displaystyle E^{\\frac{1}{2}}[\\rho_{\\mathrm{o}},(\\hat{\\rho}\\rho)^{2}]=\\frac{1}{6}\\iint\\frac{\\delta^{3}E^{\\mathrm{x}}[\\rho]}{\\delta\\rho(\\boldsymbol{\\mathbf{r}})\\delta\\rho(\\boldsymbol{\\mathbf{r}}^{\\prime})\\delta\\rho(\\boldsymbol{\\mathbf{r}}^{\\prime})}\\mathrm{d}\\boldsymbol{\\mathbf{r}}}\\\\ {{\\displaystyle~\\qquad\\times\\delta\\rho(\\boldsymbol{\\mathbf{r}})\\delta\\rho(\\boldsymbol{\\mathbf{r}}^{\\prime})\\mathrm{d}\\boldsymbol{\\mathbf{r}}\\boldsymbol{\\mathbf{\\rho}}(\\boldsymbol{\\mathbf{r}}^{\\prime})\\mathrm{d}\\boldsymbol{\\mathbf{r}}\\mathrm{d}^{\\prime}}.}\\end{array}\n$$  \n\nwith XC being the exchange correlation energy and potential. Several DFTB models have been implemented, starting from the first order non-self-consistent DFTB13,4 [originally called DFTB or non-SCC DFTB], the second order DFTB2 (originally called SCC-DFTB),5 and the more recent extension to third order, DFTB3.9–12  \n\n# 2. DFTB1  \n\nThe first order DFTB1 method is based on three major approximations: (i) it takes only $E^{0}[\\rho_{0}]$ and $E^{1}[\\rho_{0},\\delta\\rho]$ from Eq. (2) into account, (ii) it is based on a valence-only minimal basis set $(\\phi_{\\mu})$ within a linear combination of atomic orbitals (LCAO) ansatz,  \n\n$$\n\\psi_{i}=\\sum_{\\mu}c_{\\mu i}\\phi_{\\mu},\n$$  \n\nfor the orbitals $\\psi_{i}$ , and (iii) it applies a two-center approximation to the hamiltonian operator $\\hat{H}[\\rho_{0}]$ .  \n\na. Minimal atomic basis set. The atomic orbital basis set $\\phi_{\\mu}$ is explicitly computed from DFT by solving the atomic Kohn– Sham equations with an additional (usually harmonic) confining potential,  \n\n$$\n\\biggl[-\\frac{1}{2}{\\boldsymbol{\\nabla}}^{2}+V^{\\mathrm{\\scriptsize~eff}}\\bigl[\\rho^{\\mathrm{\\scriptsize~atom}}\\bigr]+\\biggl(\\frac{r}{r_{0}}\\biggr)^{n}\\biggr]\\phi_{\\mu}=\\epsilon_{\\mu}\\phi_{\\mu}.\n$$  \n\nThis leads to slightly compressed atomic-like orbitals for describing the density in bonding situations. The actual values for $r_{0}$ are usually given in the publications describing the specific parameterization. The operator ${\\hat{H}}[\\rho^{0}]$ also depends on the superposition of atomic densities, $\\rho_{A}$ (or potentials, $V_{A}^{\\mathrm{eff}}.$ ) of neutral atoms, $\\{A\\}$ , in the geometry being modeled. This density is usually determined from the same atomic KS equations using a slightly different confinement radius, $r_{0}^{\\mathrm{d}}$ .  \n\nb. DFTB matrix elements. The hamiltonian can be represented in an LCAO basis as  \n\n$$\nH_{\\mu\\nu}^{0}=\\bigl\\langle\\phi_{\\mu}\\bigl|\\hat{H}\\bigl[\\rho_{0}\\bigr]\\bigr|\\phi_{\\nu}\\bigr\\rangle\\approx\\bigl\\langle\\phi_{\\mu}\\bigl|-\\frac{1}{2}\\nabla^{2}+V\\bigl[\\rho_{A}+\\rho_{B}\\bigr]\\bigr|\\phi_{\\nu}\\bigr\\rangle,\\quad\\mu\\in A,\\nu\\in B,\n$$  \n\nwhere the neglect of the three center terms and pseudo-potential contributions lead to a representation, which can be easily computed by evaluating the Kohn–Sham equations for dimers. These matrix elements are computed once as a function of inter-atomic distance for all element pairs. The Slater–Koster13 combination rules are applied for the actual orientation of these “dimers” within a molecule or solid.  \n\nc. Total energy. ${\\cal E}^{0}[\\rho_{0}]$ depends only on the reference density, so is universal in the sense that it does not specifically depend on the chemical environment (which would determine any charge transfer (CT), $\\delta\\rho_{;}$ , occurring). It can, therefore, be determined for a “reference system” and then applied to other environments. This is the key to transferability of the parameters. In DFTB, ${\\boldsymbol E}^{0}[\\rho_{0}]$ is approximated as a sum of pair potentials called repulsive energy terms,  \n\n$$\n{\\cal E}^{0}[\\rho_{0}]\\approx E_{\\mathrm{rep}}=\\frac{1}{2}\\sum_{A B}V_{A B}^{\\mathrm{rep}}\n$$  \n\n(see Ref. 14), which are either determined by comparison with DFT calculations4 or fitted to empirical data.15 Forces are calculated with the Hellmann–Feynman theorem and derivatives of the repulsive energy.  \n\n# 3. DFTB2 and DFTB3  \n\nTo approximate the $E^{2}$ and $\\boldsymbol{E}^{3}$ terms in Eq. (2), the density fluctuations are written as a superposition of atomic contributions, taken to be exponentially decaying spherically symmetric charge densities  \n\n$$\n\\delta\\rho(\\mathbf{r})=\\sum_{A}\\delta\\rho_{A}(\\mathbf{r}-\\mathbf{R}_{A})\\approx\\frac{1}{\\sqrt{4\\pi}}\\sum_{A}\\Biggl(\\frac{\\tau_{A}^{3}}{8\\pi}\\mathrm{e}^{-\\tau_{A}|\\mathbf{r}-\\mathbf{R}_{A}|}\\Biggr)\\Delta q_{A}.\n$$  \n\nBy neglecting the XC-contributions for the moment, the second order integral $E^{2}$ leads to an analytical function, $\\gamma_{A B}$ , with energy,5  \n\n$$\nE^{2}(\\tau_{A},\\tau_{B},R_{A B})=\\frac{1}{2}\\sum_{A B(\\neq A)}\\gamma_{A B}(\\tau_{A},\\tau_{B},R_{A B})\\Delta q_{A}\\Delta q_{B}.\n$$  \n\nThe energy depends on the Mulliken charges $\\{q_{A}\\}$ (where the atomic charge fluctuation, $\\Delta q_{A}=q_{A}-Z_{A}$ , is with respect to the neutral atom), which are, in turn, dependent on the molecular orbital coefficients, $c_{\\mu i}$ . Thus, the resulting equations have to be solved selfconsistently. At large distances, $\\gamma_{A B}$ approaches $1/R_{A B}$ , while at short distances, it represents electron–electron interactions within one atom. For the limit $R_{A B}\\rightarrow0$ , one finds $\\begin{array}{r}{\\tau_{A}=\\frac{16}{5}U_{A}}\\end{array}$ , i.e., the so-called Hubbard parameter $U_{A}$ →(twice the chem c=al hardness) is inversely proportional to the width of the atomic charge density $\\tau_{A}$ . This relation is intuitive in that more diffuse atoms (or anions) have a smaller chemical hardness. For DFTB, the chemical hardness is computed from DFT, not fitted.  \n\nThe third order terms describe the change of the chemical hardness of an atom and are also computed from DFT. A function $\\Gamma_{A B}$ results as the derivative of the $\\gamma$ -function with respect to charge, and the DFTB3 total energy is then given by  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\cal E}^{\\mathrm{DFTB3}}=\\sum_{i}\\sum_{A B}\\sum_{\\mu\\in A}\\sum_{\\nu\\in B}n_{i}c_{\\mu i}c_{\\nu i}H_{\\mu\\nu}^{0}+\\frac12\\sum_{A B}\\Delta q_{A}\\Delta q_{B}\\gamma_{A B}^{h}}}\\\\ {{\\displaystyle~+~\\frac13\\sum_{A B}\\Delta q_{A}^{2}\\Delta q_{B}\\Gamma_{A B}+\\frac12\\sum_{A B}V_{A B}^{\\mathrm{rep}}.}}\\end{array}\n$$  \n\nThe third order terms become important when local densities deviate significantly from the reference, i.e., $\\Delta q_{A}$ is large. Apart from including the third order terms, DFTB3 also modifies $\\gamma_{A B}$ for the interactions between hydrogen and first row elements,9 where the deviation from the relation between the charge width and the chemical hardness, as formulated above, is most pronounced.  \n\nThe resulting DFTB3 hamiltonian takes the form  \n\n$$\nH_{\\mu\\nu}={H}_{\\mu\\nu}^{0}+{H}_{\\mu\\nu}^{2}[\\gamma^{h},\\Delta q]+{H}_{\\mu\\nu}^{3}[\\Gamma,\\Delta q],\\qquad\\mu\\in{\\cal A},\\nu\\in{\\cal B},\n$$  \n\n$$\nH_{\\mu\\nu}^{2}={\\frac{S_{\\mu\\nu}}{2}}\\sum_{C}\\Bigl(\\gamma_{B C}^{h}+\\gamma_{A C}^{h}\\Bigr)\\Delta q_{C},\n$$  \n\n$$\nH_{\\mu\\nu}^{3}=S_{\\mu\\nu}\\sum_{C}\\biggl(\\frac{\\Delta q_{A}\\Gamma_{A C}}{3}+\\frac{\\Delta q_{B}\\Gamma_{B C}}{3}+\\bigl(\\Gamma_{A C}+\\Gamma_{B C}\\bigr)\\frac{\\Delta q_{c}}{6}\\biggr)\\Delta q_{C},\n$$  \n\nwhere $S_{\\mu\\nu}$ is the overlap matrix between orbitals $\\phi_{\\mu}$ and $\\phi_{\\nu}$ , and $\\gamma^{h}$ is the modified DFTB2 interaction.  \n\n# 4. Spin  \n\nAnalogous to DFTB2, expanding the energy with respect to spin fluctuations16–18 leads to the spin-polarized expressions for DFTB. By introducing the magnetization density $m(\\mathbf{\\bar{r}})=\\rho^{\\uparrow}({\\bf r})-\\rho^{\\downarrow}({\\bf r})$ as difference of the densities of spin-up and spin-down elec−trons and its corresponding fluctuations $\\left[{\\delta m({\\bf r})}\\right]$ around the spin-unpolarized reference state $[|m({\\bf r})|=0]$ , a spin dependent term is added to the spin-independent $E^{2}$ of Eq. (2),  \n\n$$\n\\begin{array}{c}{{\\displaystyle{\\cal E}^{2}\\big[\\rho_{0},big(\\delta\\rho)^{2},\\big(\\delta m\\big)^{2}\\big]={\\cal E}^{2}\\big[\\rho_{0},\\big(\\delta\\rho\\big)^{2}\\big]+\\frac{1}{2}\\int\\left.\\frac{\\delta^{2}{\\cal E}^{\\mathrm{xc}}[\\rho,m]}{\\delta m({\\bf r})^{2}}\\right\\vert_{\\rho_{0},m=0}}}\\\\ {{\\times\\left.\\delta m({\\bf r})^{2}\\mathrm{d}{\\bf r},\\qquad(1\\mathrm{~}}}\\end{array}\n$$  \n\nwhere a local or semi-local $E^{\\mathrm{xc}}$ has been assumed.  \n\nIdentifying the spin density fluctuations with up- and downspin Mulliken charge differences, $\\Delta p_{A l},$ for angular momentum shell $l$ at atom $A$ , and approximating the second derivative of $E^{\\mathrm{xc}}[\\rho,m]$ as an atomic constant $W_{A l l^{\\prime}}$ (similar to the Hubbard $U_{A}$ ) lead to an on-site energy contribution  \n\n$$\nE_{s p i n}^{2}=\\frac{1}{2}\\sum_{A}\\sum_{l\\in A}\\sum_{l^{\\prime}\\in A}W_{A l l^{\\prime}}\\Delta p_{A l}\\Delta p_{A l^{\\prime}}.\n$$  \n\nThis term in Eq. (14) is to be added to Eq. (8). It captures the spin-polarization contribution to the total energy and couples different atomic angular momentum shells via magnetic interaction. The $W_{A l l^{\\prime}}$ are usually an order of magnitude less than the $U_{A}$ and are multiplied with a (typically) small $\\Delta p_{A l};$ hence, inclusion of spinpolarization via Eq. (14) gives only a small energy contribution. If there is a net imbalance of up- and down-spin electrons in the system, the occupation of electronic states alone carries most of the effect of the unpaired electron(s) without including Eq. (14). The use of Mulliken charges leads to an additional hamiltonian contribution17 to the (now) shell resolved form of Eq. (10),  \n\n$$\n\\begin{array}{c}{{H_{\\mu\\nu}^{s p i n\\pm}=\\pm\\displaystyle\\frac{S_{\\mu\\nu}}{2}\\Bigg(\\sum_{l^{\\prime\\prime}\\in A\\atop l\\not\\in l\\in A}W_{A l l^{\\prime\\prime}}\\Delta p_{A l^{\\prime\\prime}}+\\sum_{l^{\\prime\\prime}\\in B}W_{B l^{\\prime}l^{\\prime\\prime}}\\Delta p_{B l^{\\prime\\prime}}\\Bigg),}}\\\\ {{\\mu\\in l\\in A,\\nu\\in l^{\\prime}\\in B,}}\\end{array}\n$$  \n\nwhere the spin up (down) hamiltonian has this term added (subtracted).  \n\nExpanding further to local (not global) up and down spin populations via Pauli spinors gives the non-collinear spin model.19 Equation (14) becomes  \n\n$$\nE_{s p i n}^{2}=\\frac{1}{2}\\sum_{A}\\sum_{l\\in A}\\sum_{l^{\\prime}\\in A}W_{A l l^{\\prime}}\\Delta\\vec{p}_{A l}\\cdot\\Delta\\vec{p}_{A l^{\\prime}},\n$$  \n\nand the wave-function generalizes to two component spinors. The hamiltonian contributions take the form  \n\n$$\n\\left(H_{\\mu\\nu}^{0}+H_{\\mu\\nu}^{2}+H_{\\mu\\nu}^{3}\\right)\\otimes{\\binom{1~0}{0~1}}+\\sum_{i=1}^{3}H_{\\mu\\nu}^{\\sigma_{i}}\\otimes\\sigma_{i},\n$$  \n\nwhere $\\sigma_{i}$ is the Pauli matrix for spin component $i(=x,\\ y,\\ z)$ and $H^{\\sigma_{i}}$ is constructed from the ith spin component of $\\Delta\\vec{p}$ . This spinblock two component hamiltonian then also enables spin–orbit cou$\\mathrm{pling}^{19,20}$ to be included in DFTB $^+$ . The spin-block hamiltonian addition is  \n\n$$\nH_{\\mu\\nu}^{L\\cdot S}=\\frac{S_{\\mu\\nu}}{2}\\otimes\\left(\\xi_{A l}{\\left(L_{z}\\begin{array}{l}{L^{-}}\\\\ {L^{+}-L_{z}}\\end{array}\\right)}_{l}+\\xi_{B l^{\\prime}}{\\left(L_{z}\\begin{array}{l}{L^{-}}\\\\ {L^{+}}\\end{array}\\right)}_{l^{\\prime}}\\right),\n$$  \n\n$$\n\\mu\\in l\\in A,\\nu\\in l^{\\prime}\\in B,\n$$  \n\nwhere $\\xi_{A l}$ is the spin orbit coupling constant for shell $l$ of atom $A$ with $L^{\\pm}$ and $L_{z}$ being the angular momentum operators for atomic shells.  \n\n# 5. Limitations of the core DFTB-model  \n\nDFTB is an approximate method, and as such shows limitations, which can be traced back to the different approximations applied. However, the fitting of Eq. (6) can compensate for some of the inaccuracies. Since until now, only bonding contributions are addressed by the two-center nature of the repulsive potentials, bond-lengths, bond-stretch frequencies, and bond-energies can be targeted (properties such as bond angles or dihedral angles cannot be influenced by repulsive pair parameterization). This is the reason why DFTB performs better than a fixed minimal basis DFT method, which would be only of limited use in most of the applications. In some cases, DFTB can even perform better than double-zeta (DZ) DFT using generalized gradient approximation (GGA) functionals, as shown, e.g., in Ref. 12. This accuracy definitely can be traced back to the parameterization.  \n\na. Integral approximations. There are some approximations in DFTB that cannot be compensated by parameterization, effecting, e.g., bond angles and dihedrals, which on average show an accuracy slightly less than DFT/DZ. Furthermore, the integral approximation leads to an imbalanced description of bonds with different bond order. For example, C–O single, double, and triple bonds have to be covered by a single repulsive potential, which shows only a limited transferability over the three bonding situations. This is the reason why both good atomization energies and vibrational frequencies cannot be covered with a single fit. Hence, in that work, two parameterizations were proposed, one for obtaining accurate energies and one for the vibrational frequencies. Similarly, description of different crystal phases with the same chemical composition but with very different coordination numbers can be challenging. Recent examples show,21,22 however, that it is possible to reach a reasonable accuracy if special care is taken during the parameterization process.  \n\nb. Minimal basis set. The minimal basis set used has several clear limitations, which show up in the overall DFTB performance: First, for a good description of hydrogen in different bonding situations, relatively diffuse wave functions have to be chosen. For this atomic wave-function, however, the $\\mathrm{H}_{2}$ atomization energy is in error, which is dealt with by an ad hoc solution, again providing a special repulsive parameter set.12 Furthermore, nitrogen hybridization and proton affinities require at least the inclusion of $d$ -orbitals in the basis set: this again can be compensated by a special parameter set, which has to be applied under certain conditions.12 A similar problem occurs for highly coordinated phosphorus containing species.23 The minimal basis can also become problematic when describing the high lying (conduction band) states in solids. For example, silicon needs $d$ -orbitals in order to describe the conduction band minimum properly. The valence band, on the other hand, can be reasonably described with an $\\boldsymbol{s p}$ -only basis.  \n\nc. Basis set confinement. As a result of the orbital confinement, Pauli repulsion forces are underestimated, which leads to DFTB non-bonding interactions being on average too short by $10\\%-15\\%$ . This has been investigated in detail for liquid water, where a different repulsive potential has been suggested.24 A related problem concerns molecular polarizabilities, which are underestimated using a minimal basis set. Approaches to correct for this shortcoming have been summarized recently in Ref. 25. The too-confined range of basis functions also impairs the calculation of electron-transfer couplings. Here, unconfined basis sets have to be used.26 Similarly, it can be challenging to find a good compromise for the basis confinement when describing 2D-layered materials. As the interlayer distances are significantly longer than the intra-layer ones, the binding between the layers often becomes weaker compared to DFT.  \n\nd. DFT inherited weaknesses. DFTB is derived from DFT and uses standard DFT functionals, which also come with some wellknown limitations. There, several strategies applied within DFT are also viable for DFTB, as discussed below in more detail.  \n\n# B. Density matrix functionals  \n\nThe typical behavior of the SCC-DFTB ground state resembles local-density approximation (LDA) or GGA,27 i.e., a mean-field (MF) electronic structure method with associated self-interaction errors and, for some systems, qualitatively incorrect ground states. This is in contrast to non-SCC DFTB, which gives the correct linearity of total energy and step-wise chemical potentials28 for fractionally charged systems. However, non-SCC can also produce MF-DFT limits, such as in the case of dimer dissociation29,30 due to self-interaction errors in the underlying atomic DFT potentials.  \n\n${\\mathrm{DFTB}}+$ now also supports long-range corrected hybrid functionals for exchange and correlation. With respect to conventional local/semi-local functionals, these are known to provide a better description of wave function localization and significantly reduce self-interaction.31 In the longer term, ${\\mathrm{DFTB}}+$ will continue to develop post-DFT based methods with the aim of making large $(\\gtrsim1000$ atom) correlated systems tractable via methods with correlated self-energies or wave-functions.  \n\n# 1. Onsite corrections  \n\nDFTB2 neglects on-site hamiltonian integrals of the type $(\\mu\\nu|\\mu\\nu)$ , where $\\phi_{\\mu}$ and $\\phi_{\\nu}$ are two different atomic orbitals of the same atom [both Eq. (5) and the use of Mulliken charges give onsite elements only for $\\delta_{\\mu\\nu}=1\\mathrm{\\ddot{\\Omega}}$ ]. A generalized dual population32 can be introduced as  \n\n$$\nQ_{\\mu\\nu}^{A,l}=\\frac{1}{2}\\sum_{\\kappa}\\bigl(\\rho_{\\mu\\kappa}\\ensuremath{S_{\\kappa\\nu}}+\\ensuremath{S_{\\mu\\kappa}}\\rho_{\\kappa\\nu}\\bigr),\\qquadl\\in A;\\mu,\\nu\\in l,\n$$  \n\nwhere $Q_{\\mu\\nu}^{A,l}$ is a population matrix for shell $l$ of atom $A$ and the diagonal of each block represents the conventional Mulliken charges for orbitals in the lth shell. Based on this population, all fluctuations of the atomic parts of the density matrix from the reference can be included, not only the diagonal (charge) elements. These must then be treated self-consistently during the calculation. This generalization leads, for example, to an improved description of hydrogen bonds in neutral, protonated, and hydroxide water clusters as well as other water-containing complexes.33  \n\nThe onsite-corrected DFTB requires additional atomic parameters; these are not tunable but computed numerically using DFT (see Ref. 34 for details of their evaluation). The onsite parameter for some chemical elements can be found in the ${\\mathrm{DFTB}}+$ manual. The calculation requires convergence in the dual density populations. This is a somewhat heavier convergence criterion than just charge convergence, and thus, the computational time is moderately affected.  \n\n# 2. $D F T B+U$ and mean-field correlation corrections  \n\nFor correlated materials such as NiO, a popular correction choice in DFT is the $\\mathrm{LDA+U}$ family of methods,35 which add a contribution to the energy of the specified local orbitals obtained from the Hubbard model. The rotationally invariant36 form of $\\mathrm{LDA+U}$ can be written in terms of several choices of local projections of the density matrix.32 Likewise, the double-counting between the Hubbard-model and the density functional mean-field functional take several limiting cases.37 In ${\\mathrm{DFTB}}+$ , the fully localized limit of this functional was implemented early in the code’s history27 using the populations of Eq. (19). Originally applied for rare-earth systems,38 $\\mathrm{DFTB+U}$ gives excellent agreement with $\\mathrm{GGA+U}$ .39 A closely related correction, pseudo-SIC,40 where the local part of the self-interaction is removed, modifying only the occupied orbitals, is also available. These approximations lower the energies of occupied atomic orbitals within the specified atomic shells with the aim of removing self-interaction or more accurately representing selfenergy. However, as with its use in DFT, this approximation suffers from three main drawbacks. First, the form of the correction depends on the choice of double counting removal.41 The correlation is also mean-field in nature; hence, all equally filled orbitals within a shell receive the same correction, and therefore, cases not well described by a single determinant are not systematically improved. Finally, the choice of the $U$ (and $J_{.}$ ) values is not necessarily obvious, with a number of different empirical, linear response, and self-consistent choices possible. Specific to DFTB,42 the $U$ values may also require co-optimization with the repulsive parameters, in particular, for systems where the electronic structure is geometrically sensitive.  \n\n# 3. Long-range corrected hybrid functionals  \n\na. Single determinant formulation. To correct longer range errors, the electron–electron interactions can be split into short and long range components based on a single parameter $\\omega$ ,  \n\n$$\n\\frac{1}{r}=\\frac{\\exp\\bigl(-\\omega r\\bigr)}{r}+\\frac{\\bigl(1-\\exp\\bigl(-\\omega r\\bigr)\\bigr)}{r}.\n$$  \n\nThe short range contribution is treated in a local or semi-local density functional approximation, while the long range term gives rise to a Hartree–Fock-like exchange term in the hamiltonian.31 The necessary adaptions for the DFTB method (termed LC-DFTB) were introduced in Refs. 43 and 44. Note that quite generally for ${\\mathrm{DFTB}}+$ , the exchange-correlation functional is effectively chosen by loading the appropriate Slater–Koster files created for the desired level of theory. This also holds for LC-DFTB, where different values for the range-separation parameter, $\\omega$ , lead to different Slater–Koster files. The database at www.dftb.org currently hosts the ob2 set45 for the elements O, N, C, and H with $\\overline{{\\omega}}=0.3~a_{0}^{-1}$ .  \n\nLC-DFTB calculations can also be performed for spin-polarized systems, enabling evaluation of triplet excited states and their corresponding relaxed geometries. It also paves the way for a rational determination and tuning31 of the range-separation parameter $\\omega$ , which amounts to total energy evaluations for neutral and singly ionized species. Note that the required atomic spin constants are functional specific. The spin parameters for the ob2 Slater–Koster set are available in the manual.  \n\n$b$ . Spin restricted ensemble references. Instead of single determinants, the spin-restricted ensemble-referenced Kohn–Sham (REKS) method and its state-interaction state-averaged variant (SISA-REKS or $\\mathsf{S S R})^{46-51}$ based on ensemble density functional theory are now available in ${\\mathrm{DFTB}}+$ . SSR can describe electronic states with multi-reference character and can accurately calculate excitation energies between them (see Sec. II C 2). The SSR method is formulated in the context of the LC-DFTB method (LC-DFTB/SSR)52 since a long-range corrected functional is crucial to correctly describe the electronic structure particularly for the excited states (see Ref. 52 for details of the formalism). Spin-polarization parameters are also required to describe open-shell microstates. It was observed that LC-DFTB/SSR sometimes gives different stability of the open-shell singlet microstates from the conventional SSR results, depending on excitation characters. In such a case, a simple scaling of atomic spin constants is helpful to account for correct excitation energies (see Ref. 52 for the required scaling of spin constants). The LC-DFTB/SSR method can be extended in the future by using larger active spaces or with additional corrections such as the onsite or DFTB3 terms.  \n\n# 4. Non-covalent interactions  \n\nIn large systems, non-covalent interactions (van der Waals/vdW forces) between molecules and between individual parts of structures become of key importance. The computational performance of DFTB makes these systems accessible, but large errors are observed for these weaker interactions. Being derived from (semi-)local density-functional theory, DFTB naturally shares the shortcomings of these approximations. This includes the lack of long-range electron correlation that translates to underestimated or missing London dispersion. An accurate account of vdW forces is essential in order to reliably describe a wide range of systems in biology, chemistry, and materials science. DFTB has already been successfully combined with a range of different correction schemes53–58 to account for these weaker interactions, but here we outline some newer methods available in DFTB $^+$ .  \n\na. H5 correction for hydrogen bonds. The H5 correction59 addresses the issue of hydrogen bonding at the level of the electronic structure. For DFTB2 and DFTB3, interaction energies of H-bonds are severely underestimated for two main reasons: most importantly, the monopole approximation does not allow on-atom polarization; even if this limitation is lifted, the use of minimal basis does not allow polarization of hydrogen. In the H5 correction, the gamma function (Sec. II A 3) is multiplied by an empirical term enhancing the interactions at hydrogen bonding distances between hydrogen atoms and electronegative elements (N, O, and S). The H5 correction is applied within the SCC cycle, thus including many-body effects (the source of the important cooperativity in H-bond networks). The H5 correction was developed for DFTB3 with the 3OB parameters and a specific version of the DFT- $\\mathrm{D}3^{60,61}$ dispersion correction. Note that this D3 correction also includes an additional term augmenting hydrogen–hydrogen repulsion at short range (necessary for an accurate description of aliphatic hydrocarbons62,63).  \n\n$b$ . DFT-D4 dispersion correction. The D4 model64,65 is now available in ${\\mathrm{DFTB}}+$ as a dispersion correction. Like D3, pairwise $C_{6}^{A B}$ dispersion coefficients are obtained from a Casimir–Polder integration of effective atomic polarizabilities $\\alpha_{A/B}^{\\mathrm{eff}}(i u)$ ,  \n\n$$\nC_{6}^{A B}=\\frac{3}{\\pi}\\int_{0}^{\\infty}\\alpha_{A}^{\\mathrm{eff}}(i u)\\alpha_{B}^{\\mathrm{eff}}(i u)\\mathrm{d}u.\n$$  \n\nThe influence of the chemical environment is captured by using a range of reference surroundings, weighted by a coordination number. D4 improves on its predecessor by also including a charge scaling based on atomic partial charges determined as either Mulliken64 or classical electronegativity equilibration.65 Especially for metalcontaining systems, the introduced charge dependence improves thermochemical properties.66 Large improvements can also be observed for solid-state polarizabilities of inorganic salts.67 For a full discussion on the methodology behind D4, we refer the reader to Ref. 65, and the implementation details are presented in Ref. 67. The damping parameters for several Slater–Koster sets are provided in the supplementary material.  \n\nTo investigate the performance of the DFTB-D4 parameterizations, we evaluate the association energies for the S30L benchmark set.68,69 DFTB-D4 is compared to DFTB3(3ob)-D3(BJ),54 GFN1-xTB,70 and GFN2-xTB;71 additionally, we include the dispersion corrected $\\mathrm{SCAN}^{72}$ functional in comparison to DFT. The deviation from the reference values is shown in Fig. 1. For the mio parameterization, complexes 4, 15, and 16 were excluded due to missing Slater–Koster parameters. The direct comparison of DFTB3(3ob)-D3(BJ) with a MAD of $7.1\\mathrm{kcal/mol}$ to the respective D4 corrected method with a MAD of $6.5\\mathrm{\\kcal/mol}$ shows a significant improvement over its predecessor. The DFTB2(mio)-D4 gives an improved description with a MAD of $4.5\\mathrm{kcal/mol}$ , which is better than GFN1-xTB with a MAD of $5.5\\mathrm{kcal/mol}$ . The best performance is reached with GFN2-xTB due to the anisotropic electrostatics and the density dependent D4 dispersion, giving a MAD of $3.6\\mathrm{kcal/mol}$ .  \n\n![](images/fd9834939fafc7cd7a95abaedd3ca90441c42e58d13d9c60264532f8d315594d.jpg)  \nFIG. 1. Performance of different dispersion corrected tight binding methods on the S30L benchmark set, and the values for SCAN-D4 are taken from Ref. 65.  \n\nc. Tkatchenko–Scheffler (TS) dispersion. The Tkatchenko– Scheffler $\\mathrm{(TS)}^{73}$ correction includes vdW interactions as Londontype atom-pairwise $C_{6}/{R}^{6}$ -potentials with damping at short interatomic separations, where the electronic structure method already captures electron correlation. The suggested damping parameters for the mio and 3ob parameter sets are listed in the supplementary material. In the TS approach, all vdW parameters including the static atomic dipole polarizability, $\\alpha$ , and $C_{6}$ -dispersion coefficients depend on the local electronic structure and the chemical environment.73 High-accuracy in vacuo reference values (labeled by vac) are rescaled via  \n\n$$\nx^{2}=\\left(\\frac{\\alpha_{A}^{\\mathrm{eff}}}{\\alpha_{A}^{\\mathrm{vac}}}\\right)^{2}=\\frac{C_{6,\\mathrm{eff}}^{A A}}{C_{6,\\mathrm{vac}}^{A A}}.\n$$  \n\nIn the case of DFT, $x$ is approximated based on the Hirshfeld atomic volumes.74 When combined with DFTB, a fast yet accurate alternative has been proposed,58 which does not require evaluating a real-space representation of the electron density. Instead, the ratio between atom-in-molecule and in vacuo net atomic electron populations [i.e., $\\operatorname{tr}(\\rho)_{A}/Z_{A}]$ is used to define $x$ .  \n\nd. Many-body dispersion (MBD). Going beyond pairwise interactions, many-body dispersion $\\mathrm{(MBD)}^{75,76}$ accounts for manyatom interactions in a dipolar approximation up to infinite order in perturbation theory. This is achieved by describing the system as a set of coupled polarizable dipoles75 with rescaled in vacuo reference polarizabilities [as in Eq. (22)]. At short-ranges, this model switches, via a Fermi-like function with a range of $\\beta$ , to the local atomic environment as accounted for by solving a Dyson-like selfconsistent screening equation. $^{76}\\beta$ represents a measure for the range of dynamic correlation captured by the underlying electronic structure method, so it depends on the density functional or DFTB parameterization. The recommended $\\beta$ -values for the mio and 3ob parameter sets are listed in the supplementary material.  \n\n# C. Excited states and property calculations  \n\n# 1. Time dependent DFTB with Casida formalism  \n\nElectronic excited states are accessible in DFTB $^+$ through time dependent DFTB methods (see Ref. 86 for a review and detailed discussion of this formalism). In a linear response treatment in the frequency domain, excitation energies are obtained by solving an eigenvalue problem known as Casida or RPA (random phase approximation) equations. Compared to first-principles time dependent DFT, the computational scaling can be reduced in DFTB from $N^{6}$ to $N^{3}$ . This is due to the Mulliken approximation for two-electron integrals,87 which uses transition charges $q_{A}^{p q\\sigma}$ ,  \n\n$$\nq_{A}^{p q\\sigma}=\\frac{1}{4}\\sum_{\\mu\\in A}\\sum_{\\nu}\\Bigl(c_{\\mu p}^{\\sigma}\\tilde{c}_{\\nu q}^{\\sigma}+c_{\\mu q}^{\\sigma}\\tilde{c}_{\\nu p}^{\\sigma}+c_{\\nu p}^{\\sigma}\\tilde{c}_{\\mu q}^{\\sigma}+c_{\\nu q}^{\\sigma}\\tilde{c}_{\\mu p}^{\\sigma}\\Bigr),\\tilde{\\mathbf{c}}_{p}=\\mathbf{c}_{p}\\cdot\\mathbf{S},\n$$  \n\nfor transitions from the Kohn–Sham orbital $p\\sigma$ to $q\\sigma$ .  \n\nFor fixed geometry, ${\\mathrm{DFTB}}+$ provides a user defined number of low lying excitation energies, oscillator strengths, and orbital participations. In another mode of operation, the code computes excited state charges, eigenvectors of the Casida equation, and energy gradients for a specific state of interest, which can be combined with  \n\n![](images/874b73f1cc463e99b5f89d98885309926d0824ec5faeba06cba9d0f45fac2715.jpg)  \nFigure 2 and Ref. 58 demonstrate that DFTB and MBD represent a promising framework to accurately study long-range correlation forces and emergent behavior at larger length- and timescales. Recently, the DFT $\\ensuremath{\\mathrm{\\3+MBD}}$ approach has allowed the study of organic molecular crystals55 and solvated biomolecules, revealing the complex implications of many-body vdW forces for proteins and their interaction with aqueous environments.82 Further improvements of TS and MBD, including a better description of charge transfer effects83 and variational self-consistency,84 may also be incorporated into DFTB in the future. Both methods are formulated independently of the underlying electronic-structure methods. As a result, DFTB $^+$ outsources the evaluation of the MBD and TS interactions to Libmbd,85 an external open-source library.   \nFIG. 2. Mean absolute errors (MAEs) and mean absolute relative errors (MAREs) in inter-molecular interaction energies of bare DFTB and with different van der Waals models in comparison to high-level reference data. S66 and $S66\\times8$ : small organic dimers and their dissociation curves,77,78 SMC13: set of 13 supra-molecular complexes.79–81  \n\nMD or geometry relaxation. For spin-unpolarized calculations, the response matrix is block diagonal for the singlet and triplet channels to speed up the computation. ${\\mathrm{DFTB}}+$ allows for the computation of the excited state properties of systems with general fractional occupation of the KS orbitals. This is useful, for example, for the simulations of metals and semi-metals at a finite temperature. For a detailed discussion on spin-polarization and fractional occupation within time dependent (TD) DFTB, see Ref. 34. The onsite correction, discussed in Sec. II B 1, is also possible for excited state calculations and was shown to lead to marked improvements.34  \n\nDue to their improved treatment of charge-transfer transitions, range-separated functionals are also relevant in the context of excited states. ${\\mathrm{DFTB}}+$ implements the time dependent long range corrected (TD-LC) DFTB method, as described in Ref. 88. Compared to the conventional TD-DFTB, the lower symmetry of the response matrix leads to a non-Hermitian eigenvalue problem, which we solve by the algorithm of Stratmann and co-workers.89 Somewhat surprisingly, it turns out that TD-LCDFTB calculations are, in practice, not significantly slower than TD-DFTB calculations (see Ref. 88 for a deeper discussion). Gradients can also be calculated with TD-LC-DFTB, making it possible to perform geometry optimizations and MD simulations in singlet excited states.  \n\nNote that energetically high lying states and Rydberg excitations are clearly outside of the scope of TD(-LC)-DFTB since their description generally requires very diffuse basis sets. Apart from this class, the photochemically more relevant set of low energy valence excitations are predicted with similar accuracy to first principles TD-DFT, as several benchmarks indicate.34,90,91 As mentioned above, charge-transfer excitations can now be also treated using TD-LC-DFTB.88  \n\n# 2. SSR and excitations  \n\nCurrently, the SSR method implemented in ${\\mathrm{DFTB}}+$ is formulated for active spaces including two electrons in two fractionally occupied orbitals [i.e., SSR(2,2)], which is suitable for a singlet ground state and the lowest singlet excited state as well as a doubly excited state.52 In addition, since the SSR method is based on an ensemble representation and includes the electronic correlation, it can give correct state-interactions among nearly degenerate electronic states. Thus, the SSR approach is useful to investigate conical intersections. The LC-DFTB/SSR method with scaled spin constants can accurately describe the ground and excited states including $\\pi/\\pi^{*}$ or $n/\\pi^{*}$ character, undergoing bond cleavage/bond formation reactions as well as the conical intersections where the conventional (TD)DFTB fails to obtain the electronic properties. Analytic energy gradients as well as non-adiabatic couplings are also available.  \n\n# 3. Time-independent excited states from ΔDFTB  \n\nThe linear response approach to excited-state properties in DFTB is efficient and powerful, but there exist circumstances where a more direct route to the excited states is desirable. For example, the excited-state properties obtained from linear response theory require an additional order of derivatives relative to the ground state. As noted in Sec. II C 1, linear-response TD-DFTB (like its parent method TD-DFT)92 should invoke range-separation to achieve a qualitatively correct picture of charge-transfer excitations and related long-range phenomena.88  \n\nAs an alternative to the time-dependent linear-response approach, it is possible to variationally optimize certain electronically excited states directly. The ΔDFTB method, modeled on the $\\Delta$ -self-consistent-field (ΔSCF) approach to excited states in DFT,93,94 involves solving the SCC-DFTB equations subject to an orbital occupation constraint that forces the adoption of a non-aufbau electronic configuration consistent with the target excited state. This method is implemented for the lowest-lying singlet excited state of closedshell molecules in ${\\mathrm{DFTB}}+$ .95 The converged, non-aufbau SCC-DFTB determinant is a spin-contaminated or “mixed” spin state, but the excitation energy can be approximately spin-purified through the Ziegler sum rule, which extracts the energy of a pure singlet from the energies of the mixed state and the triplet ground state.  \n\nA significant advantage of the ΔDFTB approach is that excitedstate gradients and hessians are quite straightforward to compute, both mathematically and in terms of computational cost, relative to linear response approaches. Benchmarks of ΔDFTB excitedstate geometries and Stokes shifts95 demonstrate the suitability of the method for simulating excited-state energetics and dynamics of common organic chromophores along the $S_{1}$ potential energy surface.  \n\n# 4. Real-time propagation of electrons and Ehrenfest dynamics  \n\nIt is often desirable to study time dependent properties outside the linear response regime, e.g., under strong external fields. The numerical propagation of the electronic states enables the simulation of such phenomena, and its coupling to the nuclear dynamics in a semi-classical level can be included to the lowest order within the Ehrenfest method. Purely electronic (frozen-nuclei) dynamics as well as Ehrenfest dynamics are included in ${\\mathrm{DFTB}}+$ . We solve the equation of motion of the reduced density matrix $\\rho$ given by the Liouville-von Neumann equation  \n\n$$\n\\dot{\\rho}=-\\mathrm{i}\\Bigl({\\cal S}^{-1}H[\\rho]\\rho-\\rho H[\\rho]{\\cal S}^{-1}\\Bigr)-\\Bigl({\\cal S}^{-1}D\\rho+\\rho D^{\\dagger}{\\cal S}^{-1}\\Bigr),\n$$  \n\nwith $D$ being the non-adiabatic coupling matrix $D_{\\mu\\nu}=\\dot{\\bf R}_{B}\\cdot\\nabla_{B}S_{\\mu\\nu}$ and $\\dot{\\bf R}_{B}$ being the velocity of atom $B$ . The on-site blocks can be calculated taking the $\\mathbf{R}_{B}\\rightarrow0$ limit, although neglecting those does not introduce significant ch→anges to the dynamics.96  \n\nUnitary evolution of $\\rho$ with no change in its eigenvalues would require $D^{\\dagger}=-D_{:}$ , which is normally not the case. Therefore, nuclear dynamics can induce electronic transitions leading to thermalization.97 Unitary evolution is recovered when all nuclear velocities are equal (frozen-nuclei dynamics) and the second term in Eq. (24) vanishes.  \n\nThe force in the Ehrenfest-dynamics can be expressed as96,98  \n\n$$\n\\begin{array}{r l r}{\\lefteqn{\\mathbf{F}_{A}=-\\mathrm{Tr}\\Bigg\\{\\rho\\Bigg(\\bigtriangledown_{A}H^{0}+\\bigtriangledown_{A}S\\sum_{B}\\gamma_{A B}\\Delta q_{B}+\\bigtriangledown_{A}S S^{-1}H+H S^{-1}\\nabla_{A}S\\Bigg)\\Bigg\\}}}\\\\ &{}&{\\quad-\\mathrm{~i~}\\mathrm{Tr}\\big\\{\\rho\\nabla_{A}S S^{-1}D+\\mathrm{h.c.}\\big\\}+\\mathrm{i}\\ \\sum_{\\mu\\nu}\\big\\{\\rho_{\\nu\\mu}\\big\\langle\\nabla_{A}\\phi_{\\mu}\\big|\\nabla_{B}\\phi_{\\nu}\\big\\rangle\\cdot\\dot{\\mathbf{R}}_{B}+\\mathrm{h.c.}\\big\\}}\\\\ &{}&{\\quad-\\Delta q_{A}\\sum_{B}\\nabla_{A}\\gamma_{A B}\\Delta q_{B}-\\bigtriangledown_{A}E_{r e p}-\\Delta q_{A}\\mathbf{E}(t),\\quad\\quad\\quad(25)}\\end{array}\n$$  \n\nwhere $\\mathbf{E}(t)$ is the external electric field. In the present implementation, the velocity dependent terms have been neglected, and they would vanish for a complete basis96 and are necessary for momentum, but not for energy conservation.98 When the system is driven externally by an electric field, a dipole coupling term is added in the time-dependent hamiltonian in Eq. (24).  \n\nSome applications that have been enabled by the speedup over time-dependent DFT are the simulation of the plasmon-driven breathing-mode excitation in silver nanoparticles of $1{-}2\\ \\mathrm{nm}$ in diameter99 and the simulation of transient absorption pump–probe spectra in molecules.100,101  \n\nWhenever a time propagation approach is used for the calculation of absorption spectra in the linear regime, this method is equivalent to calculations using the Casida formalism and shares its strengths and limitations. Specific pitfalls of the time dependent approach come into play whenever simulating the response to intense external fields. In these cases, the poor description of highly lying excited states due to the use of a minimal basis set would likely be inaccurate if these states are populated during the dynamics.  \n\n# 5. pp-RPA  \n\nAn approximate particle–particle RPA scheme, the so-called pp-DFTB,88 is now implemented in ${\\mathrm{DFTB}}+$ . Particle–particle RPA, based on the pairing matrix fluctuation formalism, has been shown to be an efficient approach for the accurate description of double and charge-transfer (CT) excitations involving the highest occupied molecular orbital (HOMO) (see Ref. 102 for details). In Ref. 88, we compare against TD-LC-DFTB for CT excitation energies of donor– acceptor complexes. TD-LC-DFTB has the advantage that transitions do not necessarily have to involve the HOMO of the system. Alternatively, pp-DFTB does not require parameter tuning and is efficient for the lowest lying excitations.  \n\nAlthough one of the strengths of the original pp-RPA formulation lies on the accurate description of Rydberg excitations, our approximate formalism based on DFTB fails to describe these kinds of transitions, as explained in Sec. II C 1.  \n\n# 6. Coupled perturbed responses  \n\n${\\mathrm{DFTB}}+$ supports several types of response calculations for second-order derivatives. The general form of the response evaluation is via standard perturbation theory,  \n\n$$\nP_{i j}=\\Big\\langle c_{i}\\Big|H_{i j}^{(1)}-\\epsilon_{j}S_{i j}^{(1)}\\Big|c_{j}\\Big\\rangle,\n$$  \n\n$$\n\\epsilon_{i}^{(1)}=P_{i j}\\delta_{i j},\n$$  \n\n$$\nU_{i j}=P_{i j}/{\\left(\\epsilon_{j}-\\epsilon_{i}\\right)},\n$$  \n\n$$\nc_{i}^{(1)}=\\sum_{j}U_{i j}c_{j}^{(0)},\n$$  \n\n$$\n\\rho^{(1)}=\\sum_{i}n_{i}^{(1)}\\Big\\vert{c^{(0)}}\\Big\\rangle\\Big\\langle{c^{(0)}}\\Big\\vert+\\sum_{i}n_{i}^{(0)}\\Big(\\Big\\vert{c^{(1)}}\\Big\\rangle\\Big\\langle{c^{(0)}}\\Big\\vert+\\mathrm{~c.c.~}\\Big),\n$$  \n\nwhere the sums for the states that $U$ mixes together may be over all states or only the virtual space (parallel gauge) depending on application. $U$ is anti-symmetric (anti-Hermitian) or has no symmetry depending on whether the derivative of the overlap matrix is non-zero.  \n\nIn the case of systems with degenerate levels, a unitary transformation, $Z,$ , that diagonalizes the block of $P$ associated with that manifold can be applied to the states; note that this sub-block is always symmetric (Hermitian), leading to orthogonality between states in the perturbation operation,  \n\n$$\n\\tilde{P}_{i j}=z_{i k}P_{k l}z_{l i}^{\\dagger},\n$$  \n\n$$\n\\tilde{c}_{i}=c_{j}z_{j i}.\n$$  \n\nFor fractionally occupied levels, the derivatives of the occupations for $\\mathbf{q}=0$ perturbations (where the change in the Fermi energy should be included) are then evaluated.103  \n\nTime dependent perturbations at an energy of hω can be written as  \n\n$$\nU_{i j}^{\\pm}={P_{i j}}/{\\left({{\\epsilon_{j}}-{\\epsilon_{i}}\\pm\\hbar\\omega+i\\eta}\\right)},\n$$  \n\n$$\nc_{i}^{(1)\\pm}=\\sum_{j}U_{i j}^{\\pm}c_{j}^{(0)},\n$$  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\rho^{(1)}=\\sum_{i}n_{i}^{(1)}\\Big\\vert c^{(0)}\\Big\\rangle\\Big\\langle c^{(0)}\\Big\\vert+\\sum_{i}n_{i}^{(1)}\\Big\\vert c^{(0)}\\Big\\rangle\\Big\\langle c^{(0)}\\Big\\vert}}}\\\\ {{\\displaystyle{+\\sum_{\\pm}\\sum_{i}n_{i}^{(0)}\\Big(\\Big\\vert c^{(1)\\pm}\\Big\\rangle\\Big\\langle c^{(0)}\\Big\\rangle+\\mathrm{c.c.}\\Big)}.}}\\end{array}\n$$  \n\nHere, the small constant $\\eta$ prevents divergence exactly at excitation poles.  \n\nDerivatives with respect to external electric fields and potentials are included (giving polarizabilities and dipole excitation energies), with respect to atom positions (at $\\mathbf{q}=0$ , providing Born charges and electronic derivatives for the hessian) and with respect to $k$ in periodic systems (effective masses and also the Berry connection via $\\left\\langle u\\right|{\\partial u}/{\\partial\\mathbf{k}}\\right\\rangle$ ). In the longer term, perturbation with respect to magnetic fields, boundary conditions (elastic tensors), and alternative approaches (Sternheimer equations for $\\mathbf{q}\\neq0$ , and also lower computationally scaling density matrix perturbation theory) are planned.  \n\n# D. Non-equilibrium Green’s function based electron transport  \n\nElectron transport in the steady-state regime is described in ${\\mathrm{DFTB}}+$ within a non-equilibrium Green’s function (NEGF) method,104,105 as implemented in the code-independent libNEGF106 library. The density matrix is evaluated in terms of the electron– electron correlation matrix G<,105  \n\n$$\n\\rho=\\frac{1}{2\\pi\\mathrm{i}}\\int_{-\\infty}^{+\\infty}G^{<}(E)\\mathrm{d}E.\n$$  \n\nOpen boundary conditions are included in terms of electron baths with an arbitrary spectrum and chemical potential, allowing for a seamless description of charge injection from electrodes with an applied bias. The density matrix is then used to evaluate a real-space electron density distribution, which is coupled self-consistently with a Poisson solver. We perform a full band integration of Eq. (36), utilizing a complex contour integral to reduce the number of integration points.104 This allows for an implicit description of dielectric properties, which is crucial for an accurate modeling of ultra-scaled electron devices.107,108 After self-consistency is achieved, the total current flowing in the system is calculated with the Landauer/Caroli formula for the non-interacting case or with the Meir–Wingreen formula for the interacting case.105 A detailed description of the numerical algorithms and self-consistent coupling is presented in Ref. 109. Here, we summarize the main features that might differentiate ${\\mathrm{DFTB}}+$ from other nano-device simulation packages: (i) support for $N\\geqslant1$ electrodes (enabling structures from surfaces and semi-infinite wires to multiple terminal geometries), (ii) $O(L)$ memory and time scaling (where $L$ is the system length) via a block-iterative algorithm, (iii) a real space Poisson solver with support for gates and dielectric regions, and (iv) evaluation of local currents. Being a parameterized tight binding method, its usage is bounded by the availability of good parameters for the system under investigation.  \n\nCarbon-based materials and molecular junctions have been a typical use-case since the first integration of DFTB and NEGF.110–112 In Fig. 3, we show a non-SCC calculation example of transmission in linear response for a multi-terminal device. The simulated system is a cross-junction between two (10,10) Carbon nanotubes (CNTs). One CNT is tilted by ${60}^{\\circ}$ with respect to the second, and the transmission is calculated by displacing one CNT along the axis of the other. The transmission follows, as expected, a periodic pattern in accordance with the lattice repeat of $0.25~\\mathrm{{nm}}$ along the axis of the CNT.  \n\nCurrently, we are working on extending transport functionality in ${\\mathrm{DFTB}}+$ with electron–phonon coupling ,113–116 electron–photon coupling, spin polarized transport, and phonon transport.117–120  \n\nOverall, DFTB-NEGF shares many similarities with DFT based implementations, and it also inherits some shortcomings the less experienced users should be aware of. For example, the open boundary treatment demands that external and non-equilibrium potentials are screened at the boundaries.105 Therefore, the simulated system should be large enough compared to the screening length. This condition is easily achieved with bulk metallic electrodes, but it can be difficult with low dimensional systems that exhibit poor screening.  \n\nWhen this condition is not fulfilled, unphysical discontinuities in the potential may be obtained. In addition, compared to band structure calculations, NEGF tends to converge with more difficulty.121 Aside from these common challenges, it is important that for DFTB-NEGF calculations, any set of parameters should be evaluated by verifying at the least band structure properties in the energy range of interest. DFTB parameters fitted to reproduce total energies and forces might be excellent in those applications but lack the necessary accuracy in the band structure. Depending on the degree of accuracy required, an ad hoc fitting for transport calculations could also be necessary, for example, in the case of silicon.122  \n\n# E. Extended Lagrangian Born–Oppenheimer dynamics  \n\nThe Extended Lagrangian Born–Oppenheimer molecular dynamics (XLBOMD) framework allows123,124 molecular dynamics on the Born–Oppenheimer surface with only one hamiltonian diagonalization per time step without the need for self-consistency cycles. The basic idea is based on a backward error analysis, i.e., instead of calculating approximate forces through an expensive non-linear iterative optimization procedure for an underlying exact potential energy surface, XL-BOMD calculates exact forces for an approximate “shadow” potential energy surface, ${\\cal U}({\\bf R},n)$ . This is approximated from a constrained minimization of a linearized Kohn–Sham energy functional.124,125 The functional is linearized around an approximate ground state density, $n$ . This density is included as a dynamical field variable driven by an extended harmonic oscillator centered on an approximate ground state, $q[n]$ , which is given by the minimization of the linearized Kohn–Sham functional. The harmonic well is defined in terms of a metric tensor, $\\boldsymbol{T}=\\boldsymbol{K}^{T}\\boldsymbol{K},$ where the kernel $K$ is assumed to be the inverse Jacobian of the residual function, $q[n]-n$ .124 The equations of motion are given by  \n\n![](images/66beac2baea0fede9e4b40c89637bc408e6db2e618574cb5eb809f8587984ca6.jpg)  \nFIG. 3. Transmission across two (10,10) CNTs as a function of the displacement of the top CNT along the axis of the bottom CNT. The two curves represent the transmission resolved between electrode 1 of the bottom CNT, and, respectively, electrodes 2 and 3 of the top CNT (as labeled in the inset).  \n\n$$\nM_{I}\\ddot{\\mathbf{R}}_{I}=-\\frac{\\partial U(\\mathbf{R},n)}{\\partial R_{I}}\\bigg|_{n}\\quad\\mathrm{and}\\quad\\ddot{n}=-\\omega^{2}K(q[n]-n).\n$$  \n\nHere, $M_{I}$ are the atomic masses, $\\mathbf{R}_{I}$ are the nuclear coordinates, $\\omega$ is the frequency of the harmonic oscillator, $q[n]$ are the net Mulliken charge vectors (from an optimized linearized energy expression), and $n$ is the extended dynamical variable that is set to the optimized ground state net Mulliken charge vector in the initial time step. The details of the DFTB $^+$ implementation are given in Ref. 126.  \n\nWe currently approximate the kernel by a scaled identity matrix,  \n\n$$\nK=-c I,c\\in[0,1].\n$$  \n\nFor many problems, this is a sufficiently accurate approximation. However, for the most challenging problems including simulations of reactive chemical systems or metals, the scaled delta function is not a sufficiently stable approximation. Improved approximations have been developed124 and will be implemented in the ${\\mathrm{DFTB}}+$ program in the near future.  \n\n# F. Objective geometries  \n\nObjective structures127 (OSs) describe geometries consisting of a set of identical cells, where the corresponding atoms in different cells can be mapped onto each other by orthogonal transformation(s). Both finite and infinite OSs are possible. Currently, we describe structures127–129 possessing $C_{n}$ rotational symmetry and a $C_{m}\\otimes T$ screw axis, where $\\boldsymbol{n}\\in\\mathbb{N}^{*}$ and $m\\in\\mathbb{R}^{+}$ ,  \n\n$$\n{\\bf X}_{i,\\zeta,\\xi}=\\left(C_{n}\\right)^{\\xi}\\left(C_{m}\\right)^{\\zeta}{\\bf X}_{i}+T^{\\zeta},\\quad i\\in N,\n$$  \n\nwith $N$ atoms in the reference objective cell $(\\{{\\bf X}_{i}\\})$ and $\\{\\zeta,\\xi\\}\\in\\mathbb{N},$ where $-\\infty<\\zeta<\\infty$ and $0<\\xi<n$ . Exploiting the objec i{ve b}ou∈ndary conditions (OBCs) can introduce substantial computational savings, for example, irrational values of $m$ lead to structures with a small OS cell, but an infinitely long one dimensional periodic boundary condition (PBC), i.e., intractable purely as a $T$ operation. OBCs generalize symmetry-adapted Bloch sums for orbitals. As with molecular and periodic structures, most expressions in ${\\mathrm{DFTB}}+$ can be performed in real space via the boundary-condition agnostic and sparse representation of matrices in real space, the solution of the hamiltonian only requires dense matrices and $k$ -points. For the long-range Coulombic and dispersion interactions in DFTB, we also require lattice sums that are generalized to these boundary conditions.130  \n\nFurther examples can be found in Refs. 131–133, but here we demonstrate the bending of a BN bi-layer. Figure 4 shows a doublewalled tubular OS with a curvature of $1/R$ (from the tube radius) that represents the bent bi-layer. Bending along the a ${\\bf(b)}$ direction of the sheet is an “armchair” (“zig-zag”) tube with a $C_{n}$ proper axis, described as an eight atom objective cell in which we select $\\mathbf{T}=\\mathbf{a}$ $(\\mathbf{T}=\\mathbf{b})$ , with no tube twist. The bi-layer bends as a plate, with the outer wall stretching and the inner wall compressing along their circumferential directions; its energy change is interpreted as bending strain $(E_{\\mathrm{bend}})$ . It is important to note that the corresponding curvature is not an imposed constraint, but a result of the calculation: R is the average tube radius. Figure 4(b) demonstrates linearity with bending, and fitting to $E_{\\mathrm{bend}}=(1/2)D(|\\mathbf{a}||\\mathbf{b}|)(1/R)^{2}$ gives a bi-layer bending constant of $D=120\\mathrm{eV}$ .  \n\n![](images/eb064aedafac13dbcff61852cf18bae33bf0d7c14e4bf86dbf0969940f17b458.jpg)  \nFIG. 4. (a) OS of a BN bi-layer tube with a $B_{4}N_{4}$ unit (red and blue atoms). Angular, but not translational, objective images are shown in gray. (b) Bending energy (circles) vs curvature with a linear fit.  \n\nA wider range of OSs will be made available in later ${\\mathrm{DFTB}}+$ releases, along with adapted electrostatic evaluation for these structures.  \n\n# G. Extended tight binding hamiltonian  \n\nThe extended tight binding $\\left(\\mathbf{x}\\mathbf{\\mathrm{T}}\\mathbf{B}\\right)$ methods were primarily designed for the fast calculation of structures and non-covalent interaction energies for finite systems with a few thousand atoms. The main parameterizations, GFNn-xTB, target molecular geometries, frequencies, and non-covalent interactions follow mostly a global and element-specific parameter only strategy. The historically first parameterization, GFN1-xTB, covers all elements up to $Z=86$ and is now supported in ${\\mathrm{DFTB}}+$ . Its successor, GFN2-xTB,71 will also be made available in the future.  \n\nWe briefly outline the $\\mathbf{xTB}$ methods; for a more detailed discussion and comparison to other methods, we refer to Refs. 70 and 71. The xTB core hamiltonian is constructed in a partially polarized STO- $_{\\cdot n G}$ basis set with diagonal terms made flexible by adding a dependence on the local chemical environment according to a coordination number (CN), similar to that used in DFT-D3,60  \n\n$$\nH_{\\lambda\\lambda}=H_{A}^{l}-H_{C N_{A}}^{l}C N_{A}.\n$$  \n\nThe off-diagonal terms are approximated as an average of the diagonal terms proportional to the overlap between the corresponding basis functions.  \n\nBoth GFN1-xTB and GFN2-xTB include density fluctuation up to third order diagonal terms, while the distance dependence of the Coulomb interaction within the isotropic second order term is described by a generalized form of the Mataga–Nishimoto–Ohno– Klopman134–136 expression. In GFN2-xTB, the expansion of the second order density fluctuations goes beyond the usual isotropic energy terms and includes interactions up to $\\boldsymbol{R}^{-3}$ , i.e., charge–dipole, dipole–dipole, and charge–quadrupole interactions, which significantly improves the description of inter-molecular interactions, such as halogen bonds and hydrogen bonds, without the need to include force-field-like corrections as in DFTB or GFN1-xTB. It is planned to implement full multipole electrostatics with Ewald summation in DFTB $^+$ to enable GFN2-xTB and other generalized DFTB models.137  \n\nGFN1-xTB and GFN2-xTB have been extensively tested for their target properties,71 and further studies regarding structures for lanthanoid complexes138 and transition metal complexes66 have shown xTB methods to be robust for all its parameterized elements. Errors in these methods are very systematic, which can be used to devise correction schemes for off-target properties such as reaction enthalpies.139  \n\n# H. DFTB parameterization  \n\n# 1. Parameterization workflow  \n\na. Electronic parameters. The electronic parameterization for DFTB involves two principal steps. First, the compressed atomic densities and the atomic basis functions have to be determined (a one-center problem), followed by the calculation of the hamiltonian and overlap elements at various distances (a two-center problem). The compressed densities and wave-functions come from solving a Kohn–Sham-problem for a single atom with an additional confinement potential (usually a power function), as shown in Eq. (4). One may use different compression radii (and make separate calculations) to obtain the compressed density and the compressed atomic wave-functions for a given atom. The atomic calculations are currently carried out with a code implementing the Hartree– Fock theory based atomic problem140,141 extended with the possibility of including DFT exchange-correlation potentials via the libxc library142 and scalar relativistic effects via the zero-order relativistic approximation (ZORA).143 The resulting densities and atomic wave-functions are stored on a grid. The two-center integration tool reads those grid-based quantities and calculates the hamiltonian and overlap two-center integrals for various distances using the Becke-method.144  \n\nb. Repulsive parameters. Once the electronic parameters for certain species have been determined, the first three terms of Eq. (9) can be calculated for any systems composed of those species. The missing fourth term, the repulsive energy, is composed of pairwise contributions, $V_{A B}^{\\mathrm{rep}}$ , between all possible atomic pairs of $A$ and $B$ in the system [see Eq. (6)]. During the parameterization process, one aims to determine repulsive potentials between the atomic species as a function of the distance between the atoms $R_{A B}$ so that $V_{A B}^{\\mathrm{rep}}\\ =f_{\\mathrm{sp(}A\\mathrm{),sp(B)}}\\big(R_{A B}\\big)$ , where $\\operatorname{sp}(X)$ refers to the species of atom $X$ . I  contrast to the electronic parameters, which are determined by species-specific parameters only, the repulsive functions must be defined for each combination of species pairs separately. They are usually determined by minimizing the difference between the reference (usually ab initio) total energies and the DFTB total energies for a given set of atomic geometries. If one uses only one (or a few simple) reference system(s), the optimal repulsive function can be determined manually, while for more complex scenarios, usually semi-automatic approaches15,21,145–147 a re used.  \n\n# 2. Outlook  \n\nIn recent years, machine learning has been utilized with ${\\mathrm{DFTB}}+$ , usually to enhance the generation and description of the repulsive potentials148–152 or try to improve on electronic parameters.151,153 Related $\\Delta$ -machine learning154 methodologies based on neural network corrections for DFTB energies and forces have been also reported recently.155,156 We are currently in the process of developing a new unified machine-learning framework, which for a target system allows optimal adaption of both the electronic and the repulsive contributions. Given the predicted DFTB model, one would still have to solve it in order to obtain the system properties. On the other hand, changing external conditions (temperature, electric field, and applied bias) would not require additional training in this approach, and long range effects (e.g., metallic states) could also be described easily.  \n\n# III. TECHNICAL ASPECTS OF THE DFTB $^+$ PACKAGE  \n\n# A. Parallel scaling  \n\nIn large-scale simulations, the solution of the DFTB hamiltonian to obtain the density matrix eventually becomes prohibitively expensive, scaling cubically with the size of the system being simulated. The diagonalization infrastructure in ${\\mathrm{DFTB}}+$ has undergone a major upgrade, including distributed parallelism and GPU accelerated solutions to address this cost. If instead the density matrix is directly obtained from the hamiltonian, circumventing diagonalization, then linear or quadratic scaling can now be obtained, depending on the chosen method. ${\\mathrm{DFTB}}+$ will continue to benefit from developments in these advanced solvers as we move into the era of exascale computing.  \n\n# 1. The ELSI interface and supported solvers  \n\n$\\mathrm{ELSI}^{157}$ features a unified software interface that simplifies the use of various high-performance eigensolvers (ELPA158 EigenExa,159 SLEPc,160 and MAGMA161) and density matrix solvers (libOMM,162 PEXSI,163 and $\\mathrm{NTPoly}^{164})$ ). We convert the sparse $\\mathrm{DFTB}+H$ and $s$ structures8 into either standard 2D block-cyclic distributed dense matrices or sparse 1D block distributed matrices compatible with the ELSI interface. All $k$ -points and spin channels are then solved in parallel.  \n\nThe ELSI-supported solvers, when applied in appropriate cases, can lead to a substantial speedup over the default distributed parallel diagonalization method in ${\\mathrm{DFTB}}+.$ , i.e., eigensolvers in the ScaLAPACK library.165–167 Figure 5 demonstrates two examples: non-self-consistent-charge, spin-non-polarized, Γ-point calculations for a $\\mathrm{C}_{64000}$ nanotube (CNT) and a $\\mathrm{Si}_{6750}$ supercell, with 25 600 and $27000$ basis functions, respectively. Figure $5(\\mathrm{c})$ shows the time to build the density matrix for the CNT model with three solvers, the pdsyevr eigensolver in the MKL implementation of ScaLAPACK, the ELPA2 eigensolver, and the PEXSI density matrix solver. Here, both the MKL’s version of pdsyevr eigensolver and the ELPA2 eigensolver adopt a two-stage tri-diagonalization algorithm.158,168,169 In terms of performance, ELPA2 and MKL pdsyevr are similar, while both are outperformed by the PEXSI solver by more than an order of magnitude. The PEXSI163 method directly constructs the density matrix from the hamiltonian and overlap matrices with a computational complexity of $O(N^{(d+1)/2})$ for $d=1\\ldots3\\mathrm{D}$ systems. This reduced scaling property stems from sparse linear algebra, not the existence of an energy gap. Therefore, for any low-dimensional system, regardless of the electronic structure, PEXSI can be used as a powerful alternative to diagonalization. A similar comparison of solver performance for the silicon supercell model is shown in Fig. 5(d), where the NTPoly density matrix solver shows greater performance than the MKL pdsyevr and ELPA2 eigensolvers. Around its massively parallel sparse matrix multiplication routine, NTPoly implements various linear scaling density matrix purification methods, including the 2nd order trace-resetting purification method (TRS2)170 used here. While PEXSI is not particularly suited for 3D systems, NTPoly offers an alternative as long as the system has a non-trivial energy gap.  \n\n![](images/61232cd60f7248630bb45805c750f919809675d84b5846c70ac5c050a53e6262.jpg)  \nFIG. 5. Atomic structures of (a) the carbon nanotube (CNT) model (6400 atoms) and (b) the silicon supercell model (6750 atoms). The length of the actual CNT model is 16 times that of the structure shown in (a). (c) and (d) show the time to compute the density matrix for models (a) and (b), respectively. Calculations are performed on the NewRiver computer. MKL pdsyevr and ELPA2 first compute all the eigenvalues and eigenvectors of the eigensystem of $H$ and S and then build the density matrix. PEXSI and NTPoly directly construct the density matrix from $H$ and S.  \n\n# B. Order-N scaling with the SP2 solver  \n\nThe SP2 (second-order recursive spectral projection expansion),170 which is valid at zero electronic temperature when $1/k_{B}T\\rightarrow\\infty$ , recursively expands a Heaviside step function to project the (occupied) density matrix,  \n\n$$\n\\rho=\\operatorname*{lim}_{n\\to\\infty}F_{n}\\big(F_{n-1}\\big(\\ldots{}F_{0}\\big(H^{\\bot}\\big)\\ldots\\big)\\big),\n$$  \n\nwhere $H^{\\perp}$ is the hamiltonian transformed into an orthogonalized basis, given by the congruence transformation, $H^{\\perp}=Z^{T}H Z$ . Each iteration of the SP2 Fermi-operator expansion consists of a generalized matrix–matrix multiplication that can be performed using thresholded sparse matrix algebra. In this way, the computational complexity in each iteration can be reduced to $O(N)$ for sufficiently large sparse matrices. Note that we cannot expect linear scaling complexity for metals, since the inter-atomic elements of the density matrix decay algebraically instead of exponentially.171 The spectral projection functions in the SP2 expansion can be chosen to correct ${\\mathrm{Tr}}(\\rho)$ such that the step is formed automatically around the chemical potential separating the occupied from the unoccupied states.170 Obtaining the congruence matrix, $Z,$ introduces a potential $O(N^{3})$ bottleneck. To avoid this, the sparsity of S can be exploited and the $Z$ matrix can be obtained recursively with linear scaling complexity applying the “ZSP method” developed in Refs. 172 and 173.  \n\nSeveral versions of the SP2 algorithm can be found in the PROGRESS library,174 which uses the Basic Matrix Library $\\mathbf{\\left(BML\\right)}^{175,176}$ for the thresholded sparse matrix–matrix operations. The matrix data structure is based on the ELLPACK-R sparse matrix format, which allows efficient shared memory parallelism on a single node.177 The ${\\mathrm{DFTB}}+$ code was modified to use the LANL PROGRESS library and, in particular, the SP2 and ZSP algorithms. In combination with XL-BOMD, this allows efficient energyconserving, molecular dynamics simulations, where the computational cost scales only linearly with the system size. Figure 6 shows the performance of the SP2 algorithm compared to regular diagonalization.  \n\n# C. GPU computing  \n\nGraphics processing unit (GPU) acceleration is implemented in ${\\mathrm{DFTB}}+$ . Given the nature of the underlying theory, the timelimiting step in routine calculations corresponds to the diagonalization of the hamiltonian matrix, taking in the order of $90\\%-95\\%$ of the total running time. The hybrid CPU–GPU implementation in ${\\mathrm{DFTB}}+$ replaces the LAPACK-based eigensolver with a GPU eigensolver based on the divide-and-conquer algorithm as implemented in MAGMA.178  \n\n![](images/0a3c3f351f8d13111a4c40aedae487536498ffbffc30b00f71b9293b0ac8e361.jpg)  \nFIG. 6. CPU time for the density matrix construction for different varying sizes of water box systems. Regular diagonalization (black curve) was compared to the SP2 method (red curve). A numerical threshold of $10^{-5}$ was used in the sparse matrix–matrix multiplications of the SP2 algorithm.  \n\n![](images/cf1c711af4a8145f45c269284f3e826bffe242a5f3f56f705a85197ee5601ee9.jpg)  \nFIG. 7. Wall clock running times for total energy calculations of water clusters (with 6 basis functions/water molecule). The black curve shows timings obtained using the LAPACK compatible ESSL eigensolver on the CPU, and the red/green curves show timings obtained using the MAGMA and the ESSL libraries without/with ESSL-CUDA off-loading. Timings have been recorded on the Summit machine using 42 threads for 42 physical cores.  \n\nBenchmarking of the code shows that at least 5000 basis functions are necessary to exploit the power of the GPUs and to produce an observable speedup with respect to the CPU-only code. For systems spanning a vector space comprised of 70 000 basis functions, speedups of $17\\times$ have been observed in a system with 6 NVIDIA® Tesla® V100 w×ith respect to the multi-threading CPU-only implementation (see Fig. 7).  \n\n# IV. INTERFACING DFTB $\\uplus$ WITH OTHER SOFTWARE PACKAGES  \n\n${\\mathrm{DFTB}}+$ can be currently interfaced with other software packages using three different ways of communications: file communication, socket based, or direct connection via the DFTB $^+$ API as a library. The first one is very easy to implement but comes with an overhead for the file I/O, while the latter two enable a more efficient coupling at the price of somewhat higher complexity in implementation.  \n\n# A. File based communication  \n\nWhen using file based communication, the external driver creates necessary input files and starts an individual ${\\mathrm{DFTB}}+$ program for each of the inputs. After ${\\mathrm{DFTB}}+$ has finished, the driver analyses the created output files and extracts the necessary information from those. ${\\mathrm{DFTB}}+$ had been interfaced using file based communication to, among others, the phonopy179 code for finite difference harmonic and anharmonic phonon calculations and the Atomic Simulation  \n\nEnvironment (ASE) package180 (a set of tools and Python modules for setting up, manipulating, running, visualizing, and analyzing atomistic simulations).  \n\n# B. Socket interface  \n\nThe $\\mathrm{i{-}P I^{\\mathrm{181}}}$ interface for communication with external driving codes is supported by DFTB $^+$ . ${\\mathrm{DFTB}}+$ can then be driven directly instead of using file $\\mathrm{I}/\\mathrm{O}$ . The initial input to ${\\mathrm{DFTB}}+$ specifies the boundary conditions, type of calculation, and chemical information for atoms, and the code then waits to be externally contacted. This kind of communication with DFTB $^+$ can be used by, among others, the i-PI universal force engine package181 and ASE.180  \n\n# C. DFTB $^+$ library, QM/MM simulations  \n\n# 1. Gromacs integration  \n\nDFTB quantum-chemical models may be utilized as a QM engine in hybrid quantum-mechanical/molecular mechanical (QM/MM) approaches. This allows, for example, efficient simulations of chemical processes taking place in bio-molecular complexes. The ${\\mathrm{DFTB}}+$ library interface has been connected to the Gromacs182 MM-simulation software package. (The Gromacs part of the integration is contained in a fork of the Gromacs main branch.183) At the start of the simulation, the DFTB $^+$ input file is read in, and a DFTB calculation environment is created, containing all of the necessary information (parameters), but no atomic coordinates yet. In every step of MD simulation or energy minimization, the calculation of forces starts with a call to the ${\\mathrm{DFTB}}+$ API, passing the coordinates of QM atoms and the values of electrostatic potentials induced by the MM atoms at the positions of the QM atoms. DFTB $^+$ then returns QM forces and QM charges back to Gromacs, where the QM/MM forces are calculated in the QM/MM routines. Gromacs then continues by calculating the MM forces, integration of equations of motion, etc.  \n\nSometimes the electrostatic interactions cannot be represented as an external potential but also depend on the actual values of the QM-charges (i.e., polarizable surroundings). In those cases, a callback function can be passed to ${\\mathrm{DFTB}}+$ , which is then invoked at every SCC iteration to update the potential by the driver program whenever the QM charges change. In the DFTB $^+$ /Gromacs integration, we use this technique to calculate the QM–QM electrostatic interactions in periodic systems with the highly efficient particle mesh Ewald method184 implemented in Gromacs.  \n\n# 2. DL_POLY_4 integration with MPI support  \n\nDL_POLY_4 is a general-purpose package for classical molecular dynamics (MD) simulations.185 In conjunction with the recent extension of $\\mathrm{DFTB}{+}^{\\prime}{\\bf s}$ API, DL_POLY_4.10 supports the use of ${\\mathrm{DFTB}}+$ for self-consistent force calculations in place of empirical force fields for Born–Oppenheimer molecular dynamics.  \n\nThe interface fully supports passing MPI communicators between the programs, allowing users to run simulations in parallel, across multiple processes. The MPI parallelization schemes of DL_POLY_4 and ${\\mathrm{DFTB}}+$ differ considerably. DL_POLY_4 utilizes domain decomposition to spatially distribute the atoms that comprise the system across multiple processes, whereas ${\\mathrm{DFTB}}+$ distributes the hamiltonian matrix elements using BLACS decomposition. This does not impose any serious restrictions as DL_POLY_4 and ${\\mathrm{DFTB}}+$ run sequentially, with ${\\mathrm{DFTB}}+$ being called once per MD time step.  \n\nThe DL_POLY_4–DFTB $^+$ interface works by gathering the atoms from each DL_POLY_4 process such that all processes have a complete copy of all the atoms. Coordinates, species types, and the atomic ordering are then passed to ${\\mathrm{DFTB}}+$ . The calculated forces are returned to DL_POLY_4, which redistributes them according to its domain decomposition, and the atomic positions are propagated one time step.  \n\nSpatial decomposition means that atoms can propagate between processes. Because atoms are gathered sequentially according to their process id (or rank), when atoms propagate between processes, their ordering effectively changes. The ${\\mathrm{DFTB}}+$ API facilitates this and is, therefore, able to support any molecular dynamics software that implements domain decomposition parallelization; however, the total number of atoms (and atom types) must be conserved during the simulation.  \n\n# D. Meta-dynamics using PLUMED  \n\nMolecular dynamics is often plagued by high energy barriers that trap the nuclear ensemble in one or several local minima. This leads to inefficient or inadequate sampling of the ensemble and thus inaccurate predictions of physicochemical properties.186–188 This “timescale” problem is typical for rare-event systems or those in which ergodicity of a particular state is impeded by the local topology of the potential energy surface. A variety of methods have been conceived to circumvent this, including umbrella sampling189 and meta-dynamics.190  \n\nUmbrella sampling and meta-dynamics can now be performed using ${\\mathrm{DFTB}}+$ via its interface to the PLUMED plugin.191,192 Using PLUMED, MD trajectories generated in ${\\mathrm{DFTB}}+$ can be analyzed, sampled, and biased in a variety of ways along user-defined collective variables (CVs), enabling accelerated MD simulations and determination of the free energy surface. A CV is a subspace of the full potential energy surface that can be arbitrarily defined to sample atomic dynamics along dimensions/pathways of physicochemical interest. PLUMED also includes bias functions such as the upper and lower wall biases, enabling a constraint of MD configurations to specific areas on the potential energy surface. The utility of the DFTB $^+$ /PLUMED interface has been demonstrated on several challenging systems, including malonaldehyde intra-molecular proton transfer (Fig. 8), corannulene bowl inversion, and the diffusion of epoxide groups on graphene.192  \n\n# E. DFTB $^+$ in Materials Studio  \n\n${\\mathrm{DFTB}}+$ is included as a module in the commercial modeling and simulation software package, BIOVIA Materials Studio (MS).193 ${\\mathrm{DFTB}}+$ runs as an in-process energy server, supplying energies, forces, and stresses to drive the MS in-house simulations tools. Supported tasks include energy calculation, geometry optimization, molecular dynamics, electron transport calculation, mechanical properties, and parameterization. The module also supports calculation and visualization of standard electronic properties, such as band structure, density of states, orbitals, and so on. The ${\\mathrm{DFTB}}+$ module integrates closely with the data model and the Materials  \n\n![](images/4ca72bca11a31420cc80131873e8f1d800c6dd1aebeacdfc28c221aa729c1785.jpg)  \nFIG. 8. Intra-molecular proton transfer in malonaldehyde at $298\\mathsf{K}.$ Contours show the DFTB3-D3/3ob free energy surface of malonaldehyde obtained using welltempered meta-dynamics, with collective variables $d(0_{1}-H)$ and $d(\\mathsf{O}_{2}{-}\\mathsf{H})$ . Each point is colored according to its sampling frequency during the meta-dynamics simulation, yellow (blue) indicating high (low) sampling frequency. The DFTB3- D/3ob free energy surface yields a proton transfer barrier of $13.1\\pm0.4$ kJ $\\mathsf{m o l}^{-1}$ .  \n\nStudio Visualizer, allowing the user to construct structures and start calculations quickly, with fully automated creation of the ${\\mathrm{DFTB}}+$ input file. The ${\\mathrm{DFTB}}+$ module is also supported in the MS MaterialsScript interface and the Materials Studio Collection for Pipeline Pilot,194 allowing creation of more complicated workflows.195,196  \n\nThe ${\\mathrm{DFTB}}+$ parameterization workflow in MS supports fitting of both electronic parameters and repulsive pair potentials using DFT calculations with the DMol3 module197,198 as a reference. The ${\\mathrm{DFTB}}+$ module includes scripts for validation of parameters in terms of band structure, bond length, bond angles, and so on, as well as visualization for the hamiltonian, overlap matrix elements, and the repulsive pair potentials. The parameterization tools allow extension of existing parameters or incremental development of a parameter set. Parameters developed using the ${\\mathrm{DFTB}}+$ module can, after conversion, be used outside MS. Several default ${\\mathrm{DFTB}}+$ parameter sets, generated using these parameterization tools, are also included. In 2019, MS introduced a new parameter set that includes the Li, C, H, N, O, P, and F elements and is aimed toward Li-ion battery modeling.  \n\n# F. Outlook  \n\nIn order to enable flexible general communication with various types of external components (external drivers, QM/MM, and machine learning models), we are in the process of developing a communication library,199 which allows for data exchange between mixed language (e.g., Fortran and C) components via API-bindings as well as between different processes via socket communications. After engagement with other stakeholders, this will be released as a set of BSD-licensed tools and a library.  \n\n# V. SOFTWARE ENGINEERING IN DFTB+  \n\nThis section presents a few aspects of our software development, which may have some interest beyond the ${\\mathrm{DFTB}}+$ software package.  \n\n# A. Modern Fortran wrappers for MPI and ScaLAPACK functions  \n\nModern scientific modeling packages must be able to run on massive parallel architectures to utilize high performance computing, often using the Message Passing Interface (MPI) framework. While the MPI offers a versatile parallelization framework, its application interface was designed to support C and Fortran 77-like interfaces. This requires the programmer to explicitly pass arguments to the MPI-routines, which should be automatically deduced by the compiler for languages with higher abstraction levels $\\scriptstyle\\sum++$ or Fortran 95 and newer versions). In order to eliminate developer need to pass redundant information (and to reduce associated programming bugs), we have developed modern Fortran wrappers around the MPI-routines. These have been collected in the MPIFX-library,200 which is an independent software project outside of the ${\\mathrm{DFTB}}+$ software suite, being licensed under the more permissive BSD-license. It enables shorter MPI-calls by automatically deducing data types and data sizes from the call signature. Additionally, several MPI parameters have been made optional using their most commonly used value as a default value. For example, in order to broadcast a real array from the master process to all other process, one would have to make the following MPI-call:  \n\ncall mpi_bcast(array, size(array), MPI_FLOAT, 0, comm, err)  \n\nwhile MPIFX-wrappers reduce it to a much shorter and less errorprone line:  \n\ncall mpifx_bcast(comm, array)  \n\nwhere comm is an MPIFX derived type containing the MPIcommunicator. The type (MPI_FLOAT) and number of broadcasted items [size(array)] are automatically deduced. The process initiating the broadcasting has been assumed to be process 0 (master process), as this is probably the most common use case but can be customized when needed with an optional parameter. The error argument is optional as well, if it is not passed (as in the example above), the routine would stop the code in the case of any errors.  \n\nLikewise, the commonly used parallel linear algebra library ScaLAPACK uses Fortran 77-type interfaces. The open source SCALAPACKFX library201 offers higher level modern Fortran wrappers around routines used by DFTB $^+$ .  \n\n# B. Fortran meta-programming using Fypp  \n\nAlthough the latest Fortran standard (Fortran 2018) offers many constructs to support modern programming paradigms, it does not allow for generic template based programming. This would avoid substantial code duplication and offer useful metaprogramming capabilities for Fortran programmers. We have developed the Python based pre-processor, Fypp,202 which offers a workaround for the missing features. Fypp is used during the build process to turn the meta-programming constructs into the standard Fortran code. The Fypp project is independent of the ${\\mathrm{DFTB}}+$ software package and is licensed under the BSD-license, being also used by other scientific software packages, for example, by the CP2K code203 and both the MPIFX and the SCALAPACKFX libraries.  \n\n# VI. SUMMARY  \n\n${\\mathrm{DFTB}}+$ is an atomistic quantum mechanical simulation software package allowing fast and efficient simulations of large systems for long timescales. It implements the DFTB- and the xTBmethods and various extensions of those, such as range-separated functionals, multiple methods of excited state calculations, and electron transport simulations. It can be used either as a standalone application or as a library and has been already interfaced to several other simulation packages. ${\\mathrm{DFTB}}+$ is a community developed open source project under the GNU Lesser General Public License, which can be freely used, modified, and extended by everybody.  \n\n# SUPPLEMENTARY MATERIAL  \n\nSee the supplementary material for the damping parameters for the D4, the TS, and the MBD dispersion models.  \n\n# ACKNOWLEDGMENTS  \n\nThe authors, especially B. Hourahine and B. Aradi, are thankful to Gotthard Seifert for his suggestions and insights into density functional tight binding throughout the development of the DFTB $^+$ code. B. Hourahine acknowledges the EPSRC (Grant No. EP/P015719/1) for financial support. B. Aradi and T. Frauenheim acknowledge the research training group DFG-RTG 2247 (QM3). A. Buccheri acknowledges the EPSRC (Grant No. EP/P022308/1). C. Camacho acknowledges financial support from the Vice-Rectory for research of the University of Costa Rica (Grant No. 115-B9-461) and the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory (ORNL), which is managed by UT-Battelle, LLC, for DOE under Contract No. DE-AC05-00OR22725. S. Irle acknowledges support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, Geoscience Program. M. Y. Deshaye and T. Kowalczyk acknowledge support from a National Science Foundation RUI Award (No. CHE-1664674) and a CAREER Award (No. DMR-1848067). T. Kowalczyk is a Cottrell Scholar of the Research Corporation for Science Advancement. T. Dumitrica˘ was supported by the National Science Foundation (Grant No. CMMI-1332228). R. J. Maurer acknowledges support via a UKRI Future Leaders Fellowship (Grant No. MR/S016023/1). A. M. N. Niklasson and C. 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    "id": "10.1002_adma.202001537",
    "DOI": "10.1002/adma.202001537",
    "DOI Link": "http://dx.doi.org/10.1002/adma.202001537",
    "Relative Dir Path": "mds/10.1002_adma.202001537",
    "Article Title": "Weighted Mobility",
    "Authors": "Snyder, GJ; Snyder, AH; Wood, M; Gurunathan, R; Snyder, BH; Niu, CN",
    "Source Title": "ADVANCED MATERIALS",
    "Abstract": "Engineering semiconductor devices requires an understanding of charge carrier mobility. Typically, mobilities are estimated using Hall effect and electrical resistivity meausrements, which are are routinely performed at room temperature and below, in materials with mobilities greater than 1 cm(2) V-1 s(-1). With the availability of combined Seebeck coefficient and electrical resistivity measurement systems, it is now easy to measure the weighted mobility (electron mobility weighted by the density of electronic states). A simple method to calculate the weighted mobility from Seebeck coefficient and electrical resistivity measurements is introduced, which gives good results at room temperature and above, and for mobilities as low as 10(-3) cm(2) V-1 s(-1), mu w=331cm2Vs(m omega cm rho) (T300 K)-3/2[ exp[ |S|kB/e-2]1+exp[-5(|S|kB/e-1) ]+3 pi 2|S|kB/e1+exp[5(|S|kB/e-1) ] ]Here, mu(w) is the weighted mobility, rho is the electrical resistivity measured in m omega cm, T is the absolute temperature in K, S is the Seebeck coefficient, and k(B)/e = 86.3 mu V K-1. Weighted mobility analysis can elucidate the electronic structure and scattering mechanisms in materials and is particularly helpful in understanding and optimizing thermoelectric systems.",
    "Times Cited, WoS Core": 640,
    "Times Cited, All Databases": 656,
    "Publication Year": 2020,
    "Research Areas": "Chemistry; Science & Technology - Other Topics; Materials Science; Physics",
    "UT (Unique WOS ID)": "WOS:000532658500001",
    "Markdown": "# Weighted Mobility  \n\nG. Jeffrey Snyder,\\* Alemayouh H. Snyder, Maxwell Wood, Ramya Gurunathan, Berhanu H. Snyder, and Changning Niu  \n\nEngineering semiconductor devices requires an understanding of charge carrier mobility. Typically, mobilities are estimated using Hall effect and electrical resistivity meausrements, which are are routinely performed at room temperature and below, in materials with mobilities greater than $\\mathsf{1c m^{2}V^{-1}s^{-1}}$ . With the availability of combined Seebeck coefficient and electrical resistivity measurement systems, it is now easy to measure the weighted mobility (electron mobility weighted by the density of electronic states). A simple method to calculate the weighted mobility from Seebeck coefficient and electrical resistivity measurements is introduced, which gives good results at room temperature and above, and for mobilities as low as $\\mathsf{I}0^{-3}\\mathsf{c m}^{2}\\mathsf{V}^{-1}\\mathsf{s}^{-1},$  \n\n$$\n\\mu_{\\mathrm{w}}=331\\frac{\\mathrm{cm}^{2}}{\\mathrm{Vs}}\\left(\\frac{\\mathrm{m}\\Omega\\mathrm{cm}}{\\rho}\\right)\\left(\\frac{T}{300\\mathrm{K}}\\right)^{-3/2}\\left[\\frac{\\exp\\left[\\frac{|S|}{k_{\\mathrm{B}}/e}-2\\right]}{1+\\exp\\left[-5\\left(\\frac{|S|}{k_{\\mathrm{B}}/e}-1\\right)\\right]}+\\frac{\\frac{3}{\\pi^{2}}\\frac{|S|}{k_{\\mathrm{B}}/e}}{1+\\exp\\left[5\\left(\\frac{|S|}{k_{\\mathrm{B}}/e}-1\\right)\\right]}\\right]\n$$  \n\nHere, $\\mu_{\\mathrm{w}}$ is the weighted mobility, $\\rho$ is the electrical resistivity measured in $\\mathsf{m}\\Omega\\mathsf{c m}$ , $\\tau$ is the absolute temperature in K, S is the Seebeck coefficient, and $k_{\\mathrm{{B}}}/\\mathrm{{e}}=86.3~\\upmu\\mathrm{{V}}~\\upkappa^{-1}$ . Weighted mobility analysis can elucidate the electronic structure and scattering mechanisms in materials and is particularly helpful in understanding and optimizing thermoelectric systems.  \n\nCharge carrier mobility is perhaps the most important material parameter to experimentally characterize in order to understand or engineer semiconductor electronic devices. The optimization of carrier mobility is critical to research fields ranging from organic semiconductors to photovoltaics to thermoelectrics.[1–3] Mobility is usually defined with the Drude–Sommerfeld free electron model $1/\\rho=\\sigma=n e\\mu$ where $\\rho$ is the electrical resistivity, $\\sigma$ is the electrical conductivity, $n$ is the charge carrier concentration, $e$ the electronic charge, and $\\mu$ the drift mobility. Historically, the easiest way to characterize mobility is through the Hall effect, where the Hall mobility can be defined as $\\mu_{\\mathrm{H}}=$ $\\sigma R_{\\mathrm{H}}$ with $R_{\\mathrm{H}}$ as the Hall resistance.[4] In most cases where the charge transport is dominated by a single band, the Hall mobility is a good estimate (within $10{-}20\\%$ ) for the the drift mobility.[5]  \n\nWith the proliferation of Seebeck coefficient measurement systems, particularly in laboratories studying thermoelectric materials,[6,7] the weighted mobility is an independent measurement giving similar information. This can be particularly helpful when there are other magnetic effects to consider in the Hall measurements, such as the anomalous Hall effect, and the measured Hall resistance does not seem to correlate well with the actual charge carrier concentration $(R_{\\mathrm{H}}=1/e n_{\\mathrm{H}})$ or mobility.[8] The weighted mobility measurement is also practicable in regimes where the Hall resistivity measurement is difficult, such as at high temperatures or with low mobility bulk systems.[2,9]  \n\nThe weighted mobility, like the Hall mobility, can be defined as a simple function of two measured properties (Figure 1), Seebeck coefficient S and electrical conductivity $\\sigma$ (the Hall mobility is a function of  \n\nHall resistance and electrical conductivity). The function used for weighted mobility and Hall mobility are derived from the simple free electron model using a constant mean-free-path. Equation (1) is a simple analytic form for the weighted mobility that approximates the exact Drude–Sommerfeld free electron model within $3\\%$ for thermopower values $\\vert S\\vert>20\\upmu\\mathrm{V}\\mathrm{K}^{-1}$ .  \n\n$$\n\\mu_{\\mathrm{w}}=\\frac{3h^{3}\\sigma}{8\\pi e\\left(2m_{e}k_{\\mathrm{B}}T\\right)^{3/2}}\\left[\\frac{\\exp\\left[\\frac{\\left|S\\right|}{k_{\\mathrm{B}}/e}-2\\right]}{1+\\exp\\left[-5\\left(\\frac{\\left|S\\right|}{k_{\\mathrm{B}}/e}-1\\right)\\right]}+\\frac{\\frac{3}{\\pi^{2}}\\frac{\\left|S\\right|}{k_{\\mathrm{B}}/e}}{1+\\exp\\left[5\\left(\\frac{\\left|S\\right|}{k_{\\mathrm{B}}/e}-1\\right)\\right]}\\right]\n$$  \n\nAn equivalent form using the electrical resistivity $\\rho=1/\\sigma$ is given in the abstract.  \n\nIn the free electron model, the weighted mobility is a (temperature dependent) material property that is independent of doping whereas the drift mobility depends on doping through the free charge carrier concentration n. Generally, the weighted mobility $\\mu_{\\mathrm{w}}$ is related to the drift mobility $\\mu$ by  \n\n$$\n\\mu_{\\mathrm{w}}\\approx\\mu\\Biggl(\\frac{m^{*}}{m_{e}}\\Biggr)^{3/2}\n$$  \n\n![](images/8cf56d0e527baaa944fd3b550c26c044aadf230f82f507ca8df6e98912411714.jpg)  \nFigure 1.  Mobility measurements using the Hall or Seebeck effects. Hall mobility $\\mu_{\\mathsf{H}}$ is measured using the Hall effect (in a magnetic field B) and electrical resistivity, while the weighted mobility $\\mu_{\\mathrm{w}}$ is measured using measurements of the Seebeck effect (voltage produced from a temperature difference $T_{\\mathfrak{h}}-T_{\\mathfrak{c}})$ and electrical resistivity.  \n\nwhere $m^{*}$ is the density of states effective mass and $m_{e}$ is the electron mass. Because the density of electron states is proportional to $m^{*3/2}$ , we think of $\\mu_{\\mathrm{w}}$ as the electron mobility weighted by the density of electron states.  \n\nThe measured weighted mobilities of well known thermoelectric semiconductors such as $\\mathrm{PbTe^{[10]}}$ and $\\mathrm{Mg}_{3}\\mathrm{Sb}_{2}[11]$ are essentially the same as the Hall mobility except for the density of states $m^{*3/2}$ factor in $\\mu_{\\mathrm{w}}$ (Figure 2). Indeed, combining Hall, Seebeck, and conductivity measurements to estimate a density of states effective mass is a common analysis technique.[5,12,13] For example, the weighted mobility of most good thermoelectric materials decreases with temperature because the electrons are scattered by phonons. In the simple model for acousticphonon scattering (or deformation potential phonon scattering theory), the decrease in weighted mobility occurs with temperature as $T^{-3/2}$ whereas the Hall mobility transitions between $T^{-3/2}$ for lightly doped semiconductors to $T^{-1}$ for heavily doped semiconductors or metals (Figure 2). In some materials, grain boundary,[14] disorder,[15] or ionized impurity scattering[16] might contribute, which will be observed in different temperature and carrier concentration dependencies of the mobility. The weighted mobility gives nearly the same information about charge carrier mobility as the Hall mobility and thus can be used to investigate charge carrier transport mechanisms much in the same way the Hall mobility has been used.  \n\nBecause of the relative ease of Seebeck compared to Hall effect measurements, the weighted mobility could easily become the most common method to determine the charge transport mechanism. Some 4000 samples have Seebeck and conductivity measurements readily available on StarryData2[17] whereas only relatively few samples have reported Hall effect data. Compiling the 153 samples with both Seebeck coefficient and electrical conductivity as a function of temperature (Figure  3), it is clear from the decreasing mobility with temperature that phonon scattering of the charge carriers typically dominates. A few samples show weighted mobility increasing with temperature at low temperature before following the usual trend of decreasing with temperature (phonon scattering). In several cases this increasing mobility with temperature has been attributed to thermally activated conductivity at grain boundaries,[14,18] although ionized impurity scattering can also give a qualitatively similar effect.  \n\nParticularly for materials with low mobility, the weighted mobility appears to be an even better measure of drift mobility than measurements of Hall mobility. The same data, showing weighted mobility on a logarithmic scale (Figure  3b) gives reasonable trends even for materials with weighted mobility less than $10^{-3}\\ \\mathrm{cm}^{2}\\ \\mathrm{V}^{-1}\\ \\mathrm{s}^{-1}$ . Hall mobility is not commonly reported on materials with mobility less than $1\\ \\mathrm{cm}^{2}\\ \\mathrm{V}^{-1}\\ \\mathrm{s}^{-1}$ because the signal is very small.[4] The low weighted mobilities increase with temperature suggesting defects (grain boundaries, ionized impurities) are responsible for the low mobility and higher mobilities are possible with defect or microstructure engineering.  \n\n![](images/5786b67154f726e04125948aa19587b9234dc40474b8cb890e6769a3cf5eb14a.jpg)  \nFigure 2.  Similar behavior of Hall mobility $\\mu_{\\mathsf{H}}$ and weighted mobility $\\mu_{\\mathrm{w}},$ which uses measurements of the Seebeck coefficient. A) The $\\mu_{\\mathsf{H}}$ and $\\mu_{\\mathrm{w}}$ of n-type PbTe (StarryData2 Paper ID $777^{[70]}$ ) are both monotonously decreasing with temperature as expected from increased phonon scattering and simply scaled by the density-of-states effective mass $m^{*3/2}/m_{e}^{3/2}\\approx0.25$ . The Hall mobility $\\mu_{\\mathsf{H}}$ is expected to decrease with increasing doping,[10] but the weighted mobility $\\mu_{\\mathrm{w}}$ is expected to be similar for samples with different doping as observed. B) The mobility of polycrystalline n-type $\\mathsf{M g}_{3}\\mathsf{S b}_{1.5}\\mathsf{B i}_{0.5}[71]$ is suppressed at low temperature due to electrical resistance at grain boundaries. This effect is observed equally (scaled by $m^{*3/2}/m_{e}^{3/2}\\approx7.8)$ in both $\\mu_{\\mathsf{H}}$ and $\\mu_{\\mathrm{w}}$ for samples with differing grain size.  \n\n![](images/ec6f016c93632f6c5ab4f2d840fecc67497c98ac4467164e973c67125c521431.jpg)  \nFigure 3.   Weighted mobility calculated from the reported Seebeck coefficient and electrical conductivity of 153 samples from the StarryData2 website.[17] The weighted mobilities generally decrease with temperature (linear scale above) as expected from phonon scattering of electrons. Several samples show physically reasonable values for mobility less than $\\mathsf{1~c m^{2}~V^{-1}~s^{-1}}$ (log scale below) which is very difficult to measure using the Hall effect.  \n\nThe weighted mobility, like the Hall mobility, can be used to compare the experimental properties of materials to help decide if different properties are due to differences in doping, electronic structure or scattering. An ideal material that follows the Drude–Sommerfeld model exactly with a parabolic band will have a weighted mobility that does not change with doping; any differences could be a sign of complexity in the band structure or scattering, such as nonparabolic bands or multiple bands. Scattering differences are most easily seen in the temperature dependence where departures from simple phonon scattering will give a deviation from the simple $\\bar{T^{3}}/2$ . For example, low mobility below room temperature could be a sign of grain boundary resistance[14,18]  \n\nThe weighted mobility is even more important when studying thermoelectric materials as it most directly measures the electronic qualities that make a good thermoelectric material. The thermoelectric figure of merit $z T$ is optimized with doping. The $z T$ that an optimally doped material can achieve is proportional to the thermoelectric quality factor $B^{[12,13,19]}$  \n\n$$\nB=\\left(\\frac{k_{\\mathrm{B}}}{e}\\right)^{2}\\frac{8\\pi e\\left(2m_{e}k_{\\mathrm{B}}T\\right)^{3/2}}{3h^{3}}\\cdot\\frac{\\mu_{\\mathrm{w}}}{\\kappa_{\\mathrm{L}}}T\n$$  \n\nwhich is proportional to the weighted mobility divided by the lattice thermal conductivity $\\mu_{\\mathrm{w}}/\\kappa_{\\mathrm{L}}$ . Thus any strategy to improve a material for thermoelectric use by reducing $\\kappa_{\\mathrm{L}}$ needs also to consider the effect on $\\mu_{\\mathrm{w}}.$  \n\nThe weighted mobility, $\\mu_{\\mathrm{w}},$ is a better descriptor of the inherent electronic transport properties of a thermoelectric material for thermoelectric use than $S^{2}\\sigma,$ and now with the use of Equation (1), is almost as easy to calculate from the experimental S and $\\rho$ . Historically, $S^{2}\\sigma,$ referred to as the thermoelectric “power factor” has been discussed to evaluate improvements in the electronic properties of thermoelectric materials or when thermal conductivity measurements are unavailable. However, $S^{2}\\sigma$ depends on doping and does not optimize where $z T$ does, overemphasizing more metallic doping concentrations because it ignores the impact of the electronic thermal conductivity to $z T.$ $S^{2}\\sigma$ is frequently overanalyzed to the point where it is incorrectly concluded that a high $S^{2}\\sigma$ is preferable even at the expense of lower $z T$ .[20] An optimized thermoelectric design will produce more power using a material with higher $z T$ because the higher efficiency ensures it will produce more power from the heat flowing through it.[21] Therefore, rather than comparing $S^{2}\\sigma,$ it is more useful to compare $\\mu_{\\mathrm{w}}$ values.  \n\nIt is important to note that the definition of $\\mu_{\\mathrm{w}}$ using Equation (1) from the measured quantities S and $\\rho$ does not require any assumptions. Nonparabolic, multiband, alternate scattering, and other effects in the data are not neglected. These effects will be noticed as deviations from trends not expected from the simple free electron model $(\\mu_{\\mathrm{w}}(E_{\\mathrm{F}})=\\mathrm{con}{\\cdot}$ stant) that should be explainable by incorporating the relevant physics in a model for $\\mu_{\\mathrm{w}}$ that then can be compared with the experimental values.  \n\nIndeed, prior discussions of weighted mobility, using Equation (2) as the definition,[22–24] have relied on parabolic band and scattering assumptions. This begs the question as to the validity in real materials where these assumptions are demonstrably incomplete. Here, in contrast, we define weighted mobility as an experimental value with Equation (1), and use Equation (2) only to interpret the experimental result.  \n\nEquations (1) and (3) are motivated from the ideal expression for $\\mu_{\\mathrm{w}}$ of the free electron (parabolic band) Drude–Sommerfeld model with constant mean-free-path (same result as acoustic phonon scattering, $s=1$ in ref. [25]). This requires solving the following two parametric equations for the transport coefficient $\\sigma_{\\mathrm{E}_{0}}$ as a function of S and $\\rho$ , where $\\eta$ (the reduced Fermi level) is the parameter:  \n\n$$\n1/\\rho=\\sigma=n e\\mu=\\sigma_{\\mathrm{E}_{0}}\\cdot\\ln(1+e^{\\eta})\n$$  \n\n$$\n\\sigma_{\\mathrm{E}_{0}}={\\frac{8\\pi e\\left(2m_{e}k_{\\mathrm{B}}T\\right)^{3/2}}{3h^{3}}}\\mu_{\\mathrm{w}}\n$$  \n\n$$\nS=\\frac{k_{\\mathrm{B}}}{e}\\left[\\frac{\\displaystyle\\int_{0}^{\\infty}\\frac{\\varepsilon}{1+e^{\\varepsilon-\\eta}}d\\varepsilon}{\\ln(1+e^{\\eta})}-\\eta\\right]\n$$  \n\nThe numerator on the left side of Equation (1) $\\left(\\exp\\ [\\frac{|S|}{k_{\\mathrm{B}}/e}-2\\right]\\right)$ is the analytic result in the non-degenerate limit ( $\\eta\\ll0$ , intrinsic semiconductor), while the numerator on the right side of Equation (1) $\\left(\\frac{3}{\\pi^{2}}\\frac{|S|}{k_{\\mathrm{B}}/e}\\right)$ is the analytic result for the degenerate limit ( $\\hphantom{\\eta\\gg0}$ , metal). The functions in the denominators are sigmoid functions with parameters (5 and 1) that give a maximum error of $3\\%$ to the exact Drude–Sommerfeld result. A smaller maximum error could be achieved with more precise parameters (e.g., 5.34 and 1.14) but $3\\%$ is within the typical uncertainty of Seebeck and resistivity measurements.  \n\nIn real materials, the Seebeck coefficient and electrical resistivity can be affected by many mechanisms not accounted for in the free electron model. At low temperatures and materials with high lattice thermal conductivity, for example, the Seebeck coefficient is strongly affected by phonon drag and so the Hall mobility has been a better descriptor for mobility of simple crystalline materials for low temperature physics rather than the weighted mobility.  \n\nHowever, the weighted mobility is preferable to Hall mobility for many samples under certain conditions. For example, at high temperatures and for complex materials with low mobility, the Hall resistance can be difficult or even not possible to measure, whereas the Seebeck coefficient may be easy to measure. Practical examples of such systems include organic semiconductors as well as complex solar cell and thermoelectric materials.[2,26] Many of the examples in Figure  3 have such low weighted mobilities that Hall effect measurements are not possible and often not attempted. $\\mathrm{La}_{3-x}\\mathrm{Te}_{4}[27]$ and $\\mathrm{Yb}_{14}\\mathrm{MnSb}_{11},{}^{[28]}$ for example, are good thermoelectric materials with decent mobility, on the order of $2\\ \\mathrm{cm}^{2}\\ \\mathrm{V}^{-1}\\ \\mathrm{s}^{-1};$ , but the Hall coefficient is barely measurable. Materials in Figure 3 with even lower mobility are likely not possible to measure by Hall effect in bulk form. Also, materials with magnetic elements or impurities often have Hall effect that is not interpretable as charge carrier concentration. This is likely due to the anomalous Hall effect from the internal magnetism or magnetic impurities. For example, rare-earth filled iron-based skutterudites[8] have Hall coefficients that do not correlate well with other electronic transport properties.  \n\nThe Hall mobility $\\mu_{\\mathrm{H}}~=~\\sigma R_{\\mathrm{H}}~$ and Hall concentration $n_{\\mathrm{H}}=1/e R_{\\mathrm{H}}$ as used here[4] is the simple degenerate limit of the Drude–Sommerfeld model. The Drude–Sommerfeld drift mobility and Hall mobility decreases with charge carrier concentration (as $n^{-1/3}$ at high concentrations) as observed in n-PbTe (Figure  2). The weighted mobility, as defined here, is a constant for a Drude–Sommerfeld metal and so would not depend on $n$ . The weighted mobility can be thought of as a non-degenerate (low n) value for mobility which should not change for samples with different doping and therefore better for analyzing small changes in band structures with alloying.[29] In addition, the exact, free electron, acoustic phonon scattering result would require an additional term (the Hall factor $r_{\\mathrm{H}})$ ). Since this $r_{\\mathrm{H}}$ is only a function of the reduced Fermi level $(\\eta)$ it could, in principle, be measured from the Seebeck coefficient (much like the Lorenz factor[30]). However, because the non-degenerate $r_{\\mathrm{H}}$ differs only by $18\\%$ ,[5] it is preferable not to require an additional measurement, instead simply realizing that variations of $n_{\\mathrm{H}}$ or $\\mu_{\\mathrm{H}}$ less than $20\\%$ can be expected even within a parabolic band system. If such differences in $n_{\\mathrm{H}}$ or $\\mu_{H}$ are worthy of discussion, the Hall factor should be calculated more precisely for the material in question using accurate band structures and scattering rates.  \n\nBoth the measured weighted mobility and the Hall mobility tend to be small in compensated materials with both n-type and p-type charge carriers; smaller in comparison to that of the dominant charge carrier alone. While the conductivities of both charge carriers are positive and add together, both the Hall voltage and Seebeck coefficient will reduce in magnitude as they depend on the sign of the charge carriers.[27] As a result a bipolar or composite sample (sample with compensating phases or grain boundaries) has a smaller Hall voltage and thermopower $(\\left|S\\right|)$ , and therefore the doping level appears higher than one would expect for either charge carrier. This results in a rapid decrease in $\\mu_{\\mathrm{w}}$ and $\\mu_{\\mathrm{H}}$ with increasing temperature as charge carriers are excited across the band gap. The fact that Hall voltage is weighted by the square of the partial conductivities compared to the linear weighting of Seebeck (Supporting Information) may make $\\mu_{\\mathrm{H}}$ more sensitive to unusual bipolar behavior than μw.  \n\nWeighted mobility as defined here is an experimental material parameter that can easily be computed from the measured values of the Seebeck coefficient and electrical conductivity, which needs no additional assumptions. Weighted mobility can be used like the Hall mobility to characterize drift mobility, the most basic material parameter in transport theory. The weighted mobility is actually a better descriptor of the inherent transport property than Hall mobility because it is more independent of charge carrier concentration and magnetic impurities and has a more consistent temperature dependence (above about 100 K). The weighted mobility is easy to measure with commercial instruments and is orders of magnitude more sensitive than the Hall effect for low mobility materials. In thermoelectric materials, the weighted mobility is the quantitative measure of how good a material's electronic transport properties are for thermoelectric applications by characterizing the $S^{2}\\sigma$ term in $z T$ .[1] Without the need of a high strength magnet to measure the Hall coefficient, the weighted mobility offers insight into a most basic material property with a relatively easy Seebeck measurement.  \n\n# Supporting Information  \n\nSupporting Information is available from the Wiley Online Library or from the author.  \n\n# Acknowledgements  \n\nG.J.S. and R.G. acknowledge NSF DMREF award# 1729487 and DOE Award DE-AC02-76SF00515. C.N. acknowledges DOE SBIR award DE-SC0019679.  \n\n# Conflict of Interest  \n\nThe authors declare no conflict of interest.  \n\n# Author Contributions  \n\nG.J.S. conceived the idea in discussions with M.W., A.S. fit the parameters in Equation (1). A.S., R.G., and C.N. downloaded and processed the data. B.H.S. drew figures in Figure 3 and retrieved the missing Hall data from the original source. All authors reviewed the manuscript.  \n\n# Keywords  \n\nelectrical transport, electrical measurements, mobility, organic semiconductors, photovoltaics, semiconductors, thermoelectrics  \n\nReceived: March 3, 2020 Revised: April 5, 2020 Published online:  \n\n[11]\t M.  Wood, J. J.  Kuo, K.  Imasato, G. J.  Snyder, Adv. Mater. 2019, 31, 1902337.   \n[12]\t S. D. Kang, G. J. Snyder, arXiv:1710.06896 [cond-mat.mtrl-sci],  2017.   \n[13]\t A.  Zevalkink, D. M.  Smiadak, J. L.  Blackburn, A. J.  Ferguson, M. L.  Chabinyc, O.  Delaire, J.  Wang, K.  Kovnir, J.  Martin, L. T.  Schelhas, T. D.  Sparks, S. D.  Kang, M. T.  Dylla, G. J.  Snyder, B. R. Ortiz, E. S. Toberer, Appl. Phys. Rev. 2018, 5, 021303.   \n[14]\t J. J.  Kuo, S. D.  Kang, K.  Imasato, H.  Tamaki, S.  Ohno, T.  Kanno, G. J. Snyder, Energy Environ. Sci. 2018, 11, 429.   \n[15]\t H.  Xie, H.  Wang, C.  Fu, Y.  Liu, G. J.  Snyder, X.  Zhao, T.  Zhu, Sci. Rep. 2014, 4, 6888.   \n[16]\t H.  Wang, Y.  Pei, A. D.  LaLonde, G. J.  Snyder, Material Design Considerations Based on Thermoelectric Quality Factor, Springer, Berlin/ Heidelberg, Germany 2013, pp. 3–32.   \n[17]\t Y. Katsura, M. Kumagai, T. Kodani, M. Kaneshige, Y. Ando, S. Gunji, Y. Imai, H. Ouchi, K. Tobita, K. Kimura, K. Tsuda, Sci. Technol. Adv. Mater. 2019, 20, 511.   \n[18]\t J.  de  Boor, T.  Dasgupta, H.  Kolb, C.  Compere, K.  Kelm, E.  Mueller, Acta Mater. 2014, 77, 68.   \n[19]\t Y. Pei, H. Wang, G. J. Snyder, Adv. Mater. 2012, 24, 6125.   \n[20]\t W. Liu, H. S. Kim, Q. Jie, Z. Ren, Scr. Mater. 2015, 111, 3.   \n[21]\t L. L. Baranowski, G. J. Snyder, E. S. Toberer, J. Appl. Phys. 2014, 115, 126102.   \n[22]\t G. D.  Mahan, In Solid State Physics, Vol. 51 (Eds: F. Spaepen, H. Ehrenreich), Academic Press, New York 1998, p. 81.   \n[23]\t G. A. Slack, New Materials and Performance Limits for Thermoelectric Cooling, CRC Press, Boca Raton, FL, USA 1995.   \n[24]\t H. J. Goldsmid, Introduction to Thermoelectricity, Springer, New York 2016.   \n[25]\t S. D. Kang, G. J. Snyder, Nat. Mater. 2017, 16, 252.   \n[26]\t N. C. Greenham, S. Tiwari, Opt. Quantum Electron. 2009, 41, 69.   \n[27]\t A. F. May, J.-P. Fleurial, G. J. Snyder, Phys. Rev. B 2008, 78, 125205.   \n[28]\t E. S.  Toberer, S. R.  Brown, T.  Ikeda, S. M.  Kauzlarich, G. J.  Snyder, Appl. Phys. Lett. 2008, 93, 062110.   \n[29]\t Y. Xiao, D. Wang, Y. Zhang, C. Chen, S. Zhang, K. Wang, G. Wang, S. J. Pennycook, G. J. Snyder, H. Wu, L.-D. Zhao, J. Am. Chem. Soc. 2020, 142, 4051.   \n[30]\t H.-S. Kim, Z. M. Gibbs, Y. Tang, H. Wang, G. J. Snyder, APL Mater. 2015, 3, 041506.  ",
    "Target": null
  },
  {
    "id": "10.1038_s41467-020-15926-1",
    "DOI": "10.1038/s41467-020-15926-1",
    "DOI Link": "http://dx.doi.org/10.1038/s41467-020-15926-1",
    "Relative Dir Path": "mds/10.1038_s41467-020-15926-1",
    "Article Title": "Quantifying and understanding the triboelectric series of inorganic non-metallic materials",
    "Authors": "Zou, HY; Guo, LT; Xue, H; Zhang, Y; Shen, XF; Liu, XT; Wang, PH; He, X; Dai, GZ; Jiang, P; Zheng, HW; Zhang, BB; Xu, C; Wang, ZL",
    "Source Title": "NATURE COMMUNICATIONS",
    "Abstract": "Contact-electrification is a universal effect for all existing materials, but it still lacks a quantitative materials database to systematically understand its scientific mechanisms. Using an established measurement method, this study quantifies the triboelectric charge densities of nearly 30 inorganic nonmetallic materials. From the matrix of their triboelectric charge densities and band structures, it is found that the triboelectric output is strongly related to the work functions of the materials. Our study verifies that contact-electrification is an electronic quantum transition effect under ambient conditions. The basic driving force for contact-electrification is that electrons seek to fill the lowest available states once two materials are forced to reach atomically close distance so that electron transitions are possible through strongly overlapping electron wave functions. We hope that the quantified series could serve as a textbook standard and a fundamental database for scientific research, practical manufacturing, and engineering. The mechanism of contact electrification remains a topic of debate. Here, the authors present a quantitative database of the triboelectric charge density and band structure of many inorganic materials, verifying that contact electrification between solids is an electron quantum transition effect.",
    "Times Cited, WoS Core": 626,
    "Times Cited, All Databases": 647,
    "Publication Year": 2020,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000531855500023",
    "Markdown": "# Quantifying and understanding the triboelectric series of inorganic non-metallic materials  \n\nHaiyang Zou1,5, Litong Guo1,2,5, Hao Xue1,3,5, Ying Zhang1, Xiaofang Shen3, Xiaoting Liu3, Peihong Wang1, $\\mathsf{X u\\Pi}\\mathsf{H e}^{1}$ , Guozhang Dai1, Peng Jiang1, Haiwu Zheng1, Binbin Zhang1, Cheng $\\mathsf{X}\\mathsf{u}^{1,2}$ & Zhong Lin Wang1,4✉  \n\nContact-electrification is a universal effect for all existing materials, but it still lacks a quantitative materials database to systematically understand its scientific mechanisms. Using an established measurement method, this study quantifies the triboelectric charge densities of nearly 30 inorganic nonmetallic materials. From the matrix of their triboelectric charge densities and band structures, it is found that the triboelectric output is strongly related to the work functions of the materials. Our study verifies that contact-electrification is an electronic quantum transition effect under ambient conditions. The basic driving force for contactelectrification is that electrons seek to fill the lowest available states once two materials are forced to reach atomically close distance so that electron transitions are possible through strongly overlapping electron wave functions. We hope that the quantified series could serve as a textbook standard and a fundamental database for scientific research, practical manufacturing, and engineering.  \n\nT hneomceontoanct-healtecotrcicfiucrastifonr (llCEm) efrfieaclts iswhai uhnrievfersaltopthweo- materials that are electrically charged after physical contact. However, CE is generally referred to as triboelectrification (TE) in conventional terms. In fact, TE is a convolution of CE and tribology, while CE is a physical effect that occurs only due to the contact of two materials without rubbing against each other, and tribology refers to mechanical rubbing between materials that always involves debris and friction1.  \n\nThe key parameters for CE, the surface charge density, the polarity, and the strength of the charges, are strongly dependent on the materials2–5. The triboelectric series describes materials’ tendency to generate triboelectric charges. The currently existing forms of triboelectric series are mostly measured in a qualitative method in the order of the polarity of charge production. Recently, a standard method6 has been established that allows this material “gene” of triboelectric charge density (TECD) to be quantitatively measured by contacting a tested material with a liquid metal using the output of a triboelectric nanogenerator (TENG) under fixed conditions. A table has been set for over 55 different types of organic polymer films. In comparison, inorganic materials have different atomic structures and band structures from polymers; therefore, it is necessary to quantify the triboelectric series for a wide range of common solid inorganic materials and study their triboelectric series in order to establish a fundamental understanding about their underlying mechanisms.  \n\nOne of the oldest unresolved problems in physics is the mechanism of $\\mathrm{CE^{7,8}}$ . Many studies have been done on the analysis of the amount of the generated charges, including the correlation of charge amount with chemical nature2, electrochemical reactions9, work function10, ion densities11, thermionic emission9, triboemission12,13, charge affinity14, surface conditions and circumstances15, and flexoelectricity16. These studies focus on certain samples and quantitative data measured under various environmental conditions. The sample difference and the variance in the measurement conditions would cause large errors, and the mechanism studies based on a small dataset may not be reliable enough to derive a general understanding of the phenomenon. A systematic analysis based on a high-quality quantified database acquired in a universal standard method with a large volume of samples would provide more accurate data and facilitate a comprehensive understanding of the relationship between CE and the materials’ intrinsic properties.  \n\nHere, we applied a standard method to quantify the triboelectric series for a wide range of inorganic non-metallic materials. Nearly 30 common inorganic materials have been measured, and the triboelectric series is listed by ranking the TECDs. By comparing the work functions of these materials, we find that the polarity of the triboelectric charges and the amount of charge transfer are closely related to their work functions. The triboelectric effect between inorganic materials and a metal is mainly caused by electronic quantum mechanical transitions between surface states, and the driving force of CE is electrons seeking to fill the lowest available states. The only required condition for CE is that the two materials are forced into the atomically close distance so that electronic transitions are possible between strongly overlapping wave functions.  \n\n# Results  \n\nThe principles of measurement and experimental setup. Nonmetallic inorganics are mostly synthesized at high temperature, they are hard materials with high surface roughness, and it is a challenge to make an accurate measurement of the TECD between solid–solid interfaces due to poor intimacy with inaccurate atomicscale contact. To avoid this limitation, we measured the TECD of the tested materials with liquid metal (mercury) as the contacting counterpart as we used for organic polymer materials6. The basic principle for measuring the TECD relies on the mechanism of TENG, which is shown in Fig. 1a–d. Details about the measurement technique and the experimental design as well as the standard experimental conditions have been reported previously6. The measurement method relies on the principle of TENG in contactseparation mode (Fig. 1b)3,17. When the two materials are separated, the negative surface charges would induce positive charges at the copper electrode side (Fig. 1c). When the gap distance reaches an appropriate distance $d_{1}$ , charges fully transfer to balance the potential difference (Fig. 1d). When the tested material is pushed back in contact with liquid mercury, the charges flow back (Fig. 1e). The TECD is derived from the amount of charge flow between the two electrodes.  \n\nThe tested materials were purchased from vendors or synthesized through a pressing and sintering process in our lab (Supplementary Table 1). The tested materials were carefully cleaned with isopropyl alcohol by cleanroom wipers and dried by an air gun. Then, the specimens were deposited by a layer of Ti 1 $(15\\mathrm{nm})$ ) and a thick layer of Cu (above $300\\mathrm{nm},$ at the back as an electrode, and have a margin size of $2\\mathrm{mm}$ to avoid a short circuit when the sample contacts with mercury.  \n\nThe measured TECD. One group of typical signals measured for mica–mercury are shown in Fig. 2. The open-circuit voltage reached up to $145.4\\mathrm{V}$ (Fig. 2a). A total of $69.6\\mathrm{nC}$ electrons (Fig. 2b) flowed between the two electrodes. For each type of material, at least three samples were measured to minimize the measurement errors. The results were recorded after the measured value reached its saturation level. This will eliminate the initial surface charges on the samples. Figure 2c shows the output of three samples of mica measured at different times, and the measured values have good repeatability (Fig. 2d) and stability.  \n\nThe TECD refers to the transferred triboelectric charges per unit area of the CE surface. Nearly 30 kinds of common inorganic non-metallic materials were measured, and their triboelectric series is presented in Fig. 3. The quantified triboelectric series shows the materials’ capabilities to obtain or release electrons during the CE with the liquid metal. We have introduced a normalized TECD $\\alpha$ in our previous study  \n\n$$\n\\alpha=\\frac{\\sigma}{|\\sigma_{\\mathrm{PTFE}}|},\n$$  \n\nwhere $\\sigma$ is the measured TECD of material. Here, we keep using the same standard for these inorganic materials for reference, so that the values are comparable. The average TECD values and the normalized TECDs $\\alpha$ of the measured materials are both listed in Table 1. The more negative the $\\alpha$ value is, the more negative charges it will get from mercury, and vice versa. If two materials have a large difference of $\\alpha$ values, they will produce higher triboelectric charges when rubbed together (Supplementary Fig. 1). In contrast, the less difference of $\\alpha$ values, the fewer charges exchange between them. The triboelectric series is validated by cross-checking (Supplementary Figs. 2 and 3).  \n\nMechanism of CE for inorganic non-metallic materials. The standard measurement quantifies the TECD of various materials, the obtained values are only dependent on the materials. It remains to be systematically investigated, such as why different materials have a different amount of charges transferred; why some materials will become positively charged, but others were negatively charged after contact and separation with the same material; why the polarity of charge can be switched when they were contacted with different materials.  \n\n![](images/df6c906c4d4a80822200a3b519ba11bab402c422c23c9d7bbafe5f3f40db1f21.jpg)  \nFig. 1 Experimental setup and the working mechanism of the measurement technique. a Schematic diagram of the measurement system for the triboelectric charge density. b–e Schematic diagram of the mechanism for measuring the surface charge density. b Charges transferred between the two materials owing to the contact-electrification effect. There is no potential difference between the two materials when they are fully contacted with each other. c When the two materials are separated, the positive charges in mercury flow into the copper side in order to keep the electrostatic equilibrium. d When the gap goes beyond a specific distance $L,$ there is no current flow between two electrodes. e When the material is in contact with mercury again, the positive charges flow from copper to mercury due to the induction of the negative charges on the surface of the inorganic material.  \n\nHere, we compare the TECD values with the relative work functions of the two contacting materials. In this study, all inorganic non-metallic materials were contacted with mercury. The work function of mercury is $\\emptyset_{\\mathrm{Hg}}=4.475\\mathrm{eV}^{11}$ . The work functions of the tested materials are listed in Supplementary Table 2. The work functions of inorganic non-metallic materials are determined by materials themselves, but can be modified by crystallographic orientation, surface termination and reconstruction, and surface roughness, and so on. Therefore, some materials have a wide range of work functions in the literature. As shown in Fig. 4, as the work functions of materials decrease, the TECD values increase from $-62.66$ to $61.80\\upmu\\mathrm{C}\\mathrm{cm}^{-2}$ . The work function is related to the minimum thermodynamic energy needed to remove an electron from a solid to a point just outside the solid surface. Our results show that electron transfer is the main origin of CE between solids and metal18. In addition, the polarity of the CE charges is determined by the relative work functions of materials. When the work function of the tested material A is smaller than the work function of mercury, $\\begin{array}{r}{\\varnothing_{\\mathrm{A}}<\\varnothing_{\\mathrm{Hg}},}\\end{array}$ the tested materials will be positively charged after intimate contact with mercury; when the work functions of tested material B are close to the work function of mercury, $\\emptyset_{\\mathrm{B}}\\approx\\emptyset_{\\mathrm{Hg}}$ the tested material $\\mathbf{B}$ will be little electrically charged; when the work functions of tested material C are larger than the work function of mercury, $\\varnothing_{\\mathrm{C}}{>}\\varnothing_{\\mathrm{Hg}},$ the tested materials will be negatively charged. The TECDs of tested materials are strongly dependent on the work function difference. If the two materials have a larger difference of work functions, they will have more electrons transferred. These results show that electron transfer during CE is related to the band structure and energy level distribution. The electrons flow from the side that has high energy states to the side having low energy states.  \n\n![](images/ae253db3712199bbfa9a673e7daa69d94555bbdfc9a1444cc9064d7831205af7.jpg)  \nFig. 2 A set of typical measured signals of tested samples. a Open-circuit voltage of mica during the processes of contact and separation with mercury. b Curve of transferred charge between the two electrodes under short-circuit condition. c Measured charge transferred for three different samples of mica. d Stability of the measured values for many cycles of operation. Source data are provided as a Source Data file.  \n\nThe quantum mechanical transition model is proposed to explain the CE of inorganic non-metallic materials. Suppose we have a material A, which has a higher Fermi level than the Fermi level of the metal. The disruption of the periodic-potential function results in a distribution of allowed electric energy states within the bandgap, shown schematically in Fig. 5a, along with the discrete energy states in the bulk material. When the material is brought into intimate contact with the metal, the Fermi levels must be aligned (Fig. 5b), which causes the energy bands to bend and the surface states to shift as well. Normally, the energy states below the Fermi level of material A— $E_{\\mathrm{FA}}$ are filled with electrons and the energy states above $E_{\\mathrm{FA}}$ are mostly empty if the temperature is relatively low. Therefore, the electrons at the surface states above $E_{\\mathrm{FA}}$ will flow into the metal, thus the metal gets negatively charged, and the originally neutralized material A becomes positively charged for losing electrons. The electrons that flowed from semiconductors or insulators to metals are mainly from the surface energy states. If the work functions of two materials (B and metal) are equal, there will be little electron transfer (Fig. 5c, d); therefore, it would have no electrification. When the work function of tested material C is lower than the work function of the metal (Fig. 5e), the Fermi levels tend to level, surface energy states shift down, and electrons flow reversely from metal to fill the empty surface states in material C to reach the aligned Fermi level (Fig. 5f). Thus, the tested material will be negatively charged and the metal becomes positively charged.  \n\nIf two materials have a large difference of work functions, there are many discrete allowed surface states that electrons are able to transit; the surface is able to carry more charges after contact or friction. If the difference is low, few discrete surface states exist for electrons transition; the surface will be less charged. The surface charge density can be changed by contact with different materials, due to the different levels of work functions. The polarity of surface charges can be switched as well, since they have different directions of electron transition.  \n\n![](images/ca8c9e037bbb7fc32f39cb19faec8af766693486a34b6253190425b988eca6fb.jpg)  \nFig. 3 Quantified triboelectric series of some common inorganic nonmetalic materials. The error bar indicates the range within a standard deviation. Source data are provided as a Source Data file.  \n\nFor inorganic non-metallic materials, the dielectric constant is an important parameter. We have analyzed the relationship between dielectric constant and TECD. From the Gauss theorem, if we ignore the edge effect, the ideal induced short circuit transferred charge in the inorganic material–mercury TENG process is given by6,17:  \n\n$$\nQ_{\\mathrm{SC}}=\\frac{S\\sigma_{c}x(t)}{\\frac{d_{1}\\varepsilon_{0}}{\\varepsilon_{1}}+x(t)},\n$$  \n\nwhere $\\varepsilon_{1}$ is the dielectric permittivity of the inorganic material, $d_{1}$ is the thickness, $x(t)$ is the separation distance over time $t,$ and $\\sigma_{\\mathrm{c}}$ is the surface charge density. From Eq. (1), under the measured conditions, $d_{1}\\ll\\ x(t)$ , and the part of $\\frac{d_{1}\\varepsilon_{0}}{\\varepsilon_{1}}$ can be ignored. Therefore, the dielectric constant will not influence the charge transfer $Q_{\\mathrm{SC}}$ and the surface charge density $\\sigma_{\\mathrm{c}}$ . As expected, the relation of TECD and dielectric constant of these materials is shown in Fig. 4; the measured TECDs are not affected by the dielectric constant of materials.  \n\n# Discussion  \n\nA quantum mechanical transition always describes an electron jumping from one state to another on the nanoscale, while CE between solids is a macroscopic quantum transition phenomenon. Materials have a large scale of surface states to store or lose electrons, and charge transfer between two triboelectric materials is based on the capacitive model, so it can reach a significantly high voltage $(>100\\mathrm{\\V})^{19}$ , which is different from the contact potential (mostly ${<}1\\mathrm{V}$ )20. The quantum transition model between the surface energy states explains how electrons are accumulated or released at the surfaces of inorganic dielectric materials and how the surface becomes charged, while the contact potential model only explains carrier diffusion inside semiconductors24. The surface modification technologies, including impurity and doping elements, surface termination and reconstruction21, surface roughness22, and curvature effect23 can tune the TECD. Based on the proposed model, it is suggested that the fundamental driving force of CE is that electrons fill the lowest available energy levels if there is little barrier. When the two materials have reached atomically close distance, electron transition is possible between strongly overlapping electron wave functions25,26.  \n\n<html><body><table><tr><td colspan=\"3\">Table 1 Triboelectric series of materials and their TECD.</td></tr><tr><td>Materials (μCm-2)</td><td>Average TECD STDEV</td><td>α</td></tr><tr><td>Mica</td><td>61.80 1.63</td><td>0.547</td></tr><tr><td>Float glass 40.20</td><td>0.85</td><td>0.356</td></tr><tr><td>Borosilicate glass 38.63</td><td>1.18</td><td>0.342</td></tr><tr><td>BeO 9.06</td><td>0.21</td><td>0.080</td></tr><tr><td>PZT-5 8.82</td><td>0.16</td><td>0.078</td></tr><tr><td>MgSiO3 2.72</td><td>0.07</td><td>0.024</td></tr><tr><td>CaSiO3 2.38</td><td>0.15</td><td>0.021</td></tr><tr><td>Bi4Ti3O12 2.02</td><td>0.21</td><td>0.018</td></tr><tr><td>Bio.5Nao.5TiO3 1.76</td><td>0.05</td><td>0.016</td></tr><tr><td>NiFe2O4 1.75</td><td>0.07</td><td>0.0155</td></tr><tr><td>Bao.65Sro.35TiO3 1.28</td><td>0.11</td><td>0.011</td></tr><tr><td>BaTiO3 1.27</td><td>0.08</td><td>0.0112</td></tr><tr><td>PZT-4</td><td>1.24 0.12</td><td>0.011</td></tr><tr><td>ZnO 0.86</td><td>0.04</td><td>0.008</td></tr><tr><td>NiO 0.53</td><td>0.05</td><td>0.005</td></tr><tr><td>SnO2 0.46</td><td>0.02</td><td>0.004</td></tr><tr><td>SiC 0.31</td><td>0.07</td><td>0.003</td></tr><tr><td>CaTiO3 0.24</td><td>0.02</td><td>0.002</td></tr><tr><td>ZrO2 0.09</td><td>0.07</td><td>0.001</td></tr><tr><td>Cr2O</td><td>0.02 0.01</td><td>0.00013</td></tr><tr><td>FeO3</td><td>0.00 0.02</td><td>0.000</td></tr><tr><td>AlO3</td><td>-1.58 0.14</td><td>-0.014</td></tr><tr><td>TiO2</td><td>-6.41 0.18</td><td>-0.057</td></tr><tr><td>AIN -13.24</td><td>1.35</td><td>-0.117</td></tr><tr><td>BN -16.90</td><td>0.97</td><td>-0.149</td></tr><tr><td>Clear very high-</td><td>-39.95 2.04</td><td>-0.353</td></tr><tr><td>temperature glass ceramic Ultra-high-temperature -62.66</td><td>0.47</td><td>-0.554</td></tr><tr><td>quartz glass</td><td></td><td></td></tr></table></body></html>\n\nSTDEV, standard deviation. Note: The $\\alpha$ refers to the measured triboelectric charge density of tested materials over the absolute value of the measured triboelectric charge density of the reference material (PTFE).  \n\nThe work functions are determined by the compositions of compounds, chemical valence state, electronegativity15, crystallographic orientation27, temperature19, defects28,29, and so on. Accordingly, the calculation of work functions can be used as a comparison of a materials’ property of TE and to estimate their triboelectric output. In addition, the work functions can be modified to improve the TE for enhancing the triboelectric effect for energy harvesting30–33 and sensing34,35, or reduce the electrical discharge due to CE to improve safety.  \n\nIn summary, we have quantitatively measured the triboelectric series of some common inorganic non-metallic materials under defined conditions. The TECD data obtained depends only on the nature of the material. This serves as a basic data source for investigating the relevant mechanism of CE, and a textbook standard for many practical applications such as energy harvesting and self-powered sensing. The study verifies that the electron transfer is the origin of CE for solids, and that CE between solids is a macroscopic quantum mechanical transition effect that electrons transit between the surface states. The driving force for CE is that electrons tend to fill the lowest available surface states. Furthermore, the TE output could be roughly estimated and compared by the calculation of work functions, and ajusted by the modification of the material's work function through a variety of methods.  \n\n![](images/97362ae1c69acfa0696d6bd8e11da8092dff9e0c328063b270c9b2630fd9be07.jpg)  \nFig. 4 The influence of work function and dielectric constant on contactelectrification. a Relationship between the triboelectric charge density and work functions of materials. b Relationship between the triboelectric charge density and dielectric constant. Source data are provided as a Source Data file.  \n\n# Methods  \n\nSample preparation. The tested materials were purchased from vendors or synthesized through a pressing and sintering process. Some of the ceramic specimens, such as $\\mathrm{Mg}\\mathrm{Si}\\mathrm{O}_{3}$ , $\\mathrm{CaSiO}_{3}$ , $\\mathrm{Bi}_{4}\\mathrm{Ti}_{3}\\mathrm{O}_{12}$ $\\mathrm{Bi}_{0.5}\\mathrm{Na}_{0.5}\\mathrm{TiO}_{3}.$ , $\\mathrm{NiFe}_{2}\\mathrm{O}_{4},$ $\\mathbf{Ba}_{0.65}\\mathbf{Sr}_{0.35}\\mathrm{TiO}_{3}.$ ${\\mathrm{BaTiO}}_{3}.$ and $\\mathrm{CaTiO}_{3}$ , were prepared using a conventional solid-state reaction and solid-phase sintering. Some materials, such as $\\mathrm{znO}$ , NiO, $\\mathrm{SnO}_{2}$ , $\\mathrm{Cr}_{2}\\mathrm{O}_{3}$ $\\mathrm{Fe}_{2}\\mathrm{O}_{3}$ , and $\\mathrm{TiO}_{2}$ , were prepared by solid-phase sintering method using commercial ceramic powders. The details were described below.  \n\nFor $\\mathrm{Mg}\\mathrm{SiO}_{3}$ , the high-purity $\\mathrm{MgO}$ $(99.5\\%)$ and $\\mathrm{SiO}_{2}$ $(99.5\\%)$ ) powders were baked at $80~^{\\circ}\\mathrm{C}$ for $^{5\\mathrm{h}}$ to remove hygroscopic moisture and mixed in an ethanol medium by ball milling for $^{8\\mathrm{h}}$ according to the stoichiometric formula. The slurry was dried at $110^{\\circ}\\mathrm{C}$ for $^{10\\mathrm{h}}$ and the dried powder was calcined at $1100^{\\circ}\\mathrm{C}$ for $^{3\\mathrm{h},}$ and then ball-milled in an ethanol medium for $^{8\\mathrm{h}}$ . After drying again, the obtained powders were granulated with polyvinyl alcohol as a binder and pressed into green disks with a diameter of 2 in. and a thickness of $1\\mathrm{mm}$ under a pressure of $30\\mathrm{MPa}$ Next, the green disks were heated at $600^{\\circ}\\mathrm{C}$ for $^{3\\mathrm{h}}$ to remove the binder, and then sintered at $1400^{\\circ}\\mathrm{C}$ for $^{2\\mathrm{h}}$ . After the obtained ceramic disks were polished on both sides, the gold electrode was sputtered on one side.  \n\nOther samples, including $\\mathrm{CaSiO}_{3}$ , $\\mathrm{Bi}_{4}\\mathrm{Ti}_{3}\\mathrm{O}_{12}$ , $\\mathrm{Bi}_{0.5}\\mathrm{Na}_{0.5}\\mathrm{TiO}_{3}$ , ${\\mathrm{NiFe}}_{2}{\\mathrm{O}}_{4}$ , $\\mathrm{Ba}_{0.65}\\mathrm{Sr}_{0.35}\\mathrm{TiO}_{3}$ , $\\mathrm{BaTiO}_{3}$ , and $\\mathrm{CaTiO}_{3}$ , are prepared similarly to $\\mathrm{Mg}\\mathrm{Si}\\mathrm{O}_{3}$ , except that there are differences in the temperature and holding time of powder calcination and ceramic sintering. Specific parameters for different samples are listed in the Supplementary Table 1.  \n\nFor single element oxide, including $\\mathrm{znO}$ , NiO, $\\mathrm{SnO}_{2}$ $\\mathrm{Cr}_{2}\\mathrm{O}_{3}$ , ${\\mathrm{Fe}}_{2}{\\mathrm{O}}_{3},$ and $\\mathrm{TiO}_{2}$ , the samples are directly prepared by solid-phase sintering method using commercial powders as the raw materials. Taking zinc oxide as an example, the high-purity $\\mathrm{znO}$ powders $(99.5\\%)$ were granulated with polyvinyl alcohol as a binder and pressed into green disks with a diameter of 2 in. and a thickness of $1\\mathrm{mm}$ under a pressure of 30  \n\n![](images/98bd1795a9a26f93f3d72dd9ec9e1aea1aec8362ca0182255f8eb169494f8456.jpg)  \nFig. 5 Electronic quantum transition model of contact-electrification between a dielectric and metal. a When a dielectric A is brought into contact with the metal as shown in the figures, some electrons on the surface states flow into metal to seek the lowest energy states. b The energy bands bend to align the Fermi levels. Most electrons at the surface energy states above the balanced Fermi level flow into metal and left an equal amount of holes at the surface (as shown in green box). Thus, the original neutrally charged dielectric A turns to have positive charges on the surfaces due to the electrons lose. c, d When a dielectric B is brought into contact with the metal, the Fermi levels are balanced, the surface energy states equal. There are no quantum transitions between the two materials. e When a dielectric C contacts the metal, electrons on the surface of the metal flow into the dielectric C to seek the lowest energy levels. f The energy bands shift to align the Fermi levels. Electrons flow from metal to dielectric C to fill the empty surface states due to the difference of energy levels (as shown in the green box). The original neutrally charged dielectric C turns to carry negative charges on the surfaces by obtaining electrons.  \n\nMPa. Next, the green disks were heated at $600^{\\circ}\\mathrm{C}$ for $^{3\\mathrm{h}}$ to remove the binder, and then sintered at $\\mathrm{i}200^{\\circ}\\mathrm{C}$ for $1.5\\mathrm{h}$ . After the obtained ceramic disks were polished on both sides, the gold electrode was sputtered on one side.  \n\nSamples, such as AlN, $\\mathrm{{Al}}_{2}\\mathrm{{O}}_{3}$ , BeO, mica, float glass, borosilicate glass, PZT-5, SiC, $\\mathrm{ZrO}_{2}$ , BN, clear very high-temperature glass ceramic, and ultra-hightemperature quartz glass, were directly purchased from different companies, which were also listed in the Supplementary Table 1.  \n\nThe materials were washed with isopropyl alcohol, cleaned with cleanroom wipers, and dried by an air gun. Then, the materials were deposited with a layer of Ti $\\mathrm{{[10\\nm)}}$ and a thick layer of copper (above $300\\mathrm{nm}$ ) with a margin size of $2\\mathrm{mm}$ by E-beam evaporator (Denton Explorer).  \n\nThe measurement of TECDs. The samples were placed on the linear motor and moved up and down automatically with the help of the linear motor control program and system. For some inorganic compounds, the TECDs are relatively small; the turbulent caused by the motion of tested samples would cause some noise because of the friction between the platinum wire and mercury. Therefore, the platinum wire was then designed to go through the bottom of the Petri dish and fully immersed in the liquid metal, and sealed by epoxy glue. In this way, there is no contact and separation between them; therefore, the noise is minimized.  \n\nThe sample’s surfaces were carefully adjusted to ensure the precisely right contact between the tested material and the liquid mercury. The position and angles were adjusted by a linear motor, a high load lab jack (Newport 281), and a two-axis tilt and rotation platform (Newport P100-P). The short-circuit charge $Q_{\\mathrm{SC}}$ and open-circuit voltage $V_{\\mathrm{OC}}$ of the samples were measured by a Keithley 6514 electrometer in a glove box with an ultra-pure nitrogen environment (Airgas, $99.999\\%$ ). The environmental condition was fixed at $20\\pm1^{\\circ}\\mathrm{C}$ , 1 atm with an additional pressure of $1{-}1.5\\mathrm{in}$ . height of $\\mathrm{H}_{2}\\mathrm{O}$ and $0.43\\%$ relative humidity. In addition, samples were kept in the glove box overnight to eliminate the water vapor on the surface of the samples.  \n\n# Data availability  \n\nThe datasets generated during and/or analyzed during the current study are available from the corresponding author. The source data underlying Figs. 2a–d, 3, and 4a–b are provided as a Source Data file.  \n\nReceived: 12 January 2020; Accepted: 30 March 2020; Published online: 29 April 2020  \n\n# References  \n\n1. Wang, Z. L. On the first principle theory of nanogenerators from Maxwell’s equations. Nano Energy 104272 (2019).   \n2. Henniker, J. Triboelectricity in polymers. Nature 196, 474 (1962).   \n3. Zi, Y. L. et al. Standards and figure-of-merits for quantifying the performance of triboelectric nanogenerators. Nat. 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Mater. 28, 1805540–1805540 (2018).  \n\n# Acknowledgements  \n\nThis work was financially supported by the Hightower Chair Foundation in Georgia Tech  \n\n# Author contributions  \n\nZ.L.W. supervised and guided the project; H. Zou, L.G., H.X., and X.H. fabricated the devices; H.X., X.S., and X.L. synthesized the materials; H. Zou, Y.Z., and P.W. designed the measurement; L.G., H.X., H. Zou, G.D., P.J., B.Z., C.X., and H.Z. performed the experiments; L.G., H. Zou, and H.X. analyzed the data, H. Zou proposed the model; the manuscript was prepared with input from all authors.  \n\n# Competing interests  \n\nThe authors declare no competing interests.  \n\n# Additional information  \n\nSupplementary information is available for this paper at https://doi.org/10.1038/s41467- 020-15926-1.  \n\nCorrespondence and requests for materials should be addressed to Z.L.W.  \n\nPeer review information Nature Communications thanks Junghyo Nah and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.  \n\nReprints and permission information is available at http://www.nature.com/reprints  \n\nPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\nOpen Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.  \n\n$\\circledcirc$ The Author(s) 2020  ",
    "Target": null
  },
  {
    "id": "10.1038_s41467-019-14278-9",
    "DOI": "10.1038/s41467-019-14278-9",
    "DOI Link": "http://dx.doi.org/10.1038/s41467-019-14278-9",
    "Relative Dir Path": "mds/10.1038_s41467-019-14278-9",
    "Article Title": "Quantifying electron-transfer in liquid-solid contact electrification and the formation of electric double-layer",
    "Authors": "Lin, SQ; Xu, L; Wang, AC; Wang, ZL",
    "Source Title": "NATURE COMMUNICATIONS",
    "Abstract": "Contact electrification (CE) has been known for more than 2600 years but the nature of charge carriers and their transfer mechanisms still remain poorly understood, especially for the cases of liquid-solid CE. Here, we study the CE between liquids and solids and investigate the decay of CE charges on the solid surfaces after liquid-solid CE at different thermal conditions. The contribution of electron transfer is distinguished from that of ion transfer on the charged surfaces by using the theory of electron thermionic emission. Our study shows that there are both electron transfer and ion transfer in the liquid-solid CE. We reveal that solutes in the solution, pH value of the solution and the hydrophilicity of the solid affect the ratio of electron transfers to ion transfers. Further, we propose a two-step model of electron or/and ion transfer and demonstrate the formation of electric double-layer in liquid-solid CE.",
    "Times Cited, WoS Core": 537,
    "Times Cited, All Databases": 554,
    "Publication Year": 2020,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000511940400003",
    "Markdown": "# Quantifying electron-transfer in liquid-solid contact electrification and the formation of electric double-layer  \n\nShiquan Lin1,2, Liang Xu 1,2, Aurelia Chi Wang3 & Zhong Lin Wang 1,2,3\\*  \n\nContact electrification (CE) has been known for more than 2600 years but the nature of charge carriers and their transfer mechanisms still remain poorly understood, especially for the cases of liquid–solid CE. Here, we study the CE between liquids and solids and investigate the decay of CE charges on the solid surfaces after liquid–solid CE at different thermal conditions. The contribution of electron transfer is distinguished from that of ion transfer on the charged surfaces by using the theory of electron thermionic emission. Our study shows that there are both electron transfer and ion transfer in the liquid–solid CE. We reveal that solutes in the solution, pH value of the solution and the hydrophilicity of the solid affect the ratio of electron transfers to ion transfers. Further, we propose a two-step model of electron or/and ion transfer and demonstrate the formation of electric double-layer in liquid–solid CE.  \n\nontact electrification (CE) (or triboelectrification in general terms) is a universal but complicated phenomenon, which has been known for more than 2600 years. The solid-solid CE has been studied using various methods and different mechanisms were proposed (Electron transfer1,2, ion transfer3 and material transfer4–6 were used to explain different types of CE phenomena for various materials). In parallel, CE between liquid–solid is rather ubiquitous in our daily life, such as flowing water out of a pipe is charged, which is now the basis of many technologies and physical chemical phenomena, such as the liquid–solid triboelectric nanogenerators $(\\mathrm{TENGs})^{7-10}$ , hydrophobic and hydrophilic surfaces, and the formation of electric double-layer (EDL)11–14. However, understanding about the liquid–solid CE is rather limited and the origin about the formation of EDL remains ambiguous owing to the lacking of fundamental understanding about charge transfer at interfaces. The most important issue in the CE mechanism is the identity of charge carriers (electrons or/and ions), which has been debated for decades in the solid-solid $\\mathrm{CE}^{15,16}$ . Most recently, charge carriers have been identified as electrons for solid-solid CE based on temperature dependent effect and photoexcitation effect on the charged surfaces, and the ion transfer is out of consideration17–19.  \n\nAs for the case of liquid–solid CE, it is usually assumed to be ion transfer without any detailed studies, simply because ions are often present in liquids, such as $\\mathrm{H^{+}}$ and $\\mathrm{OH^{-}}$ in water. Regarding the nature of EDL, the charging of the isolated surfaces in a liquid is considered to be induced by ionization or dissociation of surface groups and the adsorption or binding of ions from liquid onto the solid surface20. From these points of view, the charge carriers in liquid–solid CE is naturally assumed to be ions and transfer of electrons has not been even considered. However, Wang et al. has proposed a “electron-cloud-potential-well” model for explaining CE in a general case, in which the electron transfer in CE is considered to be induced by the overlap of electron clouds as a result of mechanically forced contact18. At a liquid–solid interface, the molecules in a liquid collide with atoms on the solid surface owing to liquid pressure, which may lead to the overlap of electron clouds and result in electron transfer. Hence, there is still dispute about the identity of charge carriers in the liquid–solid CE, which is one of the most fundamental questions in CE and physical chemistry as well. Such a question can now be answered using the surface charge decay experiments at different temperatures for distinguishing electron transfer from ion transfer in liquid–solid contact17,18. This is because electrons are easily emitted from the solid surface as induced by thermionic emission, while ions usually bind with the atoms on the solid surface, and they are rather hard to be removed from the surface in comparison to electrons especially when the temperature is not too high.  \n\nHere we show the CE in liquid–solid and the charge density on solid surfaces after the contact measured using Kelvin probe force microscopy $(\\mathrm{KPFM})^{21-24}$ . We investigate the decay of CE charges on the solid surfaces at different temperatures. We particularly study the effects of solutes in the aqueous solution, $\\mathrm{\\pH}$ value of the aqueous solution and the hydrophilicity of the solid surfaces on the liquid–solid CE. We have analyzed the ratio of electron transfers to ion transfers in the liquid–solid CE for the first time according to the thermionic emission theory, to the best of our knowledge. Lastly, we propose a model about the formation of the EDL based on the understanding of the charge transfer at liquid–solid interface, providing a distinct mechanism from the general understanding in classical physical chemistry.  \n\n# Results  \n\nThe CE between the DI water and the $\\mathbf{SiO}_{2}$ . Here, flat insulating ceramic thin films, such as $\\mathrm{SiO}_{2}$ , ${\\mathrm{Al}}_{2}{\\mathrm{O}}_{3}$ , $\\mathrm{MgO}$ , ${\\mathrm{Ta}}_{2}{\\mathrm{O}}_{5},$ , $\\mathrm{HfO}_{2}$ , AlN, and $\\mathrm{Si}_{3}\\mathrm{N}_{4}.$ , deposited on highly doped silicon wafers, were used as solid samples. The liquids were chosen as deionized water (DI water) and different aqueous solutions, including NaCl, HCl and NaOH solutions. In the experiments, the liquid dropped from a grounded needle and slid across the ceramic surface, as shown in Fig. 1a. After the liquid being vaporized, the surface charge densities on the ceramic surfaces were measured by using KPFM at different substrate temperatures. According to previous studies, ions will be produced by ionization reaction on the oxide and nitride surfaces when they contact the aqueous solutions25–29. For example, $\\mathrm{O^{-}}$ ions will be generated by the ionization reaction on the $\\mathrm{SiO}_{2}$ surface as shown below (The hydroxyl on the $\\mathrm{SiO}_{2}$ surface is usually produced by adsorbing water molecules in the air or contacting with water)29:  \n\n$$\n\\equiv\\mathrm{Si}-\\mathrm{OH}+\\mathrm{OH}^{-}\\Leftrightarrow\\equiv\\mathrm{Si}-\\mathrm{O}^{-}+\\mathrm{H}_{2}\\mathrm{O}\n$$  \n\nAs introduced above, electrons may be another type of charge carrier on the $\\mathrm{SiO}_{2}$ surface after contacting with aqueous solutions. Hence, we assume that there are both $\\mathrm{O^{-}}$ ions and electrons on the liquid sliding trace on the $\\mathrm{SiO}_{2}$ surface, as shown in the inset in the Fig. 1a. When the $\\mathrm{SiO}_{2}$ sample is heated by the sample heater, the electrons will be thermally excited and emitted from the surface, as shown in Fig. 1b, while the $\\mathrm{O^{-}}$ ions may stay on the surface since they formed covalent bonds with the Si atoms on the $\\mathrm{SiO}_{2}$ surface. (As shown in the ab initio molecular dynamics simulations in the Supplementary Note 1, Supplementary Figs. 1, 2, and the simulation results are shown in Supplementary Movies 1–7). This means that, if heating can induce obvious decay of CE charges on the $\\mathrm{SiO}_{2}$ surface, it may be mainly caused by thermal emission of electrons.  \n\nIn the experiments, the CE between the $\\mathrm{SiO}_{2}$ and the DI water was first performed, and Fig. 1c gives the results of the temperature effect on the decay of CE charges on the $\\mathrm{SiO}_{2}$ surfaces. It is obvious that the $\\mathrm{SiO}_{2}$ is negatively charged and the charge density on the $\\mathrm{SiO}_{2}$ surface is about $-810\\upmu\\mathrm{Cm}^{-2}$ (negative sign means that the charges are negative) after the contact with the DI water. In Fig. 1c, the temperature affects the decay of the negative charges on the $\\mathrm{SiO}_{2}$ surface significantly. The surface charge density on $\\mathrm{SiO}_{2}$ remains almost unchanged at $313\\mathrm{K}$ and slight decay of the surface charge density is observed at $343\\mathrm{K}$ . As the sample temperature continues to rise, the decay rate of the surface charges increases. But some charges (about $-180\\upmu\\mathrm{Cm}^{-2})$ cannot be removed even when the temperature rises up to $434\\mathrm{K}$ and $473\\mathrm{K}$ (these charges can be called as “sticky” charges that remain on surfaces even when the temperature is raised). For the removable charges, the decay behaviors are consistent with the thermionic emission theory, in which the electrons are considered to obtain more energy and the electron density decay faster at higher temperatures. Moreover, it is found that the charge density decay exponentially and the decay curve follows the electron thermionic emission model as described by the following equation, which was proposed in our previous studies (The curve fitting results are shown in Supplementary Fig. 3)17,18. Hence, the removable charges in the CE between the $\\mathrm{SiO}_{2}$ and the DI water can be identified as electrons.  \n\n$$\n\\sigma=e^{-a t}\\sigma_{e}+\\sigma_{s}\n$$  \n\nwhere $\\sigma$ denotes the CE charge density on the sample surface, $\\sigma_{e}$ denotes the initial density of charges on the sample surface, which can be removed by thermal excitation, $\\sigma_{s}$ denotes the density of the “sticky” charges, which cannot be removed by heating and $t$ denotes the decay time.  \n\nFor the “sticky” charges, charging and heating cycle tests were performed to observe their behaviors, as shown in Fig. 1d. In every cycle of the testes, the $\\mathrm{SiO}_{2}$ sample contacts with the DI water first, and then it is heated to $513\\mathrm{K}$ and maintains for 10 min to remove the electrons on the surface. In the first cycle, the $\\mathrm{SiO}_{2}$ is negatively charged when it contacts with the DI water, and the density of the “sticky” charges is $-180\\upmu\\mathrm{Cm}^{-2}$ as expected. It is found that the density of the “sticky” charges increases to $-300$ $\\upmu\\mathrm{Cm}^{-2}$ in the second cycle and it continuously increases with the number of the cycles. After five cycles of experiments, the density of the “sticky” charges reaches a saturation value, and there are not removable charges on the $\\mathrm{SiO}_{2}$ surface. These behaviors suggest that the “sticky” charges should be ions, such as $\\mathrm{O^{-}}$ ions, instead of electrons. As shown in Supplementary Fig. 4, in each contact with the DI water, both electrons and $0^{-}$ ions are attached on the surface. Electrons are emitted as temperature rises, while the $\\mathrm{O^{-}}$ ions cannot be removed in the subsequent heating if the temperature is not too high. In the next cycle of introducing water droplet, more $0^{-}$ ions are produced in the ionization reaction and accumulate on the $\\mathrm{SiO}_{2}^{-}$ surface since it has not reached saturation. With the increase of cycles on introducing water droplets, the concentration of $\\dot{\\mathrm{~O~}^{-}}$ ions continues to rise and more “available charge positions and densities” are filled, thus, it becomes harder for the $\\mathrm{SiO}_{2}$ to gain more electrons in the CE, resulting in a decrease of the electron density on the surface. A few cycles later, the density of the ions reaches a saturated value, which remains stable even in the followed heating process.  \n\n![](images/093d5c2ecb2fadadecde1b1b28c3a9cac0ee0132780fdd586427fbb18f386420.jpg)  \nFig. 1 Temperature effect on the CE between the DI water and the $\\mathbf{SiO}_{2}$ . (a) The setup of the charging experiments, where the negative charges generated on the $\\mathsf{S i O}_{2}$ surface could be electrons and $\\mathsf{O}^{-}$ ions induced by surface ionization reaction. ( $'0^{\\prime}$ is the Oxygen atom, ‘Si’ is the silicon atom and $'0^{-\\prime}$ is the $\\circled{\\scriptsize{1}}$ xygen ion). (b) The setup of AFM platform for the thermionic emission experiments. (c) The decay of the CE charge (induced by contacting with the DI water at room temperature) on the $\\mathsf{S i O}_{2}$ surface at different substrate temperatures. (d) The CE charge density on the $\\mathsf{S i O}_{2}$ sample surface in the charging (contacting with the DI water at room temperature) and heating (at $513\\mathsf{K}$ for $10\\mathsf{m i n}.$ ) cycle tests. (Error bar are defined as s. d.).  \n\nBased on the analysis, it turns out that electrons can be distinguished from ions in the CE by performing the thermionic emission experiments. The removable and the “sticky” charges in the experiments are identified as electrons and ions, respectively. And the results suggest that there are both electron and ion transfers in the CE between the $\\mathrm{SiO}_{2}$ and the DI water. The density of transferred electrons is measured to be $-630\\upmu\\mathrm{Cm}^{-2}$ and the density of transferred ions is about $-180\\upmu\\mathrm{Cm}^{-2}$ . It means that the electron transfer, which account for $77\\%$ of the total charges, is dominant in the CE between $\\mathrm{SiO}_{2}$ and DI water in very first contact.  \n\nEffects of the solutes and the liquid pH value on the CE. Further, the effects of the solutes in the liquid and $\\mathrm{\\pH}$ value of the liquid on the liquid–solid CE were studied. The CE between the $\\mathrm{SiO}_{2}$ and different aqueous solutions, including NaCl, HCl and NaOH solutions, was performed and the electron transfer and ion transfer in the CE were separated by the heat-induced charge decay experiments. Figure 2a gives the effect of the NaCl concentration on the transferred charge density in the CE between the $\\mathrm{SiO}_{2}$ and the $\\mathrm{\\DeltaNaCl}$ solution. It is found that the charge density on the $\\mathrm{SiO}_{2}$ surfaces decreases with the increase of the NaCl concentration. This result is consistent with the previous studies about the liquid–solid TENG, in which the salt solution is the liquid and the output of the TENG decreases with the increase of the salt concentration30,31. The effect was not clearly explained before, because there was no method to identify the charge carriers. Here, the decay of the charge density on the $\\mathrm{SiO}_{2}$ surfaces is performed at $433\\mathrm{K}$ after the CE between the $\\mathrm{SiO}_{2}$ and the NaCl solutions, and the results are shown in Fig. 2b. It can be seen that the charge density decays exponentially, which is the same as the CE between the $\\mathrm{SiO}_{2}$ and the DI water as introduced above. The density of removable charges (electrons) on the $\\mathrm{SiO}_{2}$ surfaces decreases with the increase of the NaCl concentration, while the “sticky” charges (ions) density remains almost unchanged when the $\\mathrm{SiO}_{2}$ contacts with the NaCl solutions of different concentrations. It implies that the decrease of the charge density on the $\\mathrm{SiO}_{2}$ induced by the increase of the NaCl concentration is mainly due to the decrease of the electron transfer, which might be caused by the increase of the dielectric constant of the NaCl solution that facilitate the discharge after charging. Different from electron transfer, the ion transfer will not be significantly affected by the NaCl concentration in the first contact (Fig. 2b). This result is easy to understand, because there are no $\\mathrm{Na^{+}}$ or $\\mathrm{Cl^{-}}$ in the ionization reaction (chemical formula 1), which produce the required $\\mathrm{O^{-}}$ ions on the $\\mathrm{SiO}_{2}$ surface. Figure 2c gives the CE charge density on the $\\mathrm{SiO}_{2}$ sample surface in the charging (contacting with $0.4\\mathrm{M}\\mathrm{NaCl}$ solution) and heating ( $.513\\mathrm{K}$ for 10 min) cycle tests. The results show that the saturated ion density in the CE between $0.4\\mathrm{M}$ NaCl solution and $\\mathrm{SiO}_{2}$ is slightly lower than that between DI water and $\\mathrm{SiO}_{2}$ (Fig. 1d). The difference in the saturated ion density may be caused by the covering of the crystallized NaCl on the $\\mathrm{SiO}_{2}$ surface in the subsequent heating processes, which blocks the progress of ionization reaction.  \n\n![](images/a77b89984c55a72cb2fdd2da5ac5987e4c029ee9e3900b0b58d1848cfc344af6.jpg)  \nFig. 2 Temperature effect on the CE between the $\\sin O_{2}$ and aqueous solutions. (a) The effects of the NaCl concentration on the CE between the $\\mathsf{S i O}_{2}$ and the ${\\mathsf{N a C l}}$ solutions. (b) The decay of the CE charge at $433\\mathsf{K}$ which is induced by contacting with the NaCl solutions. (c) The CE charge density on the $\\mathsf{S i O}_{2}$ sample surface in the charging (contacting with 0.4 M NaCl solution at room temperature) and heating (513 K for $10\\mathsf{m i n}.$ cycle tests. (d) The decay of the CE charge at $433\\mathsf{K},$ which is induced by contacting with the pH 11 HCl solution and the $\\mathsf{p H3N a O H}$ solution. The charging and heating cycle testes when the liquids are (e) the pH 11 NaOH solution and $(\\pmb{\\uparrow})$ the pH 3 HCl solution. (Error bar are defined as s. d.).  \n\nDifferent from $\\mathrm{Na^{+}}$ or $\\mathrm{Cl}^{-}$ , it can be seen that the $\\mathrm{OH^{-}}$ plays an important role in the generation of the $0^{-}$ ions on the $\\mathrm{SiO}_{2}$ surface from the chemical formula 1. Hence, the density of the transferred ions on the $\\mathrm{SiO}_{2}$ surface may be affected by the pH value of the solutions. Figure 2d shows the decay of the surface charge density on the $\\mathrm{SiO}_{2}$ surface, which is produced by contacting with the $\\mathrm{pH}\\ 11\\ \\mathrm{NaOH}$ solution and $\\mathrm{pH}\\ 3\\ \\mathrm{H}\\dot{\\mathrm{Cl}}$ solution. When the $\\mathrm{pH}$ value of the solution increases to 11, the electron transfers decrease, and the density of transferred ions (about $-230\\upmu\\mathrm{Cm}^{-2},$ is slightly higher than that when the pH value of liquid is 7 (DI water). And the difference can also be observed in the charging and heating tests, in which the saturated ion density on the $\\mathrm{SiO}_{2}$ surface when the liquid is the $\\mathrm{pH}11$ $\\mathrm{\\DeltaNaOH}$ solution is higher than that when the liquid is the DI water, as shown in Fig. 2e. This is caused by the increase of the $\\mathrm{OH^{-}}$ concentration in the solution, which promotes the ionization reaction (chemical formula 1). When the $\\mathsf{p H}$ value of the solution changes to 3, the electron transfer direction and the polarity of the transferred ions on the $\\mathrm{SiO}_{2}$ surface reverse from negative to positive (Fig. 2d). And the charging and heating cycle tests in Fig. 2 f show that the positive ions also accumulate on the $\\mathrm{SiO}_{2}$ surface. In this case, the positive ions on the $\\mathrm{SiO}_{2}$ surface are produced by another ionization reaction, as shown below26–28.  \n\n$$\n\\equiv\\mathrm{Si}-\\mathrm{OH}+\\mathrm{H}^{+}\\Leftrightarrow\\equiv\\mathrm{Si}-\\mathrm{OH}_{2}^{+}\n$$  \n\nThe effects of $\\mathrm{\\tt{pH}}$ value on the CE between liquids and various ceramics are shown in Supplementary Fig. 5. The results are similar to the pH effects on the $\\mathrm{SiO}_{2}$ surface. When the $\\mathsf{p H}$ value of the solution was 11, the transferred ions on the ceramic surfaces are negative as shown in Supplementary Fig. 5a–c. When the $\\mathsf{p H}$ value of the solution changes to 3, the polarity of the transferred ions also reverses to be positive as shown in Supplementary Fig. 5d–f. This means that the effects of $\\mathrm{\\pH}$ value on the ionization reaction for different ceramics are consistent.  \n\nThese results show that no matter what the aqueous solution is, there are always both electron transfer and ion transfer in liquid–solid CE. The electron transfers between aqueous solution and solid is sensitive to solutes in the liquids, such as $\\mathrm{{Na^{+}}}$ , $\\mathrm{Cl^{-}}$ , $\\mathrm{OH^{-}}$ and $\\mathrm{H^{+}}$ etc. While the ion transfer is mainly affected by the $\\mathrm{\\pH}$ value of the solution, which dominates the ionization reactions on the insulator surfaces.  \n\nSolid effects on the liquid–solid CE. As another side in the liquid–solid CE, different solids were also tested in the thermionic emission experiments. As shown in Fig. 3a–f, the CE charge decay in the CE between the DI water and different insulating ceramics was performed, including $\\mathrm{MgO}$ , $\\mathrm{Si}_{3}\\mathrm{N}_{4},$ $\\mathrm{Ta}_{2}\\mathrm{O}_{5}$ , $\\mathrm{HfO}_{2}.$ , $\\bar{\\bf A l}_{2}\\bf O_{3}$ and AlN. (The surface ionization reaction equations between water molecules and these materials are shown in Supplementary  \n\n![](images/91c7e6789f65c5adbc5b8f337829bb6c4cc81d0b694ee5f7f74f889b499fc2db.jpg)  \nFig. 3 Temperature effect on the CE between the DI water and the solids. The decay of CE charges (induced by contacting with the DI water at room temperature) on a $M g O$ , b $\\mathsf{S i}_{3}\\mathsf{N}_{4},$ c $\\mathsf{T a}_{2}\\mathsf{O}_{5}$ , d ${\\mathsf{H}}{\\mathsf{f}}{\\mathsf{O}}_{2}$ , e $A l_{2}O_{3},$ and f AlN surfaces at $433\\mathsf{K},$ and the amount of the electron transfer and the ion transfer in the CE between the DI water and different insulators. g The relation between the electron transfer to the ion transfer ratio and the DI water contact angle (WCA) of the materials. h The schematic of WCA effects on the ion transfer and electron transfer in liquid–solid CE. $\\gamma_{L},\\gamma_{S},$ and $\\gamma_{L-S}$ denote the liquid–gas interfacial tension, solid–gas interfacial tension and liquid–solid interfacial tension, respectively. (Error bar are defined as s. d.).  \n\nNote 2). It is found that all of the charge decay curves follow the electron thermionic emission model, hence the removable charges are electrons and the “sticky” charges are ions as analyzed above. The electron transfer and the ion transfer are marked in Fig. 3a–f, it can be seen that the ratio of electron transfers to ion transfers (E/I) highly depends on the type of solid. For the CE between the AlN and the DI water, more than $88\\%$ of the total transferred charges are electrons. But in the CE between the $\\mathrm{Si}_{3}\\mathrm{N}_{4}$ and the DI water, electron transfer is only $31\\%$ of the total charge transfer. In order to test the interaction between a liquid and a solid at the interface, the water contact angle (WCA) of the ceramics was measured and the results are shown in Fig. 3g. It is noticed that the $\\mathrm{E/I}$ ratio slightly increases with the increase of the WCA when the WCA of materials is less than $90^{\\circ}$ . When the WCA of the materials increase to be larger than $90^{\\circ}$ , such as $92.2^{\\circ}$ for the $\\mathrm{SiO}_{2}$ and $97.0^{\\circ}$ for the AlN, the E/I ratio increases rapidly. For the $\\mathrm{SiO}_{2}$ and the AlN, the $\\mathrm{E/I}$ ratios are 3.5 and 7.5, respectively. Actually, the WCA is dependent on the liquid–solid, solid-gas, liquid-gas interfacial tensions, which are related to the interfacial energy of two phases, as shown in the Fig. 3h. The interfacial energy between a hydrophilic surface $({\\mathrm{WCA}}<90^{\\circ},$ and water is usually lager than that between a hydrophobic surface $({\\mathrm{WCA}}>90^{\\circ},$ and water. It means that the interaction between the water molecules and the solid surface with small WCA is usually stronger than that between water molecules and the solid surface with large WCA. And the Oxygen atoms or Hydrogen atoms in water molecules are more likely to form covalent bonds with the atoms on the hydrophilic surface. In other words, the surface ionization reaction is more likely to occur and leading to the generation of ions on the hydrophilic solid surface. On the contrary, the surface ionization reaction between the hydrophobic solid surfaces and water is less likely to occur, and the CE between the solid and aqueous solution is electron-dominated.  \n\nIt needs to be mentioned that the polarity of the transferred electrons and transferred ions not necessary to be the same in liquid–solid CE. As shown in Fig. 3a, the MgO obtains electrons and positive ions at the same time in the CE between $\\mathrm{MgO}$ and DI water (Supplementary Fig. 6a), and the positive charge density on the $\\mathrm{MgO}$ surface increases in the heating due to the emission of electrons. For the CE between AlN and DI water, the AlN loses electrons and obtains negative ions (Supplementary Fig. 6b). These results suggest that the electron transfer and ion transfer in liquid–solid CE are independent of each other. Furthermore, it may be possible that the electron transfer and ion transfer could be segregated on different surface areas, but remain proved experimentally.  \n\n![](images/0083083144788c3da6eee3cad97b9fa410c4e186bfe5b05392acb4822d728f72.jpg)  \nFig. 4 Mechanism of liquid–solid CE and formation of electric doublelayer. a The liquid contacts a virgin surface (before CE). b The water molecules and ions in the liquid impact the virgin surface and electron transfer between the water and the surface. c The surface is charged and the charge carriers are mainly electrons $\\mathsf{\\backslash W C A>90^{\\circ}}$ , $\\mathsf{p H}=7\\mathrm{:}$ ), some ions may be generated on the surface caused by the ionization reaction etc. d The opposite polarity ions are attracted to migrate toward the charged surface by the Coulomb force, electrically screening the first charged layer.  \n\nAccording to the results, the CE between solid and liquid can be affected by the $\\mathrm{\\pH}$ value of the aqueous solution, solutes in the aqueous solution and the hydrophilicity of the solids. Nevertheless, there is always electron transfer in the CE between liquid (aqueous solution) and solid. This result was predicted in the “two-step” model first proposed by Wang et al.32, but was not included at all in the classical explanation regarding the formation of the EDL. Combining the experiment results and the “two-step” model32, a new picture for the liquid–solid CE and the formation of the EDL is proposed, as shown in Fig. 4. In the first step, the liquid contacts a virgin solid surface (Fig. 4a), the molecules and ions, including $_\\mathrm{H}_{2}\\mathrm{O}$ , cation, anion etc., will impact the solid surface due to the thermal motion and the pressure from the liquid (Fig. 4b). During the impact, electrons will transfer between the solid atoms and water molecules owing to the overlap of the electron clouds of the solid atoms and water molecules18, and the ionization reaction may also occur simultaneously on the solid surface. Hence there will be both electrons and ions generated on the surface. As an example, the electron transfer plays a dominated role in the CE between the $\\mathrm{SiO}_{2}$ and DI water, as shown in Fig. 4c. In the second step, the opposite ions in the liquid would be attracted to migrate toward the charged surface by the electrostatic interactions, forming an $\\mathrm{EDL},$ as shown in Fig. 4d.  \n\nAn atom with extra/deficient electrons are referred as ion, therefore, the transferred electrons on the solid surface is considered as the first step to make the “neutral” atoms on solid surface become ions in the “two-step” model32. From this perspective, the ions produced by the ionization reaction in the experiments can also be considered as the “neutral” atoms with extra electrons. The difference is that the transferred electrons directly induced by the collisions between the atoms in the liquid and the atoms on the solid surface were usually trapped in the surface states, while the extra electrons of the “neutral” atoms produced in the ionization reaction were trapped in the atomic orbitals of the atoms (the atomic orbitals can be considered as the special surface states of solids generated in the ionization reaction). There is no essential difference between the electrons in the surface states and those in the atomic orbitals. However, the potential barrier of the surface states to prevent the electrons from emitting in the heating process might be lower than that of atomic orbitals. Hence, the electrons in the surface states of the solid are removable, while the electrons in the atomic orbitals are tightly bonded on the solid surfaces.  \n\nAlso, the surface charge density (electrons and ions) in the liquid–solid CE is not as dense as that appearing in text book drawing. The highest transferred electron density in our experiments is $-630\\upmu\\mathrm{C}\\mathrm{m}^{-2}$ in the CE between $\\mathrm{SiO}_{2}$ and DI water, which corresponds to ${\\sim}1$ excess electron per $250\\mathrm{nm}^{2}$ . Thus, the probability of electron transfer in liquid–solid CE is usually less than one out of $\\sim2500$ surface atoms. The transferred ion density in CE between $\\mathrm{SiO}_{2}$ and DI water is $-180\\upmu\\mathrm{C}\\mathrm{m}^{-2}$ , which corresponds to ${\\sim}10^{-}$ ion per $1000\\mathrm{nm}^{2}$ . Accordingly, the distance between two adjacent electrons on $\\mathrm{SiO}_{2}$ surface is ${\\sim}16\\mathrm{nm}$ , and the distance between two adjacent $\\mathrm{O^{-}}$ ions is $\\sim30\\mathrm{nm}$ as shown in Fig. 4c. These distances are much larger than the thickness of Stern layer, which is of the order of a few ångstroms20. Hence, the distance of two adjacent charges (electrons or/and ions) should be considered in the structure of the EDL.  \n\n# Discussion  \n\nIn conclusion, the CE between liquid and solid was performed and the temperature effect on the decay of the CE charge on the ceramic surfaces was investigated. It is revealed that there are both electron transfer and ion transfer in the liquid–solid CE. The results suggest that the solutes in the aqueous solution, such as $\\mathrm{Na^{+}}$ and $\\mathrm{Cl^{-}}$ etc., can reduce the electron transfer between aqueous solution and solid. And the ion transfers in the liquid–solid CE induced by the ionization reaction can be significantly affected by the pH value of the liquid. Besides, it is found that the CE between hydrophilic surfaces and aqueous solutions is likely dominated by ion transfer; and the CE between hydrophobic surfaces and aqueous solutions is more likely to be dominated by electron transfer. This is the first time that the “two-step” model about the formation of $\\mathrm{EDL},$ in which the electron transfer plays a dominant role in liquid–solid CE, is verified experimentally. Our results may have great implications in the studies of TENG and EDL.  \n\n# Methods  \n\nSample preparation. The $\\mathrm{SiO}_{2}$ layer was deposited on high doped silicon wafer by thermal oxidation. The $\\mathrm{Si}_{3}\\mathrm{N}_{4},$ $\\mathrm{{Al}}_{2}\\mathrm{{O}}_{3}$ , $\\mathrm{Ta}_{2}\\mathrm{O}_{5}$ , MgO, $\\mathrm{HfO}_{2}$ AlN layers were deposited on high doped silicon surfaces by magnetron sputtering, and the thickness of all the layers were $100\\mathrm{nm}$ . The DI water with a resistivity of $18.2\\mathrm{M}\\Omega$ cm used here was produced by deionizer (HHitech, China). Before the experiments, all the samples were heated for $10\\mathrm{min}$ at $513\\mathrm{K}$ to remove the charge on the surfaces. After the heat treatment, the charge density of the ceramic surfaces was measured to be about $0\\upmu\\mathrm{C}\\mathrm{m}^{-2}$ , except the $\\mathrm{MgO}$ and $\\mathrm{Si}_{3}\\mathrm{N}_{4}$ . The “sticky” charge density on the $\\mathrm{{\\bfMgO}}$ and $\\mathrm{Si}_{3}\\mathrm{N}_{4}$ surface was about $800\\upmu\\mathrm{C}\\mathrm{m}^{-2}$ and $-250\\upmu\\mathrm{C}\\mathrm{m}^{=2}$ before the CE with solutions, respectively. The “sticky” charges on the $\\mathrm{MgO}$ and $\\mathrm{Si}_{3}\\mathrm{N}_{4}$ surfaces may be the ions generated by the ionization reaction between the samples and the water molecules in the air, since the $\\mathrm{{\\calMgO}}$ and $\\mathrm{Si}_{3}\\mathrm{N}_{4}$ are most hydrophilic in these ceramics.  \n\nKPFM experiments. The experiments were performed on commercial AFM equipment Multimode 8 (Bruker, USA). NSC 18 (MikroMash, USA; Au coated; tip radius: $25\\mathrm{nm};$ spring constant: $2.8\\mathrm{Nm}^{-1}$ ) was used as the conductive tip here. The sample temperature was controlled by the sample heater and the tip temperature was controlled by the tip heater independently. In all the experiments, the temperature of the sample and the tip remained consistent. The tapping amplitude was $350\\mathrm{mV}$ , the scan size was $5\\upmu\\mathrm{m}$ and the lift height was set to $50\\mathrm{nm}$ in the KPFM measurements. In order to acquire the data from a big region, the KPFM was manual operated to scan different positions on the whole sample surface ${\\sqrt{>20}}$ positions). All the heating and charge measurements are performed in an Ar atmosphere. The changes of surface charge density were demonstrated not caused by the adsorption and desorption of the water molecules on $\\mathrm{SiO}_{2}$ surface, as shown in Supplementary Note 3 and Supplementary Fig. 7. And the observed changes in the surface potential in our experiments were not due to the temperature effects on the measurements, as shown in Supplementary Note 4 and Supplementary Fig. 8.  \n\nCalculation of surface charge density. In previous studies, the transferred charge density on the insulating surfaces was calculated by the following equation:  \n\n$$\n\\Delta\\sigma=\\frac{\\Delta V\\varepsilon_{0}\\varepsilon_{s a m p l e}}{t_{s a m p l e}}\n$$  \n\nwhere $\\Delta\\sigma$ denotes the transferred charge density, $\\Delta V$ denotes the change of surface potential after the CE, $\\scriptstyle{\\varepsilon_{0}}$ denotes the vacuum dielectric constant, $\\varepsilon_{s a m p l e}$ denotes the relative dielectric constant of the sample and $t_{s a m p l e}$ denotes the thickness of the insulating layer.  \n\nIn our experiments, the absolute charge density on the sample surface need to be calculated. In this case, the contact potential difference (CPD) between the tip and the substrate of the samples should be considered, and the absolute charge density on the insulating surfaces can be expressed as following (the calculations are shown in the Supplementary Note 5 and Supplementary Fig. 9):  \n\n$$\n\\sigma=\\frac{(V+C P D_{t i p-s a m p l e})\\varepsilon_{0}\\varepsilon_{s a m p l e}}{t_{s a m p l e}}\n$$  \n\nwhere $\\sigma$ denotes the absolute charge density on the sample surfaces, $V$ denotes the surface potential of the samples and the $C P D_{t i p-s a m p l e}$ is the CPD between the tip and the substrate of the samples.  \n\n# Data availability  \n\nAll data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Additional data related to this paper may be requested from the authors. The source data underlying all figures can be found in the Source Data file.  \n\nReceived: 22 August 2019; Accepted: 16 December 2019; Published online: 21 January 2020  \n\n# References  \n\n1. Gibson, H. Linear free energy relations. V. triboelectric charging of organic solids. J. Am. Chem. Soc. 97, 3832–3833 (1975).   \n2. Sakaguchi, M., Makino, M., Ohura, T. & Iwata, T. Contact electrification of polymers due to electron transfer among mechano anions, mechano cations and mechano radicals. J. Electrostat. 72, 412–416 (2014).   \n3. McCarty, L. & Whitesides, G. Electrostatic charging due to separation of ions at interfaces: contact electrification of ionic electrets. Angew. Chem. Int. Ed. 47, 2188–2207 (2008).   \n4. Sutka, A. et al. The role of intermolecular forces in contact electrification on polymer surfaces and triboelectric nanogenerators. Energy Environ. Sci. 12, 2417–2421 (2019).   \n5. Pandey, R., Kakehashi, H., Nakanishi, H. & Soh, S. Correlating material transfer and charge transfer in contact electrification. J. Phys. Chem. C 122, 16154–16160 (2018).   \n6. Baytekin, H. et al. The mosaic of surface charge in contact electrification. Science 333, 308–312 (2011).   \n7. Xu, M. et al. A highly-sensitive wave sensor based on liquid-solid interfacing triboelectric nanogenerator for smart marine equipment. Nano Energy 57, 574–580 (2019).   \n8. Cho, H. et al. Toward sustainable output generation of liquid-solid contact triboelectric nanogenerators: the role of hierarchical structures. Nano Energy 56, 56–64 (2019).   \n9. Lin, Z., Cheng, G., Lin, L., Lee, S. & Wang, Z. L. Water-solid surface contact electrification and its use for harvesting liquid-wave energy. Angew. Chem. Int. Ed. 52, 12545–12549 (2013).   \n10. Yang, X., Chan, S., Wang, L. & Daoud, W. Water tank triboelectric nanogenerator for efficient harvesting of water wave energy over a broad frequency range. Nano Energy 44, 388–398 (2018).   \n11. 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Mater. 31, 1808197 (2019).   \n18. Xu, C. et al. On the electron-transfer mechanism in the contact-electrification effect. Adv. Mater. 30, 1706790 (2018).   \n19. Lin, S., Xu, L., Zhu, L., Chen, X. & Wang, Z. L. Electron transfer in nanoscale contact electrification: photon excitation effect. Adv. Mater. 31, 1808197 (2019).   \n20. J. Israelachvili, Intermolecular and Surface Forces, University of California Santa Barbara, California, USA, 2011.   \n21. Terris, B. D., Stern, J. E., Rugar, D. & Mamin, H. J. Contact electrification using force microscopy. Phys. Rev. Lett. 63, 2669–2672 (1989).   \n22. Lin, S. & Shao, T. Bipolar charge transfer induced by water: experimental and first-principles studies. Phys. Chem. Chem. Phys. 19, 29418–29423 (2017).   \n23. Nonnenmacher, M., O’Boyle, M. P. & Wickramasinghe, H. K. Kelvin probe force microscopy. Appl. Phys. Lett. 58, 2921–2923 (1991).   \n24. Schonenberger, C. & Alvarado, S. F. Observation of single charge-carriers by force microscopy. Phys. Rev. Lett. 65, 3162–3164 (1990).   \n25. Raiteri, R., Martinoia, S. & Grattarola, M. pH-dependent charge density at the insulator-electrolyte interface probed by a scanning force microscope. Biosens. Bioelectron. 11, 1009–1017 (1996).   \n26. Davis, J., James, R. & Leckie, J. Surface ionization and complexation at oxidewater interface. 1. Computation of electrical double-layer properties in simple electrolytes. J. Colloid Interf. Sci. 63, 480–499 (1978).   \n27. Lagstrom, T., Gmur, T., Quaroni, L., Goel, A. & Brown, M. Surface vibrational structure of colloidal silica and its direct correlation with surface charge density. Langmuir 31, 3621–3626 (2015).   \n28. Bousse, L., Rooij, N. & Bergveld, P. Operation of chemically sensitive fieldeffect sensors as a function of the insulator-electrolyte interface. IEEE T. Electron Dev. 30, 1263–1270 (1983).   \n29. Gmur, T., Goel, A. & Brown, M. Quantifying specific ion effects on the surface potential and charge density at silica nanoparticle-aqueous electrolyte interfaces. J. Phys. Chem. C 120, 16617–16625 (2016).   \n30. Cao, S. et al. Fully-enclosed metal electrode-free triboelectric nanogenerator for scavenging vibrational energy and alternatively powering personal electronics. Adv. Eng. Mater. 21, 1800823 (2019).   \n31. Choi, D. et al. Spontaneous occurrence of liquid-solid contact electrification in nature: toward a robust triboelectric nanogenerator inspired by the natural lotus leaf. Nano Energy 36, 250–259 (2017).   \n32. Wang, Z. L. & Wang, A. On the origin of contact-electrification. Mater. Today 30, 34–51 (2019).  \n\n# Acknowledgements  \n\nWe would like to thank Prof. Ding Li, Prof. Xiangyu Chen, Dr. Fei Zhan, and Dr Jianjun Luo for helpful discussions. Research was supported by the National Key R & D Project from Minister of Science and Technology (2016YFA0202704), National Natural Science Foundation of China (Grant Nos. 51605033, 51432005, 5151101243, 51561145021), Beijing Municipal Science & Technology Commission (Z171100000317001, Z171100002017017, Y3993113DF).  \n\n# Author contributions  \n\nS.L. and Z.L.W. conceived the idea and designed the experiment. S.L. carried out the liquid–solid contact electrification experiments. A.C.W., S.L. and Z.L.W. contributed to the electric double-layer theory. S.L., L.X. and Z.L.W. wrote the manuscript. All the authors discussed the results and commented on the manuscript.  \n\n# Competing interests  \n\nThe authors declare no competing interests.  \n\n# Additional information  \n\nSupplementary information is available for this paper at https://doi.org/10.1038/s41467- 019-14278-9.  \n\nCorrespondence and requests for materials should be addressed to Z.L.W.  \n\nPeer review information Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.  \n\nReprints and permission information is available at http://www.nature.com/reprints  \n\nPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\nOpen Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.  ",
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  {
    "id": "10.1016_j.cpc.2019.106949",
    "DOI": "10.1016/j.cpc.2019.106949",
    "DOI Link": "http://dx.doi.org/10.1016/j.cpc.2019.106949",
    "Relative Dir Path": "mds/10.1016_j.cpc.2019.106949",
    "Article Title": "DScribe: Library of descriptors for machine learning in materials science",
    "Authors": "Himanen, L; Jäger, MOJ; Morooka, EV; Canova, FF; Ranawat, YS; Gao, DZ; Rinke, P; Foster, AS",
    "Source Title": "COMPUTER PHYSICS COMMUNICATIONS",
    "Abstract": "DScribe is a software package for machine learning that provides popular feature transformations (descriptors) for atomistic materials simulations. DScribe accelerates the application of machine learning for atomistic property prediction by providing user-friendly, off-the-shelf descriptor implementations. The package currently contains implementations for Coulomb matrix, Ewald sum matrix, sine matrix, Many-body Tensor Representation (MBTR), Atom-centered Symmetry Function (ACSF) and Smooth Overlap of Atomic Positions (SOAP). Usage of the package is illustrated for two different applications: formation energy prediction for solids and ionic charge prediction for atoms in organic molecules. The package is freely available under the open-source Apache License 2.0. Program summary Program Title: DScribe Program Files doi: http://dx.doLorg/10.17632/vzrs8n8pk6.1 Licensing provisions: Apache-2.0 Programming language: Python/C/C++ Supplementary material: Supplementary Information as PDF Nature of problem: The application of machine learning for materials science is hindered by the lack of consistent software implementations for feature transformations. These feature transformations, also called descriptors, are a key step in building machine learning models for property prediction in materials science. Solution method: We have developed a library for creating common descriptors used in machine learning applied to materials science. We provide an implementation the following descriptors: Coulomb matrix, Ewald sum matrix, sine matrix, Many-body Tensor Representation (MBTR), Atom centered Symmetry Functions (ACSF) and Smooth Overlap of Atomic Positions (SOAP). The library has a python interface with computationally intensive routines written in C or C++. The source code, tutorials and documentation are provided online. A continuous integration mechanism is set up to automatically run a series of regression tests and check code coverage when the codebase is updated. (C) 2019 The Authors. Published by Elsevier B.V.",
    "Times Cited, WoS Core": 520,
    "Times Cited, All Databases": 569,
    "Publication Year": 2020,
    "Research Areas": "Computer Science; Physics",
    "UT (Unique WOS ID)": "WOS:000503093400032",
    "Markdown": "# Journal Pre-proof  \n\nDScribe: Library of descriptors for machine learning in materials science  \n\nLauri Himanen, Marc O.J. Jäger, Eiaki V. Morooka, Filippo Federici Canova, Yashasvi S. Ranawat, David Z. Gao, Patrick Rinke, Adam S. Foster  \n\nPII: S0010-4655(19)30304-2   \nDOI: https://doi.org/10.1016/j.cpc.2019.106949   \nReference: COMPHY 106949  \n\nTo appear in: Computer Physics Communications  \n\nReceived date : 17 April 2019   \nRevised date : 14 August 2019   \nAccepted date : 24 August 2019  \n\nPlease cite this article as: L. Himanen, M.O.J. Jäger, E.V. Morooka et al., DScribe: Library of descriptors for machine learning in materials science, Computer Physics Communications (2019), doi: https://doi.org/10.1016/j.cpc.2019.106949.  \n\nThis is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.  \n\n$\\mathfrak{O}$ 2019 Published by Elsevier B.V.  \n\n# DScribe: Library of Descriptors for Machine Learning in Materials Science  \n\nLauri Himanen $^{\\mathrm{a},\\ast}$ , Marc O. J. J¨agera, Eiaki V. Morookaa, Filippo Federici Canova $^{\\mathrm{a,b}}$ , Yashasvi S. Ranawat $\\mathrm{a}$ , David Z. Gao $^{\\mathrm{b,c}}$ , Patrick Rinke $^{\\mathrm{a,f}}$ , Adam S. Foster $^{\\cdot\\mathrm{a},\\mathrm{d},\\mathrm{e}}$  \n\naDepartment of Applied Physics, Aalto University, P.O. Box 11100, 00076 Aalto, Espoo, Finland bNanolayers Research Computing Ltd., 1 Granville Court, Granville Road, London, N12 0HL, United Kingdom cDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway $d$ Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128, Germany eWPI Nano Life Science Institute (WPI-NanoLSI), Kanazawa University , Kakuma-machi, Kanazawa 920-1192, Japan fTheoretical Chemistry and Catalysis Research centre, Technische Universit¨at Mu¨nchen, Lichtenbergstr. 4, D-85747 Garching, Germany  \n\n# Abstract  \n\nDScribe is a software package for machine learning that provides popular feature transformations (“descriptors”) for atomistic materials simulations. DScribe accelerates the application of machine learning for atomistic property prediction by providing user-friendly, off-the-shelf descriptor implementations. The package currently contains implementations for Coulomb matrix, Ewald sum matrix, sine matrix, Many-body Tensor Representation (MBTR), Atom-centered Symmetry Function (ACSF) and Smooth Overlap of Atomic Positions (SOAP). Usage of the package is illustrated for two different applications: formation energy prediction for solids and ionic charge prediction for atoms in organic molecules. The package is freely available under the open-source Apache License 2.0.  \n\nKeywords: machine learning, materials science, descriptor, python, open source  \n\n# 1. Introduction  \n\nMachine learning of atomistic systems is a highly active, interdisciplinary area of research. The power of machine learning lies in the ability of interpolating existing calculations with surrogate models to accelerate predictions for new systems [1– 4]. The set of possible applications is very rich, including high-throughput search of stable compounds with machine learning based energy predictions for solids [5–8], accelerated molecular property prediction [9–11], creation of new force-fields based on quantum mechanical training data [12– 19], search for catalytically active sites in nanoclusters [20–24] and efficient optimization of complex structures [25–27].  \n\n![](images/f5b885e3a1ab5303a1b931a18b1285f58453dfddbcf4406d0761d32eab34f149.jpg)  \nFigure 1: Typical workflow for making machine learning based materials property predictions for atomistic structures. An atomic structure is transformed into a numerical representation called a descriptor. This descriptor is then used as an input for a machine learning model that is trained to output a property for the structure. There is also a possibility of combining the descriptor and learning model together into one inseparable step.  \n\nAtomistic machine learning establishes a relationship between the atomic structure of a system and its properties. This so called structureproperty relation is illustrated in Fig. 1. It is analogous to the structure-property relation in quantum mechanics. For a set of nuclear charges $\\{Z_{i}\\}$ and atomic positions $\\{R_{i}\\}$ of a system, the solution of the Schr¨odinger equation $\\hat{H}\\Phi=E\\Phi$ yields the properties of the system since both the Hamiltonian $\\hat{H}$ and the wave function $\\Phi$ depend only on $\\{Z_{i}\\}$ and $\\{R_{i}\\}$ . Atomistic machine learning bypasses the computationally costly step of solving the Schr¨odinger equation $^{1}$ by training a surrogate model. Once trained, the surrogate model is typically very fast to evaluate facilitating almost instant structure-property predictions.  \n\nUnlike for the Schr¨odinger equation, the nuclear charges and atomic positions are not a suitable input representation of atomistic systems for machine learning. They are, for example, not rotationally or translationally invariant. If presented with atomic positions, the machine learning method would have to learn rotational and translational invariance for every data set, which would significantly increase the amount of required training data. For this reason, the input data has to be transformed into a representation that is suitable for machine learning. This transformation step is often referred to as feature engineering and the selected features are called a descriptor [28]2. Various feature engineering approaches have been proposed [5–9, 14, 16, 33–40], and often multiple approaches have to be tested to find a suitable representation for a specific task [41]. Features are often based on the atomic structure, but it is also common to extend the input to other system properties [5, 28, 36, 42].  \n\nThere are several requirements for good descriptors in atomistic machine learning [6, 7]. We identify the following properties to be most important for an ideal descriptor:  \n\ni) Invariant with respect to spatial translation of the coordinate system: isometry of space.   \nii) Invariant with respect to rotation of the coordinate system: isotropy of space.   \niii) Invariant with respect to permutation of atomi indices: changing the enumeration of atoms does not affect the physical properties of the system.   \niv) Unique: there is a single way to construct a descriptor from an atomic structure and the descriptor itself corresponds to a single property. v) Continuous: small changes in the atomic structure should translate to small changes in the descriptor.   \nvi) Compact: the descriptor should contain sufficient information of the system for performing the prediction while keeping the feature count to minimum.   \nvii) Computationally cheap: the computation of the descriptor should be significantly faster than any existing computational model for directly calculating the physical property of interest.  \n\nIn this article we present the DScribe package that can be used to transform atomic structures into machine-learnable input features. The aim of this software is to provide a coherent and easily extendable implementation for atomistic machine learning and fast prototyping of descriptors. There already exist libraries like QML [43], Amp[44], Magpie [45], quippy [46], ChemML [47] and matminer [48] which include a subset of descriptors as a part of a bigger framework for materials data analytics. DScribe follows this spirit but specializes on providing efficient and scalable descriptor transformations and is agnostic to the framework used for doing the actual data analytics.  \n\nCurrently in the DScribe package we include descriptors that can be represented in a vectorial form and are not dependent on any specific learning model. By decoupling the descriptor creation from the machine learning model, the user can experiment in parallel with various descriptor/model combinations and has the possibility of directly applying emerging learning models on existing data. This freedom to switch between machine learning models becomes important because currently no universally best machine model exists for every problem, as stated by the “No Free Lunch Theorem” [49]. In practice this means that multiple models have to be tested to find optimal performance. Furthermore, vectorial features provide easier insight into the importance of certain features and facilitate the application of unsupervised learning methods, such as clustering and subsequent visualization with informative “materials maps” [50–52].  \n\nDescriptors that encode an atomic structure are typically designed to either depict a local atomi environment, or the structure as a whole. Global descriptors encode information about the whole atomic structure. These global descriptors can be used to predict properties related to the structure as a whole, such as molecular energies [9], formation energies [5] or band gaps [36]. In this work we cover four such global descriptors: the Coulomb matrix [9], the Ewald sum matrix [7], the sine matrix [7] and the Many-Body Tensor Representation (MBTR) [6]. Local descriptors are instead used to represent a localized region in an atomic structure, and are thus suitable for predicting localized properties, like atomic forces [13], adsorption energies[23], or properties that can be summed from local contributions. In this article we discuss two local descriptors, Atomcentered Symmetry functions (ACSFs) [16] and the Smooth Overlap of Atomic Positions (SOAP) [14].  \n\nWe first introduce the descriptors that have been implemented in the DScribe package and then we discuss the structure and usage of the package. After this we illustrate the applicability of the package by showing results for formation energy prediction of periodic crystals and partial charge prediction for molecules. We conclude, by addressing the impact and future extensions of this package.  \n\n# 2. Descriptors  \n\nHere we briefly introduce the different descriptors that are currently implemented in DScribe.  \n\n1 In some cases, we have deviated from the original   \n2 literature due to computational or other reasons,   \n3   \n3 and if so this is explicitly mentioned. For more   \n4 in-depth presentations of the descriptors we refer the reader to the original research papers. At the end of this section we also discuss methods for organizing the descriptor output so that it can be effectively used in typical machine learning applications.  \n\n# 2.1. Coulomb matrix  \n\nThe Coulomb matrix [9] encodes the atomic species and inter-atomic distances of a finite system in a pair-wise, two-body matrix inspired by the form of the Coulomb potential. The elements of this matrix are given by:  \n\n$$\nM_{i j}^{\\mathrm{Coulomb}}=\\left\\{\\begin{array}{l l}{0.5Z_{i}^{2.4}}&{\\forall i=j}\\\\ {\\frac{Z_{i}Z_{j}}{|{\\cal R}_{i}-{\\cal R}_{j}|}}&{\\forall i\\ne j}\\end{array}\\right.\n$$  \n\nwhere $Z$ is the atomic number, and $\\lvert R_{i}-R_{j}\\rvert$ is the Euclidean distance between atoms $i$ and $j$ . The form of the diagonal terms was determined by fitting the potential energy of neutral atoms [53].  \n\n# 2.2. Ewald sum matrix  \n\nThe Ewald sum matrix [7] can be viewed as a logical extension of the Coulomb matrix for periodic systems. In periodic systems each atom is infinitely repeated in the three crystal lattice vector directions, a, b and $\\mathbf{c}$ and the electrostatic interaction between two atoms becomes  \n\n$$\n\\phi_{i j}=\\sum_{\\bf n}{\\frac{Z_{i}Z_{j}}{|{\\bf R}_{i}-{\\bf R}_{j}|+\\bf n}}\n$$  \n\nwhere $\\scriptstyle\\sum_{\\mathbf{n}}$ is the sum over all lattice vectors $\\mathbf{n}=$ $h\\mathbf{a}+k\\mathbf{b}+l\\mathbf{c}$ .  \n\nFor $h,k,l\\rightarrow\\infty$ , this sum converges only conditionally and will become infinite if the system is not charge neutral. In the Ewald sum matrix, the Ewald summation technique [54, 55] and a neutralizing background charge [56] is used to force this sum to converge. One can separate the total Ewald energy into pairwise components, which will result in the following matrix:  \n\n$$\nM_{i j}^{\\mathrm{Ewald}}=\\left\\{\\begin{array}{l l}{\\phi_{i j}^{\\mathrm{real}}+\\phi_{i j}^{\\mathrm{recip}}+\\phi_{i j}^{\\mathrm{self}}+\\phi_{i j}^{\\mathrm{bg}}}&{\\forall\\:i=j}\\\\ {2\\left(\\phi_{i j}^{\\mathrm{real}}+\\phi_{i j}^{\\mathrm{recip}}+\\phi_{i j}^{\\mathrm{bg}}\\right)}&{\\forall\\:i\\neq j}\\end{array}\\right.\n$$  \n\nwhere the terms are given by  \n\n$$\n\\begin{array}{l}{{\\displaystyle{\\phi_{i j}^{\\mathrm{real}}=\\frac{1}{2}Z_{i}Z_{j}\\sum_{n^{\\prime}}\\frac{\\mathrm{erfc}\\left(\\alpha|{\\bf R}_{i}-{\\bf R}_{j}+{\\bf n}|\\right)}{|{\\bf R}_{i}-{\\bf R}_{j}+{\\bf n}|}}\\qquad(2)}}}\\\\ {{\\displaystyle{\\phi_{i j}^{\\mathrm{recip}}=\\frac{2\\pi}{V}Z_{i}Z_{j}\\sum_{{\\bf G}}\\frac{e^{-|{\\bf G}|^{2}/(2\\alpha)^{2}}}{|{\\bf G}|^{2}}\\cos\\left({\\bf G}\\cdot({\\bf R}_{i}-{\\bf R}_{j})\\right)}}}\\end{array}\n$$  \n\n$$\n\\begin{array}{r l}&{\\phi_{i j}^{\\mathrm{self}}=\\left\\{\\begin{array}{l l}{-\\frac{\\alpha}{\\sqrt{\\pi}}Z_{i}^{2}}&{\\forall~i=j}\\\\ {0}&{\\forall~i\\neq j}\\end{array}\\right.}\\\\ &{\\phi_{i j}^{\\mathrm{bg}}=\\left\\{\\begin{array}{l l}{-\\frac{\\pi}{2V\\alpha^{2}}Z_{i}^{2}}&{\\forall~i=j}\\\\ {-\\frac{\\pi}{2V\\alpha^{2}}Z_{i}Z_{j}}&{\\forall~i\\neq j}\\end{array}\\right.}\\end{array}\n$$  \n\nHere the primed notation means that when $\\mathbf{\\nabla}_{\\mathbf{n}}=\\mathbf{0}$ the pairs $i=j$ are not taken into account. $\\alpha$ is the screening parameter controlling the size of the gaussian charge distributions used in the Ewald method, $\\dot{\\bf G}$ is a reciprocal space lattice vector with an implicit $2\\pi$ prefactor and $V$ is the volume of the cell. A more detailed derivation is given in the supplementary information. By default we use the value $\\begin{array}{r}{\\alpha=\\sqrt{\\pi}\\left(\\frac{N}{V^{2}}\\right)^{1/6}}\\end{array}$ [57], where $N$ is the number of atoms in the unit cell.  \n\nIt is important to notice that the off-diagonal contribution φself + $\\begin{array}{r}{\\phi_{i j}^{\\mathrm{self}}+\\phi_{i j}^{\\mathrm{bg}}=-\\frac{\\pi}{2V\\alpha^{2}}Z_{i}Z_{j}\\ \\forall\\ i\\ \\ne\\ j}\\end{array}$ given here differs from the original work. In the original formulation this sum was defined as [7] $\\begin{array}{r}{\\phi_{i j}^{\\mathrm{self}}+\\phi_{i j}^{\\mathrm{bg}}=-\\frac{\\alpha}{\\sqrt{\\pi}}(Z_{i}^{2}+Z_{j}^{2})-\\frac{\\pi}{2V\\alpha_{*}^{2}}(Z_{i}+Z_{j})^{2}\\forall\\:i\\neq j}\\end{array}$ Our correction makes the total matrix elements independent of the screening parameter $\\alpha$ , which is not the case in the original formulation.  \n\nFor numerical purposes, the sums in eqs. 2 and 3 are cut off by $n~\\leq~n_{\\mathrm{cut}}$ and $G\\leq G_{\\mathrm{cut}}$ . By default we use the values $G_{\\mathrm{cut}}=2\\alpha\\sqrt{-\\ln A}$ and $n_{\\mathrm{cut}}={\\sqrt{-\\ln A}}/\\alpha[57]$ , where the small positive parameter $A$ controls the accuracy of the sum and can be determined by the user.  \n\n# 2.3. Sine matrix  \n\nThe Ewald sum matrix encodes the correct Coulomb interaction between atoms, but can become computationally heavy especially for large systems. The sine matrix [7] captures some important features of interacting atoms in a periodic system with a much reduced computational cost. The matrix elements are defined by  \n\n![](images/170596344c4a75f2f11f7f433b1d65cd762f99da25d2a9370abfa51adb8dba9f.jpg)  \nFigure 2: Illustration of the Coulomb matrix, Ewald sum matrix and sine matrix for a periodic diamond structure. The used atomic structure for the conventional diamond cell is shown on the left. The color scale (legend on the right) is used to illustrate the magnitude of the matrix elements.  \n\n$$\nM_{i j}^{\\mathrm{sine}}=\\left\\{\\begin{array}{l l}{0.5Z_{i}^{2.4}}&{\\forall~i=j}\\\\ {\\phi_{i j}}&{\\forall~i\\ne j}\\end{array}\\right.\n$$  \n\nwhere  \n\n$$\n\\phi_{i j}=Z_{i}Z_{j}|\\mathbf{B}\\cdot\\sum_{k=\\{x,y,z\\}}\\hat{\\mathbf{e}}_{k}\\sin^{2}\\left(\\pi\\mathbf{B}^{-1}\\cdot({\\pmb R}_{i}-{\\pmb R}_{j})\\right)|^{-1}\n$$  \n\nHere $\\mathbf{B}$ is a matrix formed by the lattice vectors and $\\hat{\\mathbf{e}}_{k}$ are the cartesian unit vectors. This functional form has no physical interpretation, but it captures some of the properties of the Coulomb interaction, such as the periodicity of the crystal lattice and an infinite energy when two atoms overlap.  \n\nThe Coulomb, Ewald sum and sine matrices for diamond are depicted in Fig. 2. Notice that the matrices given here are not unique, as different cell sizes can be used for a periodic crystal, and the indexing of the rows and columns depends on the ordering of atomic indices in the structure. Section 2.7 discusses some ways to overcome the issues related to this non-uniqueness.  \n\nBy construction the Coulomb matrix is not periodic as manifested by the unequivalent row elements in the matrix (one carbon in the system has four bonded neighbours, three carbons have two neighbours and four carbons have a single bonded neighbour). Conversely, both the Ewald sum and the sine matrix are periodic and correctly encode the identical environment of the carbon atoms in the lattice. As a result, each row and each column has the same matrix elements, but neighbouring rows and columns are shifted by one element relative to each other. Unlike the other matrices, Ewald sum matrix often contains negative elements due to the interaction of the positive atomic nuclei with the added uniform negative background charge. This energetically favourable interaction shifts the off-diagonal elements down in energy compared to the other two matrices. Moreover, the diagonal elements of the Ewald sum matrix encode the physical selfinteraction of atoms with their periodic copies, instead of the potential energy of the neutral atoms.  \n\n# 2.4. Many-body Tensor Representation  \n\nThe many-body tensor representation (MBTR) [6] encodes finite or periodic structures by breaking them down into distributions of differently sized structural motifs and grouping these distributions by the involved chemical elements. In MBTR, a geometry function $g_{k}$ is used to transform a configuration of $k$ atoms into a single scalar value representing that particular configuration. Our implementation provides terms up to $k\\mathbf{\\Psi}=$ 3, and provides the following geometry functions $g_{1}(Z_{l})$ : $\\mathcal{L}_{l}$ (atomic number), $g_{2}(R_{l},R_{m})$ : $|R_{l}-$ Rm| (distance) or R 1R (inverse distance) and $g_{3}({\\pmb R}_{l},{\\pmb R}_{m},{\\pmb R}_{n})$ : $\\angle(R_{l}-R_{m},R_{n}-R_{m})$ (angle) or $\\cos(\\angle(R_{l}-R_{m},R_{n}-R_{m}))$ (cosine of angle). These scalar values are then broadened by using  \n\n1 kernel density estimation with a gaussian kernel,   \n2 leading to the following distributions $\\mathcal{D}_{k}$  \n\n$$\n\\mathcal{D}_{1}^{l}(x)=\\frac{1}{\\sigma_{1}\\sqrt{2\\pi}}e^{-\\frac{\\left(x-g_{1}(Z_{l})\\right)^{2}}{2\\sigma_{1}^{2}}}\n$$  \n\n$$\n\\mathcal{D}_{2}^{l,m}(x)=\\frac{1}{\\sigma_{2}\\sqrt{2\\pi}}e^{-\\frac{\\left(x-g_{2}\\left(R_{l},R_{m}\\right)\\right)^{2}}{2\\sigma_{2}^{2}}}\n$$  \n\n$$\n\\mathcal{D}_{3}^{l,m,n}(x)=\\frac{1}{\\sigma_{3}\\sqrt{2\\pi}}e^{-\\frac{\\left(x-g_{3}\\left(R_{l},R_{m},\\mathbf{R}n\\right)\\right)^{2}}{2\\sigma_{3}^{2}}}\n$$  \n\nHere $\\sigma_{k}$ is the standard deviation of the gaussian kernel and $x$ runs over a predefined range of values covering the possible values for $g_{k}$ . A weighted sum of the distributions $\\mathcal{D}_{k}$ are then made separately for each possible combination of $k$ chemical species present in the dataset. For $k=1,2,3$ these distributions are given by  \n\n$$\n\\begin{array}{r l r}{\\mathrm{MBTR}_{1}^{Z_{1}}(x)}&{=\\displaystyle\\sum_{l}^{|Z_{1}|}w_{1}^{l}\\mathcal{D}_{1}^{l}(x)}&&{\\mathrm{(10)}}\\\\ {\\mathrm{MBTR}_{2}^{Z_{1},Z_{2}}(x)}&{=\\displaystyle\\sum_{l}^{|Z_{1}|}\\sum_{m}w_{2}^{l,m}\\mathcal{D}_{2}^{l,m}(x)}&&{\\mathrm{(12)}}\\\\ {\\mathrm{MBTR}_{3}^{Z_{1},Z_{2},Z_{3}}(x)}&{=\\displaystyle\\sum_{l}^{|Z_{1}|}\\sum_{m}^{|Z_{2}|}\\sum_{n}^{|Z_{3}|}w_{3}^{l,m,n}\\mathcal{D}_{3}^{l,m,n}(x)}&\\end{array}\n$$  \n\nwhere the sums for $\\it l$ , $m$ and $n$ run over all atoms with the atomic number $Z_{1}$ , $Z_{2}$ or $Z_{3}$ respectively, and $w_{k}$ is a weighting function that is used to control the importance of different terms. When calculating MBTR for periodic systems, the periodic copies of atoms in neighbouring cells are taken into account by extending the original cell with periodic copies. When a periodic system is extended in this way, certain sets of atoms may get counted multiple times due to translational symmetry. Like in the original formulation [6] we require that one of the atoms, $l$ , $m$ or $n$ , must be in the original cell. In addition, our implementation ensures that each translationally unique combination of the atoms is counted only once. This makes the MBTR output for different cells representing the same material identical up to a size extensive scalar multiplication factor. Unlike in the original formulation, we don’t include the possible correlation between chemical elements directly in equations (10)–(12). We don’t however lose any generality, as the correlation between chemical elements can be introduced as a postprocessing step that combines information from the different species.  \n\nFor $k=1$ , typically no weighting is used: $w_{1}^{l}=$ 1. In the case of $k\\stackrel{_{\\perp}}{=}2$ and $k=3$ , the weighting function can, however be used to give more importance to values that correspond to configuration where the atoms are closer together. For fully periodic systems, a weighting function must be used, as otherwise the sums in equations (10)– (12) do not converge. For $k=2,3$ we provide exponential weighting functions of the form  \n\n$$\n\\begin{array}{r l}&{w_{2}^{l,m}=e^{-s_{k}\\left|R_{l}-{\\pmb R}_{m}\\right|}}\\\\ &{w_{3}^{l,m,n}=e^{-s_{k}\\left(\\left|{\\pmb R}_{l}-{\\pmb R}_{m}\\right|+\\left|{\\pmb R}_{m}-{\\pmb R}_{n}\\right|+\\left|{\\pmb R}_{l}-{\\pmb R}_{n}\\right|\\right)}}\\end{array}\n$$  \n\nwhere the parameter $s_{k}$ can be used to effectively tune the cutoff distance. For computational purposes a cutoff of $w_{k}^{\\mathrm{min}}$ can be defined to ignore any contributions for which $w_{k}<w_{k}^{\\mathrm{min}}$ .  \n\nSome of the distributions, for example MBTR $_{\\mathrm{~2~}}^{1,2}$ and MBTR2 , contain identical information. In our implementation this symmetry is taken into consideration by only calculating the distributions for which the last atomic number is bigger or equal to the first: Z2 ≥ Z1 in the case of MBTR2Z1,Z2 or $Z_{3}\\geq Z_{1}$ in the case of $\\mathrm{MBTR_{3}^{Z_{1},Z_{2},Z_{3}}}$ . This reduces the computational time and the number of features in the final descriptor without losing information. The final MBTR output for a water molecule is illustrated in Fig. 3.  \n\nThere are multiple system-dependent parameters that have to be decided for the MBTR descriptor. At each level $k$ , the broadness of the distribution $\\sigma_{k}$ has to be chosen. A too small value for $O_{k}$ will lead to a delta-like distribution that is very sensitive to differences in system configurations. Conversely, a too large value will make it hard to resolve individual peaks as the distribution becomes broad and featureless. The choice of the weighting function also has a direct effect on the final distribution, as it controls how much importance is given to atom combinations that are physically far apart. When combining information from multiple $k$ -terms, it can be beneficial to control the contribution from different terms. As the number of features related to higher $k$ -values is bigger, the machine learning model may by default give more importance to these higher terms. For example if similarity between two MBTR outputs is measured by an Euclidean distance in the machine learning model, individually normalizing the total output for each term $k$ to unit length helps in equalizing the information content of the different terms.  \n\n![](images/986cb3fa7e1fd92a33039b61a0f42eaa36ec2deeb1303922f1bf4e7439c973b5.jpg)  \nFigure 3: MBTR output for a water molecule showing the distributions MBTR $\\mathbf{\\nabla}\\cdot\\mathbf{k}$ for $k=1,2,3$ with different combinations of chemical elements. For each $k$ -term, the distributions can be arranged into a $k$ -dimensional grid, resulting in a $k+1$ dimensional tensor. If a flattened onedimensional vector is needed by the learning model, the distributions may be concatenated together, possibly with some weighting, as shown in the lower panel.  \n\n# 2.5. Atom-centered Symmetry Functions  \n\nAtom-centered Symmetry Functions (ACSFs) [16] can be used to represent the local environment near an atom by using a fingerprint composed of the output of multiple two- and threebody functions that can be customized to detect specific structural features. ACSF encodes the configuration of atoms around a single central atom with index $i$ by using so called symmetry functions. The presence of atoms neighbouring the central atom are detected by using three different two-body symmetry functions $G_{i}^{1,Z_{1}}$ , $G_{i}^{2,Z_{1}}$ and Gi3,Z1, which are defined as  \n\n$$\n\\begin{array}{l}{{\\displaystyle G_{i}^{1,Z_{1}}=\\sum_{j}^{|Z_{1}|}\\ f_{\\mathrm{c}}\\left(R_{i j}\\right)}}\\\\ {{\\displaystyle G_{i}^{2,Z_{1}}=\\sum_{j}^{|Z_{1}|}e^{-\\eta\\left(R_{i j}-R_{\\mathrm{s}}\\right)^{2}}f_{\\mathrm{c}}\\left(r_{i j}\\right)}}\\\\ {{\\displaystyle G_{i}^{3,Z_{1}}=\\sum_{j}^{|Z_{1}|}\\cos\\left(\\kappa R_{i j}\\right)f_{\\mathrm{c}}\\left(r_{i j}\\right)}}\\end{array}\n$$  \n\nwhere the summation for $j$ runs over all atoms with atomic number $Z_{1},\\eta$ , $R_{\\mathrm{s}}$ and $\\kappa$ are userdefined parameters, $R_{i j}=|R_{i}-R_{j}|$ and $f_{\\mathrm{c}}$ is a smooth cutoff function defined as  \n\n$$\nf_{\\mathrm{c}}\\left(r\\right)=\\frac{1}{2}\\left[\\cos\\left(\\pi\\frac{r}{r_{\\mathrm{cut}}}\\right)+1\\right]\n$$  \n\nwhere $r_{\\mathrm{cut}}$ is a cutoff radius.  \n\nAdditionally, three-body functions may be used to detect specific motifs defined by three atoms, one being the central atom. These three-body functions include a dependence on the angle between triplets of atoms within cutoff, as well as their mutual distance. The package implements the following functions  \n\n$$\n\\begin{array}{r l r}{\\lefteqn{G_{i}^{4,Z_{1},Z_{2}}=2^{1-\\zeta}\\sum_{j\\neq i}^{|Z_{1}|}\\sum_{k\\neq i}^{|Z_{2}|}(1+\\lambda\\cos\\theta)^{\\zeta}}}\\\\ &{}&{\\quad\\cdot e^{-\\eta\\left(R_{i j}^{2}+R_{i k}^{2}+R_{j k}^{2}\\right)}f_{\\mathrm{c}}\\left(R_{i j}\\right)f_{\\mathrm{c}}\\left(R_{i k}\\right)f_{\\mathrm{c}}\\left(R_{j k}\\right)}\\end{array}\n$$  \n\n$$\n\\begin{array}{r}{G_{i}^{5,Z_{1},Z_{2}}=2^{1-\\zeta}\\displaystyle\\sum_{j\\neq i}^{|Z_{1}|}\\sum_{k\\neq i}^{|Z_{2}|}(1+\\lambda\\cos\\theta)^{\\zeta}}\\\\ {\\cdot e^{-\\eta\\left(R_{i j}^{2}+R_{i k}^{2}\\right)}f_{\\mathrm{c}}\\left(R_{i j}\\right)f_{\\mathrm{c}}\\left(R_{i k}\\right)}\\end{array}\n$$  \n\nwhere the summation of $j$ and $k$ runs over all atoms with atomic numbers $Z_{1}$ or $Z_{2}$ respectively, $\\zeta$ , $\\lambda$ and $\\eta$ are user-defined parameters and $\\theta$ is the angle between the three atoms ( $\\textit{i}$ -th atom in the center).  \n\nIn practice, multiple symmetry functions of each type are used in the descriptor, each with a different parametrization (ζ, λ, η, $R_{\\mathrm{s}}$ and $\\kappa$ ), encoding different portions of the chemical environment. As there is no single set of symmetry functions that optimally fits all applications, the selection is guided by the generally desired properties of a descriptor previously listed (unique, continuous and compact) and the specifics of the application. Different sets of symmetry functions used for various applications can be found in the literature [58–60].  \n\n![](images/64e5178845da4d4602f3aec3695195a231022aa1b3ca3263146c0328d2de5a6d.jpg)  \nFigure 4: Structure of the ACSF output vector. The values of the two-body symmetry functions $G^{1}$ , $G^{2}$ and $G^{3}$ are given first for each chemical species present in the dataset. Next the values of the three-body symmetry functions $G^{4}$ and $G^{5}$ are listed for each unique combination of two chemical species. All symmetry functions except $G^{1}$ may also have multiple parameterizations as indicated by the subindices.  \n\nThe final fingerprint for a single atom can be constructed by concatenating the output from differently parametrized symmetry functions with consistent ordering, as illustrated in Figure 4. The list starts with the two-body ACSFs, ordered by atom type $Z_{1}$ . For each type, $G^{1}$ appears first, bringing only one value since it has no parameter dependence. Next we find all values of $G^{2}$ calculated with different $(\\eta,R_{\\mathrm{s}})$ parameter pairs given by the user. The values of $G^{3}$ for all $\\kappa$ are found last. This sequence is repeated for each atomic type, sorted from lighter to heavier. Three-body ACSFs appear afterward: for each unique combination of chemical elements, we find the values of $G^{4}$ and $G^{5}$ given by all specified triplets of $(\\zeta,\\lambda,\\eta)$ .  \n\n# 2.6. Smooth Overlap of Atomic Orbitals  \n\nThe Smooth Overlap of Atomic Positions (SOAP) [14] can be used to encode a local environment within an atomic structure by using an expansion of a gaussian smeared atomic density based on spherical harmonics and radial basis functions. In SOAP, the atomic structure is first transformed into atomic density fields $\\rho^{Z}$ for each species by using un-normalized gaussians centered on each atom  \n\n$$\n\\rho^{Z}(\\pmb{r})=\\sum_{i}^{|Z|}e^{-\\frac{1}{2\\sigma^{2}}|\\pmb{r}-\\pmb{R}_{i}|^{2}}.\n$$  \n\nHere the summation for $i$ runs over atoms with the atomic number $Z$ to build a separate density for each atomic element and the width of the gaussian is controlled by $\\sigma$ .  \n\nWhen the origin $r\\neq0$ is chosen to be centered at the local point of interest, the atomic density may then be expanded with a set of orthonormal radial basis functions $g_{n}$ and spherical harmonics $Y_{l m}$ as  \n\n$$\n\\rho^{Z}(\\pmb{r})=\\sum_{n l m}c_{n l m}^{Z}g_{n}(r)Y_{l m}(\\theta,\\phi)\n$$  \n\nwhere the coefficients can be obtained through  \n\n$$\nc_{n l m}^{Z}=\\iiint_{\\mathcal{R}^{3}}\\mathrm{d}V g_{n}(r)Y_{l m}(\\theta,\\phi)\\rho^{Z}(\\pmb{r}).\n$$  \n\nInstead of using the complex spherical harmonics as in the original work[14], we use the real (tesseral) spherical harmonics as they are computationally preferable when expanding real-valued functions such as the atomic density defined by equation (17). The real spherical harmonics $Y_{l m}$ are defined as  \n\n$$\nY_{l m}(\\theta,\\phi)=\\left\\{\\begin{array}{l l}{\\sqrt{2}(-1)^{m}\\operatorname{Im}[Y_{l}^{|m|}(\\theta,\\phi)]}&{\\mathrm{if~}m<0}\\\\ {Y_{l}^{0}}&{\\mathrm{if~}m=0}\\\\ {\\sqrt{2}(-1)^{m}\\operatorname{Re}[Y_{l}^{m}(\\theta,\\phi)]}&{\\mathrm{if~}m>0}\\end{array}\\right.\n$$  \n\nwhere $Y_{l}^{m}$ corresponds to the complex orthonormalized spherical harmonics defined as  \n\n$$\nY_{l}^{m}(\\theta,\\phi)={\\sqrt{\\frac{(2l+1)}{4\\pi}\\frac{(l-m)!}{(l+m)!}}}P_{l}^{m}(\\cos{\\theta})e^{i m\\phi}\n$$  \n\nand $P_{l}^{m}$ are the associated Legendre polynomials.  \n\nThe final rotationally invariant output from our SOAP implementation is the partial power  \n\n1 spectra [51] vector $\\mathbf{p}$ where the individual vector   \n2 elements are defined as:  \n\n$$\np_{n n^{\\prime}l}^{Z_{1},Z_{2}}=\\pi\\sqrt{\\frac{8}{2l+1}}\\sum_{m}\\left(c_{n l m}^{Z_{1}}\\right)^{*}c_{n^{\\prime}l m}^{Z_{2}}\n$$  \n\nThe vector $\\mathbf{p}$ is constructed by concatenating the elements pnZ1n,′lZ2 for all unique atomic number pairs $Z_{1},Z_{2}$ , all unique pairs of radial basis functions $n,n^{\\prime}$ up to $n_{\\mathrm{max}}$ and the angular degree values $\\it l$ up to lmax.  \n\nSpherical harmonics are a natural orthogonal and complete set of functions for the angular degrees of freedom. For the radial degree of freedom the selection of the basis set is not as trivial and multiple approaches may be used. In our implementation we, by default, use a set of spherical primitive gaussian type orbitals $g_{n l}(r)$ as radial basis functions. These basis functions are defined as  \n\n$$\n\\begin{array}{l l r}{\\displaystyle{g_{n l}(r)=\\sum_{n^{\\prime}=1}^{n_{\\mathrm{max}}}\\beta_{n n^{\\prime}l}\\phi_{n^{\\prime}l}(r)}}\\\\ {\\displaystyle{\\phi_{n l}(r)=r^{l}e^{-\\alpha_{n l}r^{2}}.}}\\end{array}\n$$  \n\nThis basis set allows analytical integration of the $c_{n l m}$ coefficients defined by equation (19). This provides a speedup over various other radial basis functions that require numerical integration. Our current implementation provides the analytical solutions up to $l\\leq9$ , with the possibility of adding more in the future.  \n\nThe decay parameters $\\alpha_{n}$ are chosen so that each non-orthonormalized function $\\phi_{n l}$ decays to a threshold value of $10^{-3}$ at a cutoff radius taken on an evenly spaced grid from $\\mathrm{1\\AA}$ to $r_{\\mathrm{cut}}$ with a step of $\\frac{r_{\\mathrm{cut}}-1}{n_{\\mathrm{max}}}$ . Thus the parameter $\\widehat{r_{\\mathrm{cut}}}$ controls the maximum reach of the basis and a better sampling can be obtained by increasing the number of basis functions n .  \n\nThe weights $\\beta_{n n^{\\prime}l}$ are chosen so that the radial basis functions are orthonormal. For each value of angular degree $l$ , the orthonormalizing weights $\\beta_{n n^{\\prime}l}$ can be calculated with L¨owdin orthogonal  \n\n![](images/f7b5d1186cd90e85a087518a75e1f05084a9433f0da6a18f936e2a101bfbd22d.jpg)  \nFigure 5: Plot of the a) gaussian type orbital and b) polynomial radial basis functions, defined by equations (24) and (28) respectively. The basis functions here correspond to the orthonormalized set with $r_{\\mathrm{cut}}=3$ and up to $n_{\\mathrm{max}}=4$ . Notice that the polynomial basis is independent of the spherical harmonics degree $l$ , whereas the form of the gaussian type orbital basis depends on $l$ and the examples here are given for $l=0,1,2$ .  \n\nization [61]:  \n\n$$\n\\begin{array}{r}{\\beta=S^{-1/2}\\qquad(2}\\\\ {S_{n n^{\\prime}}=\\langle\\phi_{n l}|\\phi_{n^{\\prime}l}\\rangle=\\displaystyle\\int_{0}^{\\infty}\\mathrm{d}r r^{2}r^{l}e^{-\\alpha_{n l}r^{2}}r^{l}e^{-\\alpha_{n^{\\prime}l}r^{2}}}\\end{array}\n$$  \n\nwhere the matrix $\\beta$ contains the weights $\\beta_{n n^{\\prime}l}$ and $S$ is the overlap matrix.  \n\nWe also provide an option for using the radial basis consisting of cubic and higher order polynomials, as introduced in the original SOAP article  \n\n[14]. This basis set is defined as:  \n\n$$\n\\begin{array}{l}{\\displaystyle g_{n}(\\boldsymbol{r})=\\sum_{n^{\\prime}=1}^{n_{\\mathrm{max}}}\\beta_{n n^{\\prime}}\\phi_{n^{\\prime}}(\\boldsymbol{r})}\\\\ {\\phi_{n}(\\boldsymbol{r})=(r-r_{\\mathrm{cut}})^{n+2}}\\end{array}\n$$  \n\nThe calculations with this basis are performed with efficient numerical integration and currently support $l_{\\mathrm{max}}\\leq20$ .  \n\nThe two different basis sets are compared in Figure 5. Most notably the form of the gaussian type orbitals depend on the angular degree $l$ , whereas the polynomial basis is independent of this value. It is also good to notice that between these two radial basis functions the definition of $r_{\\mathrm{cut}}$ is somewhat different – whereas the polynomial basis is guaranteed to decay to zero at $r_{\\mathrm{cut}}$ , the gaussian basis only approximately decays near this value and the decay is also affected by the orthonormalization.  \n\n2.7. Descriptor usage as machine learning input In this section we discuss some of the methods for organizing the output from descriptors so that it can be efficiently used as input for machine learning.  \n\nThe descriptor invariance against permutations of atomic indices – property iii) in the introduction – is directly achieved in MBTR, SOAP and ACSF by stratifying the output according to the involved chemical elements. The output is always ordered by a predefined order determined by the chemical elements that are included in the dataset, making the output independent of the indexing of individual atoms. The three matrix descriptors – the Coulomb matrix, Ewald sum matrix, and sine matrix – are, however, not invariant with respect to permutation of atomic indices, as the matrix columns and rows are ordered by atomic indices. However, there are different approaches for enforcing invariance for these matrices. One way is to encode the matrices by their eigenvalues, which are invariant to changes in the column and row ordering [9]. Another way is to order the rows and columns by a chosen norm, typically the Euclidean norm [33]. A third approach is to augment the dataset by creating multiple slightly varying matrices for each structure.  \n\nIn this approach multiple matrices are drawn from a statistical set of sorted matrices where Gaussian noise is added to the row norms before sorting [33]. When the learning algorithm is trained over this ensemble of matrices it becomes more robust against small sorting differences that can be considered noise. All of these three approaches are available in our implementation.  \n\nMachine learning algorithms also often require constant-sized input. Once again the stratification of the descriptor output by chemical elements makes the output for MBTR, ACSF and SOAP constant size. For the matrix descriptors a common way to achieve a uniform size for geometries with different amount of atoms, is by introducing zero-padding. This means that we first have to determine the largest system in the dataset. If this system has $N_{\\mathrm{max}}$ , we allocate matrices of size $N_{\\mathrm{max}}\\times N_{\\mathrm{max}}$ or a vectors or size $N_{\\mathrm{max}}$ if using matrix eigenvectors. The descriptor for each system will fill the first $N^{2}$ or $N$ many entries, with the rest being set to zero. If the machine-learning algorithms expects a one-dimensional vector as input, the two-dimensional matrices can be flattened by concatenating the rows together into a single vector.  \n\nLocal descriptors, such as ACSF and SOAP, encode only local spatial regions and cannot be directly used as input for performing predictions related to entire structures. There are, however, various ways for combining information from multiple local sites to form a prediction for an entire structure. The descriptor output for multiple local sites can simply be averaged, a custom kernel can be used to combine information from multiple sites [51, 62] or the predicted property can in some cases be directly modeled as a sum of local contributions [13].  \n\n# 3. Software structure  \n\nWe use python as the default interfacing language through which the user interacts with the library. This decision was motivated by the existence of various python libraries, including ase [63], pymatgen [64] and quippy [46], that supply tools for creating, reading, writing and manipu  \n\n1 lating atomic structures. Our python interface   \n2 does not, however, restrict the implementation to   \n3 be made entirely in python. Python can easily   \n4 interact with software libraries written with highperformance, statically typed languages such as C, C++ and Fortran. We use this dual approach by performing some of the most computationally heavy calculations either in C or $\\mathrm{C}{+}{+}$ .  \n\n![](images/c72489cef5e1ec5d1a40994000801e7415d8433bb316291045053224adbde628.jpg)  \nFigure 6: Example of creating descriptors with DScribe. The structures are defined as ase.Atoms objects, in this case by using predefined molecule geometries. The usage of all descriptors follows the same pattern: a) a descriptor object is initialized with the desired configuration b) the number of features can be requested with get number offeatures c) the actual output is created with createmethod that takes one or multiple atomic structures and possibly other arguments, such as the number of parallel jobs to use.  \n\nAn example of creating a descriptor for an atomic structure with the library is demonstrated in Fig. 6. It demonstrates the workflow that is common to all descriptors in the package. For each descriptor we define a class, from which objects can be instantiated with different descriptor specific setups.  \n\nAll the descriptors have the sparse-parameter that controls whether the created output is a dense or a sparse matrix. The possibility for creating a sparse output is given so that large and sparsely filled output spaces can be handled, as typically encountered when a dataset contains large amounts of different chemical elements. Various machine learning algorithms can make use of this sparse matrix output with linear algebra routines specifically designed for sparse data structures.  \n\nOnce created, the descriptor object is ready to be used and provides different methods for interacting with it. All of the descriptors implement two methods: get number of features and create The get number of features-method can be used for querying the final number of features for the descriptor, even before a structure has been provided. This dimension can be used for initializing and reserving storage space for the resulting output array. create accepts one or multiple atomistic structures as an argument, and possibly other descriptor-specific arguments. It returns the final descriptor output that can be used in machine learning applications. To define atomic structures we use the ase.Atoms-object from the ase package[63]. The Atoms-objects are easy to create from structure files or build with the utilities provided by ase.  \n\nAs the creation of a descriptor for an atomic system is completely independent from the other systems, it can be easily parallelized with data parallelism. For convenience we provide a possibility of parallelizing the descriptor creation for multiple samples over multiple processes. This can be done by simply providing the number of parallel jobs to instantiate with the n jobs-parameter as demonstrated in Figure 6.  \n\nThe DScribe package is structured such that new descriptors can easily be added. We provide a python base-class that defines a standard interface for the descriptors through abstract classes. One of our design goals is to provide a codebase in which researchers can make their own descriptors available to the whole community. All descriptor implementations are accompanied by a test module that defines a set of standard tests. These tests include tests for rotational, translational and index permutation invariance, as well as other tests for checking the interface and functionality of the descriptor. We have adapted a continuous integration system that automatically  \n\n1 runs a series of regression tests when changes in 2 the code are introduced. The code coverage is simultaneously measured as a percentage of visited code lines in the python interface.  \n\nThe source code is directly available in github at https://github.com/SINGROUP/dscribe and we have created a dedicated home page at https: //singroup.github.io/dscribe/ that provides additional tutorials and a full code documentation. For easy installation the code is provided through the python package index (pip) under the name dscribe.  \n\n![](images/9185bd23a3a7f644d88a8e3c44d684cc1817cfdfd663521c7fa0be0fe790b00e.jpg)  \nFigure 7: Distribution of the formation energies together with the mean $(\\mu)$ , standard deviation ( $\\sigma$ ) and mean absolute deviation (MAD).  \n\n# 4. Results and discussion  \n\nThe applicability of the software is demonstrated by using the different descriptors in building a prediction model for formation energies of inorganic crystal structures and ionic charges of atoms in organic molecules. The used datasets are publicly available at Figshare (https://doi. org/10.6084/m9.figshare.c.4607783). These examples demonstrate the usage of the package in supervised machine learning tasks, but the output vectors can be as easily used in other learning tasks. For example the descriptors can be used as input for unsupervised clustering algorithms such as T-distributed stochastic neighbor embedding (T-SNE) [65] or Sketchmap [50] to analyse structure-property relations in structural and chem ical landscapes.  \n\nFor simplicity we here restrict the machine learning model to be kernel ridge regression (KRR) as implemented in the scikit-learn package [66]. However, the vectorial nature of the output from all the introduced descriptors does not impose any specific learning scheme, and many other regressors can be used, including neural networks, decision trees and support vector regression.  \n\n# 4.1. Formation energy prediction for inorganic crystals  \n\nWe demonstrate the use of multiple descriptors on the task of predicting the formation energy of inorganic crystals. The data comes from the Open Quantum Materials Database (OQMD) 1.1 [67]. We selected structures with a maximum of 10 atoms per unit cell and a maximum of 6 different atomic elements. Unconverged systems were filtered by removing samples which have a formation energy that is more than two standard deviations away from the mean, resulting in the removal of 96 samples. After these selections, 222 215 samples were left. The distribution of the formation energies is shown in Figure 7. The models are trained and tested on total dataset sizes of 1024, 2048, 4096, 8192 and 16384, from which 80% is used as training data and $20\\%$ as test data. These sizes are selected as they are successive powers of two making them equidistant on a logarithmic grid. For each dataset size the results are averaged over three different random selections. The resulting mean absolute errors are given in Figure 8. A full breakdown of the results for each descriptor and dataset size along with other performance metrics – including root mean square error, squared Pearson correlation coefficient and maximum error – are given in the Supplementary Information.  \n\nThe Coulomb matrix, Ewald sum matrix and sine matrix are used for the prediction with matrix rows and columns sorted by their Euclidean norm, and using the unit cell that was used for performing the formation energy calculation. The Coulomb matrix does not take the periodicity of the structure into account, but is included as a baseline for the other methods. We include MBTR with different values of $k$ and for each $k$ we individually optimize $\\sigma$ and $s_{k}$ with grid search. Figure 9 shows the error for each tested MBTR term, and the best performing one is included in Figure  \n\n![](images/9ece6060b15c6d08526f1a15260a4b3238ffe69eb7ed2b986305d078a01d5fe8.jpg)  \nFigure 8: Mean absolute error for formation energies in the test set as a function of training set size. The data consists of inorganic crystals from the OQMD database. The predictions are performed with kernel ridge regression and five different descriptors: Ewald sum matrix, Coulomb matrix, sine matrix, MBTR $\\mathcal{\\kappa}$ =1,2,3 and an averaged SOAP output for all atoms in the crystal. The figure shows an average over three randomly selected datasets, with the standard deviation shown by the shaded region.  \n\n8. To test the energy prediction by combining information from multiple local descriptors, as discussed in 2.7, we also include results using a simple averaged SOAP output for all atoms in the simulation cell. For SOAP we use the gaussian type orbital basis and fix $n_{\\mathrm{max}}=8$ and $l_{\\mathrm{max}}\\equiv8$ , but optimize the cutoff $r_{\\mathrm{cut}}$ and gaussian width $\\sigma$ individually with grid search.  \n\nThe possible descriptor hyperparameters are optimized at a subset of $2^{12}=4096$ samples with 5-fold cross-validation and 80%/20%-training/test split. The KRR kernel width and the regularization parameter are also allowed to vary on a logarithmic grid during the descriptor hyperparameter search. The use of a smaller subset allows much quicker evaluation for the hyperparameters than optimizing the hyperparameters for each size individually, but the transferability of these optimized hyperparameters to different sizes may affect the results slightly. After finding the optimal descriptor setup, it is used in training a model for all the different dataset sizes. The same cross-validation setup as for the descriptor hyperparameter optimization is used, but now with a finer grid for the KRR kernel width. The hyperparameter grids and optimal values for both the descriptors and kernel ridge regression are found in the Supplementary Information together with additional details.  \n\n![](images/9d7b1d6866315084f5adaef66444c242def0ce71ca49f1caa1361a916bda8fab.jpg)  \nFigure 9: Breakdown of the error for formation energies in the test set for different MBTR-terms. The predictions are performed with kernel ridge regression and four different MBTR configurations: MBTR $k{=}1$ , MBTR $\\scriptstyle{k=2}$ , MBTR $k{=}3$ and $\\mathrm{MBTR}_{k=1,2,3}$ which includes all three terms, each term normalized to unit length. The figure shows an average over three randomly selected datasets, with the standard deviation shown by the shaded region.  \n\n# 4.2. Ionic charge prediction for organic molecules  \n\nTo demonstrate the prediction of local properties with the DScribe package, a prediction of ionic charges in small organic molecules is performed with the different local descriptors included in the package. The dataset consists of Mulliken charges calculated at the CCSD level for the GDB9 dataset of 133 885 neutral molecules [68]. The structures contain up to nine atoms and five different chemical species: hydrogen, carbon, nitrogen, oxygen, and fluorine with 1 230 122, 846  \n\n![](images/9a6e17462fb7b8abcde676ca02bbd3e253cd89c159b6cdd2aefd30e8ba98b091.jpg)  \nFigure 10: Parity plot of ionic charge prediction results from the test set against true CCSD ionic charges. The predictions are performed with kernel ridge regression using $\\mathrm{SOAP_{gto}}$ (gaussian type orbital basis), $\\operatorname{SOAP}_{\\operatorname{poly}}$ (polynomial basis) and ACSF. The mean absolute error (MAE), root mean square error (RMSE), squared Pearson correlation coefficient $\\mathrm{\\Delta^{R^{2}}}$ ) and maximum error are also shown together with the total error distribution in the inset.  \n\n![](images/cb7d9e381ae1251d2bac17f4dd46d1b409864140bfc3f157ee38c706be8f7e2a.jpg)  \nFigure 11: Distribution of the ionic charges for each chemical species together with the mean $(\\mu)$ , standard deviation $(\\sigma)$ and mean absolute deviation (MAD).  \n\n557, 139 764, 187 996 and 3314 atoms present for each species respectively. The distribution of the ionic charges for each species is shown in Figure 11. The geometries have been relaxed at the B3LYP/6-31G(2df,p) level and no significant forces were present in the static CCSD calculation. The models are trained and tested on a subset of 10 000 samples per chemical species (except fluorine, for which only 3314 atoms were available and all are used), from which $80\\%$ is used as training data and 20% as test data. The combined parity plots for all five chemical species together with error metrics are given in Figure 10. A breakdown of the results for each species separately is given  \n\n# in the Supplementary Information.  \n\nThe prediction is performed with the two local descriptors included in the package, SOAP and ACSF. For SOAP we perform the prediction with both radial basis functions: the polynomial basis $(\\mathrm{SOAP_{poly}}$ ) and the gaussian type orbital radial basis $(\\mathrm{SOAP_{gto}}$ ). For them we fix $n_{\\mathrm{max}}=8$ and $l_{\\mathrm{max}}=8$ , but optimize the cutoff $r_{\\mathrm{cut}}$ and Gaussian width $o$ with grid search. For ACSF we use 10 radial functions $G^{2}$ and 8 angular functions $G^{3}$ . The cutoff value $r_{\\mathrm{cut}}$ is shared between the radial and angular functions and it is optimized with grid search. More details about the used ACSF symmetry functions are found in the Supplementary Information.  \n\nDescriptor hyperparameters are optimized with grid search separately for each species on a smaller set of 2500 sample atoms with 5-fold cross-validation and 80%/20%-training/test split. Both the KRR kernel width and the regularization parameter are allowed to vary on a logarithmic grid during the descriptor hyperparameter search. After finding the optimal descriptor setup, it is used in training a model for full dataset of 10 000 samples (except for fluorine with 3314 total samples). The training is done with the same cross-validation setup as for the descriptor hyperparameter optimization, but now with finer grid for the KRR kernel width. The hyperparameter grids and optimal values for both the descriptors and kernel ridge regression are found in the Supplementary Information together with additional details.  \n\n# 4.3. Discussion  \n\nThe formation energy prediction demonstrates that our implementation performs consistently and offers insight into the performance of the different descriptors. Special care must be taken in interpreting the results, as there exist different variations of the different descriptors. For example, as discussed in section 2.7, there are different ways to combine information from multiple local SOAP-outputs, and different geometry functions and cutoff types may be used for the MBTR. The learning rates also depend on the chosen machine learning model.  \n\nWith SOAPaverage and a training set of 0.8 $2^{14}=13107$ samples the best mean absolute error of 0.117 eV/atom is achieved. It has been demonstrated that a similar mean absolute error (0.09 eV/atom [5] and 0.12 eV/atom [40]) can be used for virtual screening of materials by stability. The fact that the training data contains 89 chemical elements and various structural phases makes highly accurate predictions challenging and the error is still relatively large when compared against the mean absolute deviation of 0.493 eV/atom for the labels. As shown by earlier research [29, 32, 69], the prediction error can be reduced further by using a learning model with a more intelligent scheme for combining local structural information.  \n\nOur results for the Ewald sum matrix and the sine matrix reflect the results reported earlier, where a formation energy prediction was performed for a similar set of data from the Materials Project [70]. They report MAE for the Ewald sum matrix to be 0.49 eV and for the sine matrix to be 0.37 eV [7] with a training set of 3000 samples, whereas we find MAE for the Ewald sum matrix to be 0.36 eV and for the sine matrix to be 0.24 eV with a training set of 3276 samples. The performance improvement in our results can be explained by differences in the contents of the used dataset. We, however, recover the same trend of the sine matrix performing better, even when issues in the original formulation of the Ewald sum matrix (as discussed in section 2.2) were addressed. The low performance of the more accurate charge interaction in the Ewald model and the relatively small difference between the performance of the Coulomb and sine matrix may indicate that for this task the information of the potential energy of the neutral atoms – contained on the diagonal of both the sine and Coulomb matrix – largely controls the performance.  \n\nWith respect to the individual performance of the different MBTR parts, the $k=2$ terms containing distance information performs best, whereas the angle information contained in $k~=~3$ and the simple composition information contained by $k=1$ lag behind. However, the best MBTR performance is achieved by combining the information from all of the terms. It is also surprising how well the simple averaging scheme for SOAP performs in the tested dataset range. When extrapolating the performance to larger datasets, it can however be seen that MBTR may provide better results.  \n\nThe charge prediction test illustrates that the ionic charges of different species in organic molecules may be learned accurately on the CCSD level just by observing the local arrangement of atoms up to a certain radial cutoff. On average the mean absolute error is around 0.005-0.01 e when using up to 10 000 samples for each species.  \n\nThe best mean absolute error of 0.0054 $e$ and root mean square error of 0.0100 $e$ is achieved with SOAP $\\mathrm{gto}$ . A similar root mean square error of 0.016 $e$ was achieved in a recent machine learning based partial charge prediction for druglike molecules using charges extracted from DFT electron density[71]. The machine learned partial charges offer a great balance between accuracy and computational cost, making them an attractive alternative to full quantum chemical calculations or empirical charge models. Potential applications include the parametrization of partial charges in classical molecular dynamics and quantitative structure–activity relationship (QSAR) mo els [71].  \n\nFigure 11 shows that the deviation of the charg in the dataset depends on the species, which is also transferred to a species-specific variation of the prediction error included in the Supplementary Information. As to be expected, the charge of the multi-valent species – C, N, O – varies much more in the CCSD data and is much harder to predict than the charge of the low valence species H and F. Predicting the ionic charge of carbon is most difficult and so most of the outliers correspond to carbon atoms, with a few noticeable outliers corresponding also to oxygen and nitrogen atoms.  \n\nOur comparison shows that there is little difference between the predictive performance of the two radial bases used for SOAP. With our current implementation there is, however, a notable difference in the speed of creating these descriptors. For identical settings ( $n_{\\mathrm{max}}=8$ , $l_{\\mathrm{max}}=8$ , $r_{\\mathrm{cut}}~=~5$ , and $\\sigma=0.1$ ), the gaussian type orbital basis is over four times faster to calculate than the polynomial basis. This difference originates largely from the numerical radial integration, which is required for the polynomial basis but not for the gaussian type orbital basis. The prediction performance of ACSF does not fall far behind SOAP and it might be possible to achieve the same accuracy by using a more advanced parameter calibration for the symmetry functions. The symmetry functions used in ACSF are easier to tune for capturing specific structural properties, such as certain pairwise distances or angles formed by three atoms. This tuning can, however, be done only if such intuition is available a priori, and in general consistently improving the performance by changing the used symmetry functions can be hard.  \n\n# 5. Conclusions  \n\nThe recent boom in creating machine learnable fingerprints for atomistic systems, or descriptors, has led to a plethora of available options for materials science. The software implementations for these descriptors is, however, often scattered across different libraries or missing altogether, making it difficult to test and compare  \n\ndifferent alternatives.  \n\nWe have collected several descriptors in the DScribe software library. DScribe has an easyto-use python-interface, with C/C++ extensions for the computationally intensive tasks. We use a set of regression tests to ensure the validity of the implementation, and provide the source code together with tutorials and documentation. 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Mater. 31 (9) (2019) 3564–3572.   \n[70] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, K. a. Persson, The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater. 1 (1) (2013) 011002.   \n[71] P. Bleiziffer, K. Schaller, S. Riniker, Machine Learning of Partial Charges Derived from High-Quality Quantum-Mechanical Calculations, J. Chem. Inf.  \n\n# PROGRAM SUMMARY  \n\nProgram Title: DScribe   \nProgram Files doi: http://dx.doi.org/10.17632/vzrs8n8pk6.1   \nLicensing provisions: Apache-2.0   \nProgramming language: Python/C/C++   \nSupplementary material: Supplementary Information as PDF   \nNature of problem: The application of machine learning for materials science is hindered by the lack of consistent software implementations for feature transformations. These feature transformations, also called descriptors, are a key step in building machine learning models for property prediction in materials science.   \nSolution method: We have developed a library for creating common descriptors used in machine learning applied to materials science. We provide an implementation the following descriptors: Coulomb matrix, Ewald sum matrix, sine matrix, Many-body Tensor Representation (MBTR), Atom-centered Symmetry Functions (ACSF) and Smooth Overlap of Atomic Positions (SOAP). The library has a python interface with computationally intensive routines written in $\\mathrm{C}$ or C++. The source code, tutorials and documentation are provided online. A continuous integration mechanism is set up to automatically run a series of regression tests and check code coverage when the codebase is updated.  ",
    "Target": null
  },
  {
    "id": "10.1016_j.cpc.2021.108033",
    "DOI": "10.1016/j.cpc.2021.108033",
    "DOI Link": "http://dx.doi.org/10.1016/j.cpc.2021.108033",
    "Relative Dir Path": "mds/10.1016_j.cpc.2021.108033",
    "Article Title": "VASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using VASP code",
    "Authors": "Wang, V; Xu, N; Liu, JC; Tang, G; Geng, WT",
    "Source Title": "COMPUTER PHYSICS COMMUNICATIONS",
    "Abstract": "We present the VASPKIT, a command-line program that aims at providing a robust and user-friendly interface to perform high-throughput analysis of a variety of material properties from the raw data produced by the VASP code. It consists of mainly the pre-and post-processing modules. The former module is designed to prepare and manipulate input files such as the necessary input files generation, symmetry analysis, supercell transformation, k-path generation for a given crystal structure. The latter module is designed to extract and analyze the raw data about elastic mechanics, electronic structure, charge density, electrostatic potential, linear optical coefficients, wave function plots in real space, etc. This program can run conveniently in either interactive user interface or command line mode. The command-line options allow the user to perform high-throughput calculations together with bash scripts. This article gives an overview of the program structure and presents illustrative examples for some of its usages. The program can run on Linux, macOS, and Windows platforms. The executable versions of VASPKIT and the related examples and tutorials are available on its official website vaspkit .com. Program summary Program title: VASPKIT CPC Library link to program files: https://doi.org/10.17632/v3bvcypg9v.1 Licensing provisions: GPLv3 Programming language: Fortran, Python Nature of problem: This program has the purpose of providing a powerful and user-friendly interface to perform high-throughput calculations together with the widely-used VASP code. Solution method: VASPKIT can extract, calculate and even plot the mechanical, electronic, optical and magnetic properties from density functional calculations together with bash and python scripts. It can run in either interactive user interface or command line mode. (C) 2021 Elsevier B.V. All rights reserved.",
    "Times Cited, WoS Core": 3941,
    "Times Cited, All Databases": 4028,
    "Publication Year": 2021,
    "Research Areas": "Computer Science; Physics",
    "UT (Unique WOS ID)": "WOS:000678508900017",
    "Markdown": "# VASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using VASP code ✩,✩✩  \n\nVei Wang a,∗, Nan ${\\tt X}{\\tt u}^{\\mathrm{b}}$ , Jin-Cheng Liu c, Gang Tang d, Wen-Tong Geng e  \n\na Department of Applied Physics, Xi’an University of Technology, Xi’an 710054, China   \nb College of Chemical and Biological Engineering, Zhejiang University, Hangzhou 310027, China   \nc Department of Chemistry and Key Laboratory of Organic Optoelectronics & Molecular Engineering of Ministry of Education, Tsinghua University, Beijing 100084,   \nChina   \nd Theoretical Materials Physics, Q-MAT, CESAM, University of Liège, Liège B-4000, Belgium   \ne School of Materials Science & Engineering, University of Science and Technology Beijing, Beijing 100083, China  \n\n# a r t i c l e i n f o  \n\n# a b s t r a c t  \n\nArticle history:   \nReceived 27 April 2020   \nReceived in revised form 28 April 2021   \nAccepted 5 May 2021   \nAvailable online 28 May 2021  \n\nKeywords: High-throughput Elastic mechanics Electronic properties Optical properties Molecular dynamics Wave-function  \n\nWe present the VASPKIT, a command-line program that aims at providing a robust and user-friendly interface to perform high-throughput analysis of a variety of material properties from the raw data produced by the VASP code. It consists of mainly the pre- and post-processing modules. The former module is designed to prepare and manipulate input files such as the necessary input files generation, symmetry analysis, supercell transformation, $k$ -path generation for a given crystal structure. The latter module is designed to extract and analyze the raw data about elastic mechanics, electronic structure, charge density, electrostatic potential, linear optical coefficients, wave function plots in real space, etc. This program can run conveniently in either interactive user interface or command line mode. The command-line options allow the user to perform high-throughput calculations together with bash scripts. This article gives an overview of the program structure and presents illustrative examples for some of its usages. The program can run on Linux, macOS, and Windows platforms. The executable versions of VASPKIT and the related examples and tutorials are available on its official website vaspkit com.  \n\n# Program summary  \n\nCPC Library link to program files: https://doi org 10 17632 v3bvcypg9v.1   \nLicensing provisions: GPLv3   \nProgramming language: Fortran, Python   \nNature of problem: This program has the purpose of providing a powerful and user-friendly interface to perform high-throughput calculations together with the widely-used VASP code.   \nSolution method: VASPKIT can extract, calculate and even plot the mechanical, electronic, optical and magnetic properties from density functional calculations together with bash and python scripts. It can run in either interactive user interface or command line mode.  \n\n$\\circledcirc$ 2021 Elsevier B.V. All rights reserved.  \n\n# 1. Introduction  \n\nWith the rapid development of high-performance computations and computational algorithms, high-throughput computational analysis and discovery of materials has become an emerging research field because it promises to avoid time-consuming try and error experiments and explore the hidden potential behind thousands of potentially unknown materials within short timeframes that the real experiments might take a long time. Density functional theory (DFT) is one of the most popular methods that can treat both model systems and realistic materials in a quantum mechanical way [1–5]. It is not only used to understand the observed behavior of solids, including the structural, mechanical, electronic, magnetic and optical properties, but increasingly more to predict characteristics of compounds that have not yet been determined experimentally [6–12].  \n\nThe last two decades have witnessed tremendous progress in the methodology development for first-principles calculations of materials properties. Dozens of electronic-structure computation packages have been developed based on DFT so far, such as Abinit [13], CASTEP [14], VASP [15,16], Siesta [17], Quantum Espresso [18,19], Elk [20] and WIEN2k [21], with great success in exploring material properties. One of the common features for these packages is that post-processing is required to extract and/or plot into a human-readable format from the raw data. There are two popular commercial programs, Materials Studio [22] and QuantumATK [23], providing a graphical user interface (GUI) that allows the researchers to efficiently build, visualize, and review results and calculation setup up with a set of mouse actions. However, these GUI programs become less productive when the users want to perform batch processing operations. In contrast, several open-source post-processing packages, such as Python Materials Genomics (pymatgen) [24], Atomic Simulation Environment (ASE) [25], and PyProcar [26] provide powerful command-line interfaces to efficiently extract, plot and analyze the raw data in batch mode but require the users to be proficient in Python programming language. It is worth mentioning here that both lev00 [27] and qvasp [28] are two interactive menu-driven programs written in Fortran which mainly focus on the post-processing of electronic structure calculations using VASP and other codes.  \n\nIn this article we will introduce a toolkit, referred to as VASPKIT which is developed to provide a robust and user-friendly integrated input/output environment to perform initial setup for calculations and post-processing analysis to derive various material properties from the raw data calculated using the VASP code. It is capable of calculating the elastic, electronic, optical and catalytic properties including equation of state, elastic constants, carrier effective masses, Fermi surfaces, band structure unfolding for supercell models, linear optical coefficients, joint density of states, transition dipole moment, wave functions plots in real space, thermal energy correction, etc. In addition, it also allows the users to perform high-throughput calculations with low barriers to entry. For example, we recently performed high-throughput calculations to screen hundreds of two-dimensional (2D) semiconductors from near 1000 monolayers using this program together with VASP [29]. The VASPKIT remains in development, with growing functionality, and is ready to be extended to work directly with outputs from other electronic structure packages.  \n\nThe rest of this paper is organized as follows: In Section 2 the workflow and basic features of the pre-processing module as implemented into VASPKIT are described. Section 3 presents the computational algorithms and some examples illustrating the capabilities of post-processing module in the VASPKIT code. Finally, it ends with the Summary section.  \n\n# 2. Capabilities of the pre-processing module  \n\nThe workflow of the VASPKIT package is illustrated in Fig. 1. In the pre-processing module, the program first reads the POSCAR file and then prepares the rest three input files (INCAR, POTCAR and KPOINTS) to perform DFT calculations using VASP. It can also manipulate the structure file such as building supercell, generating the suggested $k$ -path for band structure calculation, determining the crystal symmetry information, or finding the conventional/primitive cell for a given lattice by employing the symmetry analysis library Spglib [30]. Furthermore, it can convert POSCAR to several widely-used structural formats, such as XCrysDen (.xsf) [31], Crystallographic Information Framework (.cif) [32] or Protein Data Bank (.pdb) formats [33].  \n\n![](images/fe5d1c2d80fd0f39b3adaad204cdc68d19135556247ec430d464b4fb2ef5c981.jpg)  \nFig. 1. (Color online.) A structural overview of the VASPKIT package.  \n\n# 2.1. Definitions and conversions of crystal structures  \n\nThe crystal structures are often provided by basis vectors and point coordinates of labeled atoms. Lattice basis vectors A are represented by three row vectors  \n\n$$\n\\mathbf{A}={\\left(\\begin{array}{l}{\\mathbf{a}}\\\\ {\\mathbf{b}}\\\\ {\\mathbf{c}}\\end{array}\\right)}=\\left({\\begin{array}{l l l}{a_{x}}&{a_{y}}&{a_{z}}\\\\ {b_{x}}&{b_{y}}&{b_{z}}\\\\ {c_{x}}&{c_{y}}&{c_{z}}\\end{array}}\\right).\n$$  \n\nThe position of an ion is represented by a row vector either in fractional coordinates $(x,\\ y,z)$ concerning basis vector lengths or in Cartesian coordinates $(X,Y,Z)$ . The relationship of these two coordinates is written as  \n\n$$\n\\left(\\begin{array}{l}{\\boldsymbol{X}}\\\\ {\\boldsymbol{Y}}\\\\ {\\boldsymbol{Z}}\\end{array}\\right)=\\boldsymbol{\\mathsf{A}}^{T}\\left(\\begin{array}{l}{\\boldsymbol{x}}\\\\ {\\boldsymbol{y}}\\\\ {\\boldsymbol{z}}\\end{array}\\right)=\\left(\\begin{array}{l l l}{a_{x}}&{b_{x}}&{c_{x}}\\\\ {a_{y}}&{b_{y}}&{c_{y}}\\\\ {a_{z}}&{b_{z}}&{c_{z}}\\end{array}\\right)\\left(\\begin{array}{l}{\\boldsymbol{x}}\\\\ {\\boldsymbol{y}}\\\\ {\\boldsymbol{z}}\\end{array}\\right),\n$$  \n\nwhere $\\pmb{A}^{T}$ denotes the matrix transpose of lattice basis vectors A.  \n\nThe conversion from one lattice basis $(\\mathbf{a},\\mathbf{b},\\mathbf{c})$ to another choice of lattice basis $(\\ensuremath{\\mathbf{a}}^{\\prime},\\ensuremath{\\mathbf{b}}^{\\prime},\\ensuremath{\\mathbf{c}}^{\\prime})$ is given by  \n\n$$\n\\left(\\begin{array}{c}{\\mathbf{a}^{\\prime}}\\\\ {\\mathbf{b}^{\\prime}}\\\\ {\\mathbf{c}^{\\prime}}\\end{array}\\right)=\\mathbf{M}\\cdot\\left(\\begin{array}{c}{\\mathbf{a}}\\\\ {\\mathbf{b}}\\\\ {\\mathbf{c}}\\end{array}\\right),\n$$  \n\nwhere $\\mathbf{M}$ is the transformation matrix. Its determinant $|\\mathbf{M}|$ defines the ratio between the supercell and primitive cell volumes in the real space. Fig. 2 shows how to construct a supercell (SC) from the specified transformation matrix and the primitive cell (PC) lattice vectors.  \n\n# 2.2. Generation of suggested $k$ -path  \n\nIn order to plot a band structure, one needs to define a set of $k$ -points along with desired high-symmetry directions in the Brillouin zone (BZ). The $k$ -path utility automatically generates the suggested $k$ -path for a given 2D [29] or bulk [34] crystal structure. The flowchart of the algorithm to determine the suggested $k$ -path for a given crystal is shown in Fig. 3 (a). Specifically, VASPKIT first determines the space group number, crystal family and  \n\n![](images/fe6711702cd5413ae7a59ab75edfda5d5b6a9d517a0b9ee4f5594e3f5bf6cec8.jpg)  \nFig. 2. (Color online.) Schematic illustration of building a supercell from the lattice vectors of a primitive cell (PC) and the specified transformation matrix. The supercell and primitive cell are indicated by the yellow and red rhombuses.  \n\nBravais lattice type from the input structure, typically read from the POSCAR file; a standardized conventional cell is then identified and constructed by idealizing the lattice vectors based on the axial lengths and the interaxial angles, aiming to eliminate the non-unique choices in the possible shapes of BZ in certain Bravais lattices [35,34]; then the standard primitive cell is determined by transforming the basis vectors of the standard conventional cell according to Eq. (4),  \n\n$$\n\\left(\\begin{array}{c}{\\mathbf{a}_{p}}\\\\ {\\mathbf{b}_{p}}\\\\ {\\mathbf{c}_{p}}\\end{array}\\right)=\\mathbf{P}\\cdot\\left(\\begin{array}{c}{\\mathbf{a}_{c}}\\\\ {\\mathbf{b}_{c}}\\\\ {\\mathbf{c}_{c}}\\end{array}\\right),\n$$  \n\nwhere $(\\mathbf{a}_{p},\\mathbf{b}_{p},\\mathbf{c}_{p})$ and $(\\mathbf{a}_{c},\\mathbf{b}_{c},\\mathbf{c}_{c})$ are the basis vectors of primitive and conventional systems, respectively, $\\mathbf{p}$ is the transformation matrix from the standardized conventional cell to the primitive cell, as summarized in Table 3 in Ref. [34], and the subscripts $c$ and $p$ represent the primitive and conventional cells respectively. The atomic position of an ion in fractional coordinates transformed from the basis vectors of a conventional cell to those of primitive cell is written as below:  \n\n$$\n\\left(\\begin{array}{c c c}{x_{p}}\\\\ {y_{p}}\\\\ {z_{p}}\\end{array}\\right)=\\mathbf P^{-1}\\left(\\begin{array}{c c c}{x_{c}}\\\\ {y_{c}}\\\\ {z_{c}}\\end{array}\\right).\n$$  \n\nIt should be noted that the number of atoms in the PC is generally less than that in SC. This means that the transformation from SC to PC leads to some duplicated atoms, which must be removed.  \n\nIn the final step, the $k$ -path utility automatically saves the standard primitive cell and the suggested $k$ -path into the PRIMCELL.vasp and KPATH.in files respectively. In addition to the automatic generation of the suggested $k$ -path when a crystal structure is given as input, VASPKIT also provides the python script to visualize the specified $k$ -path in the first Brillouin zone using Matplotlib plotting library [36]. As illustrative examples, the recommended $k$ -paths of 2D-rectangular, 2D-oblique and face-centered cubic and hexagonal lattices are show in Fig. 3 (b)-(e) respectively.  \n\n# 3. Capabilities of the post-processing module  \n\nFig. 4 displays an overview of the post-processing features as implemented into the VASPKIT package. This module is designed to extract and analyze the raw data including elastic mechanics, electronic, charge density, electrostatic potential, optical wavefunction, catalysis and molecular dynamics related properties. We next present the computational algorithms and some examples to illustrate the capabilities of the post-processing module.  \n\n# 3.1. Elastic mechanics  \n\nThe second-order elastic constants (SOECs) play a crucial role in governing materials’ mechanical and dynamical properties, especially on the stability and stiffness. Within the linear elastic region, the stress $\\sigma=(\\sigma_{1},\\sigma_{2},\\sigma_{3},\\sigma_{4},\\sigma_{5},\\sigma_{6})$ response of solids to external loading strain $\\pmb{\\varepsilon}=(\\varepsilon_{1},\\varepsilon_{2},\\varepsilon_{3},\\varepsilon_{4},\\varepsilon_{5},\\varepsilon_{6})$ satisfies the generalized Hooke’s law and can be simplified in the Voigt notation [37],  \n\n$$\n\\sigma_{i}=\\sum_{j=1}^{6}\\mathsf C_{i j}\\varepsilon_{j},\n$$  \n\nwhere strain $\\sigma_{i}$ and stress $\\varepsilon_{j}$ are represented as a vector with 6 independent components respectively, i.e., $1\\leq i,j\\leq6.\\ C_{i j}$ is the second-order elastic stiffness tensor expressed by a $6\\times6$ symmetric matrix in units of GPa. The elastic stiffness tensor ${\\mathsf{C}}_{i j}$ can be determined using the first-order derivative of the stress-strain curves proposed by Nielsen and Martin [38,39], as expressed in Eq. (6). The number of independent elastic constants depends on the symmetry of the crystal. The lower the symmetry means the more the independent elastic constants. For example, the cubic crystals have three but the triclinic ones have 21 independent elastic constants. The classification of the different crystal system with the corresponding number of independent elastic constants for bulk materials is summarized in Table 1 [40–42].  \n\nAn alternative theoretical approach to calculate elastic constants is based on the energy variation by applying minor strains to the equilibrium lattice configuration [43]. The elastic energy $\\Delta E(V,\\{\\varepsilon_{i}\\})$ of a solid under the harmonic approximation is  \n\n![](images/68501c03c1829c09c4127aaa051297e6993377f4fa2e883d0da46e35a7c25986.jpg)  \nFig. 3. (Color online.) (a) Workflow of the algorithm used in the $k$ -path utility. The first Brillouin zone, special high symmetry points, and recommended $k$ -paths for (a) 2D rectangular, (b) 2D oblique, (c) face-centered cubic and (d) hexagonal close-packed lattices respectively.  \n\nTable 1 Classification of crystal systems, point group classes, and space-group numbers are provided with the number of independent second elastic constants for bulk materials. In the last column, several prototype materials are shown.   \n\n\n<html><body><table><tr><td>Crystal system</td><td>Point groups</td><td>Space-groups</td><td>Number of independent SOECs</td><td>Material prototypes</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Trilinlinic</td><td></td><td>1-25 </td><td>213</td><td>Zr02</td></tr><tr><td>Orthorhombic</td><td>222, mm2, 222</td><td>16-74</td><td>9</td><td>TiS2</td></tr><tr><td>Tetragonal I</td><td>， 422,4mm,42m，m</td><td>89-142</td><td>6</td><td>MgF2</td></tr><tr><td>Tetragonal II </td><td>4.4</td><td>75-88</td><td>7</td><td>CaMo04</td></tr><tr><td>Trigonal I</td><td>32.3m,32</td><td>149-167</td><td>6</td><td>α-Al03</td></tr><tr><td>Trigagal l </td><td></td><td>143-144</td><td></td><td></td></tr><tr><td></td><td>mmm</td><td></td><td>７５</td><td>CaMg(CO3 )2</td></tr><tr><td>Cubic</td><td>432.43m，3</td><td>195-230</td><td>3</td><td>Diamond</td></tr></table></body></html>  \n\n![](images/b2aab9e7e03761a4db8183d8661a454d2d51ed07917c2893d9a123a568ccad69.jpg)  \nFig. 4. (Color online.) A structural overview of the post-processing module implemented into the VASPKIT package.  \n\ngiven by  \n\n$$\n\\begin{array}{r}{\\Delta E\\left(V,\\{\\varepsilon_{i}\\}\\right)=E\\left(V,\\{\\varepsilon_{i}\\}\\right)-E\\left(V_{0},0\\right)}\\\\ {=\\displaystyle\\frac{V_{0}}{2}\\sum_{i,j=1}^{6}C_{i j}\\varepsilon_{j}\\varepsilon_{i},}\\end{array}\n$$  \n\nwhere $E\\left(V_{0},0\\right)$ and $E\\left(V,\\{\\varepsilon_{i}\\}\\right)$ are the total energies of the equilibrium and distorted lattice cells, with the volume of $V_{0}$ and $V$ , respectively. In the energy-strain method the elastic stiffness tensor is derived from the second-order derivative of the total energy versus strain curves [43]. In general, the stress-strain method requires higher computational precision to achieve the same accuracy as the energy-strain method. Nevertheless, it requires less distortion set than the latter [40,41,44,45,43]. Considering that the energy-strain relation has less stress sensitivity than the stressstrain one, the former method has been implemented into the VASPKIT package. Meanwhile, the determination of elastic stability criterion is also provided in the elastic utility based on the necessary and sufficient elastic stability conditions in the harmonic approximation [46] for various crystal systems proposed by Mouhat et al. [40,41,47].  \n\nWhen a crystal is deformed by applying strain $\\pmb\\varepsilon$ , the relation of lattice vectors between the distorted and equilibrium cells is given by  \n\n$$\n\\left(\\begin{array}{l}{\\mathbf{a}^{\\prime}}\\\\ {\\mathbf{b}^{\\prime}}\\\\ {\\mathbf{c}^{\\prime}}\\end{array}\\right)=\\left(\\begin{array}{l}{\\mathbf{a}}\\\\ {\\mathbf{b}}\\\\ {\\mathbf{c}}\\end{array}\\right)\\cdot(\\mathbf{I}+\\mathbf{\\epsilon}),\n$$  \n\n![](images/651042cb82e35e40579741214e97c5f4bc9affd4b579bafb4fb4ea47e6cf3556.jpg)  \nFig. 5. (Color online.) Workflow of the algorithm to determine the second-order elastic constants based on the energy-strain method used in the elastic utility.  \n\nwhere I is the $3\\times3$ identity matrix. The strain tensor $\\epsilon$ is defined by  \n\n$$\n\\begin{array}{r}{\\epsilon=\\left(\\begin{array}{l l l}{\\varepsilon_{1}}&{\\varepsilon_{6}/2}&{\\varepsilon_{5}/2}\\\\ {\\varepsilon_{6}/2}&{\\varepsilon_{2}}&{\\varepsilon_{4}/2}\\\\ {\\varepsilon_{5}/2}&{\\varepsilon_{4}/2}&{\\varepsilon_{3}}\\end{array}\\right).}\\end{array}\n$$  \n\nThe workflow of elastic utility is shown in Fig. 5. VASPKIT first reads the equilibrium structure from POSCAR in which both lattice parameters and atomic positions are fully relaxed. In addition, the dimensionality of material (either 2D or 3D) and the number of applied strain $\\pmb\\varepsilon$ need to be specified as input. For 2D materials, in order to avoid mirror interactions the periodic slabs are required to separate by sufficiently large vacuum layer in $c$ direction. In the second step, the space group number and the type of input structure are analyzed by using the Spglib code [30] to determine how many independent elastic constants need to be calculated. A classification of the different crystal system with the corresponding number of independent elastic constants is given in Table 1. Furthermore, a standard conventional cell needs to be adopted in the following calculations since the components of ${\\mathsf{C}}_{i j}$ are dependent on the choice of the coordinate system and lattice vectors. After that, based on the determined space group number, a series of distorted structures with specified values of strain around the equilibrium are generated via Eq. (8). Next, the elastic energies are calculated for each distorted structure by using VASP. Then, a polynomial fitting procedure is applied to calculate the second derivative at the equilibrium of the energy with respect to the strain. Finally, various mechanical properties such as bulk, shear modulus and Poisson’s ratio for polycrystalline materials are determined.  \n\nWe take the cubic structure as an example to demonstrate how to calculate its independent elastic constants by using the energystrain method. For cubic system, the three independent elastic constants ${\\mathsf{C}}_{11},{\\mathsf{C}}_{12}$ and $\\mathsf{C}_{44}$ , are expressed in an elastic stiffness tensor matrix  \n\n$$\nC_{i j}^{c u b i c}=\\left(\\begin{array}{c c c c c c}{C_{11}}&{C_{12}}&{C_{12}}&{0}&{0}&{0}\\\\ {C_{12}}&{C_{11}}&{C_{12}}&{0}&{0}&{0}\\\\ {C_{12}}&{C_{12}}&{C_{11}}&{0}&{0}&{0}\\\\ {0}&{0}&{0}&{C_{44}}&{0}&{0}\\\\ {0}&{0}&{0}&{0}&{C_{44}}&{0}\\\\ {0}&{0}&{0}&{0}&{0}&{C_{44}}\\end{array}\\right).\n$$  \n\nAfter substituting Eq. (10) into Eq. (7), the elastic energy is written as below:  \n\n$$\n\\begin{array}{c}{\\displaystyle{\\frac{\\Delta E}{V}=\\frac{1}{2}(C_{11}\\varepsilon_{1}\\varepsilon_{1}+C_{11}\\varepsilon_{2}\\varepsilon_{2}+C_{11}\\varepsilon_{3}\\varepsilon_{3}+C_{12}\\varepsilon_{1}\\varepsilon_{2}}}\\\\ {\\displaystyle{+C_{12}\\varepsilon_{1}\\varepsilon_{3}+C_{12}\\varepsilon_{2}\\varepsilon_{1}+C_{12}\\varepsilon_{2}\\varepsilon_{3}+C_{12}\\varepsilon_{3}\\varepsilon_{1}}}\\\\ {\\displaystyle{+C_{12}\\varepsilon_{3}\\varepsilon_{2}+C_{44}\\varepsilon_{4}\\varepsilon_{4}+C_{44}\\varepsilon_{5}\\varepsilon_{5}+C_{44}\\varepsilon_{6}\\varepsilon_{6}).}}\\end{array}\n$$  \n\nWhen applied the tri-axial shear strain $ \\pmb{\\varepsilon}\\mathrm{=}(0,0,0,\\delta,\\delta,\\delta)$ , Eq. (10) becomes  \n\n$$\n\\frac{\\Delta E}{V}=\\frac{3}{2}C_{44}\\delta^{2}.\n$$  \n\nSimilarly, $C_{11}{+}C_{12}$ can be obtained by using the strain $ \\varepsilon=(\\delta,\\delta,0,$ 0,0,0):  \n\n$$\n\\frac{\\Delta E}{V}=\\left(C_{11}+C_{12}\\right)\\delta^{2}.\n$$  \n\nAlso, $C_{11}+2C_{12}$ is calculated using the strain $\\pmb{\\varepsilon}\\mathrm{=}(\\delta,\\delta,\\delta,0,0,0)$ :  \n\n$$\n\\frac{\\Delta E}{V}=\\frac{3}{2}\\left(C_{11}+2C_{12}\\right)\\delta^{2}.\n$$  \n\nIn order to calculate the elastic stiffness constants given above, the elastic energies of a set of deformed configurations in the distortion range $-2\\%\\leq\\delta\\leq+2\\%$ with an increment of $0.5\\%$ are investigated using VASP. After that, the quadratic coefficients are determined by fitting the energy versus distortion relationship, and finally the second-order elastic constants $C_{i j}$ are determined by solving the equations (12)-(14) during the post-processing of elastic utility. The details of strain modes and the derived elastic constants for each crystal system based on the energy-strain approach are listed in Appendix A.  \n\nThe crystallites are randomly oriented for polycrystalline materials, and such materials can be considered quasi-isotropic or isotropic in a statistical sense. Thus, the bulk modulus $K$ and shear modulus $G$ are generally obtained by averaging the singlecrystal elastic constants. Three of the most widely used averaging approaches have been implemented into the elastic utility: Voigt [37], Reuss [48], and Hill [49] schemes. Hill has shown that the Voigt and Reuss elastic moduli are the strict upper and lower bounds [49], respectively. The arithmetic mean of the Voigt and Reuss bounds termed the Voigt-as Reuss-Hill (VRH) average, is a better approximation to a polycrystalline material’s actual elastic behavior.  \n\nThe Voigt bounds are given by the following equations:  \n\n$$\n\\left\\{\\begin{array}{l l}{9K_{\\mathrm{V}}=(C_{11}+C_{22}+C_{33})+2(C_{12}+C_{23}+C_{31})}\\\\ {15G_{\\mathrm{V}}=(C_{11}+C_{22}+C_{33})-(C_{12}+C_{23}+C_{31})}\\\\ {\\quad+4(C_{44}+C_{55}+C_{66})}\\end{array}\\right.,\n$$  \n\nwhile the Reuss bounds are given by:  \n\n$$\n\\left\\{\\begin{array}{l}{{1/K_{\\mathrm{R}}=(\\mathsf{S}_{11}+\\mathsf{S}_{22}+\\mathsf{S}_{33})+2(\\mathsf{S}_{12}+\\mathsf{S}_{23}+\\mathsf{S}_{31})}}\\\\ {{15/G_{\\mathrm{R}}=4(\\mathsf{S}_{11}+\\mathsf{S}_{22}+\\mathsf{S}_{33})-4(\\mathsf{S}_{12}+\\mathsf{S}_{23}+\\mathsf{S}_{31})}}\\\\ {{\\qquad+3(\\mathsf{S}_{44}+\\mathsf{S}_{55}+\\mathsf{S}_{66})}}\\end{array}\\right.,\n$$  \n\nwhere $S_{i j}$ are the components of compliance tensor, which correspond to the matrix elements of the inverse of the elastic tensor, namely, $\\left[S_{i j}\\right]=\\left[C_{i j}\\right]^{-1}$ . Based on the Voigt and Reuss bounds, Hill defined $K_{\\mathsf{V R H}}=1/2(K_{\\mathsf{V}}+K_{\\mathsf{R}})$ and $G_{\\mathsf{V R H}}=1/2(G_{\\mathsf{V}}+G_{\\mathsf{R}})$ , known as the Voigt-Reuss-Hill average [49]. Using the values of bulk modulus $K$ and shear modulus $G$ , the Young’s modulus $E$ and Poisson’s ratio $\\nu$ can be obtained by $\\begin{array}{r}{E=\\frac{9K G^{-}}{3K+G}}\\end{array}$ and $\\begin{array}{r}{\\nu=\\frac{3K-2G}{2(3K+G)}}\\end{array}$ , respectively.  \n\nFor 2D materials, VASPKIT assumes the crystal plane in the xy plane. Then the relation between strain and stress can be written in the following form [40,50]  \n\n$$\n\\left(\\begin{array}{l}{\\sigma_{1}}\\\\ {\\sigma_{2}}\\\\ {\\sigma_{6}}\\end{array}\\right)=\\left(\\begin{array}{l l l}{\\mathsf{C}_{11}}&{\\mathsf{C}_{12}}&{\\mathsf{C}_{16}}\\\\ {\\mathsf{C}_{21}}&{\\mathsf{C}_{22}}&{\\mathsf{C}_{26}}\\\\ {\\mathsf{C}_{61}}&{\\mathsf{C}_{62}}&{\\mathsf{C}_{66}}\\end{array}\\right)\\cdot\\left(\\begin{array}{l}{\\varepsilon_{1}}\\\\ {\\varepsilon_{2}}\\\\ {\\varepsilon_{6}}\\end{array}\\right),\n$$  \n\nwhere ${{C}_{i j}}$ $(i,j=1,2,6)$ is the in-plane stiffness tensor. The strain tensor $\\epsilon$ in Eq. (9) is simplified as  \n\n$$\n\\epsilon^{2D}=\\left(\\begin{array}{l l l}{\\varepsilon_{1}}&{\\varepsilon_{6}/2}&{0}\\\\ {\\varepsilon_{6}/2}&{\\varepsilon_{2}}&{0}\\\\ {0}&{0}&{0}\\end{array}\\right).\n$$  \n\nThen the elastic strain energy per unit area based on the strainenergy method can be expressed as [51]  \n\n$$\n\\begin{array}{r}{\\Delta E\\left({S},\\{\\varepsilon_{i}\\}\\right)=\\displaystyle\\frac{S_{0}}{2}(C_{11}\\varepsilon_{1}^{2}+C_{22}\\varepsilon_{2}^{2}+2C_{12}\\varepsilon_{1}\\varepsilon_{2}}\\\\ {+2C_{16}\\varepsilon_{1}\\varepsilon_{6}+2C_{26}\\varepsilon_{2}\\varepsilon_{6}+C_{66}\\varepsilon_{6}^{2}),}\\end{array}\n$$  \n\nwhere $S_{0}$ is the equilibrium area of the system. Clearly, the ${\\mathsf{C}}_{i j}$ is equal to the second partial derivative of strain energy $\\Delta E$ with respect to strain $\\varepsilon$ , namely, $\\mathsf{C}_{i j}=(1/S_{0})(\\partial^{2}\\Delta E/\\partial\\varepsilon_{i}\\partial\\varepsilon_{j})$ . Therefore, the unit of elastic stiffness tensor for 2D materials is force per unit length $\\left(\\mathsf{N}/\\mathsf{m}\\right)$ . The classification of the different crystal system with the corresponding number of independent elastic constants and elastic stability conditions for 2D materials are summarized in Table 2. The details of strain modes and the derived elastic constants for each 2D crystal system based on the energy-strain approach are listed in Appendix B.  \n\nIn order to provide a benchmark for computational studies, we list the calculated second-order elastic constants for bulk and 2D prototype materials belonging to different crystal systems in Tables 3 and 4 respectively, together with other theoretical values [41,52–54] for comparison purposes. It is found that the results produced with different DFT codes are in good agreement with each other.  \n\n# 3.2. Equations of state  \n\nThermodynamic equations of state (EOS) for crystalline solids describe the relationships among the internal energy E, pressure $P$ , volume $V$ and temperature $T$ . It plays a crucial role in predicting the structural and thermodynamical properties of materials under high pressure and high temperature in condensed matter sciences [55], especially in extreme conditions such as earth or planetary interiors where the properties of materials are quite different from those found at ambient conditions [56]. Various EOS formulas have been proposed. One of the most widely used isothermal EOSs in solid-state physics is the Murnaghan EOS model assuming that the bulk modulus varies linearly with pressure [57]. The resulting energy–volume relationship is given as:  \n\n$$\nE(\\nu)=E_{0}+\\frac{B V_{0}}{(C+1)}\\left(\\frac{\\nu^{-C}-1}{C}+\\nu-1\\right),\n$$  \n\nTable 2 Classification of crystal systems and independent elastic constants for 2D materials [50]. In the last column, several prototype materials are shown.   \n\n\n<html><body><table><tr><td>Crystal system</td><td>Number of independent SOECs</td><td>Independent SOECs</td><td>Material prototypes</td></tr><tr><td>Oblique</td><td>6</td><td>C11,C12,C22,C16,C26,C66</td><td>－</td></tr><tr><td>Rectangle</td><td>4</td><td>C11, C12, C22, C66</td><td>Borophene</td></tr><tr><td>Square</td><td>3</td><td>C11,C12,C66</td><td>SnO</td></tr><tr><td>Hexagonal</td><td>2</td><td>C11, C12</td><td>Graphene, MoS2</td></tr></table></body></html>  \n\nTable 3 PBE-calculated elastic stiffness constants (in units of $\\mathbf{GPa}$ ) for $Z\\mathrm{r}0_{2}$ , $\\mathrm{TiS}_{2}$ , $\\mathrm{MgF}_{2}$ , $\\mathsf{C a M o O}_{4}$ , $\\alpha{\\mathrm{-}}{\\mathrm{Al}}_{2}{\\mathrm{O}}_{3}$ , $\\mathsf{C a M g}(\\mathsf{C O}_{3})_{2}$ , Ti and Diamond. For comparison purposes, the available theoretical values from the literature are also shown [41].   \n\n\n<html><body><table><tr><td rowspan=\"2\">Cij</td><td colspan=\"2\">Zr02</td><td colspan=\"2\">TiS2</td><td colspan=\"2\">MgF2</td><td colspan=\"2\">CaMo04</td><td colspan=\"2\">α-A03</td><td colspan=\"2\">CaMg(CO3)2</td><td colspan=\"2\">Ti</td><td colspan=\"2\">Diamond</td></tr><tr><td>Calc.</td><td>Ref.</td><td>Calc.</td><td>Ref.</td><td>Calc.</td><td>Ref.</td><td>Calc.</td><td>Ref.</td><td>Calc.</td><td>Ref.</td><td>Calc.</td><td>Ref.</td><td>Calc.</td><td>Ref.</td><td>Calc.</td><td>Ref.</td></tr><tr><td>C11</td><td>334</td><td>334</td><td>314</td><td>312</td><td>134</td><td>130</td><td>130</td><td>126</td><td>452</td><td>451</td><td>192</td><td>194</td><td>184</td><td>189</td><td>1051</td><td>1052</td></tr><tr><td>C12</td><td>155</td><td>151</td><td>29</td><td>28</td><td>80</td><td>78</td><td>53</td><td>58</td><td>149</td><td>151</td><td>64</td><td>67</td><td>83</td><td>85</td><td>127</td><td>125</td></tr><tr><td>C13</td><td>82</td><td>82</td><td>78</td><td>84</td><td>59</td><td>55</td><td>47</td><td>46</td><td>108</td><td>108</td><td>54</td><td>57</td><td>78</td><td>74</td><td></td><td></td></tr><tr><td>C14</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>20</td><td>21</td><td>17</td><td>18</td><td></td><td></td><td></td><td></td></tr><tr><td>C15</td><td>26</td><td>32</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>13</td><td>12</td><td></td><td></td><td></td><td></td></tr><tr><td>C16 C22</td><td>352</td><td>356</td><td>311</td><td>306</td><td>10</td><td>10</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>C23</td><td>146</td><td>142</td><td>25</td><td>21</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>C24 C25</td><td>5</td><td>2</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>C26</td><td>263</td><td>251</td><td>404</td><td>406</td><td>192</td><td>185</td><td>112</td><td>110</td><td>455</td><td>452</td><td>107</td><td>108</td><td>197</td><td>187</td><td></td><td></td></tr><tr><td>C33 C34</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>C35 C36</td><td>2</td><td>7</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>C45 C44</td><td>78</td><td>71</td><td>73</td><td>73</td><td>52</td><td>61</td><td>30</td><td>29</td><td>133</td><td>132</td><td>37</td><td>39</td><td>46</td><td>41</td><td>560</td><td>559</td></tr><tr><td>C46</td><td>15</td><td>15</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>C55</td><td>70</td><td>71</td><td>100</td><td>106</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>C56 C66</td><td>113</td><td>115</td><td>118</td><td>117</td><td>90</td><td>83</td><td>38</td><td>34</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table></body></html>  \n\nTable 4 PBE-calculated in-plane elastic stiffness constants (in units of $\\mathrm{{N/m}}{\\mathrm{{.}}}$ ). For comparison purposes, the available theoretical or experimental values from the previous literature are also shown.   \n\n\n<html><body><table><tr><td rowspan=\"2\">Systems</td><td colspan=\"2\">C11</td><td colspan=\"2\">C22</td><td colspan=\"2\">C12</td><td colspan=\"2\">C66</td></tr><tr><td>Our work</td><td>Literature</td><td>Our work</td><td>Literature</td><td>Our work</td><td>Literature</td><td>Our work</td><td>Literature</td></tr><tr><td>Graphene</td><td>349.1</td><td>358.1 [52]</td><td></td><td></td><td>60.3</td><td>60.4 [52]</td><td></td><td></td></tr><tr><td>MoS2</td><td>128.9</td><td>131.4 [53]</td><td></td><td></td><td>32.6</td><td>32.6 [53]</td><td></td><td></td></tr><tr><td>SnO</td><td>48.14</td><td></td><td></td><td></td><td>38.9</td><td></td><td>39.0</td><td></td></tr><tr><td>Phosphorene</td><td>104.4</td><td>105.2 [54]</td><td>34.0</td><td>26.2 [54]</td><td>21.6</td><td>18.4 [54]</td><td>27.4</td><td></td></tr></table></body></html>  \n\nwhere $\\begin{array}{r}{\\nu=\\frac{V}{V_{0}}}\\end{array}$ , $V_{0}$ and $E_{0}$ are the volume and energy at zero pressure respectively. The values of bulk modulus $K$ and its pressure derivative $K^{\\prime}$ can be further deduced in terms of the fitting parameters $B$ and C. The bulk modulus $K$ is a measure of the resistance of a solid material to compression. It is defined as the proportion of volumetric stress related to the volumetric strain for any material, namely,  \n\n$$\nK=-V\\left({\\frac{\\partial P}{\\partial V}}\\right)_{T}.\n$$  \n\nThe workflow of EOS utility is similar to that of the elastic constants presented in Fig. 5. In addition to the equilibrium volume and bulk modulus, pressure and energy as functions of volume are also provided in this utility. Very recently, Latimer et al. evaluated the quality of fit for the 8 widely-used EOS models listed in Table 5 across 87 elements and over 100 compounds [55]. They pointed out that it is hard to find a universal EOS model applicable to all types of solids and accurate over the whole range of pressure. Furthermore, their results reveal that the Birch-Euler [58], Tait [59], and Vinet [60] models give the best overall quality of fit to the calculated energy-volume curves among all the equations under examination. However, the inconsistencies among these investigated equations are not significant. As a benchmark test, the calculated energy and pressure of diamond as a function of volume using different EOS models are presented in Fig. 6. One can find that the agreement among these EOS fits is very satisfactory on the whole. The calculated bulk modulus ranges from 440 GPa to $442\\mathsf{G P a}$ , in good agreement with the experimental value of 443 GPa [61].  \n\n# 3.3. Band structure and density of states  \n\nThe band structure is one of the essential concepts in solid-state physics. It provides the electronic levels in crystal structures, which are characterized by two quantum numbers, the band index n and the Bloch vector $\\mathbf{k},$ along with high symmetry directions in the BZ. Besides the band structure, the density of states (DOS) is another quantity that is defined as the number of states per interval of energy at each energy level that is available to be occupied by electrons. A high DOS at a specific energy level means that there are many states available for occupation and zero DOS means that no state can be occupied at that energy level. DOS can be used to calculate the density of free charge carriers in semiconductors, the electronic contribution to the heat capacity in metals. Moreover, it also provides an indirect description of magnetism, chemical bonding, optical absorption spectrum, etc.  \n\nTable 5 The analytic formulae of energy-volume relation and bulk modulus $K$ for several widely used EOS models based on Table 1 of Ref. [55].   \n\n\n<html><body><table><tr><td>Model</td><td>Internal energy E</td><td>Bulk modulus K (v =1)</td></tr><tr><td>Birch (Euler)[58]</td><td>E=Eo+Bvo((v--1)²+(v-²-1)3)</td><td></td></tr><tr><td>Birch (Lagrange) [58]</td><td>E=Eo+BVoC-Bvo(C-2)(1-v3)²+c(1-v)+C</td><td></td></tr><tr><td>Mie-Gruneisen [62]</td><td>E=E+(</td><td>B-9</td></tr><tr><td>Murnaghan [57]</td><td>+-1) 3c(1-v E=Eo+(+)</td><td>B</td></tr><tr><td>Pack-Evans-James [63]</td><td>E=Eo+BV</td><td>B</td></tr><tr><td>Poirier-Tarantola [64]</td><td>E =Eo+BVo(In(v))2(3-C(In(V)))</td><td>6B</td></tr><tr><td>Tait [59]</td><td>E=Eo+B (v-1+(ec(-)1))</td><td>B</td></tr><tr><td>Vinet [60]</td><td>E=Eo+BV 1-(1+c(v-1))e-c(</td><td>B-9</td></tr></table></body></html>  \n\n![](images/293ee32d2be126cf78f099e2d8d16c77009cbd3be033f659b2ac5746b77765bf.jpg)  \nFig. 6. (Color online.) The equations of states of diamond using different EOS models as listed in Table 5.  \n\nIn addition to the conventional plain band structure, VASPKIT can also deal with the projected band which provides insight into the atomic orbital contributions in each state. As illustrated examples, the projected band structures and density of states (DOS) of BiClO $\\left(P4/n m m\\right)$ and graphene monolayers are depicted in Fig. 7. To illustrate the band dispersion anisotropy of 2D materials, the 3D global band structures of the highest valence and lowest conduction bands for MoTe2 $(P\\overline{{6}}m2)$ and BiIO are shown in Fig. 8.  \n\n# 3.4. Effective masses of carriers  \n\nGenerally, the band dispersions close to conduction or valence band extrema can be approximated as parabolic for the semiconductors with low carrier concentrations. Consequently, the analytical expression of effective masses of carriers (EMC) $m^{*}$ for electrons and holes (in units of electron mass $m_{0}$ ) is given by  \n\n$$\nm^{*}=\\hbar^{2}\\left[\\frac{\\partial^{2}E(k)}{\\partial^{2}k}\\right]^{-1},\n$$  \n\nwhere $E(k)$ are the energy dispersion relation functions described by band structures, and $\\hbar$ is the reduced Planck constant. Clearly, $m^{*}$ is inversely proportional to the curvature of the electronic dispersion in reciprocal space, implying that CB and VB edges with larger dispersions result in smaller effective masses. It is noteworthy that the above expression should not be used in non-parabolic band dispersion cases, for example, the linear dispersion in the band edges of graphene [65]. Similarly, the Fermi velocity represents the group velocity of electrons traveling in the material is defined as  \n\n$$\n\\nu_{F}=\\frac{1}{\\hbar}\\frac{\\partial E}{\\partial k}.\n$$  \n\nFig. 9 (a) illustrates the determination of effective masses by fitting the band dispersion with a second-order polynomial schematically. The effective masses of carriers are calculated using an ultrafine $k$ -mesh of density uniformly distributed inside a circle of radius $k$ -cutoff. Haastrup et al. pointed out that the inclusion of third-order terms stabilizes the fitting procedure and yields the effective masses that are less sensitive to the details of the employed $k$ -mesh [53]. Thus, a third-order polynomial is also adopted to fit the band energy curvature in the EMC utility. In Table 6 we show the calculated effective masses for several typical 2D and bulk semiconductors with available effective mass data, including Phosphorene [53], $\\mathsf{M o S}_{2}$ [53], GaAs [66] and Diamond [67]. Overall, the agreement is excellent. In addition, the EMC utility can also calculate the orientation-dependent effective masses of charge carriers. Examples of this functionality are shown in Fig. 9 (b)-(e). One can find that the calculated effective masses of two investigated systems show strong anisotropy, especially for the case of bulk Si.  \n\n# 3.5. Charge density and potential manipulation  \n\nFor spin-polarized systems, the charge density $\\rho({\\bf r})$ and magnetization (spin) density $m(\\mathbf{r})$ are defined as  \n\n$$\n\\begin{array}{r}{\\rho(\\mathbf{r})=\\rho_{\\uparrow}(\\mathbf{r})+\\rho_{\\downarrow}(\\mathbf{r})}\\\\ {m(\\mathbf{r})=\\rho_{\\uparrow}(\\mathbf{r})-\\rho_{\\downarrow}(\\mathbf{r})}\\end{array},\n$$  \n\nwhere $\\rho_{\\uparrow}({\\bf r})$ and $\\rho_{\\downarrow}(\\mathbf{r})$ are the spin-up and spin-down densities. Note that the $\\rho_{\\uparrow}({\\bf r})=\\rho_{\\downarrow}({\\bf r})$ in non-spin-polarized cases. The spin density $\\rho_{\\sigma}({\\bf r})$ is expressed as  \n\n$$\n\\rho_{\\sigma}\\left(\\mathbf{r}\\right)=\\sum_{o c c}\\varphi_{i\\sigma}^{*}\\left(\\mathbf{r}\\right)\\varphi_{i\\sigma}\\left(\\mathbf{r}\\right),\n$$  \n\nwhere $\\sigma$ and $i$ are the spin- and band-index, respectively, $\\varphi_{i\\sigma}(\\mathbf{r})$ is the normalized single-particle wave-function. occ means that summation is over all occupied states.  \n\nThe charge density difference $\\Delta\\rho({\\bf r})$ can track the charge transfer and gain information of the interaction between the two parts that constitute the system. The $\\Delta\\rho_{(}\\mathbf{r})$ can be obtained  \n\n$$\n\\Delta\\rho({\\bf r})=\\rho_{\\mathrm{AB}}({\\bf r})-\\rho_{\\mathrm{A}}({\\bf r})-\\rho_{\\mathrm{B}}({\\bf r}),\n$$  \n\n![](images/d6c9a08b04b1cf6f6c5a78119f16f3193dd86e5b38b4be81ae8b390c2627ee19.jpg)  \nFig. 7. (Color online.) Projected band structure (left panel) and density of states (right panel) of (a) BiClO $(P4/n m m^{\\cdot}$ ) and (b) graphene monolayers. The Fermi energy is set zero eV.  \n\n![](images/39b616578dc982be782e31a4dd43631a51ac0ec79f04b01ac0e7763594abefd4.jpg)  \nFig. 8. (Color online.) The global band structures of the highest valence and lowest conduction bands for (a) MoTe2 ( $P\\overline{{6}}m2)$ and (b) BiIO $\\left(P4/n m m\\right)$ monolayers. The Ferm energy is set to zero.  \n\n![](images/cd4660ec748a44e9224494c3cf222b1ff4e26ff5c72623e8a9caba486bb8a8c6.jpg)  \nFig. 9. (Color online.) (a) Schematic illustration of the determination of effective masses based on second-order polynomial fitting around the conduction and valence ban extrema. Orientation-dependent effective masses (in units of electron mass $m_{0}$ ) of (b, d) hole and (c, e) electron carriers for 2D BN monolayer (b, c) and bulk Si (d, e respectively.  \n\nTable 6 The calculated effective masses of electron $m_{e}$ and hole $m_{h}$ carriers (in units of the electron mass ${\\mathfrak{m}}_{0}$ ) for typical semiconductors using PBE approach. The masses are labeled by the band extremum and the direction of the hight symmetry line along which the mass is calculated using a simple parabolic line fit. The labels of high-symmetry points are adopted from the Ref. [34].   \n\n\n<html><body><table><tr><td rowspan=\"2\">Material</td><td rowspan=\"2\">Direction</td><td colspan=\"2\">Electron mass (me)</td><td colspan=\"2\">Hole mass (mh)</td></tr><tr><td>Our work</td><td>Literature</td><td>Our work</td><td>Literature</td></tr><tr><td>Phosphorene</td><td> → X (zig-zag)</td><td>1.23</td><td>1.24 [53]</td><td>7.21</td><td>6.56 [53]</td></tr><tr><td>Phosphorene</td><td> → Y (armchair)</td><td>0.19</td><td>0.14 [53]</td><td>0.17</td><td>0.13 [53]</td></tr><tr><td>MoS2 monolayer</td><td>K→</td><td>0.47</td><td>0.42 [53]</td><td>0.56</td><td>0.53[53]</td></tr><tr><td>GaAs bulk</td><td>→X</td><td>0.06</td><td>0.07 [66]</td><td>0.35</td><td>0.34 [66]</td></tr><tr><td>Diamond bulk</td><td>→X</td><td>0.32</td><td>0.29 [67]</td><td>0.27</td><td>0.36 [67]</td></tr></table></body></html>  \n\n![](images/22100ab9aab39ae13265e8ee2ef90b60b912a4ee5ac6da994d66979ef73c02f2.jpg)  \nFig. 10. (Color online.) Calculated (a) charge density difference, planar- (blue line) and macroscopic averages (red line) of (b) charge density difference, (c) electrostati potential of a GaAs/AlAs (100) heterojunction, and (d) electrostatic potential of a GaAs (110) slab. Ga atoms are shown in purple, As are blue, and Al are red.  \n\nwhere $\\rho_{\\mathsf{A}}(\\mathbf{r})$ , $\\rho_{\\mathrm{B}}({\\bf r})$ and ${\\rho}_{\\mathsf{A B}}(\\mathbf{r})$ are the charge density of reactants A and B, and product C. VASPKIT can extract charge-density, spindensity, electrostatic potential as well as the difference of these quantities, and save them in VESTA (.vasp) [15,16,68], XCrysDen (.xsf) [31], or Gaussian (.cube) formats [69].  \n\nFrom the three-dimensional electronic charge density and electrostatic potential one can get the average one-dimensional charge density $\\overline{{n}}(z)$ and electrostatic potential $\\overline{{V}}(z)$ by calculating the planar average function $({\\overline{{f}}})$ [70]:  \n\n$$\n\\overline{{f}}(z)=\\frac{1}{S}\\int V(\\mathbf{r})d x d y,\n$$  \n\nwhere S represents the area of a unit cell in the $x-y$ plane. Generally, this planar-averaged charge density and potential exhibit periodic oscillations along the $z$ axis due to the spatial distribution of the electrons and ionic cores. These oscillations can be removed using a macroscopic averaging procedure [70]:  \n\n$$\n\\overline{{\\overline{{f}}}}(z)=\\frac{1}{L}\\int\\displaylimits_{-L/2}^{L/2}\\overline{{f}}(z)d z,\n$$  \n\nwhere $L$ is the length of the period of oscillation along $z$ . By definition, this macroscopic average would produce a constant value in the bulk. It is expected to reach a plateau value in the bulk-like regions of each layer in the superlattice. As an example, Fig. 10 shows the calculated planar and macroscopic averages of charge density difference and electrostatic potential for a (100)-oriented GaAs/AlAs heterojunction and a (110)-oriented GaAs slab, respectively.  \n\n# 3.6. Fermi surface  \n\nFermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature [71]. It is defined to be the set of $k$ -points such that $E({\\bf k})=\\mu$ for any band index $n$ , where $\\mu$ is the Fermi energy. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice, and the occupation of electronic energy bands. The knowledge of the topology of the Fermi surface is vital for characterizing and predicting the thermal, electronic and magnetic properties. To calculate the Fermi surface, one first needs to use VASPKIT to determine the $k$ -mesh $N_{1}{\\times}N_{2}{\\times}N_{3}$ based on the specified $k$ -spacing value. The $k$ -spacing is defined as the smallest allowed spacing between the $k$ -points in BZ, that is, $N_{i}=\\operatorname*{max}\\left(1,\\left|\\mathbf{b}_{i}\\right|/k\\mathrm{spacing}\\right)$ , where $\\left|\\mathbf{b}_{i}\\right|$ is the length of the reciprocal lattice vector in the $i-t h$ direction. To reduce the computational cost, only the eigenvalues at the inequivalent $k$ -points in the irreducible Brillouin zone are calculated using VASP. Then these $k$ - points with the sum of the corresponding weight can be mapped to fill the entire BZ using symmetry operations without approximation during the post-processing. The resulting Fermi surface can be visualized using the XcrysDen [31] or FermiSurfser programs [72]. To illustrate the capabilities of this utility, the calculated Fermi surfaces of copper colored by the atomic orbital projected weights are shown in Fig. 11.  \n\n![](images/2090da2b89c5db2c34e47b736dfba9c388d8754f3ac6cb78aab9629706591005.jpg)  \nFig. 11. (Color online.) (a) Plain Fermi surface of Cu. Orbital-resolved Fermi surface of (b) Cu-s, (c) $\\mathtt{C u}{-}p$ and (d) Cu-d states respectively, visualized by the FermiSurfser ackage [72]. The color denotes the weight of the states.  \n\n# 3.7. Wave-function visualization  \n\nTo visualize wave functions, VASPKIT first reads the plane wave (PW) coefficients $\\psi_{n\\mathbf{k}}(\\mathbf{k})$ of the specified wave-vector $\\mathbf{k}$ -point and band-index $n$ from the WAVECAR file, and performs a fast Fourier transform (FFT) algorithm to convert the $\\psi_{n\\mathbf{k}}(\\mathbf{k})$ from the reciprocal space to the real space, as denoted by $\\psi_{n\\mathbf{k}}(\\mathbf{r})$ . The $\\psi_{n\\mathbf{k}}(\\mathbf{r})$ can thus be obtained  \n\n$$\n\\psi_{n\\mathbf{k}}(\\mathbf{r})=\\sum_{\\mathbf{G}}C_{n\\mathbf{k}}(\\mathbf{k+G})\\mathrm{e}^{\\mathrm{i}(\\mathbf{k+G})\\cdot\\mathbf{r}},\n$$  \n\nwhere $\\textbf{\\^{G}}$ is the reciprocal lattice vector, and $C_{n\\mathbf{k}}(\\mathbf{k}+\\mathbf{G})$ is the PW coefficient of the wave vector $\\mathbf{k}+\\mathbf{G}$ and band-index $n$ in reciprocal space. Examples of the calculated wave function plots in real space are shown in Fig. 12.  \n\n# 3.8. Band structure unfolding  \n\nThe electronic structures of materials are perturbed by structural defects, impurities, fluctuations of the chemical composition, etc. In DFT calculations, these defects and incommensurate structures are usually investigated by using SC models. Nevertheless, it is difficult to compare the SC band structure directly with the PC band structure due to the folding of the bands into the smaller SC Brillouin zone (SBZ). Popescu and Zunger proposed the effective band structures (EBS) method which can unfold the SC band structures into the corresponding PC Brillouin zone (pbz) [73,74]. Such a delicate technique greatly simplifies the analysis of the results and enables direct comparisons with electronic structures of pristine materials.  \n\n![](images/84c04eadd363be7125a206a84c68bc6c5d95d7e466b13f508af178b2a457c90a.jpg)  \nFig. 12. (Color online.) Calculated isosurfaces of wave functions in real space for (a) CO molecule, (b) VBM and (c) CBM for graphene respectively, visualized by the VESTA package [68].  \n\nAs aforementioned, the SC and PC lattice vectors satisfy $\\pmb{A}=$ $\\mathbf{M}\\cdot\\mathbf{a}$ where A and a are the lattice vectors of SC and PC. The elements of transformation matrix $\\mathbf{M}$ are integers $\\left(m_{i j}\\in\\mathbb{Z}\\right)$ when building SC from PC. In the band unfolding utility, the transformation matrix is not required to be diagonal. In other words, the SC and PC lattice vectors do not need to be collinear. Following a general convention, capital and lower case letters indicate the quantities in the SC and PC, respectively, unless otherwise stated. A similar relation holds in reciprocal space:  \n\n$$\n\\mathbf{B}=\\left(\\mathbf{M}^{-1}\\right)^{T}\\cdot\\mathbf{b},\n$$  \n\nwhere $\\textbf{B}$ and $\\mathbf{b}$ are the reciprocal lattice vectors of the SC and PC, respectively. The reciprocal lattice vectors ${\\bf g}_{n}({\\bf G}_{m})$ in the pbz (SBZ) are expressed as  \n\n$$\n\\begin{array}{r l}&{\\mathbf{g}_{n}=\\sum_{i}n_{i}\\mathbf{b}_{i},\\quad n_{i}\\in\\mathbb{Z}}\\\\ &{\\mathbf{G}_{m}=\\sum_{i}m_{i}\\mathbf{B}_{i},\\quad m_{i}\\in\\mathbb{Z}}\\end{array},\n$$  \n\nwhere $\\{\\mathbf{g}_{n}\\}\\subset\\{\\mathbf{G}_{m}\\}$ , i.e., every reciprocal lattice vector of the pbz is also one of the SBZ.  \n\nFor a given $\\mathbf{k}$ in pbz, there is a $\\textbf{K}$ in the SBZ, and the two vectors are related by a reciprocal lattice vector G in the SBZ:  \n\n$$\n\\mathbf{k}=\\mathbf{K}+\\mathbf{G}_{i},i=1,\\ldots,N_{\\mathbf{K}},\n$$  \n\nwhere $N_{\\mathbf{K}}$ is the determinant $|M|$ that determines the multiplicity of the SC. When choosing plane waves as the basis functions, the projection of the SC eigenstates $|\\psi_{m\\bf{K}}^{\\mathrm{SC}}\\rangle$ on the PC eigenstates $|\\psi_{n\\bf{k}}^{\\mathrm{PC}}\\rangle$ is given by the spectra weight $P_{{\\bf K}m}$ [73,74]:  \n\n![](images/1629f4cd4044b29f280f1db994081a337201ab9f83855f43b9204d5ce8e505d0.jpg)  \nFig. 13. (Color online.) (a) Workflow of the algorithm used in the band unfolding utility. (b) Band structure of $3\\times3$ graphene SC along with the high-symmetry directions in pbz. The blue lines and red makers represent the band structure before and after applying the unfolding technique. The Fermi energy is set to zero.  \n\n$$\n\\begin{array}{l}{{\\displaystyle P_{{\\bf K}m}\\left({\\bf k}_{i}\\right)=\\sum_{n}\\left|\\left\\langle\\psi_{m\\bf K}^{\\mathrm{SC}}\\mid\\psi_{n\\bf k}^{\\mathrm{PC}}\\right\\rangle\\right|^{2}=\\sum_{\\bf g}|C_{m\\bf K}\\left({\\bf g}+{\\bf k}_{i}-{\\bf K}\\right)|^{2}}\\ ~}\\\\ {{\\displaystyle~=\\sum_{\\bf g}|C_{m\\bf K}\\left({\\bf g}+{\\bf G}_{i}\\right)|^{2}},}\\end{array}\n$$  \n\nwhere $m$ and $n$ stand for band indices at vectors $\\textbf{K}$ and $\\mathbf{k}_{i}$ in the reciprocal space of the SC and $\\mathsf{P C}$ , respectively. $C_{m\\mathbf{K}}$ is the PW coefficients given by Eq. (29) that span the eigenstates of the SC. This implies that the required information about the PC is the reciprocal lattice vectors of the primitive cell $\\textbf{g}$ only, and the knowledge of the PC eigenstates is not necessary. All the filtered $C_{m\\mathbf{K}}\\left(\\mathbf{g}+\\mathbf{G}_{j}\\right)$ coefficients only contribute to the spectral function. The quantity $P_{{\\bf K}m}$ represents the amount of Bloch character $\\mathbf{k}_{i}$ preserved in $|\\psi_{n\\bf{k}}^{\\mathrm{PC}}\\rangle$ at the same energy $E_{n}=E_{m}$ .  \n\nThe workflow of band unfolding utility is schematically shown in Fig. 13(a). Three input files including the information of SC structure, the transformation matrix M, and the selected $\\mathbf{k}_{i}$ vectors in pbz are required to provide respectively. To compare the unfolded band structure of SC with the band structure of PC directly, the $\\mathbf{k}_{i}$ vectors are generally sampled along with the highsymmetry directions in pbz and then translated in the SC reciprocal space by the transformation as described in Eq. (33)  \n\n$$\n\\mathbf{K}=\\mathbf{M}\\cdot\\mathbf{k}_{i},\n$$  \n\nwhere $\\textbf{K}$ and $\\mathbf{k}_{i}$ are the scaled coordinates with respect to the SC and PC reciprocal basis vectors, respectively. After reading PW coefficients and eigenvalue of each state from the WAVECAR obtained by performing VASP calculation, the intricate supercell states can be unfolded back into the larger pbz by applying the unfolding technique via Eq. (32). Finally, the unfolded band can be visualized with the maker size proportional to the spectral weight $P_{{\\bf K}m}$ . From Fig. 13(b), it is clear that folding the bands into the smaller SBZ gives rise to quite a sophisticated band structure. In contrast, one can gain more straightforward analysis once the supercell states are unfolded into the pbz despite the equivalence between the PC and the SC descriptions of a perfectly periodic material.  \n\nIt is well known that intrinsic defects (vacancies, self-interstitials, and antisites) and unintentional impurities have important effects on the properties of semiconductors. As a typical case, we take the $4\\times3~\\mathsf{M o S}_{2}$ monolayer SC with one neutral sulfur vacancy to demonstrate the role of intrinsic defect on the electronic structure of the pristine host. The calculated effective band structures of pristine and defective $\\mathsf{M o S}_{2}$ supercells in Fig. 14 (a) and (b), respectively. By comparing these two, one can find two nearly degenerated defect states existing in the fundamental band gap of $\\mathsf{M o S}_{2}$ . The orbital-resolved unfold band structures as shown in Figs. (c) and (d) further demonstrate that these two defect states are mainly derived from Mo- $\\cdot d$ and $S{-}p$ states respectively. Furthermore, the Bloch character close to the valence band edge is perturbed due to the presence of the sulfur vacancy.  \n\n# 3.9. Linear optical properties  \n\nThe linear optical properties of semiconductors can be obtained from the frequency-dependent complex dielectric function $\\varepsilon(\\omega)$  \n\n$$\n\\varepsilon(\\omega)=\\varepsilon_{1}(\\omega)+i\\varepsilon_{2}(\\omega),\n$$  \n\nwhere $\\varepsilon_{1}(\\omega)$ and $\\varepsilon_{2}(\\omega)$ are the real and imaginary parts of the dielectric function, and $\\omega$ is the photon frequency. Within the oneelectron picture, the imaginary part of the dielectric function $\\varepsilon_{2}(\\omega)$ is obtained from the following equation [75]:  \n\n$$\n\\begin{array}{l}{{\\displaystyle\\varepsilon_{2}(\\omega)=\\frac{4\\pi^{2}e^{2}}{\\Omega}\\operatorname*{lim}_{q\\rightarrow0}\\frac{1}{q^{2}}}\\ ~}\\\\ {{\\displaystyle~\\times\\sum_{c,\\nu,\\mathbf{k}}2w_{\\mathbf{k}}\\delta\\left(E_{c}-E_{\\nu}-\\omega\\right)\\left\\vert\\left\\langle c\\middle\\vert\\mathbf{e}\\cdot\\mathbf{q}\\right\\vert\\nu\\right\\rangle\\right\\vert^{2}},}\\end{array}\n$$  \n\nwhere $\\langle c|\\mathbf{e}\\cdot\\mathbf{q}|\\nu\\rangle$ is the integrated optical transitions from the valence states $(\\nu)$ to the conduction states (c), e is the polarization direction of the photon and $\\mathbf{q}$ is the electron momentum operator. The integration over $\\mathbf{k}$ is performed by summation over special $k$ - points with a corresponding weighting factor $w_{k}$ . The real part of the dielectric function $\\varepsilon_{1}(\\omega)$ can be determined from the KramersKronig relation given by  \n\n$$\n\\varepsilon_{1}(\\omega)=1+\\frac{2}{\\pi}P\\intop_{0}^{\\infty}\\frac{\\varepsilon_{2}\\left(\\omega^{\\prime}\\right)\\omega^{\\prime}}{\\omega^{\\prime2}-\\omega^{2}+i\\eta}d\\omega^{\\prime},\n$$  \n\nwhere $P$ denotes the principle value and $\\eta$ is the complex shift parameter. The frequency-dependent linear optical spectra, e.g., refractive index $n(\\omega)$ , extinction coefficient $\\kappa(\\omega)$ , absorption coefficient $\\alpha(\\omega)$ , energy-loss function $L(\\omega)$ , and reflectivity $R(\\omega)$ can be calculated from the real $\\varepsilon_{1}(\\omega)$ and the imaginary $\\varepsilon_{2}(\\omega)$ parts [76]:  \n\n$$\n\\begin{array}{r l}&{\\pi(\\omega)=\\left(\\frac{\\sqrt{\\varepsilon_{1}^{2}+k_{2}^{2}}+\\varepsilon_{1}}{2}\\right)^{\\frac{1}{2}},}\\\\ &{k(\\omega)=\\left(\\frac{\\sqrt{\\varepsilon_{1}^{2}+k_{2}^{2}}-\\varepsilon_{1}}{2}\\right)^{\\frac{1}{2}},}\\\\ &{\\alpha(\\omega)=\\frac{\\sqrt{2}\\omega}{\\varepsilon}\\left(\\sqrt{\\varepsilon_{1}^{2}+k_{2}^{2}}-\\varepsilon_{1}\\right)^{\\frac{1}{2}},}\\\\ &{L(\\omega)=\\ln\\left(\\frac{-1}{\\varepsilon(\\omega)}\\right)=\\frac{\\varepsilon_{2}}{k_{2}^{2}+k_{2}^{2}},}\\\\ &{R(\\omega)=\\frac{(\\pi-1)^{2}+k_{2}^{2}}{(4\\pi+1)^{2}+k_{2}^{2}}.}\\end{array}\n$$  \n\nIn Fig. 15 we present the linear optical spectra of silicon as determined by solving the Bethe-Salpeter Equation (BSE) on the top of ${\\sf G}_{0}{\\sf W}_{0}$ approximation. One can find that the absorption coefficient becomes significant only after $3.0~\\mathrm{eV}$ . This is because silicon has an indirect band gap, resulting in a low absorption coefficient in the visible region. Since the GW approximation includes the exchange and correlation effects in a self-energy term dependent on the one-particle Green’s function G and the dynamically screened Coulomb interaction W, it can correct the one electron eigenvalues obtained from DFT within a many-body quasiparticle framework [77,78]. Furthermore, the errors originated from the lack of ladder diagrams in determining W can be included through the solution of the Bethe-Salpeter equation (BSE) [79]. It could be expected that the GW-BSE calculated optical properties yield better agreement with the experiment. In the single-shot ${\\sf G}_{0}{\\sf W}_{0}$ approximation, the one-electron Green’s function G is self-consistently updated within a single iteration, while the screened Coulomb interaction W is fixed at its initial value.  \n\n![](images/5b9e3e370e86b0ee612e1eab7843d348c3a84519231d8a58af2c64c7e3b3bc68.jpg)  \nFig. 14. (Color online.) Effective band structure of $4\\times3~\\mathrm{MoS}_{2}$ SC unfolded into the PC Brillouin zone through Eq. (32) (a) without and (b) with a S vacancy. Orbital-resolved effective band structure of (c) $S{-}p$ and (d) Mo-d states in the defective SC. The Fermi energy is set to zero.  \n\n![](images/5c1dda4e7503baab94641cbbe6be4695ff5a0168f6185cd2f64b6c42a9c8242d.jpg)  \nFig. 15. (Color online.) ${\\sf G}_{0}{\\sf W}_{0}$ -BSE calculated (a) absorption coefficient, (b) refractive index, (c) reflectivity and (d) extinction coefficient of silicon. Vertical color lines highlight the visible light region.  \n\nIt should be pointed out that the Eqs. (37)-(41) are not welldefined for low-dimensional materials since the dielectric function is not straightforward and depends on the thickness of the vacuum layer when the low-dimensional systems are simulated using a periodic stack of layers with sufficiently large interlayer distance $L$ to avoid artificial interactions between the periodic images of the 2D sheet crystals in the standard DFT calculations [80,81]. To avoid the thickness problem, the optical conductivity $\\sigma_{2D}(\\omega)$ is used to characterize the optical properties of 2D sheets. Based on the Maxwell equation, the 3D optical conductivity can be expressed as [82]  \n\n$$\n\\sigma_{3D}(\\omega)=i[1-\\varepsilon(\\omega)]\\varepsilon_{0}\\omega,\n$$  \n\nwhere $\\varepsilon(\\omega)$ is the frequency-dependent complex dielectric function given in (34), $\\scriptstyle{\\varepsilon_{0}}$ is the permittivity of vacuum and $\\omega$ is the frequency of the incident wave. The in-plane 2D optical conductivity is directly related to the corresponding $\\sigma_{3D}(\\omega)$ component through the equation [82,83]  \n\n$$\n\\sigma_{2D}(\\omega)=L\\sigma_{3D}(\\omega),\n$$  \n\nwhere $L$ is the slab thickness in the simulation cell. The normalized reflectance $R(\\omega)$ , transmittance $T(\\omega)$ and absorbance $A(\\omega)$ are independent of the light polarization for a freestanding 2D crystal sheet when normal incidence is assumed [82,83],  \n\n$$\n\\begin{array}{r}{{\\cal R}=\\left|\\displaystyle\\frac{\\tilde{\\sigma}/2}{1+\\tilde{\\sigma}/2}\\right|^{2},}\\\\ {{\\cal T}=\\displaystyle\\frac{1}{|1+\\tilde{\\sigma}/2|^{2}},}\\\\ {{\\cal A}=\\displaystyle\\frac{\\mathrm{Re}\\tilde{\\sigma}}{|1+\\tilde{\\sigma}/2|^{2}},}\\end{array}\n$$  \n\nwhere $\\tilde{\\sigma}(\\omega)=\\sigma_{2\\mathrm{D}}(\\omega)/\\varepsilon_{0}c$ is the normalized conductivity ( $c$ is the speed of light). Since the interband contribution is only considered, the formula (44) is valid for semiconducting and insulating 2D crystals with a restriction of $A+T+R=1$ . Generally, the reflectance of 2D sheets is extremely small, and the absorbance can be approximated by the real part of $\\tilde{\\sigma}(\\omega)$ , namely, $A(\\omega)=\\mathrm{Re}\\sigma_{2D}(\\omega)/\\varepsilon_{0}c$ . To demonstrate this functionality, the PBEcalculated linear optical spectra of freestanding graphene and phosphorene monolayers are displayed in Fig. 16. Our results are in good agreement with the available theoretical optical curves [82–84].  \n\n# 3.10. Joint density of states  \n\nFor a semiconductor, the optical absorption in direct band-toband transitions is proportional to [85]  \n\n$$\n\\frac{2\\pi}{\\hbar}\\intop_{\\mathbb{B Z}}\\left|\\left\\langle v\\left|\\mathcal{H^{\\prime}}\\right|c\\right\\rangle\\right|^{2}\\frac{2}{(2\\pi)^{3}}\\delta\\left(E_{c}(\\mathbf{k})-E_{\\nu}(\\mathbf{k})-\\hbar\\omega\\right)d^{3}k,\n$$  \n\nwhere $\\mathcal{H}^{\\prime}$ is the perturbation associated with the light wave and $\\left\\langle\\nu\\left|\\mathcal{H}^{\\prime}\\right|c\\right\\rangle$ is the transition matrix from states in the valence band (V\u0018B) t\u0018o states in the conduction band (CB); $\\delta$ is the Dirac delta function which switches on this contribution when a transition occurs from one state to another, i.e., $E_{c}(\\mathbf{k})-E_{\\nu}(\\mathbf{k})=\\hbar\\omega$ . Factor 2 stems from the spin degeneracy. The integration is over the entire BZ. The matrix elements vary little within the BZ. Therefore, we can pull these out in front of the integral and obtain  \n\n![](images/b525400131ce45fa876b467ea1a984f9663788b1775d9582afa272321393bd94.jpg)  \nFig. 16. (Color online.) Real (blue line) and imaginary (red line) parts of frequencydependent optical conductivity $\\sigma_{2D}(\\omega)$ for (a) graphene and (c) phosphorene [in units of $\\sigma_{0}=e^{2}/(4\\hbar)]$ . Absorption spectra $A(\\omega)$ of (b) graphene and (d) phosphorene. The incident light polarized along the armchair and zigzag directions of phosphorene are presented by solid and dashed lines respectively. Vertical color lines highlight the visible light region.  \n\n$$\n\\frac{2\\pi}{\\Omega\\hbar}\\left|\\left\\langle v\\left|\\mathcal{H^{\\prime}}\\right|c\\right\\rangle\\right|^{2}\\cdot\\int\\frac{2\\Omega}{(2\\pi)^{3}}\\delta\\left(E_{c}(\\mathbf{k})-E_{\\nu}(\\mathbf{k})-\\hbar\\omega\\right)\\mathrm{d}^{3}k,\n$$  \n\nwhere $\\Omega$ is the volume of the lattice cell, and the factor $\\Omega/{(2\\pi)}^{3}$ normalizes the $\\mathbf{k}$ vector density within the Brillouin zone. The second term is the joint density of states (JDOS). After sum over all states within the first Brillouin zone and all possible transitions initiated by photons with a certain energy ¯hω between valence and conduction bands, we obtain  \n\n$$\n\\begin{array}{r}{j(\\omega)=\\displaystyle\\sum_{\\nu,c}\\frac{\\Omega}{4\\pi^{3}}\\int\\delta(E_{c}(\\mathbf{k})-E_{\\nu}(\\mathbf{k})-\\hbar\\omega)d^{3}k}\\\\ {=2\\displaystyle\\sum_{\\nu,c,\\mathbf{k}}w_{\\mathbf{k}}\\delta\\left(E_{c}(\\mathbf{k})-E_{\\nu}(\\mathbf{k})-\\hbar\\omega\\right),}\\end{array}\n$$  \n\nwhere $c$ and $\\nu$ belong respectively to the valence and conduction bands, $E({\\bf k})$ are the eigenvalues of the Hamiltonian, and $w_{\\mathbf{k}}$ are weighting factors. The Dirac Delta function in Eq. (47) can be numerically approximated using a normalized Gaussian function:  \n\n$$\nG(\\omega)=\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\left(E_{\\mathbf{k},n^{\\prime}}-E_{\\mathbf{k},n}-\\hbar\\omega\\right)^{2}/2\\sigma^{2}},\n$$  \n\nwhere $\\sigma$ is the broadening parameter. To demonstrate this functionality, we show the calculated total and partial JDOS for $\\mathrm{CH_{3}N H_{3}P b I_{3}}$ and Si in Fig. 17. The calculated JDOS for $\\mathrm{CH_{3}N H_{3}P b I_{3}}$ is in excellent agreement with previous data [86]. It should be pointed that the total JDOS includes all possible interband transitions from all the valence to all the conduction bands according to Eq. (47); while the partial JDOS considers only the interband transitions from the highest VB to the lowest CB.  \n\n# 3.11. Transition dipole moment  \n\nThe transition dipole moment (TDM) or dipole transition matrix elements ${\\mathsf{P}}_{a\\to b}$ , is the electric dipole moment associated with a transition between the initial state $a$ and the final state $b$ [87]:  \n\n![](images/177710a12addcfd5bf64220d51920d1fbccfd5578798df4358a7d1fa05abd80f.jpg)  \nFig. 17. (Color online.) Calculated joint density of states for (a) $\\mathrm{CH_{3}N H_{3}P b I_{3}}$ and (b) Si. Blue and purple lines represent the total and partial joint density of states respectively. The visible light region is highlighted by vertical color lines.  \n\n![](images/52f1899293984347e52314c3b80728313b9999aed78242fbf5dc849fbe8e1963.jpg)  \nFig. 18. (Color online.) Calculated band structure (top panel) and transition dipole moment (bottom panel) for (a) $\\mathsf{C s}_{2}\\mathsf{A g I n C l}_{6}$ and (b) ${\\mathrm{C}}s_{2}{\\mathrm{InBiCl}}_{6}$ .  \n\n$$\n\\mathrm{P}_{a\\to b}=\\langle\\psi_{b}|{\\bf r}|\\psi_{a}\\rangle={\\frac{i\\hbar}{(E_{b}-E_{a})m}}\\langle\\psi_{b}|{\\bf p}|\\psi_{a}\\rangle,\n$$  \n\nwhere $\\psi_{a}$ and $\\psi_{b}$ are energy eigenstates with energy $E_{a}$ and $E_{b};m$ is the mass of the electron. In general the TDM is a complex vector that includes the phase factors associated with the two states. Its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization, while the sum of the squares of TDM, $P^{2}$ , gives the transition probabilities between the two states. In Fig. 18 we provide some specific examples to illustrate its use. It is seen that the calculated TDM amplitude is zero for transition between the CBM and VBM at the $\\Gamma$ point in $\\mathsf{C s}_{2}\\mathsf{A g l n C l}_{6}$ , implying no optical absorption between these two states. On the other hand, the excellent optical absorption between CBM and VBM is predicted in ${\\mathrm{C}}s_{2}{\\mathrm{InBiCl}}_{6}$ when Bi substitues Ag atom. These findings are in good agreement with previous theoretical results [88].  \n\n# 3.12. $d$ -Band center  \n\nThe $d$ -band center model of Hammer and Nørskov is widely used in understanding and predicting catalytic activity on transition metal surfaces. The main idea underlying the theory is that the binding energy of an adsorbate to a metal surface is largely dependent on the electronic structure of the surface itself. In this model, the $d$ -states band participating in the interaction is approximated by the center of the $d$ -band $\\textstyle{\\varepsilon_{d}}$ [89]:  \n\n$$\n\\varepsilon_{\\mathrm{d}}=\\frac{\\int_{-\\infty}^{\\infty}n_{\\mathrm{d}}(\\varepsilon)\\varepsilon d\\varepsilon}{\\int_{-\\infty}^{\\infty}n_{\\mathrm{d}}(\\varepsilon)d\\varepsilon},\n$$  \n\nwhere $n_{\\mathrm{d}}$ and $\\varepsilon$ are projected-DOS and energy of transition metal $d$ -states. According to this model, the adsorption energy on the transition metal surface correlates with the upward shift of the dband center with respect to the Fermi energy. A more vital upward shift indicates the possibility of forming a more significant number of empty anti-bonding states, leading to a stronger binding energy [90,91,89]. It may be worth mentioning here that the $d$ -band center position linearly upshifts with increasing the number of empty states above the Fermi level. Therefore, one can specify the integral upper limit in Eq. (50) to calculate $d$ -band center by using VASPKIT.  \n\n# 3.13. Thermo energy correction  \n\nGibbs free energy plays a crucial role in catalysis reaction. The equations used to calculate thermochemical data for gases in VASPKIT are equivalent to those in the Gaussian program [92,93]. The Gibbs free energy $G$ is given by  \n\n$$\nG=H-T S,\n$$  \n\nwhere $H,\\ T$ and S represent enthalpy, temperature and entropy, respectively. The enthalpy $H$ in Eq. (51) can be written as $H=$ $U+P V$ . Both internal thermal energy $U$ and entropy S have included the contributions from translational, electronic, rotational and vibrational motions and zero-point energy (ZPE) of molecules. Moreover, to calculate correctly when the number of moles (labeled $N$ ) of a gas changes during the course of a reaction, the Gibbs free energy has also included $\\Delta P V=\\Delta N R T$ , where $R$ is the molar gas constant. It is worth mentioning that only the modes with real vibrational frequencies are considered and the model with imaginary one are ignored during the calculations of the vibration contributions. Specifically, for linear (non-linear) molecules containing $n$ atoms, the degree of vibrational freedom is $3n-5$ (3n - 6). VASPKIT neglects the smallest 5 (6) frequencies. We take oxygen molecular as an example to calculate its free energy at 298.15 K using the corrected algorithm mentioned above. It is found that the calculated correction to free energy of $0_{2}$ molecule is -0.4467 eV, which is very close to the experimental data of -0.4468 eV at $298.15\\mathrm{~K~}$ and normal atmospheric pressure [94]. Moreover, the thermal correction result from VASPKIT is exactly the same with that from Gaussian program by setting the same molecular structure and frequencies.  \n\nUnlike gas molecules, when the adsorbed molecules form chemical bonds with the substrate, their translational and rotational freedom will be constrained. Consequently, the contributions from translation and rotation to entropy and enthalpy are significantly reduced turn into vibrational modes. One standard method is to attribute the translational or rotational part of the contribution to vibration, that is, the $3n$ vibrations of the surface-adsorbing molecules (except the imaginary frequency) are all used to calculate the correction of the thermo energy [89]. Considering that a minor vibration mode makes a significant contribution to entropy. A minor vibration frequency will likely lead to abnormal entropy and free energy correction. Thus, VASPKIT allows specifying a threshold value which defines the lower limit of frequencies. For example, if a threshold value of $50~\\mathrm{cm}^{-1}$ is adopted, implying that the frequencies below $50~\\mathrm{cm}^{-1}$ are approximately equal to 50 $\\mathsf{c m}^{-1}$ during the calculations of the vibration contributions to the adsorbed molecular free energy correction.  \n\n# 3.14. Molecular dynamics  \n\nMolecular dynamics (MD) describes how the atoms in a material move as a function of time, and helps us to understand the structural, dynamical and thermodynamical properties of complex systems. It has been successfully applied to gases, liquids, and ordered and disordered solids. In addition to the equation of state, mean square displacement (MSD), velocity auto-correlation function (VACF), phonon vibrational density of states (VDOS) and pair correlation function (PCF) are the most critical quantities enabling us to determine various properties of interest in MD simulations.  \n\nThe MSD is a measure of the deviation of the position of a particle with respect to a reference position over time. It can help to determine whether the ion is freely diffusing, transported, or bound. It is defined as  \n\n$$\nM S D(m)=\\frac{1}{N_{\\mathrm{particles}}}\\sum_{i=1}^{N_{\\mathrm{particles}}}\\frac{1}{N-m}\\sum_{k=0}^{N-m-1}\\left({\\bf r}_{i}(k+m)-{\\bf r}_{i}(k)\\right)^{2},\n$$  \n\nwhere ${\\bf r}_{i}(t)$ is the position of atom i after $t$ time of simulation. $N.$ particles and $N$ are the total number of atoms and total frames respectively. According to this definition, the MSD is averaged over all windows of length $m$ and over all selected particles. An alternative method that can efficiently calculate MSD was proposed based on the Fast Fourier Transform (FFT) algorithm in Refs. [95,96] and references therein. If the system stays in the solid state, the MSD oscillates around a constant value. This means that all the atoms are confined to specific positions. However, for a liquid, atoms will move indefinitely and the MSD continues to increase linearly with time. This implies that sudden changes in the MSD with time indicate melting, solidification, phase transition, and so on. In addition, the calculation of MSD is the standard way to estimate the parameters of movement, such as the diffusion coefficients from MD simulations.  \n\nThe VACF is another way of checking the movement type of atoms. It is a value that basically tells until when the particle remembers its previous movements. Like the MSD, it is a timeaveraged value, defined over a delay domain. The normalized VACF is defined as  \n\n$$\nc(t)=\\frac{\\sum_{i=1}^{N}\\left\\langle\\mathbf{v}_{i}(t)\\cdot\\mathbf{v}_{i}(0)\\right\\rangle}{\\sum_{i=1}^{N}\\left(\\mathbf{v}_{i}(0)\\right)^{2}},\n$$  \n\nwhere ${\\bf v}_{i}(t)$ is the velocity of the i-th atom at time t. The bracket represents a time average over the history of the particle, i.e., all the values of t. The total velocity autocorrelation function C(t) is defined as the mass-weighted sum of the atom velocity autocorrelation functions [97]  \n\n$$\nC(t)=\\sum_{j=1}^{N}m_{j}c_{j}(t),\n$$  \n\nwhere $c_{j}(t)$ is the velocity autocorrelation of atom $j$ . The optical and thermodynamical properties of materials depend on VDOS which can be obtained from the Fourier transform of the VACF under the harmonic approximation [98,97],  \n\n$$\nf(\\omega)=\\mathcal{F}[\\gamma(t)]=\\frac{1}{k_{B}T}\\intop_{-\\infty}^{\\infty}\\gamma(t)e^{-i\\omega t}d t,\n$$  \n\nwhere $\\omega$ is the vibrational frequency, $\\mathcal{F}$ is the Fourier transform operator, $k_{B}$ is the Boltzmann constant and $T$ is the absolute temperature.  \n\n![](images/9cef053a8d746a68dd0aebbbf438425c592f04a1b7b6057deb345cec1797e460.jpg)  \nFig. 19. (Color online.) Calculated (a) MSD, (b) VACF, (c) VDOS and (d) PCF of liquid water at $400~\\mathrm{K}$ obtained from MD simulations.  \n\nThe PCF $\\mathbf{g}(\\boldsymbol{r})$ describes how atoms are distributed in a thin shell at a radius $r$ from an arbitrary atom in the material. It is useful not only for studying the details of the system but also to obtain accurate values for the macroscopic quantities such as the potential energy and pressure. This quantity can be obtained by summing the number of atoms found at a given distance in all directions from a particular atom:  \n\n$$\ng(r)=\\frac{d N/N}{d V/V}=\\frac{1}{4\\pi r^{2}}\\frac{1}{N\\rho}\\sum_{i=1}^{N}\\sum_{j\\neq i}^{N}\\left<\\delta\\left(r-\\left|\\mathbf{r}_{i}-\\mathbf{r}_{j}\\right|\\right)\\right>,\n$$  \n\nwhere $r$ is the radial distance. $\\rho$ is the average density of the entire material. The normalization via the density ensures that for large distances the radial distribution approaches unity. The partial radial distribution between two elements is calculated as  \n\n$$\ng_{A B}(r)={\\frac{1}{4\\pi r^{2}}}{\\frac{N}{\\rho N_{A}N_{B}}}\\sum_{i\\in A}\\sum_{j\\in B,j\\neq i}^{N}\\left\\langle\\delta\\left(r-\\left|\\mathbf{r}_{i}-\\mathbf{r}_{j}\\right|\\right)\\right\\rangle.\n$$  \n\nAs an illustrated example, Fig. 19 shows the PBE-calculated MSD, VACF, VDOS and PCF for liquid water at $400\\mathrm{~K~}$ processed by the MD utility. Overall, our result is in good agreement with available experimental and theoretical results [99,100].  \n\n# 4. High-throughput capabilities  \n\nVASPKIT also provides a light-weight high-throughput interface. As such it can advantageously be part of bash scripts, taking full advantage of bash capabilities (variables, loops, conditions, etc.) to batch performing pre- and post-processing. An easy-to-follow user manual is available at https://vaspkit com/tutorials html. The syntax is designed as simply as possible. For instance, to generate KPOINTS files in a series of subfolders, the syntax is  \n\nfor d i r in $^*$   \ndo echo \\$dir cd \\$RootPath / \\$dir vaspkit task 102 kpr 0.04   \ndone  \n\n# 5. Limitations and future capabilities  \n\nCurrently, VASPKIT only deals with the raw data calculated using the VASP code. This program will be extended to support other  \n\nab-initio packages in the future version. In addition, the data visualization and plotting utility based on Python and Matplotlib will be also implemented.  \n\n# 6. Summary  \n\nIn summary, VASPKIT is a user-friendly toolkit that can be easily employed to perform initial setup for calculations and postprocessing analysis to derive many material properties from the raw data generated by VASP code. We have demonstrated its capability through illustrative examples. VASPKIT provides a commandline interface to perform high-throughput calculations. It remains under development, and further functionality, including closer support for other codes, is readily to be implemented. With new features being added, we hope that VASPKIT will become an even more attractive toolkit contributing to efficient development and utilization of electronic structure theory.  \n\n# Declaration of competing interest  \n\nThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.  \n\n# Acknowledgements  \n\nWe acknowledge other contributors (in no particular order) including Peng-Fei Liu, Xue-Fei Liu, Zhao-Fu Zhang, Tian Wang, DaoXiong Wu, Ya-Chao Liu, Jiang-Shan Zhao, Yue Qiu and Qiang Li. We gratefully acknowledge helpful discussions with Zhe-Yong Fan, QiJing Zheng and Ming-Qing Liao. We also thank various researchers worldwide for reporting bugs and suggesting features, which have led to significant improvements in the accuracy and robustness of the package. V.W. gratefully appreciates Yoshiyuki Kawazoe and Shigenobu Ogata for their invaluable support. V.W. also thanks The Youth Innovation Team of Shaanxi Universities.  \n\n# Appendix A. Elastic stiffness tensor matrix and strain modes for bulk crystal systems  \n\n# 1. Triclinic system (space group numbers: 1-2)  \n\nThere are 21 independent elastic constants. C11, $C_{12}$ , C13, C14, C15, C16, C22, $C_{23}$ , $C_{24}$ , C25, C26, C33, C34, C35, C36, C44, C45, C46, $C_{55},C_{56}$ and $C_{66}$ . (See Table A.1.)  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{C_{11}}&{C_{12}}&{C_{13}}&{C_{14}}&{C_{15}}&{C_{16}}\\\\ {C_{12}}&{C_{22}}&{C_{23}}&{C_{24}}&{C_{25}}&{C_{26}}\\\\ {C_{13}}&{C_{23}}&{C_{33}}&{C_{34}}&{C_{35}}&{C_{36}}\\\\ {C_{14}}&{C_{24}}&{C_{34}}&{C_{44}}&{C_{45}}&{C_{46}}\\\\ {C_{15}}&{C_{25}}&{C_{35}}&{C_{45}}&{C_{55}}&{C_{56}}\\\\ {C_{16}}&{C_{26}}&{C_{36}}&{C_{46}}&{C_{56}}&{C_{66}}\\end{array}\\right).\n$$  \n\n# 2. Monoclinic system (space group numbers: 3-15)  \n\nThere are 13 independent elastic constants: $C_{11,~}C_{12},~C_{13},~C_{15}$ $C_{22},C_{23},C_{25},C_{33},C_{35},C_{44},C_{46},C_{55}$ and $C_{66}$ . (See Table A.2.) The elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{{C_{11}}}&{{C_{12}}}&{{C_{13}}}&{{0}}&{{C_{15}}}&{{0}}\\\\ {{C_{12}}}&{{C_{22}}}&{{C_{23}}}&{{0}}&{{C_{25}}}&{{0}}\\\\ {{C_{13}}}&{{C_{23}}}&{{C_{33}}}&{{0}}&{{C_{35}}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{C_{44}}}&{{0}}&{{C_{46}}}\\\\ {{C_{15}}}&{{C_{25}}}&{{C_{35}}}&{{0}}&{{C_{55}}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{0}}&{{C_{46}}}&{{C_{66}}}\\end{array}\\right).\n$$  \n\nTable A.1 List of strain modes and the derived elastic constants for triclinic system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy #</td></tr><tr><td>1</td><td>(8, 0,0, 0,0, 0)</td><td>C118²</td></tr><tr><td>2</td><td>(0,8,0,0,0, 0)</td><td>C2282</td></tr><tr><td>3</td><td>(0, 0,8,0,0, 0)</td><td>C3²</td></tr><tr><td>4</td><td>(0, 0, 0,8,0,0)</td><td>C4482</td></tr><tr><td>5</td><td>(0, 0,0,0,8,0)</td><td>#C552</td></tr><tr><td>6</td><td>(0, 0, 0,0,0,8)</td><td>C6682</td></tr><tr><td>7</td><td>(8,8,0,0,0,0)</td><td>++</td></tr><tr><td>8</td><td>(8,0,8,0,0,0)</td><td>+(1+</td></tr><tr><td>9</td><td>(8,0,0,8,0,0)</td><td>+C1+</td></tr><tr><td>10</td><td>(8,0,0,0,8,0)</td><td>+C15+</td></tr><tr><td>11</td><td>(8,0,0,0,0,8)</td><td>(+C1+</td></tr><tr><td>12</td><td>(0,8,8,0,0,0)</td><td>+C2+</td></tr><tr><td>13</td><td>(0,8,0,8,0,0)</td><td>+C+#</td></tr><tr><td>14</td><td>(0,8,0,0,8,0)</td><td>+C25+</td></tr><tr><td>15</td><td>(0,8,0,0,0,8)</td><td>(+</td></tr><tr><td>16</td><td>(0,0,8,8,0,0)</td><td>++</td></tr><tr><td>17</td><td>(0,0,8,0,8,0)</td><td>+C3+</td></tr><tr><td>18</td><td>(0, 0,8,0,0,8)</td><td>+C36+C</td></tr><tr><td>19</td><td>(0,0,0,8,8,0)</td><td>(+4+</td></tr><tr><td>20</td><td>(0,0,0,8,0,8)</td><td>(+C4+#</td></tr><tr><td>21</td><td>(0,0,0,0,8,8)</td><td>+C5</td></tr></table></body></html>  \n\nTable A.2 List of strain modes and the derived elastic constants for monoclinic system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy </td></tr><tr><td>1</td><td>(8,0,0, 0, 0,0)</td><td>C182</td></tr><tr><td>2</td><td>(0,8,0,0,0, 0)</td><td>##C2</td></tr><tr><td>3</td><td>(0, 0, 8,0,0, 0)</td><td>#C3²</td></tr><tr><td>4</td><td>(0,0,0,8,0,0)</td><td>C4482</td></tr><tr><td>5</td><td>(0,0,0,0,8,0)</td><td>C5582</td></tr><tr><td>6</td><td>(0, 0, 0, 0,0, 8)</td><td>C6682</td></tr><tr><td>7</td><td>(8,8,0,0,0,0)</td><td>++ # 82</td></tr><tr><td>8</td><td>(8,0,8,0,0,0)</td><td>(+ +</td></tr><tr><td>9</td><td>(8,0,0,0,8,0)</td><td>(+C1+</td></tr><tr><td>10</td><td>(0,8,8,0,0,0)</td><td>+C+</td></tr><tr><td>11</td><td>(0,8,0,0,8,0)</td><td>(+C28</td></tr><tr><td>12</td><td>(0,0,8,0,8,0)</td><td></td></tr><tr><td>13</td><td>(0,0,0,8,0,8)</td><td>+C46+ 82</td></tr></table></body></html>  \n\n# 3. Orthorhombic system (space group numbers: 16-74)  \n\nThere are 9 independent elastic constants: $C_{11},C_{12},C_{13},C_{22},$ $C_{23},C_{33},C_{44},C_{55}$ and $C_{66}$ . (See Table A.3.)  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{C_{11}}&{C_{12}}&{C_{13}}&{0}&{0}&{0}\\\\ {C_{12}}&{C_{22}}&{C_{23}}&{0}&{0}&{0}\\\\ {C_{13}}&{C_{23}}&{C_{33}}&{0}&{0}&{0}\\\\ {0}&{0}&{0}&{C_{44}}&{0}&{0}\\\\ {0}&{0}&{0}&{0}&{C_{55}}&{0}\\\\ {0}&{0}&{0}&{0}&{0}&{C_{66}}\\end{array}\\right).\n$$  \n\n# 4. Tetragonal II system (space group numbers: 75-88)  \n\nThere are 7 independent elastic constants: $C_{11},\\ C_{12},\\ C_{13},\\ C_{16},$ $C_{33},C_{44}$ and $C_{66}$ . (See Table A.4.)  \n\nTable A.3 List of strain modes and the derived elastic constants for orthorhombic system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy</td></tr><tr><td>1</td><td>(8,0,0, 0,0,0)</td><td>C1182</td></tr><tr><td>2</td><td>(0,8,0,0,0,0)</td><td>#C282</td></tr><tr><td>3</td><td>(0,0,8,0,0,0)</td><td>#33</td></tr><tr><td>4</td><td>(0,0,0,8,0,0)</td><td>#C482</td></tr><tr><td>5</td><td>(0,0,0,0,8,0)</td><td>C552</td></tr><tr><td>6</td><td>(0,0,0,0,0,8)</td><td>#6682</td></tr><tr><td>7</td><td>(8,8,0,0,0,0)</td><td>1+</td></tr><tr><td></td><td></td><td></td></tr><tr><td>８９</td><td>(6. 0 6.0.0.0)</td><td></td></tr></table></body></html>  \n\nTable A.4 List of strain modes and the derived elastic constants for tetragonal I system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy F</td></tr><tr><td>1</td><td>(8,8,0,0,0,0)</td><td>(C11 +C12)82</td></tr><tr><td>２3</td><td></td><td></td></tr><tr><td></td><td>(0. 0, 0. 0 0.0)</td><td>#C</td></tr><tr><td>4</td><td>(0,0,0,8,8,0)</td><td>C4482</td></tr><tr><td>5</td><td>(8,8,8,0,0,0)</td><td>C11+C12+2C13+33 82</td></tr><tr><td>6</td><td>(0,8,8,0,0,0)</td><td>+C13+ 82</td></tr><tr><td>7</td><td>(8,0,0,0,0,8)</td><td>+C16+ 82</td></tr></table></body></html>  \n\nTable A.5 List of strain modes and the derived elastic constants for tetragonal  system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy</td></tr><tr><td>1</td><td>(8,8,0,0, 0, 0)</td><td>(C11 +C12)82</td></tr><tr><td>2</td><td>(0, 0, 0,0,0,8)</td><td>C662</td></tr><tr><td>3</td><td>(0, 0, 8,0, 0, 0)</td><td>#C332</td></tr><tr><td>4</td><td>(0,0,0,8,8,0)</td><td>C4482</td></tr><tr><td>5</td><td>(8,8,8,0,0,0)</td><td>(C11 +C12 +2C13 +) 82</td></tr><tr><td>6</td><td>(0,8,8,0,0,0)</td><td>+C1+）8 2 2</td></tr></table></body></html>  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{C_{11}}&{C_{12}}&{C_{13}}&{0}&{0}&{C_{16}}\\\\ {C_{12}}&{C_{11}}&{C_{13}}&{0}&{0}&{-C_{16}}\\\\ {C_{13}}&{C_{13}}&{C_{33}}&{0}&{0}&{0}\\\\ {0}&{0}&{0}&{C_{44}}&{0}&{0}\\\\ {0}&{0}&{0}&{0}&{C_{44}}&{0}\\\\ {C_{16}}&{-C_{16}}&{0}&{0}&{0}&{C_{66}}\\end{array}\\right).\n$$  \n\n# 5. Tetragonal I system (space group numbers: 89-142)  \n\nThere are 6 independent elastic constants: $C_{11},\\ C_{12},\\ C_{13},\\ C_{33},$ $C_{44}$ and $C_{66}$ . (See Table A.5.)  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{C_{11}}&{C_{12}}&{C_{13}}&{0}&{0}&{0}\\\\ {C_{12}}&{C_{11}}&{C_{13}}&{0}&{0}&{0}\\\\ {C_{13}}&{C_{13}}&{C_{33}}&{0}&{0}&{0}\\\\ {0}&{0}&{0}&{C_{44}}&{0}&{0}\\\\ {0}&{0}&{0}&{0}&{C_{44}}&{0}\\\\ {0}&{0}&{0}&{0}&{0}&{C_{66}}\\end{array}\\right).\n$$  \n\n# 6. Trigonal II system (space group numbers: 143-148)  \n\nThere are 7 independent elastic constants: $C_{11},C_{12},C_{13},C_{14}$ $C_{15},C_{33}$ and $C_{44}$ . (See Table A.6.)  \n\nTable A.6 List of strain modes and the derived elastic constants for trigonal I system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector </td><td>Elastic energy 令</td></tr><tr><td>1</td><td>(8,8,0,0,0, 0)</td><td>(C11 +C12)82</td></tr><tr><td>2</td><td>(0,0,0,0,0,8)</td><td>4(C11 - C12)82</td></tr><tr><td>3</td><td>(0,0,8,0,0,0)</td><td>#C32</td></tr><tr><td>4</td><td>(0,0,0,8,8,0)</td><td>C4482</td></tr><tr><td>5</td><td>(8,8,8,0,0,0)</td><td>(C11 + C12 +2C13 + 3 82</td></tr><tr><td>6</td><td>(0,0,0,0,8,8)</td><td>(-+C1+）²</td></tr><tr><td>7</td><td>(0, 0,0,8,0,8)</td><td>1+ 82 4 2</td></tr></table></body></html>  \n\nTable A.7 List of strain modes and the derived elastic constants for trigonal  system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy </td></tr><tr><td>1</td><td>(8,8, 0,0,0, 0)</td><td>(C11 +C12)82</td></tr><tr><td>2</td><td>(0, 0, 0,0,0,8)</td><td>4(C11 -C12)82</td></tr><tr><td>3</td><td>(0, 0, 8,0, 0, 0)</td><td>C33²</td></tr><tr><td>4</td><td>(0, 0,0,8,8,0)</td><td>C4482</td></tr><tr><td>5</td><td>(8,8,8,0,0,0)</td><td>(C11 +C12 +2C13 + C3 82</td></tr><tr><td>6</td><td>(0, 0, 0,0,8,8)</td><td>-+C1+ 82</td></tr></table></body></html>  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{C_{11}}&{C_{12}}&{C_{13}}&{C_{14}}&{C_{15}}&{0}\\\\ {C_{12}}&{C_{11}}&{C_{13}}&{-C_{14}}&{-C_{15}}&{0}\\\\ {C_{13}}&{C_{13}}&{C_{33}}&{0}&{0}&{0}\\\\ {C_{14}}&{-C_{14}}&{0}&{C_{44}}&{0}&{-C_{15}}\\\\ {C_{15}}&{-C_{15}}&{0}&{0}&{C_{44}}&{C_{14}}\\\\ {0}&{0}&{0}&{-C_{15}}&{C_{14}}&{\\frac{C_{11}-C_{12}}{2}}\\end{array}\\right).\n$$  \n\n# 7. Trigonal I system (space group numbers: 149-167)  \n\nThere are 6 independent elastic constants: $C_{11},\\ C_{12},\\ C_{13},\\ C_{14},$ $C_{33}$ and $C_{44}$ . (See Table A.7.)  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{C_{11}}&{C_{12}}&{C_{13}}&{C_{14}}&{0}&{0}\\\\ {C_{12}}&{C_{11}}&{C_{13}}&{-C_{14}}&{0}&{0}\\\\ {C_{13}}&{C_{13}}&{C_{33}}&{0}&{0}&{0}\\\\ {C_{14}}&{-C_{14}}&{0}&{C_{44}}&{0}&{0}\\\\ {0}&{0}&{0}&{0}&{C_{44}}&{C_{14}}\\\\ {0}&{0}&{0}&{0}&{C_{14}}&{\\frac{C_{11}-C_{12}}{2}}\\end{array}\\right).\n$$  \n\n# 8. Hexagonal system (space group numbers: 168–194)  \n\nThere are 5 independent elastic constants: $C_{11},\\ C_{12},\\ C_{13},\\ C_{33}$ and $C_{44}$ . (See Table A.8.)  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c c}{C_{11}}&{C_{12}}&{C_{13}}&{0}&{0}&{0}\\\\ {C_{12}}&{C_{11}}&{C_{13}}&{0}&{0}&{0}\\\\ {C_{13}}&{C_{13}}&{C_{33}}&{0}&{0}&{0}\\\\ {0}&{0}&{0}&{C_{44}}&{0}&{0}\\\\ {0}&{0}&{0}&{0}&{C_{44}}&{0}\\\\ {0}&{0}&{0}&{0}&{0}&{\\frac{C_{11}-C_{12}}{2}}\\end{array}\\right).\n$$  \n\n# 9. Cubic system (space group numbers: 195–230)  \n\nThere are 3 independent elastic constants: $C_{11},\\ C_{12}$ and $C_{44}$ . (See Table A.9.)  \n\nTable A.8 List of strain modes and the derived elastic constants for hexagonal system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy 令</td></tr><tr><td>1</td><td>(8,8,0,0,0,0)</td><td>(C11 +C12)82</td></tr><tr><td>2</td><td>(0,0, 0, 0,0,8)</td><td>4(C11- C12)82</td></tr><tr><td>3</td><td>(0, 0, 8,0, 0, 0)</td><td>C33²</td></tr><tr><td>4</td><td>(0,0,0,8,8,0)</td><td>C4482</td></tr><tr><td>5</td><td>(8,8,8,0,0,0)</td><td>(C11 +C12 +2C13 +3 82</td></tr></table></body></html>  \n\nTable A.9 List of strain modes and the derived elastic constants for cubic system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy </td></tr><tr><td>1</td><td>(0,0,0,8,8,8)</td><td>CAA82</td></tr><tr><td>2</td><td>(8,8,0,0,0,0)</td><td>(C11 + C12)82</td></tr><tr><td>3</td><td>(8,8,8,0,0,0)</td><td>(C11 +2C12)82</td></tr></table></body></html>  \n\nThe elastic stiffness tensor matrix is expressed by  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c c c c}{{C_{11}}}&{{C_{12}}}&{{C_{12}}}&{{0}}&{{0}}&{{0}}\\\\ {{C_{12}}}&{{C_{11}}}&{{C_{12}}}&{{0}}&{{0}}&{{0}}\\\\ {{C_{12}}}&{{C_{12}}}&{{C_{11}}}&{{0}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{C_{44}}}&{{0}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{0}}&{{C_{44}}}&{{0}}\\\\ {{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{C_{44}}}\\end{array}\\right).\n$$  \n\n# Appendix B. Elastic stiffness tensor matrix and strain modes for 2D crystal systems  \n\n# 1. 2D oblique system  \n\nThere are 6 independent elastic constants: $C_{11,~}C_{12,~}C_{16,~}C_{22,~}$ $C_{26}$ and $C_{66}$ . (See Table B.1.)  \n\n$$\nC_{i j}=\\left(\\begin{array}{l l l}{{C_{11}}}&{{C_{12}}}&{{C_{16}}}\\\\ {{C_{12}}}&{{C_{22}}}&{{C_{26}}}\\\\ {{C_{16}}}&{{C_{26}}}&{{C_{66}}}\\end{array}\\right)\n$$  \n\n# 2. 2D rectangular system  \n\nThere are 4 independent elastic constants: $C_{11},C_{12},C_{22}$ and $C_{66}$ . (See Table B.2.)  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c}{{C_{11}}}&{{C_{12}}}&{{0}}\\\\ {{C_{12}}}&{{C_{22}}}&{{0}}\\\\ {{0}}&{{0}}&{{C_{66}}}\\end{array}\\right)\n$$  \n\n# 3. 2D square system  \n\nTable B.1 List of strain modes and the derived elastic constants for 2D oblique system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy </td></tr><tr><td>1</td><td>(8,0,0)</td><td>C1182</td></tr><tr><td>2</td><td>(0,8,0)</td><td>##28</td></tr><tr><td>3</td><td>(0,0,8)</td><td>#C6682</td></tr><tr><td>4</td><td>(8,8,0)</td><td>+(1+）</td></tr><tr><td>5</td><td>(8,0,8)</td><td>(+ 1</td></tr><tr><td>6</td><td>(0,8,8)</td><td>+C26+ 8²</td></tr></table></body></html>  \n\nTable B.2 List of strain modes and the derived elastic constants for 2D rectangular system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy F</td></tr><tr><td>1</td><td>(8,0,0)</td><td>C112</td></tr><tr><td>2</td><td>(0,8,0)</td><td>C22²</td></tr><tr><td>3</td><td>(0,0,8)</td><td>C6682</td></tr><tr><td>4</td><td>(8,8,0)</td><td>+C1+）8</td></tr></table></body></html>  \n\nTable B.3 List of strain modes and the derived elastic constants for 2D square system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy</td></tr><tr><td>1</td><td>(8,0,0)</td><td>C1182</td></tr><tr><td>2</td><td>(0,0,8)</td><td>C6682</td></tr><tr><td>3</td><td>(8,8,0)</td><td>(C11 +C12)82</td></tr></table></body></html>  \n\nTable B.4 List of strain modes and the derived elastic constants for 2D hexagonal system used in VASPKIT based on energy-strain approach.   \n\n\n<html><body><table><tr><td>Strain index</td><td>Strain vector ε</td><td>Elastic energy V</td></tr><tr><td>１</td><td>(8,0,0)</td><td>C1182</td></tr><tr><td>2</td><td>(8,8,0)</td><td>(C11 +C12)82</td></tr></table></body></html>  \n\nThere are 3 independent elastic constants: $C_{11},\\ C_{12}$ and $C_{66}$ . (See Table B.3.)  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c}{{C_{11}}}&{{C_{12}}}&{{0}}\\\\ {{C_{12}}}&{{C_{11}}}&{{0}}\\\\ {{0}}&{{0}}&{{C_{66}}}\\end{array}\\right)\n$$  \n\n# 4. 2D hexagonal system  \n\nThere are 2 independent elastic constants: $C_{11}$ and $C_{12}$ . (See Table B.4.)  \n\n$$\nC_{i j}=\\left(\\begin{array}{c c c}{{C_{11}}}&{{C_{12}}}&{{0}}\\\\ {{C_{12}}}&{{C_{11}}}&{{0}}\\\\ {{0}}&{{0}}&{{\\frac{C_{11}-C_{12}}{2}}}\\end{array}\\right)\n$$  \n\n# References  \n\n[1] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864–B871, https://doi org 10. 1103 PhysRev.136 B864.   \n[2] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133–A1138, https://doi org 10. 1103 PhysRev.140 A1133.   \n[3] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoulos, Rev. Mod. Phys. 64 (1992) 1045–1097, https://doi org 10 1103 RevModPhys 64 1045.   \n[4] R.O. Jones, O. Gunnarsson, Rev. Mod. Phys. 61 (1989) 689–746, https://doi org 10 1103 RevModPhys 61.689.   \n[5] R.O. Jones, Rev. Mod. 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    "Target": null
  },
  {
    "id": "10.1016_j.soilbio.2021.108211",
    "DOI": "10.1016/j.soilbio.2021.108211",
    "DOI Link": "http://dx.doi.org/10.1016/j.soilbio.2021.108211",
    "Relative Dir Path": "mds/10.1016_j.soilbio.2021.108211",
    "Article Title": "The microplastisphere: Biodegradable microplastics addition alters soil microbial community structure and function",
    "Authors": "Zhou, J; Gui, H; Banfield, CC; Wen, Y; Zang, HD; Dippold, MA; Charlton, A; Jones, DL",
    "Source Title": "SOIL BIOLOGY & BIOCHEMISTRY",
    "Abstract": "Plastics accumulating in the environment, especially microplastics (defined as particles <5 mm), can lead to a range of problems and potential loss of ecosystem services. Polyhydroxyalkanoates (PHAs) are biodegradable plastics used in mulch films, and in packaging material to minimize plastic waste and to reduce soil pollution. Little is known, however, about the effect of microbioplastics on soil-plant interactions, especially soil microbial community structure and functioning in agroecosystems. For the first time, we combined zymography (to localize enzyme activity hotspots) with substrate-induced growth respiration to investigate the effect of PHAs addition on soil microbial community structure, growth, and exoenzyme kinetics in the microplastisphere (i.e. interface between soil and microplastic particles) compared to the rhizosphere and bulk soil. We used a common PHAs biopolymer, poly (3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) and showed that PHBV was readily used by the microbial community as a source of carbon (C) resulting in an increased specific microbial growth rate and a more active microbial biomass in the microplastisphere in comparison to the bulk soil. Higher ss-glucosidase and leucine aminopeptidase activities (0.6-5.0 times higher Vmax) and lower enzyme affinities (1.5-2.0 times higher Km) were also detected in the microplastisphere relative to the rhizosphere. Furthermore, the PHBV addition changed the soil bacterial community at different taxonomical levels and increased the alpha diversity, as well as the relative abundance of Acidobacteria and Verrucomicrobia phyla, compared to the untreated soils. Overall, PHBV addition created soil hotspots where C and nutrient turnover is greatly enhanced, mainly driven by the accelerated microbial biomass and activity. In conclusion, microbioplastics have the potential to alter soil ecological functioning and biogeochemical cycling (e.g., SOM decomposition).",
    "Times Cited, WoS Core": 371,
    "Times Cited, All Databases": 412,
    "Publication Year": 2021,
    "Research Areas": "Agriculture",
    "UT (Unique WOS ID)": "WOS:000640189100034",
    "Markdown": "# The microplastisphere: Biodegradable microplastics addition alters soil microbial community structure and function  \n\nJie Zhou a,b,1, Heng Gui c,d,1, Callum C. Banfield b, Yuan Wen a, Huadong Zang a,\\*, Michaela A. Dippold b, Adam Charlton e, Davey L. Jones f,g  \n\na College of Agronomy and Biotechnology, China Agricultural University, Beijing, China   \nb Biogeochemistry of Agroecosystems, Department of Crop Sciences, University of Goettingen, Goettingen, Germany   \nc CAS Key Laboratory for Plant Diversity and Biogeography of East Asia, Kunming Institute of Botany, Chinese Academy of Science, Kunming, China   \nd Centre for Mountain Futures (CMF), Kunming Institute of Botany, Chinese Academy of Science, Kunming, Yunnan, China   \ne BioComposites Centre, Bangor University, Bangor, Gwynedd, LL57 2UW, UK   \nf School of Natural Sciences, Bangor University, Bangor, Gwynedd, LL57 2UW, UK   \ng Soils West, UWA School of Agriculture and Environment, The University of Western Australia, Perth, WA 6009, Australia  \n\n# A R T I C L E  I N F O  \n\nKeywords:   \nEnzyme activity   \nMicrobial growth   \nMicroplastic pollution   \nSoil organic matter   \nC turnover   \nSequencing  \n\n# A B S T R A C T  \n\nPlastics accumulating in the environment, especially microplastics (defined as particles $<5~\\mathrm{{mm}}$ ), can lead to a range of problems and potential loss of ecosystem services. Polyhydroxyalkanoates (PHAs) are biodegradable plastics used in mulch films, and in packaging material to minimize plastic waste and to reduce soil pollution. Little is known, however, about the effect of microbioplastics on soil-plant interactions, especially soil microbial community structure and functioning in agroecosystems. For the first time, we combined zymography (to localize enzyme activity hotspots) with substrate-induced growth respiration to investigate the effect of PHAs addition on soil microbial community structure, growth, and exoenzyme kinetics in the microplastisphere (i.e. interface between soil and microplastic particles) compared to the rhizosphere and bulk soil. We used a common PHAs biopolymer, poly (3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) and showed that PHBV was readily used by the microbial community as a source of carbon (C) resulting in an increased specific microbial growth rate and a more active microbial biomass in the microplastisphere in comparison to the bulk soil. Higher $\\upbeta$ -glucosidase and leucine aminopeptidase activities (0.6–5.0 times higher $V_{\\mathrm{max}})$ and lower enzyme affinities (1.5–2.0 times higher $K_{m})$ were also detected in the microplastisphere relative to the rhizosphere. Furthermore, the PHBV addition changed the soil bacterial community at different taxonomical levels and increased the alpha diversity, as well as the relative abundance of Acidobacteria and Verrucomicrobia phyla, compared to the un­ treated soils. Overall, PHBV addition created soil hotspots where C and nutrient turnover is greatly enhanced, mainly driven by the accelerated microbial biomass and activity. In conclusion, microbioplastics have the po­ tential to alter soil ecological functioning and biogeochemical cycling (e.g., SOM decomposition).  \n\n# 1. Introduction  \n\nSynthetic polymers are widely used in our daily lives, and more than 280 million tons of plastics are produced annually (Duis and Coors, 2016; Sintim and Flury, 2017). Despite the remarkable benefits of plastics to society, there are increasing concerns associated with the vast amount of plastic entering our environment and its resistance to degradation (Rochman, 2018). These concerns are supported by esti­ mates that ${>}30\\%$ of the world’s plastic waste is disposed of inappropriately, with most of it ultimately ending up in soil (Jambeck et al., 2015; Weithmann et al., 2018). In soil, larger plastic debris often becomes fragmented into smaller pieces by biota and physical distur­ bance known as microplastics (mean diameter $<5~\\mathrm{mm},$ . They have received increased attention globally due to their potential to cause environmental damage (Rillig, 2012; de Souza Machado et al., 2019). A promising approach to overcome the accumulation of microplastics in soil is to replace traditional petroleum-based plastics with biodegrad­ able bioplastics like polyhydroxyalkanoates (PHAs; Gross and Kalra,  \n\n2002; Volova et al., 2017). PHAs account for $5.6\\%$ of the global pro­ duction capacity for biodegradable polymers, and represent the second fastest growing group in the market sector since 2014 (Haider et al., 2019). Even though PHAs are used in an attempt to decrease micro­ plastic residues in terrestrial ecosystems, and praised as promising al­ ternatives for a diverse range of applications (e.g., mulch films for agriculture), the potential environmental consequences of PHAs have not yet been thoroughly studied.  \n\nUnlike petroleum-based microplastics, which biodegrade extremely slowly, PHAs can be broken down by a range of organisms and are not thought to produce any harmful by-products (Volova et al., 2017; Haider et al., 2019; Sander et al., 2019). Given their biological origin, they are considered C neutral (Garrison et al., 2016), although this assumes that they do not induce positive priming of soil organic matter (SOM). Furthermore, they are thought to not enhance $\\mathrm{N}_{2}\\mathrm{O}$ and $\\mathrm{CH}_{4}$ emissions which might offset these benefits. Given that PHAs are C-rich but nutrient-poor (i.e. no N and P; Gross and Kalra, 2002; Volova et al., 2017), they may alter microbial community composition and func­ tioning during degradation. Since the decomposition of C-rich residues is associated with N and P immobilization, subsequent plant growth may also be affected due to the increased competition between plants and soil microorganisms for nutrients (Qi et al., 2018, 2020b; Song et al., 2020; Zang et al., 2020). In response to the additional C supplied from PHAs breakdown, the turnover of native SOM may be stimulated due to the altered metabolic status of the microbial community (Kuzyakov, 2010; Zang et al., 2017), and thus influence soil C and nutrient cycling. PHAs are also naturally present in soil being produced as storage compounds by the bacterial community (Mason-Jones et al., 2019). Given that bacteria are more sensitive to environmental changes (e.g. increased labile C) compared to fungi (Barnard et al., 2012), soil bacteria may have a stronger response due to the increased C availability through PHAs breakdown. This will lead to significant long-term impacts on a range of soil ecosystem services (e.g., C storage, nutrient cycling, and pollutant attenuation; Zang et al., 2018). Although recent studies revealed that microplastics may have divergent influences on soil mi­ crobial communities and enzyme activities, e.g., activation (Liu et al., 2017; de Souza Machado et al., 2019), suppression (Fei et al., 2020), or remaining unchanged (Zang et al., 2020), the effect of microbioplastics on soil microorganisms remains poorly understood. Therefore, it is vital to investigate how biodegradable microplastics affect microbial func­ tions and below-ground C processes (Zang et al., 2019, 2020; Qi et al., 2020a).  \n\nSimilar to plant-soil interactions in the rhizosphere, the main pro­ cesses affected by microplastic input may occur at the soil-plastic interface (here defined as the microplastisphere). We hypothesize that these interactions are stimulated by the input of bioavailable C present in microbioplastics (i.e. increased microbial activity, attract or favor specific bacterial taxa, and interfere with belowground plantmicroorganisms interactions) leading to the formation of microbial hotspots in soil, similar to those seen in the rhizosphere (Kuzyakov and Blagodatskaya, 2015; Zang et al., 2016; Zhou et al., 2020b). Following PHAs addition, we predict that changes in the soil physico-chemical properties will only occur close to the microplastic particles, with changes in the (non-hotspot) bulk soil likely to be minor (Zettler et al., 2013; Huang et al., 2019). The specific niches of the microorganisms in the microplastisphere are of ecological relevance, given that most agricultural soils are contaminated by microplastics (Steinmetz et al., 2016; Qi et al., 2020a). However, it still remains unclear how PHAs affect soil microbial communities in hotspots and, thus, alters soil C and nutrient cycling.  \n\nHere, for the first time, we coupled zymography, a method to accurately locate microbial hotspots (Hoang et al., 2020; Zhang et al., 2020), the kinetics of exoenzyme activities involved in C, N, and P cycling, microbial growth, and bacterial community structure to evaluate microbial functions, as well as soil process in hotspots (rhizosphere and microplastisphere) and bulk soil. Poly (3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) represents a commercially available copolymer used for mulch film production. Compared to PHB, it has higher flexibility, thermal stability, and processibility due to the monomeric composition, which makes it a promising example of PHAs (Table S1; Jiang et al., 2009; Bugnicourt et al., 2014). Therefore, we aimed to 1) identify microbial hotspots in situ in soil treated with PHBV; 2) investigate the effect of biodegradable microplastics on microbial growth and enzyme kinetics; 3) evaluate changes in the bacterial community structure and function in the microplastisphere and rhizosphere. We hypothesized that: 1) the labile C in PHBV will greatly alter soil bacterial community structure and functioning compared to the rhizosphere and bulk soil, and 2) the microplastisphere contains microorganisms with a high growth rate and enzyme activity in comparison to rhizosphere and bulk soil.  \n\n# 2. Materials and methods  \n\n# 2.1. Site description and sampling  \n\nSoil samples were taken from the Ap horizon $\\mathrm{(0{-}20~c m)}$ of an experimental field at the Reinshof Research Station of the Georg-August University of G¨ottingen, Germany $(28^{\\circ}33^{\\prime}26^{\\prime\\prime}\\mathrm{N}$ , $113^{\\circ}20^{\\prime}8^{\\prime\\prime}\\mathrm{E})$ . This experimental site was established more than 40 years ago and the farming history is well documented. No plastic mulch has ever been applied, and no plastic pollution has been recorded for the site. The soil was air-dried, sieved $(<2~\\mathrm{mm})$ ), and mixed to achieve a high degree of homogeneity and to reduce the variability among replicates. Fine roots and visible plant residues were carefully removed prior to use. The soil contained $1.3\\%$ total C, $0.14\\%$ total N, and had a pH of 6.8 (Zhou et al., 2020b). Ten percent (w/w) of the soil dry weight was added as poly (3-hydroxybutyrate-co-3-hydroxyvalerate) $\\mathrm{\\langle[COCH_{2}C H(C H_{3})O]m}$ $\\mathrm{\\small{[COCH_{2}C H(C_{2}H_{5})O]_{n})}}$ (PHBV). PHBV was obtained in a pelletized form from the Tianan Biologic Materials Company Ltd., Beilun, Ningbo, China. PHBV represents one of the most widespread and best charac­ terized members of the PHA family (Bugnicourt et al., 2014). It is a $100\\%$ biobased thermoplastic linear aliphatic (co-)polyester obtained from the copolymerization of 3-hydroxybutanoic acid and 3-hydroxy­ pentanoic acid which are produced through the bacterial fermentation of sugars and lipids (Zinn et al., 2001). Most of the PHBV is composed of hydroxybutyrate, however, a small fraction of hydroxyvalerate is pre­ sent in its polymeric backbone (Rivera-Briso and Serrano-Aroca, 2018). This type and amount of highly crystalline plastic were chosen to simulate the localized disposal of bioplastics in agricultural soils (e.g., ploughing in of mulch film residues at the end of the field season) and was based on field investigations and a review of the literature (Fuller and Gautam, 2016; Qi et al., 2020a). We added very high amounts of microplastic to reflect soil hotspots with higher contamination levels $(1-20\\%)$ .  \n\n# 2.2. Experimental design  \n\nA mesocosm experiment with a completely randomized design and four replicates was set up in a climate-controlled room. For the PHBV addition treatment, $_{400\\mathrm{~g~}}$ soil and PHBV were mixed homogeneously and then put in a rhizobox ${10\\times10\\times4}$ cm; Qiangsheng Co., Ltd. Heibei, China). The control treatment contained soil $(400~\\mathrm{g})$ without PHBV, but with a comparable soil disturbance. The soil bulk density was main­ tained at $1.2\\ {\\mathrm{g}}\\ {\\mathrm{cm}}^{-3}$ for all rhizoboxes. Prior to use, the soil was preincubated under field-moist $(25\\%\\ \\mathrm{v/v})$ conditions in a greenhouse for one week to allow the soil to equilibrate. Before planting, wheat (Triti­ cum aestivum L.) seeds were sterilized in $10\\%$ $\\mathrm{H}_{2}\\mathrm{O}_{2}$ for $10\\ \\mathrm{min}$ , then rinsed with deionized water and germinated on wet filter paper. Five days after germination, seedlings were transplanted in all rhizoboxes (one seedling per rhizobox), and then moved to the climate-controlled chamber (day/night regime of $14{\\mathrm{~h~}}/24{\\mathrm{~}}^{\\circ}{\\mathrm{C}}$ and $10\\mathrm{~h~}/14^{\\mathrm{~\\circ~C~}}$ respec­ tively). The relative humidity in the chamber was kept at $40\\%$ and the plants received $800\\upmu\\mathrm{mol}\\mathrm{m}^{-2}s^{-1}$ photosynthetic active radiation (PAR) at canopy height (Zhou et al., 2020b). Plants were watered every three days and the soil moisture was maintained at a gravimetric water con­ tent of $25\\%$ throughout the experiment by weighing the rhizoboxes.  \n\n# 2.3. Hotspot identification  \n\nAt 24 days after transplanting, zymography was used to visualize the spatial distribution of three hydrolytic enzymes (Razavi et al., 2016). B-glucosidase, acid phosphatase, and leucine-aminopeptidase play major roles in cellulose, organic phosphate, and protein degradation (Lopez-Hernandez et al., 1993; Lammirato et al., 2010). They reflect key enzymes related to soil C, P and N cycle, respectively (German et al., 2011). Polyamide membrane ${\\mathrm{7}}0.45{\\upmu\\mathrm{m}}$ mesh size, Tao yuan, China) were saturated with 4-methylumbelliferyl (MUF) or 7-amido-4-methylcou­ marin (AMC) based substrate to visualize the specific enzymes. Each substrate was separately dissolved in $10\\mathrm{mM}$ MES and TRIZMA buffer for MUF and AMC, respectively. The saturated membranes were placed on soil surfaces and covered with aluminum foil to avoid evaporation and moisture changes during the incubation period (Hoang et al., 2020). After incubation for $^\\textrm{\\scriptsize1h}$ , the membranes were carefully peeled off the soil surface and any attached soil particles were gently removed with tweezers and a soft brush (Razavi et al., 2016). Enzyme detection se­ quences followed as: $\\upbeta$ -glucosidase, acid phosphatase, leucine-aminopeptidase activity, with $^\\textrm{\\scriptsize1h}$ interval after each zymog­ raphy. The gray scale values transferred to the enzyme activities was calibrated using membranes $(2\\times2~\\mathrm{cm})$ saturated with a range of con­ centrations of corresponding products, i.e. MUF and AMC (0, 0.01, 0.2,  \n\n# 0.5, 1, 2, 5 mM).  \n\nThe zymograms were transferred into a 16-bit gray scale by ImageJ with a correction for environmental variations and camera noise (Razavi et al., 2016). The calibration equation obtained for each enzyme was used to convert gray values of each zymography pixel into enzyme ac­ tivities (Hoang et al., 2020). Enzyme activities exceeding $25\\%$ of mean corresponding activity of the whole soil were defined as hotspots (Zhang et al., 2020). Specifically, soil with a high color intensities (shown here in dark red) represent microbial hotspots, while low intensities (shown here in dark blue) indicate (non-hotspot) bulk soil on the zymograms (Fig. 1; Hoang et al., 2020). Given the hotspots in the control and PHBV-treated soil were detected at a distance of $1.5{-}2~\\mathrm{mm}$ from the roots and microplastics, the hotspots in the control and PHBV-treated soil were identical to the rhizosphere and microplastisphere zones, respectively (Fig. 1). After collecting soil from hotspots and bulk soil, a total of 16 samples [2 treatments (without and with $\\mathbf{PHBV}\\times2$ micro­ sites from each treatment (hotspots and bulk soil) $\\times4$ replicates] were obtained.  \n\n# 2.4. Plant and soil sampling  \n\nAt 25 days after transplanting, the shoots were cut off at the base of the stem and the roots were collected separately. For precise localized sampling, soil particles were carefully collected using needles (tip 1.5 mm) directly from the hotspots (rhizosphere and microplastisphere) identified by zymography (Fig. 1). Bulk soil was collected in a similar way. Once collected, soil samples (hotspots and bulk soil) were sepa­ rated into two sub-samples. One sub-sample was stored at $-80~^{\\circ}\\mathrm{C}$ to analyze the bacterial community structure, while the other sub-sample was used to measure enzyme kinetics and the kinetics of substrateinduced growth respiration directly. After removal of the hotspot sam­ ples and bulk soil, the remaining soil in the rhizobox was mixed and then stored at $4^{\\circ}\\mathbf{C}$ to measure microbial biomass N, dissolved organic C and N. Shoots and roots were oven-dried ( $60^{\\circ}\\mathrm{C}$ , 5 days) and then weighed.  \n\n![](images/a3a3a04fc32c877668787fbfff5b0af6f458199397fe2e2f0749244aede0b322.jpg)  \nFig. 1. Zymograms and hotspots of $\\upbeta$ -glucosidase (BG), acid phosphatase (ACP) and leucine aminopeptidase (LAP) in untreated soil (Control) and soil to which the bioplastic poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) was added. The color intensity is proportional to the respective enzyme activity (nmol $\\mathrm{cm}^{-2}\\mathrm{h}^{-1},$ ). The zymograms are representative of 4 independent replicates. The corresponding area of hotspots relative to the total area of the rhizobox for each enzyme is shown in the right-hand panel. Values are means $(\\pm S\\mathbf{E})$ of four replicates. Different letters show significant differences between treatments $(p<0.05)$ . Here, 1, 2, 3 indicate rhizosphere, microplastisphere, and bulk soil. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)  \n\nSoil microbial biomass N (MBN) was extracted with $\\mathrm{K_{2}S O_{4}}$ $32~\\mathrm{mL}$ , $0.05~\\mathrm{M})$ ), and calculated with a corresponding ${\\mathrm{K}}_{\\mathrm{EN}}$ factor of 0.45 ac­ cording to Wen et al. (2020). Briefly, the fresh soil was homogenized and $_{8\\mathrm{~g~}}$ sub-sample of the soil were extracted with $32\\mathrm{\\mL}0.05\\mathrm{\\M\\K_{2}S O_{4}}.$ . Another $_{8\\mathrm{~g~}}$ sub-sample of the soil was fumigated with chloroform for $24\\mathrm{h}$ and then extracted in the same way. Total C and N in extracts were measured on a $2100~\\mathrm{N/C}$ analyzer (Analytik Jena GmbH, Jena, Ger­ many). The non-fumigated extractions were used as a measure for dis­ solved organic C (DOC) and N (DON).  \n\n# 2.5. Enzyme kinetics  \n\nThe activity of the exoenzymes $\\upbeta$ -1,4-glucosidase (BG) (EC 2.2.1.21), leucine aminopeptidase (LAP) (EC 3.4.11.1), and acid phosphatase (ACP) (EC 3.1.3.2) were determined by the 4-methylumbelliferyl (MUF)-based and 7-amido-4-methylcoumarin (AMC)-based artificial substrates (Marx et al., 2001; Wen et al., 2019). Briefly, $_{0.5~g}$ soil was mixed with $50\\mathrm{mL}$ sterile water and then shaking for $30\\mathrm{min}$ . After $2\\mathrm{min}$ low-energy sonication $(40\\mathrm{~J~}s^{-1})$ by ultrasonic disaggregation, $50~\\upmu\\mathrm{l}$ of the soil suspension, $50\\upmu\\mathrm{l}$ of corresponding buffer (MES or TRIZMA) and $100\\upmu\\mathrm{l}$ of the corresponding substrates at concentrations of 2, 5, 10, 20, 50, 100 and $200\\upmu\\mathrm{mol}1^{-1}$ were pipetted into 96-well black microplates (Brand $\\textsuperscript{\\textregistered}$ plates pureGrade, Sigma-Aldrich, Germany). The Victor 1420-050 Multi label Counter (PerkinElmer, USA) was used to measure the fluorescence at an excitation wavelength of $355\\mathrm{nm}$ and an emission wavelength of $460\\mathrm{nm}$ . Enzyme activities were taken at four times (0, 30 min, $^\\textrm{\\scriptsize1h}$ and $2~\\mathrm{h}\\mathrm{\\prime}$ ), and was expressed as nmol ${\\bf g}^{-1}$ soil $\\ensuremath{\\mathbf{h}}^{-1}$ .  \n\nTo calculate key parameters describing the enzyme kinetics, we fitted a Michaelis-Menten equation to the experimental data (Marx et al., 2001):  \n\n$$\nV=\\frac{V_{\\mathrm{max}}\\times[S]}{K_{m}+[S]}\n$$  \n\nwhere $V$ is the enzymatically mediated rate of reaction, $V_{\\mathrm{max}}$ is the maximal rate of reaction, $K_{\\mathrm{m}}$ (Michaelis constant) is the substrate con­ centration at $\\%V_{\\mathrm{max}}$ and $s$ is substrate concentration. The substrate turnover time ( $\\cdot\\mathbf{\\nabla}T_{\\mathrm{t}})$ was calculated according to the following equation: $T_{\\mathrm{t}}$ (hours) $=(K_{\\mathrm{m}}+S)/V_{\\mathrm{max}},$ where S is the substrate concentration (200 $\\upmu\\mathrm{mol}\\mathrm{l}^{-1}\\big)$ . The catalytic efficiency of enzymes $\\left(K_{\\mathrm{a}}\\right)$ was calculated by the ratio of $V_{\\mathrm{max}}$ and $K_{\\mathrm{m}}$ (Hoang et al., 2020). The microbial metabolic limitation was quantified by calculating the vector lengths and angles of enzymatic activity for all data based on untransformed proportional activities (e.g. (BG): $(\\mathrm{BG}+\\mathrm{LAP})$ , (BG): $(\\mathbf{BG}+\\mathbf{ACP}).$ ) (Moorhead et al., 2016).  \n\n# 2.6. Kinetics of substrate-induced growth respiration  \n\nThe substrate-induced growth respiration (SIGR) approach was used to distinguish total and active biomass fractions, as well as microbial specific growth rate and lag-time before growth (Zhang et al., 2020; Zhou et al., 2020a). It should be noted that although C substrate addition is required for the SIGR approach, all kinetic parameters analyzed by SIGR represent the intrinsic features of dominating microbial pop­ ulations before substrate addition (Blagodatskaya et al., 2010).  \n\nOne gram of fresh soil was amended with a mixture containing 10 $\\mathrm{mg}\\ \\mathbf{g}^{-1}$ glucose, $1.9\\mathrm{mg}\\mathrm{g}^{-1}$ $(\\mathrm{NH}_{4})_{2}S O_{4}.$ , $2.25\\mathrm{\\mg\\g^{-1}\\ K_{2}H P O_{4}};$ and 3.8 $\\ensuremath{\\mathbf{m}}\\ensuremath{\\mathbf{g}}\\ensuremath{\\mathrm{~\\boldmath~\\pmb{g}~}}^{-1}$ $\\mathrm{MgSO_{4}^{2}7H_{2}O}$ , and placed in a Rapid Automated Bacterial Impedance Technique bioanalyzer (RABIT; Microbiology International Ltd, Frederick, MD, USA), for measuring $\\mathsf{C O}_{2}$ production at room temperature $(22^{\\circ}\\mathrm{C})$ . Firstly, we pre-incubated 16 samples from hotspots and bulk soil with and without PHBV amendment for 2 days at $45\\%$ water holding capacity (WHC) to minimize the effect of sampling disturbance. To measure substrate-induced respiration, a mixture of glucose and nutrients was added and the samples were further incubated for 5 day at $75\\%$ WHC (Blagodatskaya et al., 2010; Zhou et al., 2020a). The evolving $\\mathsf{C O}_{2}$ was trapped in a KOH solution where the impedance of the solution was continuously measured. The average value of $\\mathsf{C O}_{2}$ emission during the $^{3\\mathrm{h}}$ before and after adding substrates were taken as basal respiration (BR), and substrate-induced growth respiration (SIGR).  \n\nMicrobial respiration in glucose amended soil was used to calculate the following kinetic parameters: the microbial maximal specific growth rate $(\\mu)$ , the growing microbial biomass (GMB) that capable for imme­ diate growth on glucose, the total microbial biomass (TMB) responding by respiration to glucose addition, and the lag period $(T_{\\mathrm{lag}})$ .  \n\nMicrobial maximal specific growth rate $\\mu$ was used as an intrinsic property of the microbial population to estimate the prevailing growth strategy of the microbial community. According to Blagodatskaya et al. (2010), higher $\\mu$ reflects relative domination or shift towards fast-growing $r$ -strategists, while lower $\\mu$ values show relative domina­ tion or shift towards slow-growing $K$ -strategists.  \n\nConsidering that PHBV is partially soluble in chloroform at $30~^{\\circ}\\mathrm{C}$ (Jacquel et al., 2007), the microbial biomass we measured by chloroform-fumigation extraction might contain a minor contribution from PHBV degradation during fumigation. Therefore, microbial biomass C (MBC) was determined using the initial rate of substrate-induced respiration after substrate addition according to the equation of Blagodatskaya et al. (2010):  \n\n$$\n\\beth(\\upmu\\mathrm{g}\\mathrm{~\\underline{{{g}}}^{-1}\\ s o i l)=(\\upmu l\\ C O_{2}\\underline{{{g}}}^{-1}\\ s o i l\\ h^{-1})\\times40.04}\n$$  \n\n# 2.7. Soil bacterial community structure  \n\n# 2.7.1. Soil genomic DNA extraction, PCR amplification and illumina sequencing  \n\nTotal DNA was extracted from $_{0.5g}$ soil for each treatment using the Mo Bio PowerSoil DNA isolation kit (Qiagen Inc., Carlsbad, CA, USA) according to the manufacturer’s instructions. After extraction, the quality and concentration of DNA were tested using a NanoDrop ND 200 spectrophotometer (Thermo Scientific, USA). According to the concen­ tration, all DNA samples were diluted to $1~\\mathrm{{ng}}~\\upmu\\mathrm{{l}}^{-1}$ before PCR amplifi­ cation. We note that the DNA extracted from control hotspots leaked out during shipping for sequencing analysis, causing the DNA concentration to drop under the detection threshold. Therefore, the samples from this treatment could not be determined.  \n\nThe V4 and V5 variable region of the bacterial 16S rRNA gene were amplified using the primers 515F ( $5^{\\prime}$ -CCATCTCATCCCTGCGTGTCTCC­ GAC- $3^{\\prime}$ ) and 907R $5^{\\prime}$ -CCTATCCCCTGTGTGCCTTGGCAGTC-3′). The polymerase chain reaction (PCR) amplification mixture was prepared with $1~\\upmu\\mathrm{l}$ purified DNA template $(10~\\mathrm{ng})$ , $5\\ \\upmu\\mathrm{l}\\ 10\\times\\mathrm{PCR}$ buffer, 2.25 mmol $1^{-1}~\\mathrm{MgCl_{2}}$ , 0.8 mmol $1^{-1}$ deoxyribonucleotide triphosphate (dNTP), $0.5~\\upmu\\mathrm{mol}~\\mathrm{l}^{-1}$ of each primer, $2.5\\mathrm{~U~}$ Taq DNA polymerase, and sterile filtered ultraclean water to a final volume of $50~\\upmu\\mathrm{l}$ . All the re­ actions were carried out in a PTC-200 thermal cycler (MJ Research Co., NY, USA). The PCR cycles included a $4\\mathrm{min}$ initial denaturation at $94^{\\circ}\\mathrm{C},$ followed by 30 cycles of denaturation at $94^{\\circ}\\mathrm{C}$ for $1~\\mathrm{min}$ , annealing at $53^{\\circ}\\mathrm{C}$ for $30~\\mathsf{s}_{:}$ , extension at $72^{\\circ}\\mathrm{C}$ for $1\\mathrm{min}$ , and a 5-min final elongation step at $72^{\\circ}\\mathrm{C}$ . The PCR products were quality-screened and purified by Qiagen Gel Extraction kit (Qiagen, Hilden, Germany). Next, all the amplicons were sequenced on the Illumina Miseq PE250 platform at Novogene Biotech Co., Ltd., Beijing, China. All the sequences have been submitted to NCBI SRA data repository under the Accession No. PRJNA648785.  \n\n# 2.7.2. 16S gene sequences processing  \n\nBriefly, de-noising and chimera analysis conducted with the Ampli­ conNoise and UCHIME algorithms were used to reduce sequence errors (Vargas-Gastelum et al., 2015). Furthermore, quality trimming was conducted to remove unwanted sequences shorter than 200 bp and reads containing ambiguous bases and with homopolymers longer than eight bases. The remaining sequences were used to identify the unique se­ quences by aligning with the SILVA reference database (v.128) (Quast et al., 2013). Within unique sequences, the UCHIME tool was applied to remove chimeras. Then, “Chloroplast”, “Mitochondria”, or “unknown” were identified and removed from the dataset. Subsequently, after calculating the pairwise distance and generating the distance matrix, a $97\\%$ identity threshold was used to cluster sequences into Operational Taxonomic Units (OTUs) according to the UCLUST algorithm (Edgar et al., 2011). The most abundant sequence in each OTU was picked as the representative sequence. For each representative sequence, the SILVA reference database (v.128) was applied to annotate the taxo­ nomic information using RDP classifier algorithm (Wang et al., 2007).  \n\n# 2.8. Statistical analysis  \n\nThe experiment was carried out with four replicates for each parameter. All values presented in the figures are means $\\pm$ standard errors of the means $(\\mathrm{mean}\\pm\\mathrm{SE})$ ). The enzyme kinetic parameters $\\boldsymbol{\\mathrm{{V}}_{\\mathrm{{max}}}}$ and $\\ensuremath{K_{\\mathrm{m}}})$ ) were fitted via the non-linear regression routine of SigmaPlot (version 12.5; Systat Software, Inc., San Jose, CA, USA). The DNA data were rarefied to an equal depth within the minimum observed sample size across all the samples. The following six parameters, namely Rich­ ness, Pielou, Chao1, Shannon, Simpson, and abundance-based coverage (ACE), were calculated to describe the alpha diversity of the soil bac­ terial community based on OTU abundance. The calculation was con­ ducted in QIIME 2 and the illustration was performed by R software (Ver. 3.2) using the packages “ggplot2” and “metacoder”.  \n\nPrior to the analysis of variance (ANOVA), the data were tested for normality (Shapiro-Wilk, $p>0.05\\$ and homogeneity of variance (Lev­ ene-test, $p>0.05\\$ ). Any dataset that was not normally distributed was root square or $\\log_{10}$ -transformed to conform with the assumption of normality before further statistical analysis. For alpha diversity indices that did not conform to the assumption of normality, the nonparametric Kruskal-Wallis H-Test was applied to determine whether there were significant differences in alpha diversity among different treatments.  \n\n# 3. Results  \n\n# 3.1. Effect of PHBV on plant and soil properties  \n\nThe mean plant biomass was $0.24\\ g\\ \\mathrm{pot}^{-1}$ without microplastics addition (Table 1). However, PHBV addition ultimately resulted in plant death after 25 days. PHBV addition greatly increased the soil microbial biomass and dissolved organic C content $(p<0.05$ , Table 1). MBC and DOC were 12 and 54 times higher in the PHBV-treated than in the control soil, respectively. Additionally, MBN was $45\\%$ higher in the PHBV-treatment in comparison to the control, whereas DON decreased by $66\\%$ compared to the control soil.  \n\nTable 1 Plant biomass, microbial biomass carbon (MBC) and nitrogen (MBN), and dis­ solved organic carbon (DOC) and nitrogen (DON) in untreated soil (Control) and soil to which the bioplastic poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) was added. Values are means $(\\pm S\\mathbf{E})$ of four replicates. Letters show significant differences between treatments $(p<0.05)$ . MBC was calculated by substrate-induced growth respiration (according to Eq. (2)), MBN was measured by chloroform-fumigation extraction, DOC and DON were determined by nonfumigated extractions.   \n\n\n<html><body><table><tr><td>Treatment</td><td>Plant biomass (g DM pot-1)</td><td>MBC (mg kg-1)</td><td>MBN (mg kg-1)</td><td>DOC (mg kg-1)</td><td>DON (mg kg-1)</td></tr><tr><td>Control</td><td>0.24 ± 0.02</td><td>131 ± 23b</td><td>20.6 ± 3.4b</td><td>163 ± 20b</td><td>93.9 ± 5.5a</td></tr><tr><td>PHBV</td><td>n.d.</td><td>1723 ± 625a</td><td>30.4 ± 5.6a</td><td>9049 ± 889a</td><td>32.3 ± 5.2b</td></tr></table></body></html>\n\nn.d.: no data due to plant death.  \n\n# 3.2. Effect of PHBV on soil enzyme activities  \n\nThe maximum potential enzyme activities $(V_{\\mathrm{max}})$ were $60\\%$ and 5- folds higher for $\\upbeta$ -glucosidase and leucine aminopeptidase in the microplastisphere than in the rhizosphere, respectively $(p<0.05$ , Fig. 2a and b). Similarly, the substrate affinities $(K_{m})$ of $\\upbeta$ -glucosidase and leucine aminopeptidase in the microplastisphere were 1.5–2 times higher in the rhizosphere, respectively $(p<0.05$ , Fig. 2b, d). The $V_{\\mathrm{max}}$ and $K_{m}$ of $\\upbeta$ -glucosidase and leucine aminopeptidase in microplasti­ sphere were significantly higher compared to those in the PHBV-treated bulk soil $(p<0.05$ , Fig. 2). In the bulk soil, however, none of the tested enzymes were affected by PHBV addition $(p>0.05$ , Fig. 2). Further­ more, the $V_{\\mathrm{max}}$ of $\\upbeta$ -glucosidase was positively correlated with active microbial biomass $(\\mathrm{R}^{2}=0.7,p<0.05$ , Fig. S4). The catalytic efficiency $(V_{\\mathrm{max}}/K_{m})$ of leucine aminopeptidase was higher in the microplasti­ sphere than in the rhizosphere $(p<0.05$ , Fig. S2c), and the turnover time was approximately 5 times shorter in the microplastisphere than in the rhizosphere soil (Fig. S2d). However, no changes in the catalytic efficiency and turnover time for all the enzymes were found in the bulk soil between the PHBV-treated and control soil $(p>0.05$ Fig. S2). Further, the vector angle was lowest in the microplastisphere compared to other soil samples $(p<0.05$ , Fig. S7d), indicating that microbial metabolisms may be N limited.  \n\n![](images/c976c5c91abe05158cf3404ee95302ec2b259cf90ae1257d25d643dc0efd42f0.jpg)  \nFig. 2. Potential enzyme activities $(V_{\\mathrm{max}})$ and substrate affinities $(K_{\\mathrm{m}})$ of $\\upbeta$ -glucosidase (BG), leucine aminopeptidase (LAP), and acid phosphatase (ACP) in bulk and hotspots in untreated soil (Control) and soil to which the bioplastic poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) was added. Values are means $(\\pm S\\mathbf{E})$ of four replicates. Different letters show significant differences between treatments $(p<0.05)$ .  \n\n# 3.3. Effect of PHBV on soil microbial growth rate  \n\nDifferent microbial growth patterns in response to substrate addition were observed among hotspots (microplastisphere and rhizosphere) and the bulk soil with and without PHBV addition (Fig. S3). The basal respiration (BR, $45{\\upmu}\\mathrm{g}\\mathrm{~C~g}^{-1}\\mathrm{~h}^{-1})$ and substrate-induced growth respi­ ration (SIGR, $58~{\\upmu\\mathrm{g}}\\subset{\\up g}^{-1}~{\\up h}^{-1})$ in the microplastisphere were 10 times and 12 times higher relative to the rhizosphere soil, respectively (Fig. 3a and b). However, the BR and SIGR in the bulk soil were not affected by PHBV addition compared to the control.  \n\nSoil respiration showed a clear response to PHBV addition both in the hotspots and in bulk soil (Fig. S3). PHBV addition decreased the maximum specific growth rate $\\mathbf{\\mu}(\\mu)$ by $22\\%$ in the microplastisphere compared to the bulk soil $(p<0.05$ ; Fig. 3c), whereas there was no difference in $\\mu$ between the microplastisphere and the PHBV-treated bulk soil $(p>0.05)$ . Despite a slower specific growth rate, a 6-fold in­ crease in the fraction of active microbial biomass, and a four times shorter lag period was observed in the microplastisphere vs. rhizosphere soil (Fig. 3d,e,f).  \n\n# 3.4. Effect of PHBV on soil bacterial community composition and diversity  \n\nThe dominant bacteria phyla were Actinobacteria, Proteobacteria, Acidobacteria, Firmicutes, Bacteroidetes, Chloroflexi, Thaumarchaeota, and Gemmatimonadetes in all treatment soils (Fig. 4A), which together encompassed ca. $96\\mathrm{-}98\\%$ of the bacterial reads. Although the dominant phyla in all soils were consistent, changes in the relative abundances of the dominant taxa were observed across the treatments. There was a higher abundance of Proteobacteria and Acidobacteria and a lower abundance of Firmicutes in soils with PHBV addition comparing with control treatment $(p<0.05$ , Fig. 4A). In the family level, the fraction of these 20 dominant families with highest relative abundance decreased after PHBV addition (Fig. 4B). Specifically, the addition of PHBV induced the decrease of Planococcaceae, Xanthomonadaceae, Bacillaceae and the increase of Chitinophagaceae, Comamonadaceae and Oxalo­ bacteraceae (Fig. 4B). The detailed family level changes of bulk and hotspot soil bacterial community induced by PHBV addition were also given in Fig. 5C. Of the 3800 OTUs detected across all samples, the major numbers of OTUs $(n=3622)$ ) were shared by control-bulk, PHBV-bulk, and PHBV-hotspots soils, while 54 OTUs were unique to PHBV-hotspots soil and 16 OTUs were unique to the PHBV-bulk soil (Fig. 5A).  \n\nThe mean values for ACE, Chao1, Richness, and Shannon indices in the PHBV-treated bulk soil increased by $10\\%$ , $11\\%$ , $16\\%$ , and $18\\%$ relative to the control soil, respectively (Fig. S5), while there were no differences between the microplastisphere and bulk soil after PHBV addition $(p>0.05)$ .  \n\n# 4. Discussion  \n\n# 4.1. Effect of PHBV on plant growth  \n\nIntact PHBV and its decomposition products are thought to be of very low cytotoxicity (Napathorn, 2014). In all the rhizoboxes amended with PHBV, however, all the plants eventually died during the 4-week experiment. This is consistent with previous reports showing that degradation of conventional and bio-based microplastics might nega­ tively affect plant growth when present in high quantities (Qi et al., 2018, 2020b; Zang et al., 2020). Given that bioplastic polymers are solely composed of C, O and H, it is likely that PHBV addition to soil induced microbial immobilization of essential nutrients (e.g., N, P) leading to increased plant stress (Volova et al., 2017; Boots et al., 2019). Such an N immobilization was further confirmed by the decreased DON but increased MBN in PHBV-added soil compared to the unamended control treatment (Table 1). This is consistent with Sander (2019) who found that microorganisms on the surface of microplastics need to ac­ quire N from the surrounding soil to fuel growth. It also suggests that PHBV may have stimulated opportunistic plant pathogens (Matavulj et al., 1992), however, more work is required to confirm this. An alternative explanation might be that PHBV induced phytotoxicity due to acidification of the soil because of the release of high quantities of 3-hydroxybutyric acid during PHBV degradation. However, this would normally affect root growth rather than shoot growth (Lucas et al., 2008). Further, based on the degradation of other biopolymers (e.g. cellulose, proteins), it is unlikely that an accumulation of the monomer will occur due to rapid microbial consumption (Jan et al., 2009). This is quite likely as it is a monomer which is naturally present as a microbial storage compound (Mason-Jones et al., 2019). However, it is possible that undisclosed additives or contaminants in the polymer might also have induced phytotoxicity (Zimmermann et al., 2019). Lastly, we cannot disregard other general changes in soil properties and microbial communities following PHBV addition which may also have inhibited plant growth, contributing to plant death (Saarma et al., 2003; Wen et al., 2020). We conclude, that contrary to expectation, commercially sourced PHBV was deleterious to plant growth, at least under higher contents of PHBV in the short term, as indicated by the lower seed germination over 7-days germination (Fig. S6). Further experiments are therefore needed to determine the mechanistic basis of this response.  \n\n![](images/d067317003264e844a274a03a1772c0aaa92201d95ee3cdf2d48d7bcf37acb1f.jpg)  \nFig. 3. Basal respiration (BR), substrate-induced growth respiration (SIGR), specific growth rate $\\mathbf{\\Pi}(\\mu)$ , total microbial biomass (TMB), the fraction of growing microbial biomass to total microbial biomass (GMB/TMB), and their lag time in bulk and hotspots in untreated soil (Control) and soil to which the bioplastic poly(3-hydroxybutyrate-co-3- hydroxyvalerate) (PHBV) was added. Values are means $(\\pm S\\mathbf{E})$ of four replicates. Letters show sig­ nificant differences between treatments $(p<0.05)$ .  \n\n![](images/251a6f1aebf040aca2b5085c3a34657303f5629bc2f021030935f47b7fd6bb38.jpg)  \nFig. 4. Stacked bar chart of the top 10 bacterial phyla with the largest mean relative abundance in untreated soil (Control-bulk), and bulk (PHBV-bulk) and hotspots (PHBV-hotspots) soils with the bioplastic poly (3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) addition (A). Stacked bar plot of the 20 families with largest mean relative abundance in all soil samples (B).  \n\n# 4.2. Effect of PHBV on soil microbial and enzymatic functional traits  \n\nSoil enzyme production is sensitive to both energy and nutrient availability (Allison et al., 2011). This notion was supported in our study where the input of bioavailable C (i.e. degradation products of PHBV) increased enzyme activities in hotspots by up to 2 times compared to the bulk soil. This increase in microbial activity is unsurprising given that poly-3-hydroxybutyrate is a common storage compound produced by a wide range of taxonomically different groups of microorganisms, particularly in response to N deficiency and cold stress (Obruca et al., 2016). Consequently, the ability to use PHBV-C is expected to be a widespread trait within the microbial community. For C- and N-degrading enzymes, the activity difference between hotspots and the bulk soil was 2–10 times larger when PHBV was added (Fig. 2a, c), demonstrating that bioplastic incorporation into the soil directly in­ fluences C and N cycling. The higher $V_{\\mathrm{max}}$ of $\\upbeta$ -glucosidase in the microplastisphere versus rhizosphere soil can be attributed to the faster growing biomass after PHBV addition (Fig. 3e). This is supported by the positive correlation between our measurement of the active microbial biomass and the $V_{\\mathrm{max}}$ of $\\upbeta$ -glucosidase $(\\mathrm{R}^{2}=0.7;$ Fig. S4). The increase in $\\upbeta$ -glucosidase also suggests that PHBV is stimulating the breakdown of other common soil polymers (i.e. cellulose). Further, PHBV could be broken down by depolymerases releasing hydroxybutyric acid mono­ mers which fuel the production of energetically expensive exoenzymes (i.e. leucine aminopeptidase; Fig. 2c) capable of degrading SOM to ac­ quire N for growth (i.e. positive priming; Zang et al., 2016; Zhou et al., 2020a, b). This was supported by a higher BR and SIGR in the micro­ plastisphere relative to the bulk soil (Fig. 3a and b), as well as the wider ratio of DOC and DON in the PHBV-treated soil (294) than in the control soil (1.77) (Table 1). In accordance with previous studies, N limitation also induced an increase in the catalytic properties $\\left(K_{\\mathrm{a}}\\right)$ of leucine aminopeptidase (Song et al., 2020). In line with this, the much shorter turnover time of substrates and higher $K_{\\mathrm{a}}$ of leucine aminopeptidase in the microplastisphere was observed compared to the rhizosphere (Figs. S2c and d), which suggests that the community was more limited by N than P in the microplastisphere. This could be supported by lower proportional activity of C- to N-cycling enzymes but higher proportional activity of C- to P-cycling enzymes in the microplastisphere versus the rhizosphere (Fig. S7). The lower vector angle in the microplastisphere further confirmed the microbial metabolisms were likely limited by soil N. We therefore hypothesize that due to N limitation the microbial community either (i) changed the intrinsic properties of their hydrolytic enzymes to adapt to the presence of the C-rich bioplastic, and/or (ii) that PHBV induced a shift in the soil microbial community and thus the types of enzymes being produced (Kujur and Patel, 2013). Overall, we conclude that N limitation is connected to microbial N immobilization due to stimulated microbial growth after C supply from PHBV. The C input stemming from the catabolism of PHBV will increase microbial biomass and intensify the N limitation. This was supported by the increased MBC and enzyme activities (especially N related), as well as the shift in enzymatic stoichiometric ratio and bacterial community.  \n\n![](images/f79538e996d9ec8c94afa7e86032678c39e19ff6a829637d7307a10c4229505d.jpg)  \nFig. 5. Venn diagram shows shared number of OTUs by untreated soil (Control-bulk), and bulk (PHBV-bulk) and hotspots (PHBV-hotspots) soils with the bioplastic poly (3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) addition (A). The taxonomical information for each node was given in an individual enlarged heatmap (B). Metacoder heatmap to family level across different treatment. Each node from the center (Kingdom) to outward (Family) represents different taxonomical levels (C). The map is weighted and colored-coded based on read abundance.  \n\nThis contrasts with C hotpots in the rhizosphere, where the supply of C is probably less and where N is also lost from root epidermal cells in the form of amino acids providing a more balanced nutrient supply to the microbial community (Jones et al., 2009).  \n\nHere we speculate that PHBV breakdown was initially limited by the availability of polyhydroxybutyrate depolymerase (Jendrossek et al., 1993). The abundance and level of expression of this enzyme in soil remains unknown, however, an NCBI search revealed its presence in a wide range of microbial taxa. Although PHB depolymerase may be internally targeted (i.e. to break down internal storage C), there is also an evidence that it can be externally targeted (i.e. be an exoenzyme; Jendrossek and Handrick, 2002), probably to degrade microbial nec­ romass (Handrick et al., 2004). Our data support the view that PHBV can be used as a sole C substrate by the bacterial community when supplied exogenously (Martinez-Tobon et al., 2018). However, we also observed a significant decrease $(22\\%)$ in microbial specific growth rate $\\mu$ in the microplastisphere compared to the rhizosphere, indicating the potential dominance of $K$ -strategy microorganisms. $K$ -strategists typically store more C in their cells and consume it slower (Nguyen and Guckert, 2001), lowering respiration rates. We therefore hypothesize that PHBV de­ graders break down PHBV exogenously into monomeric units which can then be subsequently transported into the cell where re-polymerization into PHB occurs (Shen et al., 2015). Consequently, microbial community structure in the microplastisphere shifted toward species with a lower affinity to oligosaccharides and peptides indicated by a higher $K_{\\mathrm{m}}$ of $\\upbeta$ -glucosidase and leucine aminopeptidase.  \n\n# 4.3. Effect of PHBV on soil bacterial community structure  \n\nPHBV addition was associated with an increase in the relative abundance of Acidobacteria, Proteobacteria and Chloroflexi, and a decrease in the relative abundance of Firmicutes, The latter have previ­ ously been described as fast-growing copiotrophs that thrive in envi­ ronments of high C availability (Cleveland et al., 2007; Jenkins et al., 2010). In contrast, Acidobacteria and Chloroflexi tend to dominate in oligotrophic environments where N availability is low (Ho et al., 2017). As mentioned, the release of high quantities of 3-hydroxybutyric acid during PHBV degradation may have also reduced the pH, thus favoring the growth of Acidobacteria. Nitrospirae are nitrite-oxidizing bacteria that are ubiquitous in terrestrial environments and that play a major role in biological N cycling and nitrification in agricultural soils (Xia et al., 2011). The higher abundance of Nitrospirae after PHBV addition indi­ cated a change in N cycling (Zecchin et al., 2018), which was attributed to greater nutrient limitation in the microplastisphere than in the bulk soil (as indicated by $V_{\\mathrm{max}}$ ratio of C-to-N cycling enzymes; 5.1 vs. 8.6) (Table S1). The relative proportion of Bacteroidetes also increased in the PHBV treatments. These largely copiotrophic organisms are widely distributed in soils, and are considered to be specialized in degrading complex organic matter (Huang et al., 2019). Thus, DOM pools increased in the PHBV-treated soil compared with bulk soil due to the release of monomeric compounds from PHBV degradation (Table 1). Although only bacterial communities were investigated in this study, it is likely that fungi and mesofauna populations are also greatly affected by PHBV addition and involved in its degradation. Further studies are required to gain a better insight into the complex interactions between these groups. Overall, our results highlight the potential of PHBV to trigger metabolic changes in soil microorganisms (Fig. 6), and thus potentially impact their functional role in soil (Huang et al., 2019). In addition to the microplastisphere, PHBV addition also changed the mi­ crobial community in the bulk soil, suggesting that these changes are not only confined to hotspots in the soil.  \n\n![](images/dcb1cbcc042a9e5fdc2b53645ba014d5e1fbd8eeedfaea8926cd7749e81329c5.jpg)  \nFig. 6. Conceptual diagram showing changes of microbial activities and functions in the hotspots as affected by poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV) addition. Vertical and horizontal red arrows indicate either an increase or no change of microbial exoenzyme kinetics and functions in the hotspots compared to the bulk soil, respectively. The red and orange gradient between the panels indicates the decreasing trend in enzyme activity $\\mathrm{(V_{max})}$ and substrate affinity $\\mathrm{(K_{m})}$ , respectively between the microplastisphere and the rhizosphere. The blue gradient indicates the increasing trend in microbial specific growth rate $(\\mu)$ . (For inter­ pretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)  \n\n# 5. Conclusion  \n\nMicrobial activity in agricultural soil is typically C-limited, such that even small C inputs can induce metabolic changes in the soil microbial community. Here we clearly showed that PHBV addition increased mi­ crobial activity, growth, and exoenzyme activity. This most likely leads to the enhanced mineralization of native SOM by co-metabolism, i.e. microorganisms degrade SOM by using degradable polymers as an en­ ergy source. Remarkably, greater enzyme activity and microbial biomass, and lower affinity for the substrate were observed in the microplastisphere compared to the rhizosphere, indicating a stronger and faster C and nutrient turnover with PHBV addition in hotspots. Taken together, the unique environment may benefit microbial survival in PHBV-treated soil compared with the rhizosphere, possibly altering the soil ecological functions and biogeochemical processes, which may result in a stimulation of soil C and nutrients cycling. Although bio­ plastics have been heralded as a solution to petroleum-based plastics, our research indicates that it is also important to consider the potential drawbacks of bioplastics, e.g., for plant growth and health. This is exemplified in the use of plastic microbeads in cosmetics and plastic mulch films in agriculture where the negative environmental conse­ quences were only realized decades after their introduction (Sintim and Flurt, 2017; Qi et al., 2020b). Our research was designed to understand the short-term impact of a localized PHBV hotspot in soil. It is clear, however, that longer-term field-scale studies are also required. In-field testing of biodegradation of PHBV under different scenarios (e.g., soil types, agricultural practice, climate changes) as well as using a realistic mixture of polymers over longer periods is therefore required, with particular attention to plant-soil-microbial interactions.  \n\n# Declaration of competing interest  \n\nThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.  \n\n# Acknowledgements  \n\nThis work was supported by the China Agriculture Research System (CARS07-B-5) and the UKRI Global Challenges Research Fund (GCRF) project awarded to Bangor University (NE/V005871/1). We also would like to thank the UK-China Virtual Joint Center for Agricultural Nitrogen (CINAg, BB/N013468/1), which is jointly supported by the Newton Fund, via UK Biotechnology and Biological Sciences Research Council and Natural Environment Research Council, and the Chinese Ministry of Science and Technology. Jie Zhou would like to thank the support from the China Scholarship Council (CSC). Heng Gui would like to thank the National Natural Science Foundation of China (NSFC Grant 32001296) and Yunnan Fundamental Research Projects (Grant No. 2019FB063). The authors would like to thank Karin Schmidt for her laboratory assistance. The authors also thank the editor and two anonymous re­ viewers for their insightful comments.  \n\n# Appendix A. Supplementary data  \n\nSupplementary data to this article can be found online at https://doi. org/10.1016/j.soilbio.2021.108211.  \n\n# References  \n\nAllison, S.D., Weintraub, M.N., Gartner, T.B., Waldrop, M.P., 2011. 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  {
    "id": "10.1103_PhysRevX.12.031042",
    "DOI": "10.1103/PhysRevX.12.031042",
    "DOI Link": "http://dx.doi.org/10.1103/PhysRevX.12.031042",
    "Relative Dir Path": "mds/10.1103_PhysRevX.12.031042",
    "Article Title": "Beyond Conventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry",
    "Authors": "Smejkal, L; Sinova, J; Jungwirth, T",
    "Source Title": "PHYSICAL REVIEW X",
    "Abstract": "Recent series of theoretical and experimental reports have driven attention to time-reversal symmetry -breaking spintronic and spin-splitting phenomena in materials with collinear-compensated magnetic order incompatible with conventional ferromagnetism or antiferromagnetism. Here we employ an approach based on nonrelativistic spin-symmetry groups that resolves the conflicting notions of unconventional ferromagnetism or antiferromagnetism by delimiting a third basic collinear magnetic phase. We derive that all materials hosting this collinear-compensated magnetic phase are characterized by crystal-rotation symmetries connecting opposite-spin sublattices separated in the real space and opposite-spin electronic states separated in the momentum space. We describe prominent extraordinary characteristics of the phase, including the alternating spin-splitting sign and broken time-reversal symmetry in the nonrelativistic band structure, the planar or bulk d-, g-, or i-wave symmetry of the spin-dependent Fermi surfaces, spin -degenerate nodal lines and surfaces, band anisotropy of individual spin channels, and spin-split general, as well as time-reversal invariant momenta. Guided by the spin-symmetry principles, we discover in ab initio calculations outlier materials with an extraordinary nonrelativistic spin splitting, whose eV-scale and momentum dependence are determined by the crystal potential of the nonmagnetic phase. This spin -splitting mechanism is distinct from conventional relativistic spin-orbit coupling and ferromagnetic exchange, as well as from the previously considered anisotropic exchange mechanism in compensated magnets. Our results, combined with our identification of material candidates for the phase ranging from insulators and metals to a parent crystal of cuprate superconductors, underpin research of novel quantum phenomena and spintronic functionalities in high-temperature magnets with light elements, vanishing net magnetization, and strong spin coherence. In the discussion, we argue that the conflicting notions of unconventional ferromagnetism or antiferromagnetism, on the one hand, and our symmetry-based delimitation of the third phase, on the other hand, favor a distinct term referring to the phase. The alternating spin polarizations in both the real-space crystal structure and the momentum-space band structure characteristic of this unconventional magnetic phase suggest a term altermagnetism. We point out that d-wave altermagnetism represents a realization of the long-sought-after counterpart in magnetism of the unconventional d-wave superconductivity.",
    "Times Cited, WoS Core": 349,
    "Times Cited, All Databases": 356,
    "Publication Year": 2022,
    "Research Areas": "Physics",
    "UT (Unique WOS ID)": "WOS:000865310200001",
    "Markdown": "# Beyond Conventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry  \n\nLibor Šmejkal,1,2 Jairo Sinova,1,2 and Tomas Jungwirth 2,3 1Institut für Physik, Johannes Gutenberg Universität Mainz, 55128, Mainz, Germany   \n2Institute of Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00, Praha 6, Czech Republic   \n3School of Physics and Astronomy, University of Nottingham, NG7 2RD, Nottingham, United Kingdom  \n\n![](images/bbffaff04f153d6fc44b3ab0cee97a7e457085a9d6d7fdd1932e06c568ee5257.jpg)  \n\n(Received 6 February 2022; revised 6 April 2022; accepted 11 August 2022; published 23 September 2022)  \n\nRecent series of theoretical and experimental reports have driven attention to time-reversal symmetrybreaking spintronic and spin-splitting phenomena in materials with collinear-compensated magnetic order incompatible with conventional ferromagnetism or antiferromagnetism. Here we employ an approach based on nonrelativistic spin-symmetry groups that resolves the conflicting notions of unconventional ferromagnetism or antiferromagnetism by delimiting a third basic collinear magnetic phase. We derive that all materials hosting this collinear-compensated magnetic phase are characterized by crystal-rotation symmetries connecting opposite-spin sublattices separated in the real space and opposite-spin electronic states separated in the momentum space. We describe prominent extraordinary characteristics of the phase, including the alternating spin-splitting sign and broken time-reversal symmetry in the nonrelativistic band structure, the planar or bulk $d\\cdot$ -, $g-$ , or $i$ -wave symmetry of the spin-dependent Fermi surfaces, spindegenerate nodal lines and surfaces, band anisotropy of individual spin channels, and spin-split general, as well as time-reversal invariant momenta. Guided by the spin-symmetry principles, we discover in ab initio calculations outlier materials with an extraordinary nonrelativistic spin splitting, whose eV-scale and momentum dependence are determined by the crystal potential of the nonmagnetic phase. This spinsplitting mechanism is distinct from conventional relativistic spin-orbit coupling and ferromagnetic exchange, as well as from the previously considered anisotropic exchange mechanism in compensated magnets. Our results, combined with our identification of material candidates for the phase ranging from insulators and metals to a parent crystal of cuprate superconductors, underpin research of novel quantum phenomena and spintronic functionalities in high-temperature magnets with light elements, vanishing net magnetization, and strong spin coherence. In the discussion, we argue that the conflicting notions of unconventional ferromagnetism or antiferromagnetism, on the one hand, and our symmetry-based delimitation of the third phase, on the other hand, favor a distinct term referring to the phase. The alternating spin polarizations in both the real-space crystal structure and the momentum-space band structure characteristic of this unconventional magnetic phase suggest a term altermagnetism. We point out that $d$ -wave altermagnetism represents a realization of the long-sought-after counterpart in magnetism of the unconventional $d$ -wave superconductivity.  \n\nDOI: 10.1103/PhysRevX.12.031042  \n\nSubject Areas: Condensed Matter Physics, Magnetism, Spintronics  \n\n# I. INTRODUCTION  \n\nRecent predictions of time-reversal symmetry breaking [1–3] and spin splitting [1,2,4–12] in electronic bands, typical of ferromagnets, in materials with collinearcompensated magnetic order, typical of antiferromagnets, are incompatible with the conventional division into the ferromagnetic and antiferromagnetic phases. The consequences of the intriguing electronic structure of these collinear-compensated magnets have been illustrated by predictions of odd-under-time-reversal responses, including anomalous Hall and Kerr effects [1–3,9,13–15], as well as spin current, giant and tunneling magnetoresistance, and spin-torque phenomena [6,11,12,16–18]. Some of the predictions of these unexpected responses have been already supported by experiments [9,19–22].  \n\nIn this article, we resolve the conflicting notions of unconventional ferromagnetism or unconventional antiferromagnetism by deriving that on the basic level of uncorrelated nonrelativistic nonfrustrated (collinear) magnetism, symmetry allows for three instead of two distinct phases. We employ a symmetry approach based on a nonrelativistic spin-group formalism [23–25]. To explain its merits, we first recall the conventional theory frameworks.  \n\nA traditional approach to the basic categorization of a magnetic materials phases based on models focusing on spatial and spin arrangements of magnetic atoms alone while omitting nonmagnetic atoms in the lattice can be traced back to the seminal works on Ne´el’s collinear antiferromagnetism [26]. Subsequently, the approach was employed, e.g., when discussing the competition of Ne´el’s antiferromagnetism and the spin-liquid phase in the context of high-temperature cuprate superconductors [27]. Recently, models considering clusters of magnetic atoms have underpinned the multipole theory of the anomalous Hall effect in noncollinear-compensated magnets [28], and of the nonrelativistic spin splitting in collinearcompensated magnets [5,8]. However, these models are principally incapable of providing a general classification and description of the underlying magnetic phases in prominent families of materials. A specific example is rutile crystals with the collinear-compensated magnetic order [1,4,7], in which the nonmagnetic atoms have been recognized to play a key role in the anomalous timereversal symmetry-breaking spin phenomena [1,3]. In particular, $\\mathrm{RuO}_{2}$ is a prominent room-temperature metallic member of this rutile family, in which the unconventional spin physics and spintronics have already been studied both theoretically and experimentally [1,4,16–18,20–22].  \n\nA traditional symmetry description of the full structure of magnetic crystals, including the nonmagnetic atoms, considers transformations in coupled real physical space and the space of magnetic moment vectors. In other words, the transformations acting on the coordinates of the atoms, subject to the standard crystallographic restrictions, simultaneously act on the components of the magnetic moment vectors [24,25,29] (see Supplemental Material Sec. I Fig. S1 [30]). This symmetry formalism naturally arises from the classical orbital-current model of magnetic moments [31], as well as from the relativistic quantummechanical description of coupled spin and orbital degrees of freedom of electrons [31,32]. The corresponding magnetic groups [29,31,33–35] have been broadly applied in the research of equilibrium and nonequilibrium phenomena, including their modern topological variants [36–38], and have represented the primary tool for a systematic classification of hundreds of magnetic structures in materials databases [35,39].  \n\nMagnetic groups are indispensable for the description of effects governed by relativistic physics. However, the inherent relativistic nature of the magnetic-group symmetry transformations in coupled real and spin space makes the magnetic groups generally unsuitable for the classification of nonrelativistic phenomenology, which typically plays the leading role in magnetism [23,40]. Magnetic space groups of type II describing time-reversal invariant crystals without a magnetic order, are an exception for which a transition to a nonrelativistic physics description in decoupled spin and real space can be generally performed by making a direct product with the SU(2) group of spinspace rotation transformations [24,28]. For the remaining magnetic space groups of types I, III, and IV encompassing collinear as well as noncollinear magnets [7,10], a transition to the nonrelativistic physics description is not available [5,8,28]. Therefore, the strong nonrelativistic spinsplitting phenomena are not generally described by magnetic groups augmented by spin-space transformations [7,10].  \n\nIn this article, we use an approach to rigorously and systematically classify and describe nonrelativistic magnetic materials phases and their physical properties based on the spin-group formalism [23–25] of symmetry transformations in decoupled real and spin space. The spin groups are a generalization of the conventional magnetic groups [23–25]. They consider pairs of transformations $[R_{i}\\backslash|R_{j}]$ , where the transformations on the left of the double vertical bar act only on the spin space and on the right of the double vertical bar only on the real space [23–25] (see Supplemental Material Sec. I Fig. S1 [30]). The symmetry landscape of the spin groups is much richer because, in general, different rotation transformations can simultaneously act on the spin and real space, and only the transformations in the real space are crystallographically restricted. (The same rotation transformations simultaneously acting on the spin and real space are contained in both magnetic and spin groups.) Despite their richness, studies based on the spin symmetries have appeared only sporadically in the literature. For example, in the past they were used for the classification of possible spin arrangements on crystals and spin dynamics, with an emphasis on complex noncollinear or disordered structures, while not focusing on the electronic structure [40]. Very recently, they have been applied in studies of magnons [41] or topological quasiparticles [42–45]. Overall, however, the spin-group formalism has remained largely unexploited and undeveloped [46].  \n\nThe nonrelativistic spin groups represent an example of approximate or so-called “hidden” symmetries in the sense that relativistic effects are generally present in all magnets. The key significance of the nonrelativistic spin groups is that they can offer a systematic symmetry description of physics that is commonly leading in magnetism and that arises from the strong nonrelativistic electromagnetic crystal potentials [23,40]. Here, by the electric crystal potential, we refer to the internal potential in the nonmagnetic phase of the crystal, as described, e.g., by the local density approximation of the density-functional theory (DFT); by the additional magnetic component, we refer to the modification of the internal crystal potential due to the transition to the magnetically ordered phase. Since the magnetic groups represent only a small subset of the spin groups [25], they are prone to omitting prominent magnetic phases dominated by the nonrelativistic electromagnetic crystal potentials. For example, the magnetic groups, which generally encompass collinear and noncollinear magnets, can determine only whether a net magnetization is allowed or not, but do not distinguish ferromagnets from antiferromagnets in which magnetization arises only as a weak relativistic perturbation [31]. For the band structures, the magnetic groups can be used to identify a violation of Kramers spin degeneracy [10,36,47–52]. However, both the magnetic-group formalism and Kramers theorem [53,54] entangle nonrelativistic and relativistic physics. Consequently, the nonrelativistic spin splitting in materials from the magnetic groups violating the Kramers spin degeneracy were identified by performing numerical DFT calculations with the relativistic spin-orbit coupling turned off [10].  \n\nIn our work, by employing and developing the spingroup formalism, we derive three distinct phases of nonrelativistic collinear magnetism: The first phase has one spin lattice (or opposite-spin sublattices not connected by any symmetry transformation). It corresponds to conventional ferromagnetism (ferrimagnetism) [31]. The second phase has opposite-spin sublattices connected by translation or inversion (or both), and corresponds to conventional antiferromagnetism [23,26,41]. The third phase has opposite-spin sublattices connected by rotation (proper or improper and symmorphic or nonsymmorphic) but not connected by translation or inversion. Unlike the conventional ferromagnetic phase with a nonrelativistic magnetization and spin-split bands that break time-reversal symmetry [31], and unlike the conventional antiferromagnetic phase with nonrelativistic spin-degenerate timereversal invariant bands and zero net magnetization [26,32,55–58], the third phase has split but equally populated spin-up and spin-down energy isosurfaces in the band structure that break time-reversal symmetry. The spingroup formalism allows us to provide a complete classification and description of the specifics of the spinmomentum locking in the band structure of the third phase. Our direct link of the spin groups to real material candidates establishes that the third phase is abundant. We also show that it is a strong, robust, and fundamental phase, as it does not require (but can coexist with) relativistic spin-orbit coupling, electronic correlations, or magnetic fluctuations or frustrations. We point out that our classification and description based on the spin-group formalism are universally applicable to any effective single-particle Kohn-Sham Hamiltonian, as well as for the Dyson-equation description of correlated or disordered systems.  \n\nPrinciples based on the spin-group symmetries guide us to our discovery of outlier materials hosting the third phase, with an extraordinary microscopic spin-splitting mechanism, whose eV scale and momentum dependence are determined by the electric crystal potential, i.e., by the scale and momentum dependence of the band splitting of the nonmagnetic phase. It is fundamentally distinct from the earlier-considered various internal magnetic-interaction mechanisms [4,7,8,17,59,60], such as the anisotropic spin-dependent hopping in the magnetic state [8,17]. The spin-splitting mechanism in the third magnetic phase by the electric crystal potential is nonrelativistic and accompanied by zero net magnetization. Therefore, it also starkly contrasts with the conventional mechanisms of the ferromagnetic splitting due to the nonzero net magnetization, or the relativistic spin-orbit splitting due to the broken inversion symmetry. It opens a new paradigm for designing spin quantum phases of matter based on the strong crystalpotential effects complementing the widely explored relativistic or many-body correlation phenomena [61].  \n\nThe focus of our work is on the classification and description of the nonrelativistic band structures of materials hosting the third phase and the identification of new material candidates, which opens a range of potential science and technology implications of this magnetic phase. In the Supplemental Material Sec. I [30], we briefly comment on the links to relativistic effects and noncollinear magnetism [3].  \n\n# II. DERIVATION OF SPIN-GROUP CATEGORIZATION OF NONRELATIVISTIC COLLINEAR MAGNETISM  \n\nWe start with the derivation of the three distinct spingroup types describing, respectively, the three nonrelativistic phases of collinear magnets. In general, spin groups can be expressed as a direct product $\\mathbf{r}_{s}\\times\\mathbf{R}_{s}$ of so-called spin-only group $\\mathbf{r}_{s}$ containing transformations of the spin space alone, and so-called nontrivial spin groups ${\\bf R}_{s}$ containing the elements $[R_{i}\\backslash|R_{j}]$ , but no elements of the spin-only group [24,25]. For the collinear spin arrangements on crystals, the spin-only group is given by [24,25] $\\mathbf{r}_{s}=\\mathbf{C}_{\\infty}+\\bar{C}_{2}\\mathbf{C}_{\\infty}$ . Here, $\\mathbf{C}_{\\infty}$ is a group representing all rotations of the spin space around the common axis of spins, and $\\bar{C}_{2}$ is a $180^{\\circ}$ rotation around an axis perpendicular to the spins, combined with the spin-space inversion. We recall that the spin-space inversion in the spin groups enters via the time reversal [24,25,40]. We also again emphasize here that the relativistic magnetic space groups encompass collinear, as well as general noncollinear magnets [7,10]. This implies that, e.g., a conjecture based on the spin-space $\\bar{C}_{2}$ symmetry that nonrelativistic spin splitting is generally excluded in materials belonging to the type IV magnetic space groups [7,10] is invalid.  \n\nThe form of the nonrelativistic spin-only group $\\mathbf{r}_{s}$ for collinear magnets has two basic general implications independent of the specific nontrivial spin group ${\\bf R}_{s}$ . The first implication follows from $\\mathbf{C}_{\\infty}$ . This symmetry makes spin a good quantum number with a common quantization axis independent of the crystal momentum across the nonrelativistic band structure. The electronic structure is thus strictly separated into nonmixing spin-up and spin-down channels.  \n\nThe second implication follows from the $\\bar{C}_{2}$ symmetry in the spin-only group of collinear magnets. Since the spinspace inversion enters via the time reversal [24,25,40], it is accompanied by a time reversal in the real space $(\\mathcal{T})$ . Although $\\tau$ acts as an identity on the real-space coordinates of the atoms, it flips the sign of the crystal momentum. This is important for the band-structure spin symmetries. In particular, we now use the symmetry $\\big[\\bar{C}_{2}\\big|\\big|\\mathcal{T}\\big]$ , which follows directly from the above spin-only group symmetry of the collinear magnets and from the simultaneous action of the time reversal on the spin and real (momentum) space. When applying the transformation $[\\bar{C}_{2}||\\mathcal{T}]$ on spin $(s)$ and crystal-momentum $(\\mathbf{k})$ -dependent bands $\\epsilon(s,{\\bf k})$ , we obtain $[\\bar{C}_{2}\\|\\mathcal{T}]\\epsilon(s,\\mathbf{k})=\\epsilon(s,-\\mathbf{k})$ . We see that the $[\\bar{C}_{2}||\\mathcal{T}]$ transformation acts the same way on $\\epsilon(s,{\\bf k})$ as the real-space inversion. Next, since $\\big[\\bar{C}_{2}\\big|\\big|\\mathcal{T}\\big]$ is a symmetry of nonrelativistic collinear spin arrangements on crystals, $[\\bar{C}_{2}\\|{\\cal T}]\\epsilon(s,{\\bf k})=\\epsilon(s,{\\bf k})$ , and hence, $\\epsilon(s,{\\bf k})=$ $\\epsilon(s,-\\mathbf{k})$ . We derive that the nonrelativistic bands of all collinear magnets are invariant under real-space (crystalmomentum) inversion not only in inversion-symmetric collinear magnets [1,7,8], but even if the crystals lack the real-space inversion symmetry.  \n\nWe now move on to the nontrivial spin groups. While the above spin-only group is common to all nonrelativistic collinear magnets, we derive three different types of the nontrivial spin groups corresponding, respectively, to the three distinct phases. The nontrivial spin groups are obtained by combining groups of spin-space transformations with groups of real-space crystallographic transformations [24,25]. Regarding the groups of spin-space transformations, there can be some freedom in their choice [24,25]. For the collinear spin arrangements, one of the two spin-space transformation groups is ${\\bf S}_{1}=\\{E\\}$ ; i.e., it contains just the spin-space identity [24]. We choose the second group in the form of $\\mathbf{S}_{2}=\\{E,C_{2}\\}$ which is favorable for our derivation of the categorization into the three phases of nonrelativistic collinear magnets. The group contains the spin-space identity and the $180^{\\circ}$ rotation of the spin space around an axis perpendicular to the spins. (We note that because of the above spin-only group symmetry element $\\bar{C}_{2}$ , and because the product of spinspace transformations $\\bar{C}_{2}C_{2}$ is equal to the spin-space inversion, an alternative choice [24] of $\\mathbf{S}_{2}$ contains the spin-space inversion instead of $C_{2}$ .)  \n\nAfter introducing the spin-only group and the spin-space transformations in the nontrivial spin groups, we move on to the real-space crystallographic transformations in the nontrivial spin groups. The procedure of constructing the nontrivial spin groups applies equally when considering crystallographic space groups (i.e., those containing also translations) or crystallographic point groups (i.e., those where the translations are replaced by identity). To categorize the nonrelativistic collinear magnets based on their magnetic crystal structure, we need to consider the crystallographic space groups. However, to make our manuscript concise, we do not explicitly list all nontrivial spin groups constructed from the crystallographic space groups. This is because the physical consequence of the third phase that we focus on in this work is the spin-momentum locking in the nonrelativistic band structure. In other words, we focus on determining which momenta in the Brillouin zone have spin-degenerate eigenstates protected by the spin-group symmetries, and for which momenta the spin-group symmetries allow for lifting the spin degeneracy. For all collinear spin arrangements on crystal independent of the crystal’s real-space translation symmetries, and independent of whether the crystal does or does not have the real-space inversion symmetry, the spinmomentum locking is described by the direct product of the spin-only group and nontrivial spin groups constructed from the crystallographic point groups containing the real-space inversion symmetry (crystallographic Laue groups).  \n\nThe general independence of the spin-momentum locking of translations is a consequence of the strict separation of the nonrelativistic band structure into nonmixing spin-up and spin-down channels protected by the spin-only group symmetries of the collinear magnets. The separate spin-up and spin-down channels then have equal energies at a given momentum $\\mathbf{k}$ in the Brillouin zone when the nontrivial spin group contains a symmetry element $\\big[C_{2}\\big|\\big|R\\big]$ , where $R$ transforms the momentum $\\mathbf{k}$ on itself or a momentum separated from $\\mathbf{k}$ by a reciprocal lattice vector ( $R$ belongs to the little group of $\\mathbf{k}$ ). Since $\\mathbf{k}$ is invariant under translations, the spin degeneracy at a given momentum $\\mathbf{k}$ is protected by ${\\big[}C_{2}{\\big|}|R{\\big]}$ irrespective of whether $R$ does or does not contain a translation. Note that additional band degeneracies can exist within one spin channel, i.e., degeneracies in band indices other than spin that are protected by crystallographic space-group symmetries. These features, whose systematic study is beyond the scope of our present manuscript, can be readily included in the symmetry analysis based on the nonrelativistic spin-group formalism and can be important when, e.g., exploring exotic (topological) quasiparticles near such degeneracy points [42–45].  \n\nThe general invariance of bands of nonrelativistic collinear magnets under real-space (crystal-momentum) inversion is derived above from the spin-only group symmetry $[\\bar{C}_{2}\\vert\\vert\\mathcal{T}]$ . Later in the text, we give specific examples of the inversion-symmetric spin-momentum locking in the band structures of the third phase in crystals with or without the inversion symmetry.  \n\nBy using the isomorphism theorem [24], we construct all the nontrivial spin (Laue) groups, whose elements on the left of the double vertical bar form a group of the spinspace transformations and on the right of the double vertical bar a (Laue) group of the real-space crystallographic transformations. It implies the procedure of combining all isomorphic coset decompositions of the two groups, i.e., decompositions with the same number of cosets for the two groups [24]. (A coset decomposition of a group $\\mathbf{X}$ is given by ${\\bf X}={\\bf x}+X_{1}{\\bf x}+X_{2}{\\bf x}+\\cdot\\cdot\\cdot_{3}$ , where $\\mathbf{x}$ is a subgroup of $\\mathbf{X}$ , and $X_{i}$ are elements of $\\mathbf{X}$ [24].) The details of our derivation are in the Supplemental Material Sec. II [30]. Here we summarize the result in which all the nontrivial spin Laue groups describing $\\epsilon(s,{\\bf k})$ of collinear magnets are arranged into the following three distinct types using the isomorphic coset decompositions.  \n\nThe first type of nontrivial spin Laue group is given by $\\mathbf{R}_{s}^{\\mathrm{I}}=[E\\vert\\vert\\mathbf{G}]$ , where $\\mathbf{G}$ are the crystallographic Laue groups. Because there are 11 different crystallographic Laue groups, there are also 11 different ${\\bf R}_{s}^{\\mathrm{I}}$ groups. As highlighted in Fig. 1, the ${\\bf R}_{s}^{\\mathrm{I}}$ groups do not imply spin degeneracy of $\\epsilon(s,{\\bf k})$ at any $\\mathbf{k}$ -point. They describe nonrelativistic spin-split band structures with broken time-reversal symmetry and nonzero magnetization corresponding to conventional collinear ferromagnets (ferrimagnets) whose magnetic crystal structure contains one spin lattice (or the opposite-spin sublattices are not connected by any spin-space-group transformation).  \n\nThe second type of nontrivial spin Laue group is given by $\\mathbf{R}_{s}^{\\mathrm{II}}=[E\\lVert\\mathbf{G}]+[C_{2}\\lVert\\mathbf{G}]$ . Here, the $\\big[C_{2}\\big|\\big|E\\big]$ symmetry (recall that $\\mathbf{G}$ is a group containing the real-space identity $E$ element) implies spin degeneracy of $\\epsilon(s,{\\bf k})$ for all $\\mathbf{k}$ -vectors in the Brillouin zone. The 11 different ${\\bf R}_{s}^{\\mathrm{II}}$ groups describe nonrelativistic spin-degenerate time-reversal invariant band structures with zero magnetization of conventional collinear antiferromagnets (see Fig. 1). The corresponding antiferromagnetic spin arrangements on crystals have a symmetry $\\left[C_{2}\\middle|\\middle|\\mathbf{t}\\right]$ in their spin-space group, which interchanges atoms and rotates the spin by $180^{\\circ}$ between opposite-spin sublattices. Here, t on the right side of the double vertical bar is a real-space translation. Examples [62,63] are antiferromagnets FeRh or $\\mathrm{MnBi}_{2}\\mathrm{Te}_{4}$ . The ${\\bf R}_{s}^{\\mathrm{II}}$ groups also describe nonrelativistic spin-degenerate collinear antiferromagnetism in crystals with the opposite-spinsublattice transformation symmetry $\\big[C_{2}\\big|\\big|\\bar{E}\\big],$ where $\\bar{\\boldsymbol{E}}$ on the right side of the double vertical bar is the real-space inversion. This is because of the spin-only group symmetry $[\\bar{C}_{2}\\vert\\vert\\mathcal{T}]$ that implies the inversion symmetry of the bands, i.e., that the bands in all nonrelativistic collinear magnets are invariant under the transformation $[E\\rVert\\bar{E}]$ . Symmetries $\\big[C_{2}\\big|\\big|\\bar{E}\\big]$ and $[E\\rVert\\bar{E}]$ give the $\\big[C_{2}\\big|\\big|E\\big]$ symmetry that implies the spin degeneracy across the Brillouin zone (for a more detailed derivation, see Supplemental Material Sec. II [30]). Here, the examples [50,64] are antiferromagnets CuMnAs or $\\mathrm{Mn}_{2}\\mathrm{Au}$ .  \n\n![](images/888c42fcb587980d09ecb6ab6fa5c30061ca36e0c695a5b92ed56bbf3a197ec3.jpg)  \nFIG. 1. Illustration (in columns) of the three nonrelativistic collinear magnetic phases. Top box: Illustrative collinear spin arrangements and magnetization densities on crystals. Opposite spin directions are depicted by blue and red color. Spin arrows are placed outside the real-space cartoons to highlight that the overall spin axis orientation is not related to the real space coordinates for the nonrelativistic spin-group symmetries. 1st magnetic phase (conventional ferromagnetism) crystal corresponds to Fe, 2nd magnetic phase (conventional antiferromagnetism) to MnPt, and the 3rd unconventional magnetic phase (altermagnetism) to $\\mathrm{RuO}_{2}$ . Magenta arrow and magenta label highlight opposite-spin-sublattice transformation symmetries characteristic of the 2nd magnetic phase (real-space translation or inversion) and the 3rd magnetic phase (real-space rotation). Bottom box: Cartoons of band-structures and corresponding energy iso-surfaces show ferromagnetically spin-split bands (opposite spin states depicted by blue and red color), a spin-degenerate antiferromagnetic band, and bands in the 3rd magnetic phase with alternating sign of the spin splitting. The opposite-spinsublattice transformation of the spin Laue group which maps the same-energy eigenstates with opposite spins on the same $\\mathbf{k}$ -vector in the 2nd magnetic phase and on different k-vectors in the 3rd magnetic phase is again highlighted. The remaining rows give the spin Laue group structure for the given phase with the number of different groups in brackets, and presence/absence of time-reversal-symmetry breaking, compensation and $d\\mathrm{-},g\\mathrm{-}$ , and $i$ -wave symmetries.  \n\nThe remaining third distinct type of nontrivial spin Laue group describes the third magnetic phase and is given by  \n\n$$\n\\mathbf{R}_{s}^{\\mathrm{III}}=[E\\|\\mathbf{H}]+[C_{2}\\|A][E\\|\\mathbf{H}]=[E\\|\\mathbf{H}]+[C_{2}\\|\\mathbf{G}-\\mathbf{H}].\n$$  \n\nHere, $\\mathbf{H}$ is a halving subgroup of the crystallographic Laue group $\\mathbf{G}$ and the coset $\\mathbf{G}-\\mathbf{H}=A\\mathbf{H}$ is generated by transformations $A$ that can be only real-space proper or improper rotations and cannot be real-space inversion. (Note that this implies that the real-space inversion that is always present in $\\mathbf{G}$ is contained in H.) We see from Eq. (1) that for $\\mathbf{R}_{s}^{\\mathrm{III}}$ , $\\mathbf{G}$ is expressed as a sublattice coset decomposition, where the halving subgroup H contains only the real-space transformations which interchange atoms between same-spin sublattices, and the coset $\\mathbf{G}-\\mathbf{H}$ contains only the real-space transformations which interchange atoms between opposite-spin sublattices. The third magnetic phase corresponds to the magnetic crystal structures in which opposite-spin sublattices are connected by rotation (proper or improper and symmorphic or nonsymmorphic) and are not connected by translation or inversion.  \n\nThe third-phase magnets have nonrelativistic spin-split band structures with broken time-reversal symmetry and zero magnetization [1] (see Fig. 1). The broken timereversal symmetry is seen when multiplying $\\left[C_{2}\\big|\\big|\\mathbf{G}-\\mathbf{H}\\right]$ by the spin-only group symmetry $\\begin{array}{r l}{[\\bar{C}_{2}\\|\\mathcal{T}].}\\end{array}$ , which gives $\\big[\\bar{E}\\big|\\big|\\mathcal{T}(\\mathbf{G}-\\mathbf{H})\\big]$ [or equivalently, $[T||T(\\mathbf{G}-\\mathbf{H})]]$ ; i.e., spin groups of the third type do not contain the time-reversal symmetry element. (Recall that the coset $\\mathbf{G}-\\mathbf{H}$ does not contain the identity element.) Lifted spin degeneracies in the $\\pmb{\\mathrm{R}}_{s}^{\\mathrm{III}}$ groups are allowed for crystal momenta whose little group does not contain $A\\mathbf{H}$ elements. They satisfy $A\\mathbf{H}\\mathbf{k}=$ $\\mathbf{k}^{\\prime}\\neq\\mathbf{k}$ , implying that $\\epsilon(s,\\mathbf{k})=[C_{2}\\|A\\mathbf{H}]\\epsilon(s,\\mathbf{k})=\\epsilon(-s,\\mathbf{k}^{\\prime})$ (see Fig. 1). It guarantees that the spin-up and spin-down energy isosurfaces are split, but have the same number of states. These nonrelativistic band-structure signatures of the $\\pmb{\\mathrm{R}}_{s}^{\\mathrm{III}}$ phase are unparalleled in the ${\\bf R}_{s}^{\\mathrm{I}}$ or ${\\bf R}_{s}^{\\mathrm{II}}$ phases. Simultaneously, there are ten different $\\mathbf{R}_{s}^{\\mathrm{III}}$ groups which is comparable to the number of ${\\bf R}_{s}^{\\mathrm{I}}$ or ${\\bf R}_{s}^{\\mathrm{II}}$ groups, suggesting that the third phase is abundant. The ten nontrivial spin Laue groups of the third phase are listed in Fig. 2, where we adopt Litvin’s notation of the spin groups [25], with the upper index 1 refers to the spin-space identity and the upper index 2 to the spin-space rotation $C_{2}$ . Note that they are constructed from only eight different crystallographic Laue groups. However, the third-phase spin Laue groups cannot be constructed for the three remaining crystallographic  \n\nLaue groups, namely, from $\\mathbf{G}=\\bar{1}$ , 3; or $m3$ . In the Supplemental Material Sec. II Table S1 [30], we list all 37 nontrivial spin point groups of the third magnetic phase, together with their corresponding ten nontrivial spin Laue groups.  \n\nBefore moving to the analysis of the spin-momentum locking protected by the symmetries of the third-phase spin groups, we emphasize the additional differences from the magnetic groups. The latter are constructed by combining crystallographic groups (with the same transformations acting simultaneously on coordinates of atoms and components of magnetic moment vectors) with one group containing the identity element alone, and a second group containing the identity and the time reversal. Comparing this construction to the spin-group formalism with $\\mathbf{S}_{1}$ also containing only the identity element and $\\mathbf{S}_{2}$ with again two elements, implies that for describing all magnetic structures, the relativistic symmetry formalism has the same number of different magnetic groups as is the number of different nonrelativistic spin groups describing exclusively collinear spin arrangements. For the Laue (point) groups, the total number is 32 (122). Our nonrelativistic spin groups then split into 11 (32) nontrivial spin Laue (point) groups of the ferromagnetic phase, 11 (53) of the antiferromagnetic phase, and ten (37) of the third magnetic phase (see Supplemental Material Sec. II Table S1 [30]).  \n\nWe also note that because of the crystallographic operations applied in the coupled real and spin space, there is no counterpart in the magnetic groups of the sublattice coset decomposition form of the ${\\bf R}_{s}^{\\mathrm{III}}$ spin groups (see Supplemental Material Sec. II [30]). As we further highlight below, the decomposition into same-spin- and opposite-spin-sublattice transformations in $\\pmb{{\\cal R}}_{s}^{\\mathrm{III}}$ plays a central role in understanding the third magnetic phase.  \n\n# III. SPIN-MOMENTUM LOCKING PROTECTED BY SPIN SYMMETRIES  \n\nWe now discuss the basic characteristics of the spinmomentum locking in the third magnetic phase as derived from the spin Laue group symmetries, i.e., from the symmetries of the direct product of the spin-only group and the nontrivial spin Laue groups. We derive above that the nonrelativistic collinear magnetic order described by the spin-only group symmetries implies that spin is a good quantum number with a common $\\mathbf{k}$ -independent quantization axis, and that the bands are space-inversion symmetric (symmetric with respect to the inversion of $\\mathbf{k}$ ). We also derive that the bands in the third phase described by the nontrivial spin Laue groups $\\pmb{{\\cal R}}_{s}^{\\mathrm{III}}$ break the time-reversal symmetry. The space-inversion symmetry implies that the bands are even in momentum around the $\\mathbf{\\deltaT}$ -point. Moreover, the $\\mathbf{\\deltaT}$ -point is invariant under all real-space transformations. The $\\big[C_{2}\\big|\\big|A\\big]$ symmetry present in the $\\mathbf{R}_{s}^{\\mathrm{III}}$ groups thus guarantees spin degeneracy of the $\\mathbf{\\deltaT}$ -point.  \n\n![](images/49ce204c290ff5b45fa12713d54c96dd2b36c47b72b8151b323070e69d6bdd70.jpg)  \nFIG. 2. Classification of spin-momentum locking in the third magnetic phase protected by spin-group symmetries, and material candidates. The columns describe the characteristic planar $(P)$ or bulk $(B)$ spin-momentum locking on model Hamiltonian bands with the characteristic spin-group integer and the even-parity wave form of altermagnetism, the crystallographic Laue group G, the halving subgroup $\\mathbf{H}$ of symmetry elements which interchange atoms between same-spin sublattices, a generator $A$ of symmetry elements which interchange atoms between opposite-spin sublattices, the nontrivial spin Laue group $R_{s}^{\\mathrm{III}}$ (in brackets we list the number of symmetry elements), and material candidates of the third magnetic phase. The model Hamiltonian bands on which we illustrate the spinmomentum locking character are described in Supplemental Material Sec. III [30]. References describing the materials are in the main text and Supplemental Material Secs. V–VI [30].  \n\nOn the other hand, lifted spin degeneracies in the rest of the Brillouin zone, including other time-reversal invariant momenta, are not generally excluded in the third phase.  \n\nThese basic spin-momentum locking characteristics of the third phase are in striking contrast to the spinmomentum locking in crystals with Kramers spin degeneracy lifted by the relativistic spin-orbit coupling. The relativistic spin-momentum locking has the form of a continuously varying spin texture in the momentum space, it is not symmetric with respect to the inversion of $\\mathbf{k}$ because of the required broken real-space inversion symmetry of the crystal, the bands are time-reversal invariant, and all time-reversal invariant momenta are spin degenerate. These distinct characteristics of the relativistic spin-momentum locking apply to nonmagnetic systems [65], as well as to conventional antiferromagnets with broken real-space inversion symmetry and spin-orbitcoupling effects included. An example is the relativistic time-reversal invariant band structure with Rashba spin splitting in a noncentrosymmetric antiferromagnet $\\mathbf{BiCoO}_{3}$ with the opposite-spin sublattices connected by translation [52].  \n\nOther prominent spin-momentum locking features in the third phase are protected by the specific ${\\big[}C_{2}{\\big|}{\\big|}A{\\big]}[E]{\\big|}\\mathbf{H}{\\big]}$ symmetries present in the given $\\mathbf{R}_{s}^{\\mathrm{III}}$ group. For example, a symmetry $[C_{2}||{\\cal M}_{c}].$ , where $c$ is the axis perpendicular to the a- $b$ mirror plane, defines a spin-degenerate $k_{a}-k_{b}$ nodal plane at $k_{c}=0$ , or other $k_{c}$ separated from $M_{c}k_{c}=-k_{c}$ by a reciprocal lattice vector. This is because $[C_{2}||{M}_{c}]$ transforms a wave vector from this plane on itself, or on an equivalent crystal momentum separated by the reciprocal lattice vector, while spin is reversed. Similarly, a $\\big[C_{2}\\big|\\big|C_{n,c}\\big]$ symmetry, where $C_{n,c}$ is an $n$ -fold rotation symmetry around the $c$ axis, imposes a spin-degenerate nodal line parallel to the $k_{c}$ axis for wave vectors with $k_{a}=k_{b}=0$ , or other $k_{a(b)}$ separated from $C_{n,c}k_{a(b)}$ by a reciprocal lattice vector. We note that the high-symmetry planes or lines are typically of main focus when assessing the electronic structures. This may explain why, apart from the omission by the conventional magnetic groups, the third phase remained unnoticed during the decades of DFT and experimental studies of band structures.  \n\nEach of the ten ${\\bf R}_{s}^{\\mathrm{III}}$ spin Laue groups classifying the spin-momentum locked band structures can be assigned a characteristic even integer, which we define as follows. When making a closed loop in the momentum space around the $\\mathbf{\\deltaT}$ -point in a plane orthogonal to a spin-degenerate nodal surface crossing the $\\mathbf{\\deltaT}.$ -point, the spin rotates by $360^{\\circ}$ following two discrete reversals. Each spin-degenerate nodal surface crossing the $\\mathbf{\\deltaT}$ -point that is present in the crystal momentum space generates such a spin rotation. We define the characteristic spin-group integer as a number of these spin-degenerate nodal surfaces crossing the $\\Gamma$ -point. The spin-group integer is given in Fig. 2, and it is an even number ranging from 2 to 6. As an illustration, we show in the Supplemental Material Sec. III and Fig. S2 [30] spin-degenerate nodal planes crossing the $\\mathbf{\\deltaT}$ -point corresponding to mirror-symmetry planes combined with spinspace rotation for representative $R_{s}^{\\mathrm{III}}$ groups from Fig. 2.  \n\nIn Fig. 2, we show the characteristic spin-group integer next to a spin-momentum locking depicted on top of model Hamiltonian bands. The six model Hamiltonians, with anisotropic $d$ -wave, $g\\mathrm{.}$ -wave, and $i$ -wave harmonic symmetry are listed in Supplemental Material Sec. III [30] and are derived to have the same spin-degenerate nodal planes crossing the $\\mathbf{\\deltaT}.$ -point as the nodal planes corresponding to the representative $R_{s}^{\\mathrm{III}}$ groups in Supplemental Material Fig. S2 [30]. We obtain either planar or bulk nonrelativistic spin-momentum locking, with the characteristic spin-group integer from 2 to 6. The planar spin-momentum locking is relevant for (quasi)two-dimensional and three-dimensional crystals, while the bulk spin-momentum locking only for three-dimensional crystals. We note that the earlier reported materials [1,2,4–11,16–22,66–68] $\\mathrm{FeF}_{2}$ , $\\mathbf{M}\\mathbf{n}\\mathbf{O}_{2}$ , $\\mathrm{RuO}_{2}$ , $\\kappa{\\mathrm{-Cl}}$ , $\\mathrm{{MnF}}_{2}$ , $\\mathrm{Mn}_{5}\\mathrm{Si}_{3}$ , $\\mathrm{LaMnO}_{3}$ , $\\mathrm{FeSb}_{2}$ , and ${\\mathrm{CaCrO}}_{3}$ referred to as unconventional spin-split antiferromagnets in these studies, all correspond to the third magnetic phase with the characteristic planar spin-momentum locking and spin-group integer 2.  \n\nThe presence of nonrelativistic anisotropic spindependent conductivities in the third-phase magnets and the corresponding giant-magnetoresistance and spin-torque phenomena [16,17] is symmetrywise more restrictive than the presence of the phase itself [17]. Only the ${\\bf R}_{s}^{\\mathrm{III}}$ spin  \n\nLaue groups with the characteristic spin-group integer 2 $(\\mathbf{R}_{s}^{\\mathrm{III}}={}^{2}m^{2}m^{1}m$ , ${^2}4/{^1}m$ , ${^2}4/{^1}m^{2}m^{1}m$ , and ${^2}2/{^2m})$ ) have a sufficiently low symmetry that allows for these prominent time-reversal symmetry-breaking spintronic effects in the third magnetic phase.  \n\n# IV. SPIN SPLITTING BY THE ELECTRIC CRYSTAL POTENTIAL  \n\nIn Supplemental Material Sec. IV [30], we summarize the properties of the third magnetic phase derived from the spin Laue group symmetries. Among those, we highlight here the symmetry principles which guide us to the discovery of an extraordinary spin-splitting mechanism, which we illustrate on $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ and $\\mathrm{RuO}_{2}$ . The latter example, in which the amplitude of the extraordinary spin splitting is on the eV scale, is the workhorse material in the emerging research field of time-reversal symmetry-breaking spintronic phenomena in the third magnetic phase [1,3,16–18,20–22].  \n\nThe spin-symmetry guiding principles for the extraordinary spin splitting by the electric crystal potential are as follows: (i) The magnetic crystals should be anisotropic to allow for the symmetries defining the third magnetic phase $\\left[C_{2}\\big|\\big|\\mathbf{G}-\\mathbf{H}\\right]$ , which separate opposite-spin equal-energy states in the momentum space. (ii) The symmetries interchanging atoms within the same-spin sublattice $\\big[E\\big||\\mathbf{H}\\big]$ should be low enough to generate a sufficient anisotropy in the momentum space of the bands dominated by the given sublattice. (iii) The symmetries of $\\mathbf{G}$ are high enough to allow for the orbital degeneracy at the $\\mathbf{\\deltaT}$ -point; this is fulfilled in all groups $\\mathbf{G}$ allowing for the third magnetic phase, except for $\\mathbf{G}=m m m$ or $2/m$ . (iv) The chemistry should allow for these degenerate orbitals to be present in the material and in the desired part of the energy spectrum (e.g., near the Fermi level).  \n\nWe first illustrate how these principles for identifying outlier spin splittings materialize in a pristine way in ruthenate $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ when hosting the third magnetic phase. The signature of the extraordinary microscopic spinsplitting mechanism is that its size and momentum dependence are determined by the electric crystal potential of the nonmagnetic phase. The material fulfills all the above spin-symmetry guiding principles, including its corresponding $\\mathbf{G}=4/m$ . (In contrast, $\\mathrm{LaMnO}_{3}$ , $\\kappa$ -Cl, $\\mathrm{FeSb}_{2}$ , or ${\\mathrm{CaCrO}}_{3}$ have $\\mathbf{G}=m m m$ [2,6,10,11,68] that excludes this electric-crystal-potential mechanism of the spin splitting.)  \n\nThe real-space crystal structure of ${\\mathrm{KRu}}_{4}{\\mathrm{O}}_{8}$ , as reported in earlier studies [69,70], is schematically illustrated in Fig. 3(a). The symmetry of the lattice is body-centered tetragonal (crystallographic space group $I4/m\\rangle$ ). Red and blue color in Fig. 3(a) represent the collinear antiparallel spin arrangement on the crystal. In addition, the $A$ and $B$ symbols label the real-space sublattices corresponding to the opposite spins in the third magnetic phase. The $A$ and $B$ real-space sublattices are strongly anisotropic and related by a mutual planar rotation by $90^{\\circ}(C_{4z})$ . Correspondingly, the nontrivial spin Laue group describing the spinmomentum locking in the third magnetic phase is ${^2}4/{^1}m$ . According to Eq. (1), it can be decomposed as  \n\n![](images/0cd3d97716423a3143f07304f5d2e2a83e12c44c2f5daf6b2cc5514e005092b3.jpg)  \nFIG. 3. Spin splitting by the electric crystal potential in $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ in the third magnetic phase. (a) Schematic spin arrangement on the $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ crystal with opposite-spin directions depicted by red and blue color. Magenta arrow and its label highlights the opposite-spinsublattice transformation containing a real-space fourfold rotation. (b) Calculated spin-momentum locking with the characteristic spingroup integer 2 on top of two DFT Fermi surface sheets. (c),(d) DFT band structure of the nonmagnetic phase and the third magnetic phase, respectively. Gray shading highlights the $\\mathbf{k}$ -dependent splitting by the anisotropic electric crystal potential. (e),(f) Projection of bands on the sublattices $A$ and $B$ in the nonmagnetic phase (black) and third magnetic phase (red and blue) for the upper bands and lower bands, respectively. Color shading in (f) highlights the nearly $\\mathbf{k}$ -independent magnetic splitting of the lower bands, and its opposite sign for the sublattices $A$ and $B$ bands. (g) Real-space DFT spin density around the Ru atom in sublattices $A$ and $B$ .  \n\n$$\n^{2}4/^{1}m=[E\\|2/m]+[C_{2}\\|C_{4z}][E\\|2/m].\n$$  \n\nFigure 3(b) shows the DFT calculation of the spinmomentum locking protected by the spin-group symmetries, on top of two selected $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ Fermi surface sheets. (The band structure is obtained using the DFT full-potential linearized augmented-plane-wave code ELK within the local-spin-density generalized-gradient approximation [71].) In particular, the $\\big[C_{2}\\big|\\big|C_{4z}\\big]$ symmetry leads to three spin-degenerate nodal lines parallel to the $k_{z}$ axis, $[0,0,k_{z}]$ , $[2,0,k_{z}]$ , and $[0,2,k_{z}]$ marked by gray points in Fig. 3(b) (here the wave vectors are in units of $\\pi$ divided by the lattice constant). The latter two correspond, for $k_{z}=0$ , to time-reversal invariant momenta $\\mathbf{S}_{1}$ and $\\mathbf{S}_{2}$ . On the other hand, the spin degeneracy is strongly lifted at time-reversal invariant momenta $\\mathbf{X}$ and $\\mathbf{Y}$ corresponding to the directions of the real-space anisotropy axes of the two sublattices. Consistently, the little crystallographic Laue group at the $\\mathbf{X}$ and $\\mathbf{Y}$ wave vectors is $2/m$ , which coincides with the halving subgroup of same-spin-sublattice transformations. The spin-momentum locking is planar, reflecting the realspace planar mutual rotations of the crystal anisotropies of the opposite-spin sublattices, and the characteristic spingroup integer is 2.  \n\nWe now move on to the demonstration of the spin splitting whose size and momentum dependence are determined by the electric crystal potential and compare this extraordinary microscopic mechanism to the more conventional magnetic spin-splitting mechanism. The analysis is presented in Figs. 3(c)–3(g). Energy bands in the nonmagnetic and third magnetic phase are shown in Figs. 3(c) and 3(d). The high-energy band around $0.9\\ \\mathrm{eV}$ in the depicted portion of the Brillouin zone is twofold spin degenerate in the nonmagnetic phase [upper part of Fig. 3(c)]. The magnetic component of the internal electromagnetic crystal potential in the magnetic phase generates an anisotropic $\\mathbf{k}$ -dependent spin splitting, as shown in the upper part of Fig. 3(d) where the red and blue color correspond to opposite spin states. The sign of the spin splitting alternates, following the symmetries of the spin group. This type of spin splitting belongs to a family generally referred to as internal magnetic-interaction mechanisms [4,7,8,17].  \n\nThe other bands of $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ for energies near the Fermi level also show a spin splitting within the DFT bandstructure theory. However, here the microscopic origin is fundamentally distinct from the internal magnetic-interaction mechanisms. In the nonmagnetic phase, we observe in the lower part of Fig. 3(c) a couple of twofold spindegenerate bands whose mutual splitting (highlighted by gray shading) by the electric crystal potential is $\\mathbf{k}$ dependent, merging at the fourfold degenerate Γ, $\\mathbf{S}_{1}$ , and $\\mathbf{S}_{2}$ points. Remarkably, the $\\mathbf{k}$ -dependent spin splitting in the magnetic phase in the lower part of Fig. 3(d) (highlighted again by gray shading) copies the size and k dependence of the band splitting by the electric crystal potential of the nonmagnetic phase. Its microscopic explanation is provided in Figs. 3(f) and $3(\\mathrm{g)}$ .  \n\nWe start the discussion of Fig. 3(f) from the projections of the bands on the sublattices $A$ and $B$ in the nonmagnetic phase (black lines). The projections are dominated by Ru $d_{x z}$ and $d_{y z}$ orbitals. This is in agreement with earlier report [70], which showed the presence of $\\ensuremath{\\mathrm{Ru}}\\ t_{2g}$ orbitals near the Fermi level. At the $\\mathbf{\\deltaT}$ -point, the $A$ and $B$ projected bands are degenerate, which is consistent with the octahedral environment with the tetragonal symmetry [72]. Including spin, the $\\mathbf{\\deltaT}.$ -point is then fourfold degenerate in the nonmagnetic phase.  \n\nThe band whose dominant weight is on sublattice $A$ is strongly anisotropic with respect to $\\mathbf{k}$ when moving toward the $\\mathbf{X}$ and $\\mathbf{Y}$ points [left panel of Fig. 3(f)]. The same applies to the sublattice $B$ band; however, the sense of the anisotropy reverses [right panel of Fig. 3(f)]. The band anisotropies reflect the strong crystalline anisotropy, conspiring with the favorable symmetry of the involved orbitals. By adding up the $A$ and $B$ projections, we obtain the bands shown in Fig. 3(c). They progressively split by the electric crystal potential when the $\\mathbf{k}$ -vector moves from the $\\mathbf{\\deltaT}$ -point toward, e.g., the $\\mathbf{X}$ point, with the lower band dominated by one sublattice and the upper band by the other sublattice. Along the $\\boldsymbol{\\Gamma}-\\boldsymbol{\\mathbf{Y}}$ line, the sublattice indices of the lower and upper bands switch places.  \n\nThe bands in the magnetic phase projected again on sublattices $A$ and $B$ are also plotted in Fig. 3(f). As in Fig. 3(d), the red and blue colors correspond to opposite spins. We see that for bands with dominant weight on sublattice $A$ , spin states shown in red move up in energy, while the opposite spin states shown in blue move down [left panel of Fig. 3(f)]. The magnetic component of the internal crystal potential in the magnetic phase generates in this case a splitting (highlighted by light-blue shading), which is nearly k independent. This scenario is fundamentally distinct from the strongly $\\mathbf{k}$ -dependent magnetic splitting of the high-energy band shown in Fig. 3(e). It is reminiscent of ferromagnets. However, unlike the common ferromagnetic case, the nearly $\\mathbf{k}$ -independent magnetic splitting reverses sign for the sublattice $B$ bands [right panel of Fig. 3(f)]. This locality, in which band states near the Fermi level with one spin have a dominant weight on one sublattice, is again distinct from the delocalized nature of spin states in the high-energy bands shown in Fig. 3(e). It also starkly contrasts with the conventional mechanisms of the ferromagnetic splitting of band spin states experiencing the global magnetization or the relativistic spin-orbit splitting due to the global electric inversion asymmetry. An additional illustration of the locality is shown in Fig. $3(\\mathrm{g)}$ where we plot the real-space DFT spin density around the Ru atom in sublattices $A$ and $B$ . Consistent with the spin-group symmetry and the dominant $d_{x z}$ and $d_{y z}$ orbitals near the Fermi level, the opposite-spin local densities in the two sublattices are highly anisotropic with the mutually rotated real-space anisotropy axes.  \n\nAdding up the $A$ and $B$ sublattice projections of Fig. 3(f) then explains the formation of two pairs of spin-split bands seen in Fig. 3(d). The mutual magnetic splitting between the two pairs is nearly $\\mathbf{k}$ independent, while the spin splitting within each pair is a $\\mathbf{k}$ -dependent copy of the band splitting by the anisotropic electric crystal potential of the nonmagnetic phase [Fig. 3(c)]. It also explains that the two pairs have opposite sign of the spin splitting and that, within each pair, the spin-splitting sign is opposite when moving from the $\\mathbf{\\deltaT}.$ -point toward the $\\mathbf{X}$ or $\\mathbf{Y}$ points. We see from Figs. 3(d) and 3(f) that even if the nearly $\\mathbf{k}$ -independent magnetic splitting were small, the electric crystal potential of the nonmagnetic phase would still determine the splitting between the two nearest bands with opposite spin in the magnetic phase at $\\mathbf{k}$ -vectors sufficiently close to the $\\mathbf{\\deltaT}$ -point. This is a consequence of the nearly $\\mathbf{k}$ -independent magnetic band splitting and of the spin degeneracy of the $\\Gamma$ -point in the third magnetic phase.  \n\nIn the studied ${\\mathrm{KRu}}_{4}{\\mathrm{O}}_{8}$ , the spin splitting originating from this extraordinary electric-crystal-potential mechanism reaches a $300\\mathrm{-meV}$ scale. In Supplemental Material Sec. V and Fig. S3 [30], we show that in $\\mathrm{RuO}_{2}$ , a spin splitting reaching a 1-eV scale [1,4,17] is also due to the electric-crystal-potential mechanism. These spin-splitting magnitudes are comparable to spin splittings in ferromagnets but, unlike ferromagnets, are accompanied by a zero net magnetization. They also illustrate that spin splittings in the third magnetic phase can exceed by an order of magnitude the record relativistic spin-orbit splittings in bulk crystals with heavy elements [73]. Moreover, unlike the spin-orbit split bands, the third magnetic phase preserves a common $\\mathbf{k}$ -independent spin quantization axis.  \n\nFinally, we emphasize that relativistic DFT calculations in $\\mathrm{RuO}_{2}$ and $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ presented in Supplemental Material Sec. V and Figs. S3 and S4 [30] show the expected weak effect of the spin-orbit coupling on the bands. This highlights that the apparent prominent features of the relativistic bands, including the spin-momentum locking characteristics and the electric-crystal-potential mechanism of the spin splitting, still reflect the nonrelativistic spin-group symmetries. In contrast, these prominent symmetries are omitted by the relativistic magnetic groups of $\\mathrm{RuO}_{2}$ and $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ . In general, as also illustrated in Supplemental Material Secs. III and IV and Figs. S3 and S4 [30], only the spin-group formalism facilitates the sublattice coset decomposition into transformations which interchange atoms between same-spin and opposite-spin sublattices, which plays the central role in understanding the third magnetic phase. Apart from the spin-symmetry guiding principles and DFT calculations of the spin splitting by the electric crystal potential, we also provide a description of this extraordinary mechanism by a minimal lattice model in Supplemental Material Sec. V Fig. S5 [30].  \n\n# V. CANDIDATE MATERIALS  \n\nFigure 2 lists the selected candidate materials for the third magnetic phase. In Fig. 4, we highlight $\\mathrm{CrSb}$ , a metal with the critical temperature of $705\\mathrm{~K~}$ [74]. As shown in Fig. 4(a), it crystallizes in the hexagonal NiAs-type structure (crystal space group $P6_{3}/m m c)$ [74,75]. The collinear antiparallel spin arrangement corresponds to the nontrivial spin Laue group $^{2}6/^{2}m^{2}m^{1}m$ $([E||\\bar{3}m]+$ $[C_{2}\\vert\\vert C_{6z}][E\\vert\\vert\\bar{3}m])$ . It contains the $[C_{2}||M_{z}]$ symmetry, which makes the spin-momentum locking bulklike. Additional mirror planes orthogonal to the three hexagonal crystal axes combined with the spin rotation imply that the characteristic spin-group integer is 4 (see Supplemental  \n\nMaterial Fig. S2 [30]). This is confirmed by the DFT calculations in Fig. 4(b).  \n\nCrSb has a more complex band structure than $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ , as shown in Fig. 4(c). Nevertheless, we can trace a pair of bands with opposite spin [highlighted by gray shading in Fig. 4(c)] which are degenerate at the $\\mathbf{\\deltaT}$ , $\\mathbf{L}_{1}$ , and $\\mathbf{L}_{2}$ points and split when moving away from these high-symmetry points. The spin splitting is as high as $1.2\\mathrm{eV}.$ . We also note that $\\mathrm{CrSb}$ hosts an exotic spin-polarized quasiparticle which is fourfold degenerate at the $\\Gamma$ -point and spin split away from the Γ-point.  \n\nA semiconducting MnTe, which is isostructural to $\\mathrm{CrSb}$ , also hosts an extraordinarily large spin splitting in the valence band of $1.1\\ \\mathrm{eV}.$ In Supplemental Material Sec. V Fig. S6 [30], we give a summary of the spin splittings vs critical temperature in selected materials hosting the third magnetic phase. In Supplemental Material Sec. VI and Figs. S7 and S8 [30], we discuss additional material candidates among insulators, semiconductors, and metals, and give an example illustrating the inversion symmetry of the nonrelativistic bands of the third magnetic phase even when the crystal is inversion asymmetric $\\mathrm{(VNb}_{3}\\mathrm{S}_{6})$ .  \n\nFinally, we discuss the parent cuprate ${\\mathrm{La}}_{2}{\\mathrm{CuO}}_{4}$ of a high-temperature superconductor [27,76]. The band structure for the collinear antiparallel spin arrangement on this crystal falls into the $\\mathbf{R}_{s}^{\\mathrm{III}}$ nontrivial spin Laue group $^2m^{2}m^{1}m$ $([E||2/m]+[C_{2}||C_{2y}][E||2/m])$ . The symmetry element $\\left[C_{2}\\big|\\big|C_{2y}\\right]$ generates a planar spin-momentum locking with the characteristic spin-group integer 2. Remarkably, according to our symmetry analysis based on the spingroup theory, the energy bands of ${\\mathrm{La}}_{2}{\\mathrm{CuO}}_{4}$ are spin split and break time-reversal symmetry. This is confirmed by the DFT calculations in Supplemental Material Fig. S9 [30] and is in contrast with the conventional perception of spindegenerate bands in ${\\mathrm{La}}_{2}{\\mathrm{CuO}}_{4}$ [76]. The omission of the spin-splitting physics in earlier electronic-structure studies of cuprates could be explained by the focus on highsymmetry lines or planes, such as the $k_{z}=0$ plane [61], where the states are spin degenerate (see Supplemental Material Fig. S9 [30]).  \n\n![](images/311527c786f819f6e847f19960c8fd578cfa40a4e07ae819c00cb3fe60aa6cb1.jpg)  \nFIG. 4. Metallic high critical temperature $\\mathrm{CrSb}$ with the third magnetic phase. (a) Schematic crystal structure with DFT spin densities. Cr sublattices and the respective magnetization densities with opposite orientation of the magnetic moment are depicted by red and blue color. Magenta arrow and its label highlights the opposite-spin-sublattice transformation containing a real-space mirror or sixfold rotation. (b) Calculated bulklike spin-momentum locking with the characteristic spin-group integer 4 on top of two selected DFT Fermi surface sheets. (c) DFT band structure in the third magnetic phase. Wave-vector dependence of the spin splitting between the bands highlighted by the gray shading is plotted in the lower panel.  \n\n# VI. BROAD RELEVANCE IN CONDENSED-MATTER PHYSICS  \n\nOur spin-group delimitation and description of the third magnetic phase and the discovery of the extraordinary spinsplitting mechanism by the electric crystal potential in the Ru-oxide crystals provides a unifying theory picture of recent intriguing theoretical and experimental observations of broken time-reversal symmetry transport anomalies and spintronic effects in the magnetically compensated $\\mathrm{RuO}_{2}$ . These include the large crystal (anomalous) Hall effect, charge-spin conversion and spin-torque phenomena, and giant and tunneling magnetoresistance [1,3,16–22]. Our identification of the third magnetic phase in chalcogenide $\\mathrm{CoNb}_{3}\\mathrm{S}_{6}$ , perovskite $\\mathrm{CaMnO}_{3}$ , or cuprate ${\\mathrm{La}}_{2}{\\mathrm{CuO}}_{4}$ also sheds new light on puzzling time-reversal breaking magnetotransport anomalies reported in earlier studies of these materials [77–79].  \n\nThe diversity of the material types illustrates the relevance of the third magnetic phase for a range of condensed-matter physics fields prone to generate new discoveries. Spintronics based on this phase [1,3,6,9,11,12,16–22] would circumvent the traditional prerequisites of magnetization or relativistic spin-orbit coupling in conventional ferromagnetic spintronics [80–83]. Unlike ferromagnets, the third magnetic phase eliminates stray fields and adds insensitivity to external magnetic field perturbations, while allowing for the strong nonrelativistic effects which facilitate the reading and writing functionalities in commercial spintronics. When comparing to the relativistic nonmagnetic spin-texture phases, these textures share with the third magnetic phase the zero net magnetization. However, large relativistic spin splittings require rare heavy elements. In addition, the relativistic phases suffer from spin decoherence even for small-angle elastic scattering off common isotropic impurities. We illustrate that this obstacle is diminished in the third magnetic phase by the collinearity of spins and by the possibility of a large- $\\mathbf{\\nabla}\\cdot\\mathbf{k}$ -vector separation in the Brillouin zone of the equalenergy eigenstates with the opposite spin.  \n\nOur results on the ruthenate $\\mathrm{KRu}_{4}\\mathrm{O}_{8}$ illustrate the distinct features of nonrelativistic valleytronics in the third magnetic phase, in comparison to valleytronics in nonmagnetic 2D materials [84]. Here the specific merit of the third magnetic phase are spin-split valleys at time-reversal invariant momenta. Both the spintronics and valleytronics fields can take advantage of the spin-conserving nature of the third magnetic phase, stemming from its nonrelativistic origin in the nonfrustrated collinear magnetic crystals.  \n\nUnexplored connections might also exist between the third magnetic phase and topological insulators and semimetals. In this context, we point out that, on the one hand, symmetry prohibits a realization of the third magnetic phase in one-dimensional chains; collinear antiferromagnetic spin arrangements on one-dimensional chains have the $\\big[C_{2}\\big|\\big|\\bar{E}\\big]$ (and possibly also $\\big[C_{2}\\big|\\big|\\mathbf{t}\\big]\\big)$ symmetry and, therefore, have spin-degenerate bands. On the other hand, we identify candidates of the third magnetic phase among quasi-one-dimensional, quasi-two-dimensional, and threedimensional insulators and metals. This opens the possibility of searching for unconventional spin-polarized fermion quasiparticles (cf. CrSb), topological insulators, and topological semimetals, including Chern insulators with the quantized Hall effect in high-temperature systems with vanishing internal or external magnetic dipole.  \n\nIn the field of electromagnetic multipoles, the zero magnetic dipole of the third magnetic phase opens a new route for realizing magnetic toroidal phases [8,85]. Related to this is the field of Fermi-liquid instabilities [86], where we show that the principally uncorrelated third magnetic phase represents an unprecedented example of an anisotropic ( $d$ -wave, $g\\cdot$ -wave, or $i$ -wave) instability. Certain anisotropic instabilities were expected in the past to arise in correlated systems [4,86]. Our recognition of the $d$ -wave spin-momentum locking in the parent cuprate crystals of high-temperature $d$ -wave superconductors [87] brings a new element into the research of the coexistence and interplay of magnetic and superconducting quantum orders. In addition to bulk systems, intriguing phenomena can be envisaged also in heterostructures in fields such as topological superconductivity [88].  \n\nAn extensive perspective on how the emerging third magnetic phase can enrich basic condensed-matter physics concepts and have impact on prominent condensed-matter research and application areas is given in Ref. [89].  \n\n# VII. DISCUSSION: UNCONVENTIONAL MAGNETIC PHASE  \n\nA phase of matter is commonly associated with a uniform state of a physical system and is distinguished from other phases by, among others, crystal structure, composition, or type of order (e.g., magnetic). Each phase in a material system generally exhibits a characteristic set of physical properties, and symmetry is among the fundamental guiding principles for identifying the distinct phases of matter and for describing their phenomenology [31,90]. We show in this work that on the basic level of nonrelativistic physics of nonfrustrated (collinear) magnetism, spin-group symmetries in the crystal-structure real space and electronic-structure momentum space allow, besides the conventional ferromagnetism and antiferromagnetism, for the third distinct magnetically ordered phase. As summarized in Supplemental Material Figs. S10 and S11 [30], the conventional ferromagnetism is characterized by a type of crystal structure and magnetic order with nonzero magnetization allowed by the spin-group symmetry, while the conventional antiferromagnetism is characterized by a different type of crystal structure and magnetic order with zero net magnetization protected by the spin-group symmetry. The key distinction between the nonrelativistic phenomenologies of the two conventional magnetic phases is the spin-split time-reversal symmetrybroken electronic structure and corresponding time-reversal symmetry-breaking responses in ferromagnets contrasting with the spin-degenerate time-reversal symmetric electronic structure and the absence of time-reversal symmetrybreaking responses in antiferromagnets.  \n\nThe unconventional magnetic phase classified and described in this work has a type of crystal structure and magnetic order that is distinct from the conventional ferromagnets and antiferromagnets. Its zero net magnetization is protected by the spin-group symmetries that, simultaneously, allow for spin-split time-reversal symmetrybroken electronic structure and corresponding time-reversal symmetry-breaking responses. In trying to retain the classification with only the two traditional basic phases of magnetically ordered materials, a conflict arises. Placing emphasis on the phenomenology of the spin-split timereversal symmetry-broken electronic structure and responses would lead to a notion of unconventional ferromagnetism. In contrast, emphasizing the zero net magnetization would lead to a notion of unconventional antiferromagnetism. Our work provides a resolution of the conflict by delimiting the unconventional magnetic phase of the $d$ -wave (or high even-parity wave) form as a third distinct symmetry type. The alternating spin polarizations in both real-space crystal structure and momentum-space band structure characteristic of this unconventional magnetic phase suggest a term altermagnetism.  \n\nWe note that on the basic level of nonrelativistic spingroup symmetries, altermagnetism is delimited as an exclusive separate phase next to the conventional ferromagnetism and antiferromagnetism, while ferrimagnets are a subclass of ferromagnets. Indeed, in general, a distinction between ferrimagnetic crystals characterized by oppositespin sublattices not connected by any symmetry transformation, and crystals commonly referred to as ferromagnets can be ambiguous. For example, in crystals referred to as ferrimagnets, all magnetic atoms can be the same chemical elements, and the absence of any crystallographic transformation connecting the opposite-spin sublattices then originates from different local symmetries of the sites occupied by magnetic atoms from the opposite-spin sublattices. This can be compared to crystals commonly regarded as ferromagnets, where the microscopic spin density changes in magnitude and can also change in sign as a function of the spatial coordinate within the crystal unit cell. Whether or not such variations are correlated with individual atomic species does not change the symmetry of the system.  \n\nFinally, we point out that altermagnetism is a realization of a long-sought-after counterpart in magnetism of unconventional superconductivity [91]. Magnetism and superconductivity were once regarded as the best understood fields in many-body solid-state physics. Moreover, they were connected by a striking analogy: The electronelectron Cooper pairs forming around the Fermi surface and driving the conventional $s$ -wave superconductivity have a counterpart in the majority spin electron—minority spin-hole pairs distributed isotropically around the Fermi surface in the conventional model of (s-wave) ferromagnetism [91]. The discovery of the unconventional $d$ -wave superconductivity not only opened an entirely new research landscape of this many-body phase [87] but also raised a fundamental question of whether and how an unconventional $d$ -wave counterpart could be realized in magnetism [91]. Earlier considerations focused on possible realizations of the unconventional $d.$ -wave magnetism due to strong electronic correlations [86,92,93]. In contrast, our identification is directly linked to symmetries of the crystal potential and does not require strongly correlated systems. This makes the altermagnetic materials discussed in our work realistic candidates for a robust unconventional $d$ -wave (or higher even-parity wave) magnetism that can host unconventional time-reversal symmetry-breaking responses of comparable strength to the conventional ( $\\scriptstyle{\\mathsf{\\bar{s}}}$ -wave) ferromagnets.  \n\n# ACKNOWLEDGMENTS  \n\nWe acknowledge fruitful interactions with Igor Mazin, Rafael González-Hernández, Helen Gomonay, and Roser Valentí. This work is supported by Ministry of Education of the Czech Republic Grants No. LNSM-LNSpin and No. LM2018140, the Czech Science Foundation Grant No. 19-28375X, EU Future and Emerging Technologies Open RIA Grant No. 766566, $\\mathrm{SPIN}+\\mathrm{X}$ (Grant No. DFG SFB TRR 173) and Elasto-Q-Mat (Grant No. DFG SFB TRR 288). We acknowledge the computing time granted on the supercomputer Mogon at Johannes Gutenberg University Mainz.  \n\nMagnetism in Doped $\\mathrm{FeSb}_{2}$ , Proc. Natl. Acad. Sci. 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    "Target": null
  },
  {
    "id": "10.1038_s41467-022-29837-w",
    "DOI": "10.1038/s41467-022-29837-w",
    "DOI Link": "http://dx.doi.org/10.1038/s41467-022-29837-w",
    "Relative Dir Path": "mds/10.1038_s41467-022-29837-w",
    "Article Title": "Data-driven capacity estimation of commercial lithium-ion batteries from voltage relaxation",
    "Authors": "Zhu, JG; Wang, YX; Huang, Y; Gopaluni, RB; Cao, YK; Heere, M; Mühlbauer, MJ; Mereacre, L; Dai, HF; Liu, XH; Senyshyn, A; Wei, XZ; Knapp, M; Ehrenberg, H",
    "Source Title": "NATURE COMMUNICATIONS",
    "Abstract": "Accurate capacity estimation is crucial for lithium-ion batteries' reliable and safe operation. Here, the authors propose an approach exploiting features from the relaxation voltage curve for battery capacity estimation without requiring other previous cycling information. Accurate capacity estimation is crucial for the reliable and safe operation of lithium-ion batteries. In particular, exploiting the relaxation voltage curve features could enable battery capacity estimation without additional cycling information. Here, we report the study of three datasets comprising 130 commercial lithium-ion cells cycled under various conditions to evaluate the capacity estimation approach. One dataset is collected for model building from batteries with LiNi0.86Co0.11Al0.03O2-based positive electrodes. The other two datasets, used for validation, are obtained from batteries with LiNi0.83Co0.11Mn0.07O2-based positive electrodes and batteries with the blend of Li(NiCoMn)O-2 - Li(NiCoAl)O-2 positive electrodes. Base models that use machine learning methods are employed to estimate the battery capacity using features derived from the relaxation voltage profiles. The best model achieves a root-mean-square error of 1.1% for the dataset used for the model building. A transfer learning model is then developed by adding a featured linear transformation to the base model. This extended model achieves a root-mean-square error of less than 1.7% on the datasets used for the model validation, indicating the successful applicability of the capacity estimation approach utilizing cell voltage relaxation.",
    "Times Cited, WoS Core": 342,
    "Times Cited, All Databases": 360,
    "Publication Year": 2022,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000788592600010",
    "Markdown": "# Data-driven capacity estimation of commercial lithium-ion batteries from voltage relaxation  \n\nJiangong Zhu 1,2,7, Yixiu Wang3,7, Yuan Huang1,2, R. Bhushan Gopaluni3, Yankai Cao3, Michael Heere2,4, Martin J. Mühlbauer2, Liuda Mereacre2, Haifeng Dai 1✉, Xinhua Liu $\\textcircled{1}$ 5, Anatoliy Senyshyn $\\textcircled{1}$ 6, Xuezhe Wei1, Michael Knapp 2✉ & Helmut Ehrenberg 2  \n\nAccurate capacity estimation is crucial for the reliable and safe operation of lithium-ion batteries. In particular, exploiting the relaxation voltage curve features could enable battery capacity estimation without additional cycling information. Here, we report the study of three datasets comprising 130 commercial lithium-ion cells cycled under various conditions to evaluate the capacity estimation approach. One dataset is collected for model building from batteries with $\\mathsf{L i N i}_{0.86}\\mathsf C\\mathsf{o}_{0.11}\\mathsf{A l}_{0.03}\\mathsf{O}_{2}.$ -based positive electrodes. The other two datasets, used for validation, are obtained from batteries with $\\mathsf{L i N i}_{0.83}\\mathsf{C o}_{0.11}\\mathsf{M n}_{0.07}\\mathsf{O}_{2}$ -based positive electrodes and batteries with the blend of $\\mathsf{L i}(\\mathsf{N i}\\mathsf{C o M n})\\mathsf{O}_{2}\\textrm{-}\\mathsf{L i}(\\mathsf{N i}\\mathsf{C o A l})\\mathsf{O}_{2}$ positive electrodes. Base models that use machine learning methods are employed to estimate the battery capacity using features derived from the relaxation voltage profiles. The best model achieves a root-mean-square error of $1.1\\%$ for the dataset used for the model building. A transfer learning model is then developed by adding a featured linear transformation to the base model. This extended model achieves a root-mean-square error of less than $1.7\\%$ on the datasets used for the model validation, indicating the successful applicability of the capacity estimation approach utilizing cell voltage relaxation.  \n\nithium-ion batteries have become the dominant energy storage device for portable electric devices, electric vehicles (EVs), and many other applications1. However, battery degradation is an important concern in the use of lithium-ion batteries as its performance decreases over time due to irreversible physical and chemical changes2,3. State of Health (SoH) has been used as an indicator of the state of the battery and is usually expressed by the ratio of the relative residual capacity with respect to the initial capacity4. The accurate battery capacity estimation is challenging but critical to the reliable usage of the lithium-ion battery, i.e., accurate capacity estimation allows an accurate driving range prediction and accurate calculation of the maximum energy storage capability in a vehicle. Typically, the battery capacity is gained by a full discharge process after it has been fully charged. In a real-life usage scenario, the battery full charge is often achieved while the EVs are parking with grid connection, however, the battery discharge depends on the user behavior with uncertainties in environmental and operational conditions, a complete discharge curve is seldom available for on-board battery health monitoring. The battery charging and discharging voltage, as one of the easily obtained parameters, depend on both, thermodynamic and kinetic characteristics of the battery. Thus, those methods using a charge/discharge process are proposed to estimate capacity for practical applications5,6, in which the input variables are extracted from the measured voltage curves, and the data-driven methods using statistical and machine learning techniques have been popular in battery research recently due to their strong data processing and nonlinear fitting capabilities6,7. The data-driven methods do not need a deep understanding of battery electrochemical principles, but large numbers of data are required to ensure the reliability of model8. Severson et al.9 reported a promising route using machine learning to construct models that accurately predicted graphite $\\mathrm{||LiFePO_{4}}$ (LFP) commercial cell lives using charge-discharge voltage data. Zhang et al.10 identified battery degradation patterns from impedance spectroscopy using Gaussian process machine learning models. Ding et al.11 introduced a machine learning method for the improvement of the efficiency of membrane electrode assembly design and experiment. Such data-driven methods focus on the relationships among the input and output features, and a key part of data-driven battery state estimation is the extraction of degradation features, which largely determines the estimation performance12–14.  \n\nIn practical electric transport applications, battery charging is essential and happens regularly compared to the random discharge process affected by the driving behaviors and road environments. Therefore, extracting voltage features from the charging process has attracted wide attention. Taking into account the state-of-the-art literature, three classes of voltage-based extraction methods can be defined: (I) CC (constant current) charge voltagebased, (II) CC-CV (constant current–constant voltage) charge voltage-based, and (III) rest voltage-based as listed in Supplementary Table 1. The partial charge process in a specific voltage range for feature extraction is commonly used for capacity estimation15, and the estimation accuracy of the state of art is ranging from a root-mean-square error (RMSE) of $0.39\\%$ to a RMSE of $4.26\\%$ based on in-house experiments and different public datasets5,6,16. The transformations of the partial voltage curves, i.e., differential voltage analysis17,18 and incremental capacity analysis19–21, are used for battery aging mechanism identification and capacity fade evaluation. Typically, SVR (Support Vectors Regression)22, GPPF (Gaussian Process Particle Filter)23, BPNN (Back-Propagation Neural Network)24, and linear model20 are applied to estimate battery capacity using the partial incremental capacity curve. Compared to the charge voltage-based methods, studies extracting features from the rest voltage are few.  \n\nA representative battery capacity estimation method utilizing the resting process was proposed by Baghdadi et al.25. They proposed a linear model to estimate battery capacity using the voltage after $30\\mathrm{min}$ rest when the cell is fully charged, and the capacity estimation percentage error is ranging from 0.7 to $3.3\\%$ for three different commercial batteries. Schindler et al.26 and Lüders et al.27 took the voltage relaxation for the lithium plating detection in the battery capacity fade process. Qian et al.28 used an equivalent circuit model (ECM) to describe the voltage relaxation and found that the extracted parameters provided an evaluation of the battery SoH and aging mechanisms. Attidekou et al.29 modeled the battery capacity decay during rest periods at $100\\%$ SoC using a dynamic time constant derived from the resistor-capacitor (RC) network model. However, as the amount of RC links increases, the complexity of the ECM will increase accordingly, which makes it difficult to use in an on-board application30. Besides, the accuracy and robustness of capacity estimation are difficult to evaluate because of the differences in battery types and working conditions8,9.  \n\nIt has been proven that the relaxation process including the relaxation voltage value at a specific time and the voltage curve during a specific period shows a relationship with the battery $\\mathrm{SoH}^{26-29,31}$ . From the review of battery charging studies32–34, the real-time data of $\\mathrm{EV}s^{35,36}$ , and a survey of real-world EV charging (Supplementary Note 1, Supplementary Table 2 and 3, and Supplementary Figs. 1 and 2), in addition to the CC charging strategy, the multistage current charging algorithm using a SoC dependent charging current is a promising method to maximize the charging efficiency. The start of charge for the EVs is normally distributed around intermediate SoCs as expected from the statistics35,37,38. The various multistage current charge strategies and the uncertain start of charge points bring difficulties to the acquirement of specific voltage ranges under constant current in the voltage-based methods. The relaxation after being fully charged is relatively unaffected by the charging process and is also easy to obtain since the battery is fully charged with high probability in real EV usage35,37,38, there is also no need for additional devices as the voltage data can be directly obtained from the battery management system. However, to the best of our knowledge, the relaxation voltage curve of the battery has not yet been studied systematically with machine learning methods for large-scale data from different battery types. Herein, an approach based on features extracted from the battery relaxation voltage is proposed, which focuses on short-term battery capacity estimation without any previous cycling information for on-board implementation.  \n\nIn this study, base models using machine learning methods, i.e., the linear model (ElasticNet39), and nonlinear models (XGBoost40 and Support Vector Regression $(\\mathrm{SVR})^{41}.$ ), using large datasets from three kinds of commercial lithium-ion batteries are employed. The model inputs are statistical features extracted from the voltage relaxation curve. Batteries with $\\mathrm{LiNi}_{0.86}\\mathrm{Co}_{0.11}\\mathrm{Al}_{0.03}\\mathrm{O}_{2}$ positive electrode (NCA battery) cycled at different temperatures and current rates are used for base model building, showing the best test performance with a RMSE of $1.0\\%$ . The transfer learning method is applied on batteries with $\\mathrm{LiNi}_{0.83}\\mathrm{Co}_{0.11}\\mathrm{Mn}_{0.07}\\mathrm{O}_{2}$ positive electrode (NCM battery) and batteries with 42 (3) wt. $\\%$ $\\mathrm{Li}(\\mathrm{NiCoMn})\\mathrm{O}_{2}$ blended with 58 (3) wt. $\\%$ $\\mathrm{Li}(\\mathrm{NiCoAl})\\mathrm{O}_{2}$ positive electrode $(\\mathrm{NCM}+\\mathrm{NCA}$ battery), obtaining $1.7\\%$ RMSE and $1.6\\%$ RMSE respectively, and enabling the generalizability of our approach.  \n\n# Results  \n\nData generation. Large cycling datasets on NCA battery, NCM battery, and $\\mathrm{{NCM}+\\tilde{N}C A}$ battery are created in this study. The batteries are cycled in a temperature-controlled chamber with different charge current rates. The battery specifications are listed in Supplementary Table 4. Long-term cycling is conducted on all cells with a summary of cycling conditions in Table 1. The temperatures chosen are 25, 35, and $45^{\\circ}\\mathrm{C}$ . Current rates ranging from $0.25\\mathrm{C}$ (0.875 A) to 4 C (10 A) are used. The current rate is calculated from the nominal capacity of batteries, i.e., $1\\mathrm{C}$ is equal to $3.5\\mathrm{A}$ for the NCA battery and NCM battery, and 1 C is equal to $2.5\\mathrm{A}$ for the $\\mathrm{{NCM}+N C A}$ battery. The cells are named as CYX-Y/Z according to their cycling conditions. X means the temperature, $\\mathrm{Y}/\\mathrm{Z}$ represents the charge/discharge current rate. The number of cells assigned to each cycling condition in Table 1 is aimed to obtain a dataset covering possible variations between cells. One data unit comprises a relaxation voltage curve after full charge with the following discharge capacity. Each relaxation voltage curve is transformed into six statistical features, i.e., variance (Var), skewness (Ske), maxima (Max), minima (Min), mean (Mean), and excess kurtosis (Kur). The mathematical description of the six features is depicted in Supplementary Table 5. The datasets collected from NCA, NCM, and $\\mathrm{{NCM}+N C A}$ cells are named as dataset 1, dataset 2, and dataset 3 in this study, respectively. Dataset 1 is used for base model training and test. Dataset 2 and dataset 3 are used for assessing and improving the generalizability of the proposed approach by transfer learning.  \n\n<html><body><table><tr><td colspan=\"6\"> Table 1 Cycled batteries and cycling conditions for the dataset generation.</td></tr><tr><td>Datasets</td><td>Cell type</td><td>Cycling temperature (±0.2 °℃)</td><td>Charge current rate (C)/discharge rate (C)</td><td> Number of cells</td><td>Number of data units</td></tr><tr><td>Dataset 1</td><td>NCA battery</td><td>25</td><td>0.25/1</td><td>7</td><td>1853</td></tr><tr><td rowspan=\"5\"></td><td>Type: 18,650</td><td></td><td>0.5/1</td><td>19</td><td>3278</td></tr><tr><td>Cutoffvoltage: 2.65-4.2V</td><td></td><td>1/1</td><td>9</td><td>260</td></tr><tr><td>Nominal capacity: 3.5 Ah</td><td>35</td><td>0.5/1</td><td>3</td><td>1112</td></tr><tr><td></td><td>45</td><td></td><td>28</td><td>15,775</td></tr><tr><td>Dataset2 NCM battery</td><td>25</td><td></td><td>23</td><td>5490</td></tr><tr><td rowspan=\"6\"></td><td>Type: 18,650</td><td>35</td><td></td><td>4.</td><td>4712</td></tr><tr><td>Cutoff voltage: 2.5-4.2 V</td><td>45</td><td></td><td>28</td><td>17,600</td></tr><tr><td>Nominal capacity: 3.5Ah</td><td></td><td></td><td></td><td></td></tr><tr><td>Dataset 3NCM+NCA battery</td><td>25</td><td>0.5/1</td><td>3</td><td>2843</td></tr><tr><td>Type: 18650</td><td></td><td>0.5/2</td><td>3</td><td>2913</td></tr><tr><td>Cutoff voltage: 2.5-4.2 V</td><td></td><td>0.5/4</td><td>3</td><td>2826</td></tr><tr><td></td><td>Nominal capacity: 2.5 Ah</td><td></td><td></td><td></td><td></td></tr></table></body></html>\n\nAll cells are commercial 18,650 type batteries. The cycling temperature is controlled by climate chambers $(\\pm0.2^{\\circ}\\mathsf{C})$ . The current rate is calculated from the battery nominal capacity $(1mathsf{C}=3.5\\mathsf{A}$ for the NCA battery and NCM battery, and $\\mathsf{1C}=2.5\\mathsf{A}$ for the NCM + NCA battery).  \n\nVoltage and current are the basic data recorded in these experiments, which include charging, discharging, and relaxation processes. The cell cycling is performed with constant current (CC) charging to $4.2\\mathrm{V}$ with current rates ranging from $0.25\\mathrm{C}\\left(0.875\\mathrm{A}\\right)$ to 1 C (3.5 A), followed by a constant voltage (CV) charging step at $4.2\\mathrm{V}$ until a current of $0.05\\mathrm{C}$ is reached. Constant current is then employed for the discharge to $2.65\\mathrm{V}$ for the NCA cells and $2.5\\mathrm{V}$ for the NCM and $\\mathrm{{NCM+NCA}}$ cells, respectively. One complete cycling curve using a $0.5\\mathrm{C}$ charging rate for the NCA cell is shown in Fig. 1a, which includes five processes, i.e., (I) CC charging, (II) CV charging, (III) relaxation after charging, (IV) CC discharging, and (V) relaxation after discharging. The CC discharging capacity is treated as the battery residual capacity during cycling. The relaxation time between the CV charging and CC discharging is $30\\mathrm{min}$ for the NCA battery and NCM battery with a real sampling time of $120s.$ , and it is $60\\mathrm{min}$ for the $\\mathrm{{NCM}+N C A}$ battery with a sampling time of $30\\mathrm{{s}}$ . The starting and ending voltage during the battery relaxation show a declining trend with increasing cycle number as presented in Fig. 1b.  \n\nThree datasets with capacity down to $71\\%$ of the nominal capacity are generated. The battery capacity as a function of cycle number for the NCA cells is shown in Fig. 1c. The cycle number is ranging from 50 to 800 in the $100\\mathrm{-}71\\%$ capacity window. It is evident that both, charging current and temperature have a strong influence on the capacity decay, and the battery capacity shows significant variance as depicted in the embedded plot in Fig. 1c, indicating the degradation distribution of the cycled cells. The worst scenario is the one with cells cycled at 1C charge at $25^{\\circ}\\mathrm{C}$ (CY25-1/1), only 50 cycles can be obtained until the cells reach $71\\%$ of the nominal capacity. In all, $71\\%$ capacity is reached after 125 and 600 cycles at 25 and $35^{\\circ}\\mathrm{C}$ respectively, for cells charged with $0.5\\mathrm{C}$ $(\\mathrm{CY}25–0.5/1$ , and CY35-0.5/1). In total, $71\\%$ capacity is reached after 250 cycles at $25^{\\circ}\\mathrm{C}$ with $0.25\\mathrm{C}$ charging current $(\\mathrm{CY}25–0.25/1)$ and in a range of 500–800 cycles at $45^{\\circ}\\mathrm{C}$ with $0.5\\mathrm{C}$ charging current (CY45-0.5/1). The cycling data of the NCM cells are shown in Fig. 1d. Fatigue down to $71\\%$ residual capacity is found between 250 and 500 cycles $(25^{\\circ}\\mathrm{C})$ , 1250 and 1500 cycles $(35^{\\circ}\\mathrm{C})$ , and around 1000 cycles at $45^{\\circ}\\mathrm{C}$ cycling temperature. The capacity fade results indicate that increasing the temperature to 35 and $45^{\\circ}\\mathrm{C}$ has a beneficial effect on the capacity retention and that the charging current is at the limit of what the cells can handle. For NCA and NCM cells, a capacity spread for the cells cycled under equal conditions is observed, which is speculated to be ascribed to the intrinsic manufacturing variations as this spread is already seen at the beginning of cycling42,43. The cycling data of the $\\mathrm{{NCM+NCA}}$ cells are shown in Fig. 1e, exhibiting a linear degradation trend regardless of the cycling discharge rates, and $71\\%$ residual capacity appears in a range of 750 to 850 cycles showing the influence of the cell cycling conditions.  \n\nFeature extraction. Summarizing statistics are proven to be effective to illustrate numerically the shape and position change of the voltage curve5,9. As mentioned above, the relaxation process after fully charging is taken for feature extraction because of its strong relationship with battery degradation and its easy acquisition in battery real use. Each voltage relaxation curve is converted to six statistical features, i.e., Var, Ske, Max, Min, Mean, and Kur, as displayed in Fig. 2.  \n\nThe relationship between battery capacity and the corresponding features is dependent on the cycling conditions as presented in Fig. 2. It can be seen that it is difficult to describe the relationships only by linear functions. The Var in Fig. 2a represents the distribution of the voltage points in one relaxation process, a decrease of Var versus capacity fade means that the relaxation voltages show a sharper distribution with increasing cycle number, and vice versa. Both Ske and Kur are normalized using Var, they are used to describe the shape of the corresponding voltage curve. The Ske in Fig. 2b is positive for almost all cycling conditions, indicating that more than half of the sampled voltage data are below the average voltage (Mean), which corresponds to the shape of the relaxation voltage curve, i.e., with respect to the relaxation time, the voltage drops initially fast and then gradually slows down. The Max in Fig. 2c presents a monotonous decrease of the maximum voltage versus capacity drop for all cycling conditions. The Min and Mean first increase and then decrease versus the capacity reduction as displayed in Fig. 2d, e, respectively. The Kur shown in Fig. 2f is the excess kurtosis obtained from the kurtosis of the raw data minus the kurtosis of a normal distribution. The excess kurtosis is negative for all cycling conditions, meaning that the distribution of the relaxation voltage is gentler than a normal distribution.  \n\n![](images/6bd6a0c74e23be94f47d40f7abc832208be680504629b34db3858bc37a6094f4.jpg)  \nFig. 1 Battery cycling data. Voltage and current profile in the first cycle of one CY25-0.5/1 NCA battery (a). A plot of relaxation voltage change (region III) while cycling for one NCA cell (b). NCA battery discharge capacity (until $71\\%$ of nominal capacity) versus cycle number of NCA battery (c), NCM battery (d), and $N C M+N C A$ battery (e). The embedded plots in c, d, and e are the cycle distribution of cells at around $71\\%$ of nominal capacity, the points are offset randomly in the horizontal direction to avoid overlapping.  \n\n![](images/46be618726721b1a32ea40083c1f595e43b65febb08fac3150635d6afc5251ef.jpg)  \nFig. 2 Extracted features from the voltage relaxation curves as a function of battery capacity for NCA cells. (a) Variance (Var), (b) skewness $({\\mathsf{S k e}})$ , (c) maxima (Max), (d) minima (Min), (e) mean (Mean), and (f) excess kurtosis (Kur). Feature changes between $3500~\\mathrm{mAh}$ and $2500~\\mathrm{{mAh}}$ $71\\%$ of nominal capacity) for NCA cells are shown to be consistent with the used datasets. The mathematical description of the six features is depicted in Supplementary Table 5.  \n\nCapacity estimation. Based on the features extracted from the relaxation voltage curve after charging, data-driven methods are used for battery capacity estimation. Owing to the difference in the order of magnitudes of the features, a standard normalization for battery features is performed for dataset 1. The features of dataset 2 and dataset 3 are normalized by applying the same normalizing scales as used for dataset 1. The capacity is uniformized considering the difference in the battery nominal capacity. The XGBoost40 is selected as the main machine learning method. The ElasticNet39 as the multivariate linear model is used for comparison, and the $\\mathrm{SVR^{41}}$ is a support for the verification of the transfer learning approach. For the base model training and test, different data splitting strategies are compared with dataset 1 in Supplementary Note 2 and Supplementary Tables 6–9. The best test result of the temperature dependence splitting method shows a $1.5\\%$ RMSE. A $2.3\\%$ test RMSE is obtained from the time-series data splitting method. The data random splitting and cell stratified sampling methods achieve good estimation accuracy with $1.1\\%$ RMSEs, implying that the variation of the working conditions leading to different degradation patterns is essential to improve the generalization of the model. The results of cell stratified sampling method meaning that the data from the same cell is either in the training set or in the test set are presented in this study (Strategy $\\mathrm{~D~}$ in Supplementary Note 2). The cells are approximately in a 4:1 ratio for training and test (Supplementary Table 9). In the model training process, the K-fold cross-validation with $K=5$ is used to determine the hyperparameters of the models. A feature reduction is performed by using different feature combinations to reduce the number of inputs and simplify the model complexity. The cross-validation RMSEs under different feature combinations using the XGBoost method are compared in Fig. 3. The $i$ and $j$ are used to represent different feature combinations referring the Supplementary Table 10.  \n\nIt shows that the RMSE gradually decreases as the number of features increases, and the accuracy improvement is no longer obvious after using three features in Fig. 3. The best estimation result is obtained by the input [Var, Ske, Max] in a three feature combination. The effect of the duration of the relaxation on the capacity estimation is presented in Supplementary Fig. 3, in which the RMSEs of training and test decrease as the relaxation time increases in the XGBoost method, indicating that longer relaxation time improves the model accuracy. Therefore, the Var, Ske, and Max of the voltage relaxation after $30\\mathrm{min}$ are extracted as inputs for the base model. The hyperparameters of each algorithm are available in Supplementary Table 11. The RMSEs of different estimation methods on dataset 1 are summarized in Fig. 4a. It can be concluded that the test RMSE of XGBoost and SVR all reaches $1.1\\%$ , showing better performance than the linear model, and the RMSEs of train and test are close to each other, indicating the effectiveness of data splitting. The estimated capacity versus real capacity is illustrated in Fig. 4b–d for visualization purposes.  \n\nPerformance of the proposed approach. The performance of the proposed approach is benchmarked with state-of-the-art models using voltage curves for battery capacity estimation as shown in Table 2. One representative method is selected from each class of the presented capacity estimation methods (Supplementary Table 1). Since the datasets used in the literature are different in battery material and test procedures from ours, the strategy to solve this difference is to apply their algorithms to our datasets. A detailed description of data processing and estimation results for each method is presented in Supplementary Note 3 and Supplementary Figs. 4–7. The performance of the linear model to estimate the battery capacity based on the resting voltage in Baghdadi et al.25 shows a $2.5\\%$ RMSE, which can be explained by the large data volume and variety of working conditions in our dataset 1 highlighting the difficulty of capacity estimation only with the linear model. In the CC charge voltage-based methods, the random forest regression (RFR) method16 using the voltage ranging from $3.6\\mathrm{V}$ to $3.8\\mathrm{V}$ achieves a RMSE of $1.0\\%$ on dataset 1, which is $0.1\\%$ less than our RMSE based on the voltage relaxation. A method based on the remaining electrical charge with a threshold according to the incremental capacity value is proposed in Peri et al.20. The application of the same incremental capacity transformation method on dataset 1 provides a RMSE of $1.3\\%$ , indicating that our proposed approach has better accuracy. The Gaussian process regression (GPR) method44 using a full CC-CV charge voltage curve obtains good estimation results on dataset 1 with a test RMSE of $1.1\\%$ . Compared with the current research status, especially with respect to large datasets, the proposed approach using resting voltage can achieve a good estimation accuracy. As mentioned in the introduction section, there are some challenges in the acquisition of specific charging voltage curves because the start of battery charge is usually dependent on the driver behavior and the charge modes differ significantly from the charging stations in the real application of EVs. The relaxation process of a battery being fully charged is easily obtained without the requirement of specific working conditions and voltage ranges, which offers a new sight for battery capacity estimation.  \n\n![](images/f7144892102d556a2e421ec81f8a89a7fed5126f1c6f560839e0c987e38c140a.jpg)  \nFig. 3 Cross-validation root-mean-square error (RMSE) of the XGBoost method using different feature combinations. $({\\boldsymbol{{j}}},{\\boldsymbol{{j}}})$ means different feature combinations referring the Supplementary Table 10. The $(7,1)=\\mathsf{\\Gamma}[\\mathsf{V a r}_{\\cdot}$ Ske, Max] obtains the best cross-validation $\\mathsf{R M S E}=1.0\\%$ within a three feature combination.  \n\nPhysical explanation. The alternating current (AC) electrochemical impedance provides information in the frequency domain on the degradation mechanisms of the battery as proven in ref. 45. The degradation mechanisms can be determined from the change of electrochemical impedance parameters extracted by fitting the impedance spectra with an $\\mathrm{EC}\\mathrm{\\bar{M}}^{46}$ . A schematic plot of electrochemical impedance spectra during cycling and the corresponding ECM are complemented in Supplementary Figure 8.  \n\n![](images/2123dbc77bbdd2f98db592fc04f3dd3ab1bd47cdafbd9417782294d6f9ced749.jpg)  \nFig. 4 Results of battery capacity estimation with the input of three features [Var, Ske, Max] by different estimation methods. The capacity results are uniformized by the nominal capacity for comparison. root-mean-square error (RMSE) of battery capacity estimation ${\\bf\\Pi}({\\bf a})$ , test results of estimated capacity versus real capacity by ElasticNet (b), XGBoost $(\\bullet),$ and Support Vectors Regression (SVR) (d).  \n\n<html><body><table><tr><td colspan=\"3\">Table 2 Test means root-mean-square error (RMSE) of different models using voltage-based features for battery capacity estimation.</td></tr><tr><td>Features from</td><td>Methods</td><td>Test RMSE on Dataset 1</td></tr><tr><td>Rest voltage-based</td><td>Linear model25</td><td>0.025</td></tr><tr><td>Constant current charge voltage- based</td><td>Random forest regression16</td><td>0.010</td></tr><tr><td>Incremental capacity analysis transformation</td><td>Linear model20</td><td>0.013</td></tr><tr><td>Constant current-constant voltage Gaussian process charge voltage-based</td><td>regression44</td><td>0.011</td></tr></table></body></html>  \n\nBasically, an increase of R0 is likely due to contact loss and the reduction of ionic conductivity in the electrolyte47. R1 represents the resistance associated with the anode solid electrolyte interphase (SEI) indicated by the semicircle at high frequencies46. R2 is the charge-transfer resistance describing the rate of electrochemical reaction, which is related to the loss of electrode material through particle cracking18,48. The capacity loss of the cycled cells in dataset 1 and dataset 2 has been investigated by in situ neutron powder diffraction in our previous work42, which exhibits that the decrease in lithium content in the positive and negative electrodes correlates well with the observed discharge capacity. Both positive and negative electrodes do not decompose to other crystalline phases during cycling, but the lithium loss in the electrodes leading to lithiated material loss is traced by detecting changes in the lattices of the electrodes. The lithiated material loss and the SEI formation are suspected to contribute to the lithium loss.  \n\nHerein, the dominating aging factors for each cycling group are discussed by fitted electrochemical impedance parameters in  \n\nFig. 5. The coefficient of determination $(\\mathbb{R}^{2})$ of each measured impedance spectrum between the raw and fitted data is summarized in Supplementary Table 12. All $\\mathrm{R}^{2}$ values are greater than 0.999, indicating the credible fitting accuracy. All the raw and fitted impedance data can be found from the data availability. By comparison of the resistance increment from the initial value $(\\dot{\\mathrm{R}}_{\\mathrm{init}})$ for all three type cells, the increment of R0 is minimal (Fig. 5a–c), followed by R1 (Fig. 5d–f). R2 shows the highest increase during the battery capacity fade as shown in Fig. 5g–i. The dominating degradation factors are different under different working conditions. For the NCA cell, as shown in Fig. 5a, the CY25-0.25/1 shows a steady and relatively small increase of R0, nevertheless, its R1 in Fig. 5d shows an accelerated rise, indicating the increase in the thickness of the SEI layer. The R2 of CY25- $0.25/1$ in Fig. $5\\mathrm{g}$ presents a similar increasing trend to its R0. The R0 of $\\mathrm{CY}25–0.5/1$ and $\\mathrm{CY}25–1/1$ in Fig. 5a remains the largest resistive contribution throughout, but their R1 and R2 are relatively lower than that of others, which indicates a more serious cell degradation such as electrolyte dry-out or contact loss likely caused by lithium plating47,49. For the results of NCM cells in Fig. 5b, e, h, all resistances of $\\mathrm{CY}25–0.5/1$ increase slowly, while resistances of cells cycled at 35 and $45^{\\circ}\\mathrm{C}$ exhibit a large increase rate. For the $\\mathrm{{NCA}+\\mathrm{{NCM}}}$ cells, the influence of discharge rate is mainly represented by R1 by comparing the results in Fig. 5c, f, i. The CY25-0.5/4 SEI resistance increase in Fig. 5f is significantly slower than that of other cycling conditions. The temperature influence on the degradation mechanism can be seen in Fig. 5g, h, in which the increase of R2 is associated mainly with the increase of ambient temperature. The cells cycled at 45 and $35^{\\circ}\\mathrm{C}$ mainly lead to an increase of R2, which could be associated with the positive active material loss, e.g., particle cracking and pulverization50,51. The diversity of the battery internal degradation mechanisms results in various degradation paths, which can explain the difficulty in applying a simple linear model on the battery capacity estimation. Additionally, it seems that different battery types follow to some extent similar degradation rules, e.g., the exponential rise of R2, inspiring the use of transfer learning in the following part.  \n\n![](images/ccf20298f3335ec1843de072b3e31f8a592c3be7a802cde118b2e0d6de433308.jpg)  \nFig. 5 AC electrochemical impedance variations of the lithium-ion cells during cycling. The resistance increment from the initial value $(R_{\\mathrm{init}})$ is calculated for comparison. The ohmic resistance of NCA cells (a), NCM cells $(\\pmb{\\ b})$ , and ${\\mathsf{N C A}}+{\\mathsf{N C M}}$ cells (c). SEI resistance of NCA cells (d), NCM cells (e), and NCA $+N C M$ cells (f). Charge transfer resistance of NCA cells $\\mathbf{\\sigma}(\\mathbf{g})$ , NCM cells ${\\bf\\Pi}({\\bf h})$ , and $N C A+N C M$ cells (i). Only resistances before the capacity reducing to $71\\%$ of nominal capacity are shown to be consistent with the datasets in the study. The coefficient of determination $(R^{2})$ between the raw and fitted impedance data is summarized in Supplementary Table 12. The SEI resistances are not identified in some cycles (seen in Supplementary Table 12) for the NCA battery (d) and NCM battery (e). The shared information of raw impedance data and fitted data can be found in the data availability.  \n\nApproach verification by transfer learning. The transfer learning (TL) method, which is applied to improve the learning ability by rebuilding the machine learning model using a relatively small amount of newly collected data, is proposed for easy adaption to the variation of voltage features existing in dataset 2 and dataset 3 in which different batteries and cycling conditions are used. The model weights are pre-trained through dataset 1 to obtain the base model. Then, some new data units from dataset 2 and dataset 3 are set as the input variable to re-train the TL model. Different data selection methods are discussed in Supplementary Note 4 and Supplementary Table 13, depicting that the variation of working conditions is necessary to improve the accuracy of the model estimation. One cell is randomly selected from each cycling condition in dataset 2 and dataset 3, then the data units in each cell are chosen with an interval of 100 cycles as the input variables for the re-training of TL models (Strategy D in Supplementary Note 4). The sizes of the input variable are summarized in Supplementary Table 14 (occupying $0.06\\%$ of dataset 2 and $0.35\\%$ of dataset 3). Verification on dataset 2 and dataset 3 without changing any weights of the base model is used as a zeroshot learning (ZSL) reference. The full base model is retrained using the same input variables from dataset 2 and dataset 3 as a No TL comparison. Two TL methods (TL1 and TL2) with fine-tuning strategies are activated to adjust the weights of a newly added layer, while the weights of other layers remain unchanged. TL1 means that a linear transformation layer is added before the output of capacity. TL2 means that a linear transformation layer before the base model is constructed to adapt the input features as illustrated in Supplementary Fig. 9. The test RMSEs are compared in Table 3.  \n\nThe ZSL strategy obtains more than $3.4\\%$ test RMSE on all datasets directly using the base models. The error between the estimated capacity and real capacity is quite large as shown in Supplementary Fig. 10, meaning that the differences in battery types and materials cannot be ignored. When the base model is retrained in the No TL strategy, the XGBoost reaches a $2.9\\%$ test RMSE on dataset 2 and a $2.0\\%$ test RMSE on dataset 3, and the SVR gives no obvious improvement in the accuracy (Supplementary Fig. 11 and Supplementary Table 15). When the TL1 is applied on dataset 2 and dataset 3, the test RMSE of the SVR method goes down to 2.6 and $3.5\\%$ respectively, but a high number of outliers still appears in Supplementary Fig. 12. The results of estimated capacity versus real capacity by TL2 are presented in Fig. 6. The test RMSE is reduced to $2.4\\%$ by the XGBoost using the TL2 on dataset 2, noting that the performance of XGBoost using the No TL on dataset 3 is better than that of TL, which could be ascribed to the narrow distribution of capacity fade in dataset 3. The best accuracies on dataset 2 and dataset 3 are all reached by SVR using the TL2, showing test RMSEs of 1.7 and $1.6\\%$ , respectively. It can be concluded that the use of TL2 improves the estimation accuracy, and the reason behind the accuracy improvement is that a linear transformation of the input features helps the model adapt to the differences in battery types but similarity degradation modes. Interestingly, we find that the SVR is more reliable and suitable for transfer learning than the XGBoost with a small amount of newly collected data. The possible reason is that the XGBoost is a discrete gradient boosting framework, the output of the model is limited by the base model even if a new layer is added before the base model, whereas the SVR is a kernel-based framework, in which the continuous calculation achieves a better prediction under the designed TL2. In summary, the proposed approach using the relaxation voltage curve is useful to estimate the battery capacity, and the transfer learning improves the accuracy of capacity estimation requiring little tuning to adapt to the difference in batteries.  \n\n<html><body><table><tr><td colspan=\"6\">Table 3 Test RMSEs of battery capacity estimation using zero-shot learning (ZSL) and different transfer learning (TL) methods on dataset 2 and dataset 3.</td></tr><tr><td>Methods</td><td>Dataset</td><td>ZSL</td><td>No TL</td><td>TL1</td><td>TL2</td></tr><tr><td>XGBoost</td><td>Dataset2 Dataset3</td><td>0.038 0.038</td><td>0.029 0.020</td><td>0.027</td><td>0.024</td></tr><tr><td></td><td>Dataset 2</td><td>0.034</td><td>0.039</td><td>0.034 0.026</td><td>0.024</td></tr><tr><td>Support vectors</td><td>Dataset3</td><td>0.073</td><td>0.052</td><td>0.035</td><td>0.017</td></tr><tr><td>regression</td><td></td><td></td><td></td><td></td><td>0.016</td></tr></table></body></html>  \n\n# Discussion  \n\nAccurate identification of lithium-ion battery capacity facilitates the accurate estimation of the driving range which is a primary concern for EVs. An approach without requiring information from the previous cycling to estimate battery capacity is proposed. The proposed approach uses three statistical features ([Var, Ske, Max]) extracted from the voltage relaxation curve as input to predict the capacity in the next cycle. The transfer learning embedding machine learning methods is applied on 130 cells to establish a suitable model and for the verification of the approach. The best base model achieves a root-mean-square error of $1.1\\%$ . The transfer learning adding a linear transformation layer before the base model shows good predictive ability within a RMSE of $1.7\\%$ on different batteries. The retraining of transfer learning only needs a small number of data units on the condition that a variation of the input data needs to be guaranteed to improve the applicability of the proposed approach.  \n\n![](images/10f044569c66e89142c2f8538156dc47a10e75b4e279c382361b91bca77134c7.jpg)  \nFig. 6 Test results of estimated capacity versus real capacity by transfer learning. The capacity results are uniformized by the nominal capacity for comparison. Results of TL2 embedding XGBoost method (a) and embedding SVR (b) on dataset 2. Results of TL2 embedding XGBoost method (c) and embedding SVR (d) on dataset 3. Additional results are disclosed in Supplementary Figs. 10–12.  \n\nThe relaxation process of a battery after full charge is easily obtained without the requirement of specific working conditions and voltage ranges, providing a new possibility for battery capacity estimation using data-driven methods in the system implementation of EV applications.  \n\n# Methods  \n\nCell selection and cycling. Commercially available lithium-ion batteries, i.e., LG INR18650-35E (3.5 Ah, 3.6 V), Samsung INR18650-MJ1 (3.5 Ah, 3.6 V), and Samsung INR18650-25R (2.5Ah, 3.6 V), have been tested. More battery specifications are listed in Supplementary Table 4. The positive electrode compositions of the INR18650-35E battery and INR18650-MJ1 battery are $\\mathrm{LiNi}_{0.86}\\mathrm{Co}_{0.11}\\mathrm{Al}_{0.03}\\mathrm{O}_{2}$ and $\\mathrm{Li}(\\mathrm{Ni}_{0.83}\\mathrm{Co}_{0.11}\\mathrm{Mn}_{0.07})\\mathrm{O}_{2}$ respectively, and the negative electrodes for both cell types have roughly $97\\mathrm{wt\\%}$ C and $2\\mathrm{wt\\%}$ Si as well as traces of H, N, and S from Sorensen et al.42. The positive electrode of the INR18650-25R battery is the blend of 42 (3) wt. $\\%$ $\\mathrm{Li}(\\mathrm{NiCoMn}){\\mathrm{O}}_{2}\\cdot5\\xi$ 8 (3) wt.% $\\mathrm{Li}(\\mathrm{NiCoAl})\\mathrm{O}_{2}$ , and the negative electrode is graphite from ref. 18. The INR18650-35E battery is named as NCA battery. The INR18650-MJ1 is named as NCM battery. The INR18650-25R is named as $\\mathrm{{NCM+NCA}}$ battery according to the positive electrode. A potentiostat (BioLogic BCS-815, France) is employed for cell cycling. The measurements are conducted at 25, 35, and $45^{\\circ}\\mathrm{C}$ in a climate chamber (BINDER, $\\pm0.2^{\\circ}\\mathrm{C}$ , Germany). Long-term cycling is conducted on a total of 130 cells with a summary of cycling conditions as provided in Table 1. A schematic connection of the potentiostat, chamber, and cells is shown in Supplementary Figure 13. For the NCA and NCM batteries, the metal taps are spot-welded to the cells, and the contact is soldered to the metal taps. A four-wire holder is used for the $\\mathrm{{NCM+NCA}}$ battery. For partially charged/ discharged NCA and NCM cells, the electrochemical impedance is measured in the fully charged state using a frequency range of $10\\mathrm{kHz}$ to $50~\\mathrm{mHz}$ (20 data points per decade of frequency) and a potential amplitude of $20\\mathrm{mV}$ . 30 min are set at the open circuit voltage before the electrochemical impedance tests. The electrochemical impedance is tested every 25 cycles for the NCA battery and every 50 cycles for the NCM battery. For the $\\mathrm{{NCM+NCA}}$ battery, the electrochemical impedance is conducted every 50 cycles at full charge in a range of $10\\mathrm{kHz}$ to $0.01\\mathrm{Hz}$ (6 data points per decade of frequency) with a sinusoidal amplitude of 250 mA. 60 min are set at the open circuit voltage before the electrochemical impedance tests. The NCA cells and NCM cells are tested from 2016 to 2018, and the $\\mathrm{{NCM+NCA}}$ cells are cycled in 2020. Different experimenters at different test periods are responsible for the difference in battery connection methods and experimental parameters in AC impedance tests, e.g., perturbation modes, perturbation amplitudes, and open circuit voltage time.  \n\nMachine learning methods. Two transfer learning strategies embedding the XGBoost method and SVR method are applied in our study, and an illustration of the implemented transfer learning process is shown in Supplementary Fig. 9. The algorithms of the ElasticNet method, XGBoost method, and SVR method are introduced in Supplementary Note 5.  \n\n(1) The base model is trained on all experimental data of NCA batteries (dataset 1). Firstly, the base model is directly verified on dataset 2 and dataset 3 without changing model weights as a zero-shot learning (ZSL) reference.   \n(2) The base model is retrained using some new data units (Strategy D in Supplementary Note 4) as input variables from dataset 2 and dataset 3 as a No TL comparison.   \n(3) Two transfer learning strategies (TL1 and TL2) are proposed by adding layers behind and in front of the base model. All weights in the base model are frozen in the transfer learning strategies except the newly added layer. In detail, TL1 means that a linear transformation layer is added before the output of capacity, which is described as  \n\n$$\n\\mathrm{Q}^{\\prime}=\\mathrm{wQ}+\\mathrm{b}\n$$  \n\nTL2 means that a linear transformation layer is constructed to adapt the input features, which is described as  \n\n$$\n\\left[\\begin{array}{l}{\\mathrm{Var}^{\\prime}}\\\\ {\\mathrm{Ske}^{\\prime}}\\\\ {\\mathrm{Max}^{\\prime}}\\end{array}\\right]=W\\left[\\begin{array}{l}{\\mathrm{Var}}\\\\ {\\mathrm{Ske}}\\\\ {\\mathrm{Max}}\\end{array}\\right]+b\n$$  \n\n$w,W_{;}$ , and $b$ are the weights in the added layer. The target dataset from dataset 2 and dataset 3 are selected to train the new layer weights.  \n\nThe transfer learning models are verified on the remaining dataset 2 and dataset 3 respectively. The test RMSEs are compared in Table 3, and the estimation results are presented in Fig. 6 and Supplementary Figs. 10–12 for visualization purposes.  \n\n# Data availability  \n\nThe data generated in this study have been deposited in the Zenodo database under accession code [https://doi.org/10.5281/zenodo.6379165].  \n\n# Code availability  \n\nThe data processing is performed in python and is available at [https://github.com/YixiuWang/data-driven-capacity-estimation-from-voltage-relaxation]. Code for the modeling work is available from the corresponding authors upon request.  \n\nReceived: 1 August 2021; Accepted: 1 April 2022; Published online: 27 April 2022  \n\n# References  \n\n1. Bresser, D. et al. Perspectives of automotive battery R&D in China, Germany, Japan, and the USA. J. Power Sources 382, 176–178 (2018).   \n2. Harper, G. et al. Recycling lithium-ion batteries from electric vehicles. Nature 575, 75–86 (2019).   \n3. Waag, W., Käbitz, S. & Sauer, D. U. 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Transportation Electrification 7, 410–421 (2020).   \n51. Stiaszny, B. et al. Electrochemical characterization and post-mortem analysis of aged LiMn2O4–NMC/graphite lithium ion batteries part II: Calendar aging. J. Power Sources 258, 61–75 (2014).  \n\n# Acknowledgements  \n\nThis work contributes to the research performed at CELEST (Center for Electrochemical Energy Storage Ulm-Karlsruhe) and is supported in the frame of the Alexander von Humboldt Postdoctoral Research Program. Jiangong Zhu would like to thank the foundation of the National Natural Science Foundation of China (NSFC, Grant No. 52107230) and he is supported by the Fundamental Research Funds for the Central Universities. Haifeng Dai would like to thank the foundation of the National Natural Science Foundation of China (NSFC, Grant No. U20A20310).  \n\n# Author contributions  \n\nConceptualization, writing, and original draft preparation were done by J.Z., Y.W., and H.D. The experimental studies were performed by J.Z., L.M., M.J.M., and M.H. The computational studies are performed by Y.W., J.Z., and Y.H. R.B.G., Y.C., X.L., H.D., M.K., M.H., A.S., and H.E. were involved in the writing, review, and editing of this manuscript. H.D., M.K., X.W., and H.E. supervised the work.  \n\n# Competing interests  \n\nThe authors declare no competing interests.  \n\n# Additional information  \n\nSupplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41467-022-29837-w.  \n\nCorrespondence and requests for materials should be addressed to Haifeng Dai or Michael Knapp.  \n\nPeer review information Nature Communications thanks Penelope Jones, Shunli Wang, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.  \n\n# Reprints and permission information is available at http://www.nature.com/reprints  \n\nPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\nOpen Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.  ",
    "Target": null
  },
  {
    "id": "10.1038_s41586-021-04223-6",
    "DOI": "10.1038/s41586-021-04223-6",
    "DOI Link": "http://dx.doi.org/10.1038/s41586-021-04223-6",
    "Relative Dir Path": "mds/10.1038_s41586-021-04223-6",
    "Article Title": "Deep physical neural networks trained with backpropagation",
    "Authors": "Wright, LG; Onodera, T; Stein, MM; Wang, TY; Schachter, DT; Hu, Z; McMahon, PL",
    "Source Title": "NATURE",
    "Abstract": "Deep-learning models have become pervasive tools in science and engineering. However, their energy requirements now increasingly limit their scalability(1). Deep-learning accelerators(2-9) aim to perform deep learning energy-efficiently, usually targeting the inference phase and often by exploiting physical substrates beyond conventional electronics. Approaches so far(10-22) have been unable to apply the backpropagation algorithm to train unconventional novel hardware in situ. The advantages of backpropagation have made it the de facto training method for large-scale neural networks, so this deficiency constitutes a major impediment. Here we introduce a hybrid in situ-in silico algorithm, called physics-aware training, that applies backpropagation to train controllable physical systems. Just as deep learning realizes computations with deep neural networks made from layers of mathematical functions, our approach allows us to train deep physical neural networks made from layers of controllable physical systems, even when the physical layers lack any mathematical isomorphism to conventional artificial neural network layers. To demonstrate the universality of our approach, we train diverse physical neural networks based on optics, mechanics and electronics to experimentally perform audio and image classification tasks. Physics-aware training combines the scalability of backpropagation with the automatic mitigation of imperfections and noise achievable with in situ algorithms. Physical neural networks have the potential to perform machine learning faster and more energy-efficiently than conventional electronic processors and, more broadly, can endow physical systems with automatically designed physical functionalities, for example, for robotics(23-26), materials(27-29) and smart sensors(30-32).",
    "Times Cited, WoS Core": 355,
    "Times Cited, All Databases": 386,
    "Publication Year": 2022,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000749546400022",
    "Markdown": "# Article  \n\n# Deep physical neural networks trained with backpropagation  \n\nhttps://doi.org/10.1038/s41586-021-04223-6  \n\nReceived: 19 May 2021  \n\nAccepted: 9 November 2021  \n\nPublished online: 26 January 2022  \n\nOpen access  \n\n# Check for updates  \n\nLogan G. Wright1,2,4 ✉, Tatsuhiro Onodera1,2,4 ✉, Martin M. Stein1, Tianyu Wang1, Darren T. Schachter3, Zoey Hu1 & Peter L. McMahon1 ✉  \n\nDeep-learning models have become pervasive tools in science and engineering. However, their energy requirements now increasingly limit their scalability1. Deep-learning accelerators2–9 aim to perform deep learning energy-efficiently, usually targeting the inference phase and often by exploiting physical substrates beyond conventional electronics. Approaches so far10–22 have been unable to apply the backpropagation algorithm to train unconventional novel hardware in situ. The advantages of backpropagation have made it the de facto training method for large-scale neural networks, so this deficiency constitutes a major impediment. Here we introduce a hybrid in situ–in silico algorithm, called physics-aware training, that applies backpropagation to train controllable physical systems. Just as deep learning realizes computations with deep neural networks made from layers of mathematical functions, our approach allows us to train deep physical neural networks made from layers of controllable physical systems, even when the physical layers lack any mathematical isomorphism to conventional artificial neural network layers. To demonstrate the universality of our approach, we train diverse physical neural networks based on optics, mechanics and electronics to experimentally perform audio and image classification tasks. Physics-aware training combines the scalability of backpropagation with the automatic mitigation of imperfections and noise achievable with in situ algorithms. Physical neural networks have the potential to perform machine learning faster and more energy-efficiently than conventional electronic processors and, more broadly, can endow physical systems with automatically designed physical functionalities, for example, for robotics23–26, materials27–29 and smart sensors30–32.  \n\nLike many historical developments in artificial intelligence33,34, the widespread adoption of deep neural networks (DNNs) was enabled in part by synergistic hardware. In 2012, building on earlier works, Krizhevsky et al. showed that the backpropagation algorithm could be efficiently executed with graphics-processing units to train large DNNs35 for image classification. Since 2012, the computational requirements of DNN models have grown rapidly, outpacing Moore’s law1. Now, DNNs are increasingly limited by hardware energy efficiency.  \n\nThe emerging DNN energy problem has inspired special-purpose hardware: DNN ‘accelerators’2–8, most of which are based on direct mathematical isomorphism between the hardware physics and the mathematical operations in DNNs (Fig. 1a, b). Several accelerator proposals use physical systems beyond conventional electronics8, such as optics9 and analogue electronic crossbar arrays3,4,12. Most devices target the inference phase of deep learning, which accounts for up to $90\\%$ of the energy costs of deep learning in commercial deployments1, although, increasingly, devices are also addressing the training phase (for example, ref. 7).  \n\nHowever, implementing trained mathematical transformations by designing hardware for strict, operation-by-operation mathematical isomorphism is not the only way to perform efficient machine learning. Instead, we can train the hardware’s physical transformations directly to perform desired computations. Here we call this approach physical neural networks (PNNs) to emphasize that physical processes, rather than mathematical operations, are trained. This distinction is not merely semantic: by breaking the traditional software–hardware division, PNNs provide the possibility to opportunistically construct neural network hardware from virtually any controllable physical system(s). As anyone who has simulated the evolution of complex physical systems appreciates, physical transformations are often faster and consume less energy than their digital emulations. This suggests that PNNs, which can harness these physical transformations most directly, may be able to perform certain computations far more efficiently than conventional paradigms, and thus provide a route to more scalable, energy-efficient and faster machine learning.  \n\n![](images/b63f8de339d26114efc9d083b05041dddddf9a673257dd6c45700afc0616459e.jpg)  \nFig. 1 | Introduction to PNNs. a, Artificial neural networks contain operational units (layers): typically, trainable matrix-vector multiplications followed by element-wise nonlinear activation functions. b, DNNs use a sequence of layers and can be trained to implement multi-step (hierarchical) transformations on input data. c, When physical systems evolve, they perform, in effect, computations. We partition their controllable properties into input data and control parameters. Changing parameters alters the transformation performed on data. We consider three examples. In a mechanical (electronic) system, input data and parameters are encoded into time-dependent forces (voltages) applied to a metal plate (nonlinear circuit). The controlled  \n\nmultimode oscillations (transient voltages) are then measured by a microphone (oscilloscope). In a nonlinear optical system, pulses pass through a $\\chi^{(2)}$ crystal, producing nonlinearly mixed outputs. Input data and parameters are encoded in the input pulses’ spectra, and outputs are obtained from the frequency-doubled pulses’ spectra. d, Like DNNs constructed from sequences of trainable nonlinear mathematical functions, we construct deep PNNs with sequences of trainable physical transformations. In PNNs, each physical layer implements a controllable physical function, which does need to be mathematically isomorphic to a conventional DNN layer.  \n\nPNNs are particularly well motivated for DNN-like calculations, much more so than for digital logic or even other forms of analogue computation. As expected from their robust processing of natural data, DNNs and physical processes share numerous structural similarities, such as hierarchy, approximate symmetries, noise, redundancy and nonlinearity36. As physical systems evolve, they perform transformations that are effectively equivalent to approximations, variants and/ or combinations of the mathematical operations commonly used in DNNs, such as convolutions, nonlinearities and matrix-vector multiplications. Thus, using sequences of controlled physical transformations (Fig. 1c), we can realize trainable, hierarchical physical computations, that is, deep PNNs (Fig. 1d).  \n\nAlthough the paradigm of constructing computers by directly training physical transformations has ancestry in evolved computing materials18, it is today emerging in various fields, including optics14,15,17,20, spintronic nano-oscillators10,37, nanoelectronic devices13,19 and small-scale quantum computers38. A closely related trend is physical reservoir computing (PRC)21,22, in which the transformations of an untrained physical ‘reservoir’ are linearly combined by a trainable output layer. Although PRC harnesses generic physical processes for computation, it is unable to realize DNN-like hierarchical computations. In contrast, approaches that train the physical transformations13–19 themselves can, in principle, overcome this limitation. To train physical transformations experimentally, researchers have frequently relied on gradient-free learning algorithms10,18–20. Gradient-based learning algorithms, such as the backpropagation algorithm, are considered essential for the efficient training and good generalization of large-scale DNNs39. Thus, proposals to realize gradient-based training in physical hardware have appeared40–47. These inspiring proposals nonetheless make assumptions that exclude many physical systems, such as linearity, dissipation-free evolution or that the system be well described by gradient dynamics. The most general proposals13–16 overcome such constraints by performing training in silico, that is, learning wholly within numerical simulations. Although the universality of in silico training is empowering, simulations of nonlinear physical systems are rarely accurate enough for models trained in silico to transfer accurately to real devices.  \n\nHere we demonstrate a universal framework using backpropagation to directly train arbitrary physical systems to execute DNNs, that is, PNNs. Our approach is enabled by a hybrid in situ–in silico algorithm, called physics-aware training (PAT). PAT allows us to execute the backpropagation algorithm efficiently and accurately on any sequence of physical input–output transformations. We demonstrate the universality of this approach by experimentally performing image classification using three distinct systems: the multimode mechanical oscillations of a driven metal plate, the analogue dynamics of a nonlinear electronicoscillator and ultrafast optical second-harmonic generation (SHG). We obtain accurate hierarchical classifiers that utilize each system’s unique physical transformations, and that inherently mitigate each system’s unique noise processes and imperfections. Although PNNs are a radical departure from traditional hardware, it is easy to integrate them into modern machine learning. We show that PNNs can be seamlessly combined with conventional hardware and neural network methods via physical–digital hybrid architectures, in which conventional hardware learns to opportunistically cooperate with unconventional physical resources using PAT. Ultimately, PNNs provide routes to improving the energy efficiency and speed of machine learning by many orders of magnitude, and pathways to automatically designing complex functional devices, such as functional nanoparticles28, robots25,26 and smart sensors30–32.  \n\n![](images/096af250f69968a42b25be4174d30abed0a2f72959596fb8fce76ede3b33fcb1.jpg)  \nFig. 2 | An example PNN, implemented experimentally using broadband optical SHG. a, Input data are encoded into the spectrum of a laser pulse (Methods, Supplementary Section 2). To control transformations implemented by the broadband SHG process, a portion of the pulse’s spectrum is used as trainable parameters (orange). The physical computation result is obtained from the spectrum of a blue (about $390{\\mathsf{n m}},$ ) pulse generated within $\\mathbf{a}\\chi^{(2)}$   \nmedium. b, To construct a deep PNN, the outputs of the SHG transformations are used as inputs to subsequent SHG transformations, with independent trainable parameters. c, d, After training the SHG-PNN (see main text, Fig. 3), it classifies test vowels with $93\\%$ accuracy. c, The confusion matrix for the PNN on the test set. d, Representative examples of final-layer output spectra, which show the SHG-PNN’s prediction.  \n\n# An example PNN based on nonlinear optics  \n\nFigure 2 shows an example PNN based on broadband optical pulse propagation in quadratic nonlinear media (ultrafast SHG). Ultrafast SHG realizes a physical computation roughly analogous to a nonlinear convolution, transforming the input pulse’s near-infrared spectrum (about $800\\cdot\\mathrm{{nm}}$ centre wavelength) into the blue (about $400\\mathsf{n m}$ ) through a multitude of nonlinear frequency-mixing processes (Methods). To control this computation, input data and parameters are encoded into sections of the spectrum of the near-infrared pulse by modulating its frequency components using a pulse shaper (Fig. 2a). This pulse then propagates through a nonlinear crystal, producing a blue pulse whose spectrum is measured to read out the result of the physical computation.  \n\nTo realize vowel classification with SHG, we construct a multilayer SHG-PNN (Fig. 2b) where the input data for the first physical layer consist of a vowel-formant frequency vector. After the final physical layer, the blue output spectrum is summed using a digital computer into seven spectral bins (Fig. 2b, d, Supplementary Figs. 21, 22). The predicted vowel is identified by the bin with the maximum energy (Fig. 2c). In each layer, the output spectrum is digitally renormalized before being passed to the next layer (via the pulse shaper), along with a trainable digital rescaling. Mathematically, this transformation is given by $\\mathbf{X}^{[l+1]}=\\frac{a\\mathbf{y}^{[l]}}{\\operatorname*{max}(\\mathbf{y}^{[l]})}+b$ , where $\\mathbf{x}^{[l]}$ and $\\mathbf{y}^{[l]}$ are the inputs and outputs of the [l]th layer, respectively, and $a$ and $b$ are scalar parameters of the transformation.  Thus, the SHG-PNN’s computations are carried out almost entirely by the trained optical transformations, without digital activation functions or output layers.  \n\n![](images/1d3a141ddf7736a9391277596d8b8451509b7e1a4f4a170932cdd35b3580541c.jpg)  \nFig. 3 | Physics-aware training. a, PAT is a hybrid in situ–in silico algorithm to apply backpropagation to train controllable physical parameters so that physical systems perform machine-learning tasks accurately even in the presence of modelling errors and physical noise. Instead of performing the training solely within a digital model (in silico), PAT uses the physical systems to compute forward passes. Although only one layer is depicted in a, PAT   \ngeneralizes naturally to multiple layers (Methods). b, Comparison of the validation accuracy versus training epoch with PAT and in silico training, for the experimental SHG-PNN depicted in Fig. 2b. c, Final experimental test accuracy for PAT and in silico training for SHG-PNNs with increasing numbers of physical layers. The length of error bars represent two standard errors.  \n\nDeep PNNs essentially combine the computational philosophy of techniques such as $\\mathsf{P R C}^{21,22}$ with the trained hierarchical computations and gradient-based training of deep learning. In PRC, a physical system, often with recurrent dynamics, is used as an untrained feature map and a trained linear output layer (typically on a digital computer) combines these features to approximate desired functions. In PNNs, the backpropagation algorithm is used to adjust physical parameters so that a sequence of physical systems performs desired computations physically, without needing an output layer. For additional details, see Supplementary Section 3.  \n\n# Physics-aware training  \n\nTo train the PNNs’ parameters using backpropagation, we use PAT (Fig. 3). In the backpropagation algorithm, automatic differentiation determines the gradient of a loss function with respect to trainable parameters. This makes the algorithm $N$ -times more efficient than finite-difference methods for gradient estimation (where N is the number of parameters). The key component of PAT is the use of mismatched forward and backward passes in executing the backpropagation algorithm. This technique is well known in neuromorphic computing48–53, appearing recently in direct feedback alignment52 and quantization-aware training48, which inspired PAT. PAT generalizes these strategies to encompass arbitrary physical layers, arbitrary physical network architectures and, more broadly, to differentially programmable physical devices.  \n\nPAT proceeds as follows (Fig. 3). First, training input data (for example, an image) are input to the physical system, along with trainable parameters. Second, in the forward pass, the physical system applies its transformation to produce an output. Third, the physical output is compared with the intended output to compute the error. Fourth, using a differentiable digital model, the gradient of the loss is estimated with respect to the controllable parameters. Finally, the parameters are updated according to the inferred gradient. This process is repeated, iterating over training examples, to reduce the error. See Methods for the intuition behind why PAT works and the general multilayer algorithm.  \n\nThe essential advantages of PAT stem from the forward pass being executed by the actual physical hardware, rather than by a simulation. Our digital model for SHG is very accurate (Supplementary Fig. 20) and includes an accurate noise model (Supplementary Figs. 18, 19). However, as evidenced by Fig. 3b, in silico training with this model still fails, reaching a maximum vowel-classification accuracy of about $40\\%$ . In contrast, PAT succeeds, accurately training the SHG-PNN, even when additional layers are added (Fig. 3b, c).  \n\n# Diverse PNNs for image classification  \n\nPNNs can learn to accurately perform more complex tasks, can be realized with virtually any physical system and can be designed with a variety of physical network architectures. In Fig. 4, we present three PNN classifiers for the MNIST (Modified National Institute of Standards and Technology database) handwritten digit classification task, based on three distinct physical systems. For each physical system, we also demonstrate a different PNN architecture, illustrating the variety of physical networks possible. In all cases, models were constructed and trained using PyTorch54.  \n\n![](images/316d218227be5006f7f1e8a812b5d528849e7e1abf9744e5fd19d347f1e1875f.jpg)  \nFig. 4 | Image classification with diverse physical systems. We trained reference model where the physical transformations implemented by the PNNs based on three physical systems (mechanics, electronics and optics) to speaker are replaced by identity operations. d, Confusion matrix for the classify images of handwritten digits. a, The mechanical PNN: the multimode mechanical PNN after training. e–h, The same as a–d, respectively, but for a oscillations of a metal plate are driven by time-dependent forces that encode nonlinear analogue-electronic PNN. i–l, The same as a–d, respectively, for a the input image data and parameters. b, The mechanical PNN multilayer hybrid physical–digital PNN based on broadband optical SHG. The final test architecture. c, The validation classification accuracy versus training epoch for accuracy is $87\\%$ , $93\\%$ and $97\\%$ for the mechanical, electronic and optics-based the mechanical PNN trained using PAT. The same curves are shown also for a PNNs, respectively.  \n\nIn the mechanical PNN (Fig. 4a–d), a metal plate is driven by time-varying forces, which encode both input data and trainable parameters. The plate’s multimode oscillations enact controllable convolutions on the input data (Supplementary Figs. 16, 17). Using the plate’s trainable transformation sequentially three times, we classify 28-by-28 (784 pixel) images that are input as an unrolled time series. To control the transformations of each physical layer, we train element-wise rescaling of the forces applied to the plate (Fig. 4b, Methods). PAT trains the three-layer mechanical PNN to $87\\%$ accuracy, close to a digital linear classifier55. When the mechanical computations are replaced by identity operations, and only the digital rescaling  \n\n# Article  \n\noperations are trained, the performance of the model is equivalent to random guessing $(10\\%)$ . This shows that most of the PNN’s functionality comes from the controlled physical transformations.  \n\nAn analogue-electronic PNN is implemented with a circuit featuring a transistor (Fig. 4e–h), which results in a noisy, nonlinear transient response (Supplementary Figs. 12, 13). The usage and architecture of the electronic PNN are mostly similar to that of the mechanical PNN, with several minor differences (Methods). When trained using PAT, the analogue-electronic PNN performs the classification task with $93\\%$ test accuracy.  \n\nUsing broadband SHG, we demonstrate a physical–digital hybrid PNN (Fig. 4i–l). This hybrid PNN involves trainable digital linear input layers followed by trainable ultrafast SHG transformations. The trainable SHG transformations boost the performance of the digital baseline from roughly $90\\%$ accuracy to $97\\%$ . The classification task’s difficulty is nonlinear with respect to accuracy, so this improvement typically requires increasing the number of digital operations by around one order of magnitude55. This illustrates how a hybrid physical–digital PNN can automatically learn to offload portions of a computation from an expensive digital processor to a fast, energy-efficient physical co-processor.  \n\nTo show the potential for PNNs to perform more challenging tasks, we simulated a multilayer PNN based on a nonlinear oscillator network. This PNN is trained with PAT to perform the MNIST task with $99.1\\%$ accuracy, and the Fashion-MNIST task, which is considered significantly harder56, with $90\\%$ accuracy, in both cases with simulated physical noise, and with mismatch between model and simulated experiment of over $20\\%$ (Supplementary Section 4).  \n\n# Discussion  \n\nOur results show that controllable physical systems can be trained to execute DNN calculations. Many systems that are not conventionally used for computation appear to offer, in principle, the capacity to perform parts of machine-learning-inference calculations orders of magnitude faster and more energy-efficiently than conventional hardware (Supplementary Section 5). However, there are two caveats to note. First, owing to underlying symmetries and other constraints, some systems may be well suited for accelerating a restricted class of computations that share the same constraints. Second, PNNs trained using PAT can only provide significant benefits during inference, as PAT uses a digital model. Thus, as in the hybrid network presented in Fig. 4i–l, we expect such PNNs to serve as a resource, rather than as a complete replacement, for conventional general-purpose hardware (Supplementary Section 5).  \n\nTechniques for training hardware in situ7,40–47 and methods for reliable in silico training (for example, refs. 57–60) complement these weaknesses. Devices trained using in situ learning algorithms will perform learning entirely in hardware, potentially realizing learning faster and more energy-efficiently than current approaches. Such devices are suited to settings in which frequent retraining is required. However, to perform both learning and inference, these devices have more specific hardware requirements than inference-only hardware, which may limit their achievable inference performance. In silico training can train many physical parameters of a device, including ones set permanently during fabrication12–16. As the resulting hardware will not perform learning, it can be optimized for inference. Although accurate, large-scale in silico training has been implemented4–6,57–60, this has been achieved with only analogue electronics, for which accurate simulations and controlled fabrication processes are available. PAT may be used in settings where a simulation–reality gap cannot be avoided, such as if hardware is designed at the limit of fabrication tolerances, operated outside usual regimes or based on platforms other than conventional electronics.  \n\nImprovements to PAT could extend the utility of PNNs. For example, PAT’s backward pass could be replaced by a neural network that directly estimates parameter updates for the physical system. Implementing this ‘teacher’ neural network with a PNN would allow subsequent training to be performed without digital assistance.  \n\nThis work has focused so far on the potential application of PNNs as accelerators for machine learning, but PNNs are promising for other applications as well, particularly those in which physical, rather than digital, data are processed or produced. PNNs can perform computations on data within its physical domain, allowing for smart sensors30–32 that pre-process information before conversion to the electronic domain (for example, a low-power, microphone-coupled circuit tuned to recognize specific hotwords). As the achievable sensitivity, resolution and energy efficiency of many sensors is limited by conversion of information to the digital electronic domain, and by processing of that data in digital electronics, PNN sensors should have advantages. More broadly, with PAT, one is simply training the complex functionality of physical systems. 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In 2021 IEEE 3rd International Conference on Artificial Intelligence Circuits and Systems (AICAS) (IEEE, 2021).  \n\nPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\nOpen Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.  \n\n$\\circledcirc$ The Author(s) 2022  \n\n# Article Methods  \n\n# Physics-aware training  \n\nTo train the PNNs presented in Figs. 2–4, we used PAT to enable us to perform backpropagation on the physical apparatuses as automatic differentiation (autodiff) functions within PyTorch54 (v1.6). We used PyTorch Lightning61 (v0.9) and Weights and Biases62 (v0.10) during development as well. PAT is explained in detail in Supplementary Section 1, where it is compared with standard backpropagation, and training physical devices in silico. Here we provide only an overview of PAT in the context of a generic multilayer PNN (Supplementary Figs. 2, 3).  \n\nPAT can be formalized by the use of custom constituent autodiff functions for the physically executed submodules in an overall network architecture (Supplementary Fig. 1). In PAT, each physical system’s forward functionality is provided by the system’s own controllable physical transformation, which can be thought of as a parameterized function $f_{\\mathfrak{p}}$ that relates the input $\\mathbf{x}$ , parameters θ, and outputs y of the transformation via $\\mathbf{y}=f_{\\mathsf{p}}$ (x,θ). As a physical system cannot be auto-differentiated, we use a differentiable digital model $f_{\\mathrm{m}}$ to approximate each backward pass through a given physical module. This structure is essentially a generalization of quantization-aware training48, in which low-precision neural network hardware is approximated by quantizing weights and activation values on the forward pass, but storing weights and activations, and performing the backward pass with full precision.  \n\nTo see how this works, we consider here the specific case of a multilayer feedforward PNN with standard stochastic gradient descent. In this case, the PAT algorithm with the above-defined custom autodiff functions results in the following training loop:  \n\nPerform forward pass:  \n\n$$\n\\mathbf{x}^{[l+1]}{=}\\mathbf{y}^{[l]}{=}f_{\\mathrm{p}}(\\mathbf{x}^{[l]},\\mathbf{\\boldsymbol{\\Theta}}^{[l]})\n$$  \n\nCompute (exact) error vector:  \n\n$$\n{\\displaystyle{g}_{{\\bf{y}}^{[N]}}=\\frac{\\partial L}{\\partial{\\bf{y}}^{[N]}}=\\frac{\\partial\\ell}{\\partial{\\bf{y}}^{[N]}}({\\bf{y}}^{[N]},{\\bf{y}}_{\\mathrm{target}})}\n$$  \n\nPerform backward pass  \n\n$$\n{\\displaystyle{\\boldsymbol g}_{{\\bf y}^{[l-1]}}=\\left[\\frac{\\partial f_{\\mathrm{m}}}{\\partial{\\bf x}}({\\bf x}^{[l]},{\\bf\\pmb\\theta}^{[l]})\\right]^{\\mathrm{T}}}{\\bf g}_{{\\bf y}^{[l]}}}\n$$  \n\n$$\n\\boldsymbol{g}_{\\mathbf{\\pmb{\\theta}}^{[l-1]}}=\\left[\\frac{\\partial f_{\\mathrm{m}}}{\\partial\\mathbf{\\pmb{\\theta}}}(\\mathbf{x}^{[l]},\\mathbf{\\pmb{\\theta}}^{[l]})\\right]^{\\top}\\boldsymbol{g}_{\\mathbf{y}^{[l]}}\n$$  \n\nUpdate parameters:  \n\n$$\n{\\Theta^{[l]}\\to\\Theta^{[l]}-\\eta\\frac{1}{N_{\\mathrm{data}}}\\sum_{k}g_{\\L_{\\L}^{[l]}}^{(k)}}\n$$  \n\nwhere $g_{\\pmb{\\uptheta}^{[l]}}$ and $g_{\\mathbf{v}^{[l]}}$ are estimators of the physical systems’ exact gradients, $\\frac{\\partial{\\cal L}}{\\partial\\pmb{\\theta}^{[l]}}$ and ∂∂yL[ , respectively for the [ ]th layer, obtained by autodifferentiation of the model, $\\iota$ is the loss, $\\ell$ is the loss function (for example, cross-entropy or mean-squared error), $\\mathbf{y_{target}}$ is the desired (target) output, $N_{\\mathrm{data}}$ is the size of the batch and $\\eta$ is the learning rate. $\\mathbf{x}^{[l+\\mathrm{i}]}$ is the input vector to the $[l+1]$ th layer, which for the hidden layers of the feedforward architecture is equal to the output vector of the previous layer, $\\mathbf{x}^{[l+1]}{=}\\mathbf{y}^{[l]}{=}f_{\\mathrm{p}}(\\mathbf{x}^{[l]},\\bar{\\mathbf{\\Theta}}^{[l]})$ , where ${\\pmb\\theta}^{[l]}$ is the controllable (trainable) parameter vector for the [ ]th layer. For the first layer, the input data vector x [1] is the data to be operated on. In PAT, the error vector is exactly estimated $(g_{\\mathbf{y}_{\\bot}^{[M]}}=\\frac{\\partial L}{\\partial\\mathbf{y}^{[M]}})$ as the forward pass is performed by the physical system. This error vector is then backpropagated via equation (3), which involves Jacobian matrices of the differential digital model evaluated at the correct inputs at each layer (that is, the actual physical inputs) ∂xm (x[ ], θ[ ]) , where T represents the transpose operation. Thus, in addition to utilizing the output of the PNN $(\\mathbf{\\dot{y}}^{[N]})$ via physical computations in the forward pass, intermediate outputs $(\\mathbf{y}^{[l]})$ are also utilized to facilitate the computation of accurate gradients in PAT.  \n\nAs it is implemented just by defining a custom autodiff function, generalizing PAT for more complex architectures, such as multichannel or hybrid physical–digital models, with different loss functions and so on is straightforward. See Supplementary Section 1 for details.  \n\nAn intuitive motivation for why PAT works is that the training’s optimization of parameters is always grounded in the true optimization landscape by the physical forward pass. With PAT, even if gradients are estimated only approximately, the true loss function is always precisely known. As long as the gradients estimated by the backward pass are reasonably accurate, optimization will proceed correctly. Although the required training time is expected to increase as the error in gradient estimation increases, in principle it is sufficient for the estimated gradient to be pointing closer to the direction of the true gradient than its opposite (that is, that the dot product of the estimated and true gradients is positive). Moreover, by using the physical system in the forward pass, the true output from each intermediate layer is also known, so gradients of intermediate physical layers are always computed with respect to correct inputs. In any form of in silico training, compounding errors build up through the imperfect simulation of each physical layer, leading to a rapidly diverging simulation–reality gap as training proceeds (see Supplementary Section 1 for details). As a secondary benefit, PAT ensures that learned models are inherently resilient to noise and other imperfections beyond a digital model, as the change of loss along noisy directions in parameter space will tend to average to zero. This makes training robust to, for example, device– device variations, and facilitates the learning of noise-resilient (and, more speculatively, noise-enhanced) models8.  \n\n# Differentiable digital models  \n\nTo perform PAT, a differentiable digital model of the physical system’s input–output transformation is required. Any model, $f_{\\mathrm{m^{\\prime}}}$ of the physical system’s true forward function, $f_{\\mathfrak{p^{\\prime}}}$ can be used to perform PAT, so long as it can be auto-differentiated. Viable approaches include traditional physics models, black-box machine-learning models13,63,64 and physics-informed machine-learning65 models.  \n\nIn this work, we used the black-box strategy for our differentiable digital models, namely DNNs trained on input–output vector pairs from the physical systems as $f_{\\mathrm{m}}$ (except for the mechanical system). Two advantages of this approach are that it is fully general (it can be applied even to systems in which one has no underlying knowledge-based model of the system) and that the accuracy can be extremely high, at least for physical inputs, $({\\bf x},\\pmb\\theta)$ , within the distribution of the training data (for out-of-distribution generalization, we expect physics-based approaches to offer advantages). In addition, the fact that each physical system has a precise corresponding DNN means that the resulting PNN can be analysed as a network of DNNs, which may be useful for explaining the PNN’s learned physical algorithm.  \n\nFor our DNN differentiable digital models, we used a neural architecture search66 to optimize hyperparameters, including the learning rate, number of layers and number of hidden units in each layer. Typical optimal architectures involved 3–5 layers with 200–1,000 hidden units in each, trained using the Adam optimizer, mean-squared loss function and learning rates of around $10^{-4}$ . For more details, see Supplementary Section 2D.1.  \n\nFor the nonlinear optical system, the test accuracy of the trained digital model (Supplementary Fig. 20) shows that the model is remarkably accurate compared with typical simulation–experiment agreement in broadband nonlinear optics, especially considering that the pulses used exhibit a complex spatiotemporal structure owing to the pulse shaper. The model is not, however, an exact description of the physical system: the typical error for each element of the output vector is about 1– $2\\%$ . For the analogue electronic circuit, agreement is also good, although worse than the other systems (Supplementary Fig. 23), corresponding to around $5-10\\%$ prediction error for each component of the output vector. For the mechanical system, we found that a linear model was sufficient to obtain excellent agreement, which resulted in a typical error of about $1\\%$ for each component of the output vector (Supplementary Fig. 26).  \n\n# In silico training  \n\nTo train PNNs in silico, we applied a training loop similar to the one described above for PAT except that both the forward and backward passes are performed using the model (Supplementary Figs. 1, 3), with one exception noted below.  \n\nTo improve the performance of in silico training as much as possible and permit the fairest comparison with PAT, we also modelled the input-dependent noise of the physical system and used this within the forward pass of in silico training. To do this, we trained, for each physical system, an additional DNN to predict the eigenvectors of the output vector’s noise covariance matrix, as a function of the physical system’s input vector and parameter vector. These noise models thus provided an input- and parameter-dependent estimate of the distribution of noise in the output vector produced by the physical system. We were able to achieve excellent agreement between the noise models’ predicted noise distributions and experimental measurements (Supplementary Figs. 18, 19). We found that including this noise model improved the performance of experiments performed using parameters derived from in silico training. Consequently, all in silico training results presented in this paper make use of such a model, except for the mechanical system, where a simpler, uniform noise model was found to be sufficient. For additional details, see Supplementary Section 2D.2.  \n\nAlthough including complex, accurate noise models does not allow in silico training to perform as well as PAT, we recommend that such models be used whenever in silico training is performed, such as for physical architecture search and design and possibly pre-training (Supplementary Section 5), as the correspondence with experiment (and, in particular, the predicted peak accuracy achievable there) is significantly improved over simpler noise models, or when ignoring physical noise.  \n\n# Ultrafast nonlinear optical pulse propagation experiments  \n\nFor experiments with ultrafast nonlinear pulse propagation in quadratic nonlinear media (Supplementary Figs. 8–10), we shaped pulses from a mode-locked titanium:sapphire laser (Spectra Physics Tsunami, centred around $780\\mathsf{n m}$ and pulse duration around 100 fs) using a custom pulse shaper. Our optical pulse shaper used a digital micromirror device (DMD, Vialux V-650L) and was inspired by the design in ref. 67. Despite the binary modulations of the individual mirrors, we were able to achieve multilevel spectral amplitude modulation by varying the duty cycle of gratings written to the DMD along the dimension orthogonal to the diffraction of the pulse frequencies. To control the DMD, we adapted code developed for ref. 68, which is available at ref. 69.  \n\nAfter being shaped by the pulse shaper, the femtosecond pulses were focused into a $0.5{\\cdot}\\mathsf{m m}$ -long beta-barium borate crystal. The multitude of frequencies within the broadband pulses then undergo various nonlinear optical processes, including sum-frequency generation and SHG. The pulse shaper imparts a complex phase and spatiotemporal structure on the pulse, which depend on the input and parameters applied through the spectral modulations. These features would make it impossible to accurately model the experiment using a one-dimensional pulse propagation model. For simplicity, we refer to this complex, spatiotemporal quadratic nonlinear pulse propagation as ultrafast SHG.  \n\nAlthough the functionality of the SHG-PNN does not rely on a closed-form mathematical description or indeed on any form of mathematical isomorphism, some readers may find it helpful to understand the approximate form of the input–output transformation realized in this experimental apparatus. We emphasize that the following model is idealistic and meant to convey key intuitions about the physical transformation: the model does not describe the experimental transformation in a quantitative manner, owing to the numerous experimental complexities described above.  \n\nThe physical transformation of the ultrafast SHG setup is seeded by the infrared light from the titanium:sapphire laser. This ultrashort pulse can be described by the Fourier transform of the electric field envelope of the pulse, $A_{0}(\\omega)$ , where $\\omega$ is the frequency of the field detuned relative to the carrier frequency. For simplicity, consider a pulse consisting of a set of discrete frequencies or frequency bins, whose spectral amplitudes are described by the discrete vector $\\mathbf{A}_{0}=[A_{0}(\\omega_{1}),A_{0}(\\omega_{2}),...,A_{0}(\\omega_{N})]^{\\mathrm{T}}$ . After passing through the pulseshaper, the spectral amplitudes of the pulse are then given by  \n\n$$\n\\mathbf{A}=[\\sqrt{x_{1}}A_{0}(\\omega_{1}),\\sqrt{x_{2}}A_{0}(\\omega_{2}),...,\\sqrt{\\theta_{1}}A_{0}(\\omega_{N_{x}+1}),\\sqrt{\\theta_{2}}A_{0}(\\omega_{N_{x}+2}),...]^{\\mathrm{T}},\n$$  \n\nwhere $N_{x}$ is the dimensionality of the data vector, $\\theta_{i}$ are the trainable pulse-shaper amplitudes and $x_{i}$ are the elements of the input data vector. Thus, the output from the pulse shaper encodes both the machine-learning data as well as the trainable parameters. Square roots are present in equation (5) because the pulse shaper was deliberately calibrated to perform an intensity modulation.  \n\nThe output from the pulse shaper (equation (5)) is then input to the ultrafast SHG process. The propagation of an ultrashort pulse through a quadratic nonlinear medium results in an input–output transformation that roughly approximates an autocorrelation, or nonlinear convolution, assuming that the dispersion during propagation is small and the input pulse is well described by a single spatial mode. In this limit, the output blue spectrum $B(\\omega_{i})$ is mathematically given by  \n\n$$\nB(\\omega_{i})=k\\sum_{j}A(\\omega_{i}+\\omega_{j})A(\\omega_{i}-\\omega_{j}),\n$$  \n\nwhere the sum is over all frequency bins $j$ of the pulsed field. The output of the trainable physical transformation $\\scriptstyle\\mathbf{y}=f_{\\mathrm{n}}\\left(\\mathbf{x},\\mathbf{\\boldsymbol{\\mathsf{0}}}\\right)$ is given by the blue pulse’s spectral power, $\\mathbf{y}=[|B_{\\omega_{1}}|^{2},|B_{\\omega_{2}}|^{2},...,|B_{\\omega_{N}}|^{2}]^{\\mathrm{T}}$ , where $N$ is the length of the output vector.  \n\nFrom this description, it is clear that the physical transformation realized by the ultrafast SHG process is not isomorphic to any conventional neural network layer, even in this idealized limit. Nonetheless, the physical transformation retains some key features of typical neural network layers. First, the physical transformation is nonlinear as the SHG process involves the squaring of the input field. Second, as the terms within the summation in equation (6) involve both parameters and input data, the transformation also mixes the different elements of the input data and parameters to product an output. This mixing of input elements is similar, but not necessarily directly mathematically equivalent to, the mixing of input vector elements that occur in the matrix-vector multiplications or convolutions that appear in conventional neural networks.  \n\n# Vowel classification with ultrafast SHG  \n\nA task often used to demonstrate novel machine-learning hardware is the classification of spoken vowels according to formant frequencies10,11. The task involves predicting the spoken vowels given a 12-dimensional input data vector of formant frequencies extracted from audio recordings10. Here we use the vowel dataset from ref. 10, which is based on data originally from ref. 70; data available at https:// homepages.wmich.edu/\\~hillenbr/voweldata.html. This dataset consists of 273 data input–output pairs. We used 175 data pairs as the training  \n\n# Article  \n\nset—49 for the validation and 49 for the test set. For the results in Figs. 2, 3, we optimized for the hyperparameters of the PNN architecture using the validation error and only evaluated the test error after all optimization was conducted. In Fig. 3c, for each PNN with a given number of layers, the experiment was conducted with two different training, validation and test splits of the vowel data. In Fig. 3c, the line plots the mean over the two splits, and the error bars are the standard error of the mean.  \n\nFor the vowel-classification PNN presented in Figs.  2, 3, the input vector to each SHG physical layer is encoded in a contiguous short-wavelength section of the spectral modulation vector sent to the pulse shaper, and the trainable parameters are encoded in the spectral modulations applied to the rest of the spectrum. For the physical layers after the first layer, the input vector to the physical system is the measured spectrum obtained from the previous layer. For convenience, we performed digital renormalization of these output vectors to maximize the dynamic range of the input and ensure that inputs were within the allowed range of 0 to 1 accepted by the pulse shaper. Relatedly, we found that training stability was improved by including additional trainable digital re-scaling parameters to the forward-fed vector, allowing the overall bias and amplitude scale of the physical inputs to each layer to be adjusted during training. These digital parameters appear to have a negligible role in the final trained PNN (when the physical transformations are replaced by identity operations, the network can be trained to perform no better than chance, and the final trained values of the scale and bias parameters are all very close to 1 and 0, respectively). We hypothesize that these trainable rescaling parameters are helpful during training to allow the network to escape noise-affected subspaces of parameter space. See Supplementary Section 2E.1 for details.  \n\nThe vowel-classification SHG-PNN architecture (Supplementary Fig. 21) was designed to be as simple as possible while still demonstrating the use of a multilayer architecture with a physical transformation that is not isomorphic to a conventional DNN layer, and so that the computations involved in performing the classification were essentially all performed by the physical system itself. Many aspects of the design are not optimal with respect to performance, so design choices, such as our specific choice to partition input data and parameter vectors into the controllable parameters of the experiment, should not be interpreted as representing any systematic optimization. Similarly, the vowel-classification task was chosen as a simple example of multidimensional machine-learning classification. As this task can be solved almost perfectly by a linear model, it is in fact poorly suited to the nonlinear optical transformations of our SHG-PNN, which are fully nonlinear (Supplementary Figs. 9, 10). Overall, readers should not interpret this PNN’s design as suggestive of optimal design strategies for PNNs. For initial guidelines on optimal design strategies, we instead refer readers to Supplementary Section 5.  \n\n# MNIST handwritten digit image classification with a hybrid physical–digital SHG-PNN  \n\nThe design of the hybrid physical–digital MNIST PNN based on ultrafast SHG for handwritten digit classification (Fig. 4i–l) was chosen to demonstrate a proof-of-concept PNN in which substantial digital operations were co-trained with substantial physical transformations, and in which no digital output layer was used (although a digital output layer can be used with PNNs, and we expect such a layer will usually improve performance, we wanted to avoid confusing readers familiar with reservoir computing, and so avoided using digital output layers in this work).  \n\nThe network (Supplementary Fig. 29) involves four trainable linear input layers that operate on MNIST digit images, whose outputs are fed into four separate channels in which the SHG physical transformation is used twice in succession (that is, it is two physical layers deep). The output of the final layers of each channel (the final SHG spectra) are concatenated, then summed into ten bins to perform a classification. The structure of the input layer was chosen to minimize the complexity of inputs to the pulse shaper. We found that the output second-harmonic spectra produced by the nonlinear optical process tended towards featureless triangular spectra if inputs were close to a random uniform distribution. Thus, to ensure that output spectra varied significantly with respect to changes in the input spectral modulations, we made sure that inputs to the pulse shaper would exhibit a smoother structure in the following way. For each of 4 independent channels, 196-dimensional input images (downsampled from 784-dimensional $28\\times28$ images) are first operated on by a 196 by 50 trainable linear matrix, and then (without any nonlinear digital operations), a second 50 by 196 trainable linear matrix. The second 50 by 196 matrix is identical for all channels, the intent being that this matrix identifies optimal ‘input modes’ to the SHG process. By varying the middle dimension of this two-step linear input layer, one may control the amount of structure (number of ‘spectral modes’) allowed in inputs to the pulse shaper, as the middle dimension effectively controls the rank of the total linear matrix. We found that a middle dimension below 30 resulted in the most visually varied SHG output spectra, but that 50 was sufficient for good performance on the MNIST task. In this network, we also utilized skip connections between layers in each channel. This was done so that the network would be able to ‘choose’ to use the linear digital operations to perform the linear part of the classification task (for which nearly $90\\%$ accuracy can be obtained55) and to thus rely on the SHG co-processor primarily for the harder, nonlinear part of the classification task. Between the physical layers in each channel, a trainable, element-wise rescaling was used to allow us to train the second physical layer transformations efficiently. That is, $x_{i}=a_{i}y_{i}+b_{i}$ where $b_{i}$ and $a_{i}$ are trainable parameters, and $x_{i}$ and $y_{i}$ are the input to the pulse shaper and the measured output spectrum from the previous physical layer, respectively.  \n\nFor further details on the nonlinear optical experimental setup and its characterization, we refer readers to Supplementary Section 2A. For further details on the vowel-classification SHG-PNN, we refer readers to Supplementary Section 2E.1, and for the hybrid physical–digital MNIST handwritten digit-classification SHG-PNN, we refer readers to Supplementary Section 2E.4.  \n\n# Analogue electronic circuit experiments  \n\nThe electronic circuit used for our experiments (Supplementary Fig. 11) was a resistor-inductor-capacitor oscillator (RLC oscillator) with a transistor embedded within it. It was designed to produce as nonlinear and complex a response as possible, while still containing only a few simple components (Supplementary Figs. 12, 13). The experiments were carried out with standard bulk electronic components, a hobbyist circuit breadboard and a USB data acquisition (DAQ) device (Measurement Computing USB-1208-HS-4AO), which allowed for one analogue input and one analogue output channel, with a sampling rate of 1 MS $\\mathsf{\\pmb{s}}^{-1}$ .  \n\nThe electronic circuit provides only a one-dimensional time-series input and one-dimensional time-series output. As a result, to partition the inputs to the system into trainable parameters and input data so that we could control the circuit’s transformation of input data, we found it was most convenient to apply parameters to the one-dimensional input time-series vector by performing trainable, element-wise rescaling on the input time-series vector. That is, $x_{i}=a_{i}y_{i}+b_{i}$ , where $b_{i}$ and $a_{i}$ are trainable parameters, $y_{i}$ are the components of the input data vector and $x_{i}$ are the re-scaled components of the voltage time series that is then sent to the analogue circuit. For the first layer, $y_{i}$ are the unrolled pixels of the input MNIST image. For hidden layers, $y_{i}$ are the components of the output voltage time-series vector from the previous layer.  \n\nWe found that the electronic circuit’s output was noisy, primarily owing to the timing jitter noise that resulted from operating the DAQ at its maximum sampling rate (Supplementary Fig. 23). Rather than reducing this noise by operating the device more slowly, we were motivated to design the PNN architecture presented in Fig. 4 in a way that allowed it to automatically learn to function robustly and accurately, even in the presence of up to $20\\%$ noise per output vector element (See Supplementary Fig. 24 for an expanded depiction of the architecture). First, seven, three-layer feedforward PNNs were trained together, with the final prediction provided by averaging the output of all seven, three-layer PNNs. Second, skip connections similar to those used in residual neural networks were employed71. These measures make the output of the network effectively an ensemble average over many different subnetworks71, which allows it to perform accurately and train smoothly despite the very high physical noise and multilayer design.  \n\nFor further details on the analogue electronic experimental setup and its characterization, we refer readers to Supplementary Section 2B. For further details on the MNIST handwritten digit-classification analogue electronic PNN, we refer readers to Supplementary Section 2E.2.  \n\n# Oscillating mechanical plate experiments  \n\nThe mechanical plate oscillator was constructed by attaching a 3.2 cm by $3.2\\mathrm{cm}$ by $1\\mathsf{m m}$ titanium plate to a long, centre-mounted screw, which was fixed to the voice coil of a commercial full-range speaker (Supplementary Figs. 14, 15). The speaker was driven by an audio amplifier (Kinter ${\\bf K}2020{\\bf A}^{+}.$ ) and the oscillations of the plate were recorded using a microphone (Audio-Technica ATR2100x-USB Cardioid Dynamic Microphone). The diaphragm of the speaker was completely removed so that the sound recorded by the microphone is produced only by the oscillating metal plate.  \n\nAs the physical input (output) to (from) the mechanical oscillator is a one-dimensional time series, similar to the electronic circuit, we made use of element-wise trainable rescaling to conveniently allow us to train the oscillating plate’s physical transformations.  \n\nThe mechanical PNN architecture for the MNIST handwritten digit classification task was chosen to be the simplest multilayer PNN architecture possible with such a one-dimensional dynamical system (Supplementary Fig. 27). As the mechanical plate’s input–output responses are primarily linear convolutions (Supplementary Figs. 16, 17), it is well suited to the MNIST handwritten digit classification task, achieving nearly the same performance as a digital linear model55.  \n\nFor further details on the oscillating mechanical plate experimental setup and its characterization, we refer readers to Supplementary Section 2C. For further details on the MNIST handwritten digit-classification oscillating mechanical plate PNN, we refer readers to Supplementary Section 2E.3.  \n\n# Data availability  \n\nAll data generated during and code used for this work are available at https://doi.org/10.5281/zenodo.4719150.  \n\n# Code availability  \n\nAn expandable demonstration code for applying PAT to train PNNs is available at https://github.com/mcmahon-lab/Physics-Aware-Training. All code used for this work is available at https://doi.org/10.5281/ zenodo.4719150.  \n\n61.\t Falcon, W. et al. PyTorch Lightning (2019); https://github.com/PyTorchLightning/ pytorch-lightning   \n62.\t Biewald, L. Experiment Tracking with Weights and Biases (2020); https://www.wandb. com/   \n63.\t Kasim, M. F. et al. Building high accuracy emulators for scientific simulations with deep neural architecture search. Preprint at https://arxiv.org/abs/2001.08055 (2020).   \n64.\t Rahmani, B. et al. Actor neural networks for the robust control of partially measured nonlinear systems showcased for image propagation through diffuse media. Nat. Mach. Intell. 2, 403–410 (2020).   \n65.\t Karniadakis, G. E. et al. Physics-informed machine learning. Nat. Rev. Phys. 3, 422–440 (2021).   \n66.\t Akiba, T., Sano, S., Yanase, T., Ohta, T. & Koyama, M. Optuna: a next-generation hyperparameter optimization framework. In Proc. 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining 2623–2631 (2019).   \n67.\t Liu, W. et al. Programmable controlled mode-locked fiber laser using a digital micromirror device. Opt. Lett. 42, 1923–1926 (2017).   \n68.\t Matthès, M. W., del Hougne, P., de Rosny, J., Lerosey, G. & Popoff, S. M. Optical complex media as universal reconfigurable linear operators. Optica 6, 465–472 (2019).   \n69.\t Popoff, S. M. & Matthès, M. W. ALP4lib: q Python wrapper for the Vialux ALP-4 controller suite to control DMDs. Zenodo https://doi.org/10.5281/zenodo.4076193 (2020).   \n70.\t Hillenbrand, J., Getty, L. A., Wheeler, K. & Clark, M. J. Acoustic characteristics of American English vowels. J. Acoust. Soc. Am. 97, 3099–3111 (1995).   \n71.\t Veit, A.,Wilber, M. & Belongie, S. Residual networks behave like ensembles of relatively shallow networks Preprint at https://arxiv.org/abs/1605.06431 (2016).  \n\nAcknowledgements We thank NTT Research for their financial and technical support. Portions of this work were supported by the National Science Foundation (award CCF-1918549). L.G.W. and T.W. acknowledge support from Mong Fellowships from Cornell Neurotech during early parts of this work. P.L.M. acknowledges membership of the CIFAR Quantum Information Science Program as an Azrieli Global Scholar. We acknowledge discussions with D. Ahsanullah, M. Anderson, V. Kremenetski, E. Ng, S. Popoff, S. Prabhu, M. Saebo, H. Tanaka, R. Yanagimoto, H. Zhen and members of the NTT PHI Lab/NSF Expeditions research collaboration, and thank P. Jordan for discussions and illustrations.  \n\nAuthor contributions L.G.W., T.O. and P.L.M. conceived the project and methods. T.O. and L.G.W. performed the SHG-PNN experiments. L.G.W. performed the electronic-PNN experiments. M.M.S. performed the oscillating-plate-PNN experiments. T.W., D.T.S. and Z.H. contributed to initial parts of the work. L.G.W., T.O., M.M.S. and P.L.M. wrote the manuscript. P.L.M. supervised the project.  \n\nCompeting interests L.G.W., T.O., M.M.S. and P.L.M. are listed as inventors on a US provisional patent application (number 63/178,318) on physical neural networks and physics-aware training. The other authors declare no competing interests.  \n\n# Additional information  \n\nSupplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41586-021-04223-6.   \nCorrespondence and requests for materials should be addressed to Logan G. Wright, Tatsuhiro Onodera or Peter L. McMahon.   \nPeer review information Nature thanks Tayfun Gokmen and Damien Querlioz for their contribution to the peer review of this work. Peer reviewer reports are available.   \nReprints and permissions information is available at http://www.nature.com/reprints.  ",
    "Target": null
  },
  {
    "id": "10.1038_s41467-023-36329-y",
    "DOI": "10.1038/s41467-023-36329-y",
    "DOI Link": "http://dx.doi.org/10.1038/s41467-023-36329-y",
    "Relative Dir Path": "mds/10.1038_s41467-023-36329-y",
    "Article Title": "Learning local equivariant representations for large-scale atomistic dynamics",
    "Authors": "Musaelian, A; Batzner, S; Johansson, A; Sun, LX; Owen, CJ; Kornbluth, M; Kozinsky, B",
    "Source Title": "NATURE COMMUNICATIONS",
    "Abstract": "A simultaneously accurate and computationally efficient parametrization of the potential energy surface of molecules and materials is a long-standing goal in the natural sciences. While atom-centered message passing neural networks (MPNNs) have shown remarkable accuracy, their information propagation has limited the accessible length-scales. Local methods, conversely, scale to large simulations but have suffered from inferior accuracy. This work introduces Allegro, a strictly local equivariant deep neural network interatomic potential architecture that simultaneously exhibits excellent accuracy and scalability. Allegro represents a many-body potential using iterated tensor products of learned equivariant representations without atom-centered message passing. Allegro obtains improvements over state-of-the-art methods on QM9 and revMD17. A single tensor product layer outperforms existing deep MPNNs and transformers on QM9. Furthermore, Allegro displays remarkable generalization to out-of-distribution data. Molecular simulations using Allegro recover structural and kinetic properties of an amorphous electrolyte in excellent agreement with ab-initio simulations. Finally, we demonstrate parallelization with a simulation of 100 million atoms. The paper presents a method that allows scaling machine learning interatomic potentials to extremely large systems, while at the same time retaining the remarkable accuracy and learning efficiency of deep equivariant models. This is obtained with an E(3)- equivariant neural network architecture that combines the high accuracy of equivariant neural networks with the scalability of local methods.",
    "Times Cited, WoS Core": 253,
    "Times Cited, All Databases": 265,
    "Publication Year": 2023,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000955633400011",
    "Markdown": "# Learning local equivariant representations for large-scale atomistic dynamics  \n\nReceived: 16 June 2022  \n\nAccepted: 23 January 2023  \n\nPublished online: 03 February 2023  \n\n# Check for updates  \n\nAlbert Musaelian1,3, Simon Batzner 1,3 , Anders Johansson 1, Lixin Sun1, Cameron J. Owen 1, Mordechai Kornbluth $\\textcircled{\\bullet}^{2}$ & Boris Kozinsky 1,2  \n\nA simultaneously accurate and computationally efficient parametrization of the potential energy surface of molecules and materials is a long-standing goal in the natural sciences. While atom-centered message passing neural networks (MPNNs) have shown remarkable accuracy, their information propagation has limited the accessible length-scales. Local methods, conversely, scale to large simulations but have suffered from inferior accuracy. This work introduces Allegro, a strictly local equivariant deep neural network interatomic potential architecture that simultaneously exhibits excellent accuracy and scalability. Allegro represents a many-body potential using iterated tensor products of learned equivariant representations without atom-centered message passing. Allegro obtains improvements over state-of-the-art methods on QM9 and revMD17. A single tensor product layer outperforms existing deep MPNNs and transformers on QM9. Furthermore, Allegro displays remarkable generalization to out-of-distribution data. Molecular simulations using Allegro recover structural and kinetic properties of an amorphous electrolyte in excellent agreement with ab-initio simulations. Finally, we demonstrate parallelization with a simulation of 100 million atoms.  \n\nMolecular dynamics (MD) and Monte-Carlo (MC) simulation methods are a core pillar of computational chemistry, materials science, and biology. Common to a diverse set of applications ranging from energy materials1 to protein folding2 is the requirement that predictions of the potential energy and atomic forces must be both accurate and computationally efficient to faithfully describe the evolution of complex systems over long timescales. While first-principles methods such as density functional theory (DFT), which explicitly treat the electrons of the system, provide an accurate and transferable description of the system, they exhibit poor scaling with system size and thus limit practical applications to small systems and short simulation times. Classical force fields based on simple functions of atomic coordinates are able to scale to large systems and long timescales but are inherently limited in their fidelity and can yield unfaithful dynamics. Descriptions of the potential energy surface (PES) using machine learning (ML) have emerged as a promising approach to move past this trade-off3–24. Machine learning interatomic potentials (MLIPs) aim to approximate a set of high-fidelity energy and force labels with improved computational efficiency that scales linearly in the number of atoms. A variety of approaches have been proposed, from shallow neural networks and kernel-based approaches3–6 to more recent methods based on deep learning14,15,20,25,26. In particular, a class of MLIPs based on atom-centered message-passing neural networks (MPNNs) has shown remarkable accuracy9,11,14,15,26,27. In interatomic potentials based on MPNNs, an atomistic graph is induced by connecting each atom (node) to all neighboring atoms inside a finite cutoff sphere surrounding it. Information is then iteratively propagated along this graph, allowing MPNNs to learn many-body correlations and access non-local information outside of the local cutoff. This iterated propagation, however, leads to large receptive fields with many effective neighbors for each atom, which impedes parallel computation and limits the length scales accessible to atom-centered message-passing MLIPs. MLIPs using strictly local descriptors such as Behler-Parrinello neural networks5, $\\mathbf{GAP^{6}}$ , SNAP7, DeepMD20, Moment Tensor Potentials8, or $\\mathsf{A C E}^{12}$ do not suffer from this obstacle due to their strict locality. As a result, they can be easily parallelized across devices and have been successfully scaled to extremely large system sizes28–31. Approaches based on local descriptors, however, have so far fallen behind in accuracy compared to state-of-the-art equivariant, atom-centered message passing interatomic potentials15.  \n\n# Message-passing interatomic potentials  \n\nMessage-passing neural networks (MPNNs) which learn atomistic representations have recently gained popularity in atomistic machine learning due to advantages in accuracy compared to hand-crafted descriptors. Atom-centered message-passing interatomic potentials operate on an atomistic graph constructed by representing atoms as nodes and defining edges between atoms that are within a fixed cutoff distance of one another. Each node is then represented by a hidden state $\\mathbf{h}_{i}^{t}\\in\\mathbb{R}^{c}$ representing the state of atom $i$ at layer $t,$ and edges are represented by edge features $\\mathbf{e}_{i j},$ for which the interatomic distance $r_{i j}$ is often used. The message-passing formalism can then be concisely described $\\mathbf{a}\\mathbf{s}^{32}$ :  \n\n$$\n\\mathbf{m}_{i}^{t+1}{=}\\sum_{j\\in\\mathcal{N}(i)}M_{t}\\Big(\\mathbf{h}_{i}^{t},\\mathbf{h}_{j}^{t},\\mathbf{e}_{i j}\\Big)\n$$  \n\n$$\n\\ensuremath{\\mathbf{\\mathsf{h}}}_{i}^{t+1}{=}{\\cal U}_{t}\\left(\\ensuremath{\\mathbf{\\mathsf{h}}}_{i}^{t},\\ensuremath{\\mathbf{\\mathsf{m}}}_{i}^{t+1}\\right)\n$$  \n\nwhere $M_{t}$ and $U_{t}$ are an arbitrary message function and node update function, respectively. From this propagation mechanism, it is immediately apparent that as messages are communicated over a sequence of $t$ steps, the local receptive field of an atom $i,$ i.e., the effective set of neighbors that contribute to the final state of atom $i$ grows approximately cubically with the effective cutoff radius $r_{c,e}$ . In particular, given a MPNN with $N_{\\mathrm{layer}}$ message-passing steps and local cutoff radius of ${\\dot{\\boldsymbol{r}}}_{c,l},$ the effective cutoff is $r_{c,e}{=}N_{\\mathrm{layer}}r_{c,l}.$ Information from all atoms inside this receptive field feeds into a central atom’s state $\\mathbf{h}_{i}$ at the final layer of the network. Due to the cubic growth of the number of atoms inside the receptive field cutoff $r_{c,e},$ parallel computation can quickly become unmanageable, especially for extended periodic systems. As an illustrative example, we may take a structure of 64 molecules of liquid water at pressure $P{=}1$ bar and temperature $T{=}300\\mathsf{K}$ For a typical setting of $N_{t}{=}6$ message-passing layers with a local cutoff of $r_{c,l}=6\\mathring{\\mathbf{A}}$ this would result in an effective cutoff of $r_{c,e}=36\\mathring{\\mathbf{A}}$ . While each atom only has approximately 96 atoms in its local $6\\mathring{\\mathbf{A}}$ environment (including the central atom), it has 20,834 atoms inside the extended $36\\mathring{\\mathbf{A}}$ environment. Due to the atom-centered message-passing mechanism, information from each of these atoms flows into the current central atom. In a parallel scheme, each worker must have access to the high-dimensional feature vectors $\\mathbf{h}_{i}$ of all 20,834 nodes, while the strictly local scheme only needs to have access to approximately $6^{3}=216$ times fewer atoms’ states. From this simple example, it becomes obvious that massive improvements in scalability can be obtained from strict locality in machine learning interatomic potentials. It should be noted that conventional, atom-centered message passing allows for the possibility, in principle, to capture long-range interactions (up to $r_{c,e})$ and can induce many-body correlations. The relative importance of these effects in describing molecules and materials is an open question, and one of the aims of this work is to explore whether many-body interactions can be efficiently captured without increasing the effective cutoff.  \n\n# Equivariant neural networks  \n\nThe physics of atomic systems is unchanged under the action of a number of geometric symmetries—rotation, inversion, and translation —which together comprise the Euclidean group $E(3)$ (rotation alone is $S O(3)$ , and rotation and inversion together comprise $O(3){\\mathrm{.}}$ ). Scalar quantities such as the potential energy are invariant to these symmetry group operations, while vector quantities such as the atomic forces are equivariant to them and transform correspondingly when the atomic geometry is transformed. More formally, a function between vector spaces $f{:}X{\\to}Y$ is equivariant to a group $G$ if  \n\n$$\nf(D_{X}[g]x)=D_{Y}[g]f(x)\\quad\\forall g\\in G,\\forall x\\in X\n$$  \n\nwhere $D_{\\lambda}[g]\\in G L(\\lambda)$ is the representation of the group element $g$ in the vector space $\\chi.$ The function $f$ is invariant if $D_{\\mathrm{{\\smash~\\left[g\\right]~}}}$ is the identity operator on $r\\mathrm{:}$ in this case, the output is unchanged by the action of symmetry operations on the input $x$  \n\nMost existing MLIPs guarantee the invariance of their predicted energies by acting only on invariant inputs. In invariant, atom-centered message-passing interatomic potentials in particular, each atom’s hidden latent space is a feature vector consisting solely of invariant scalars25. More recently, however, a class of models known as equivariant neural networks33–36 have been developed which can act directly on non-invariant geometric inputs, such as displacement vectors, in a symmetry-respecting way. This is achieved by using only $E(3)$ -equivariant operations, yielding a model whose internal features are equivariant with respect to the 3D Euclidean group. Building on these concepts, equivariant architectures have been explored for developing interatomic potential models. Notably, the NequIP model15, followed by several other equivariant implementations26,27,37–39, demonstrated unprecedentedly low error on a large range of molecular and materials systems, accurately describes structural and kinetic properties of complex materials, and exhibits remarkable sample efficiency. In both the present work and in NequIP, the representation $D_{\\mathrm{{\\scriptscriptstyle{X}}}}[g]$ of an operation $g\\in O(3)$ on an internal feature space $\\chi$ takes the form of a direct sum of irreducible representations (commonly referred to as irreps) of $O(3)$ . This means that the feature vectors themselves are comprised of various geometric tensors corresponding to different irreps that describe how they transform under symmetry operations. The irreps of $O(3)$ , and thus the features, are indexed by a rotation order $\\ell\\geq0$ and a parity $p\\in(-1,1)$ . A tensor that transforms according to the irrep $\\ell,p$ is said to “inhabit” that irrep. We note that in many cases one may also omit the parity index to instead construct features that are only $S E(3)$ -equivariant (translation and rotation), which simplifies the construction of the network and reduces the memory requirements.  \n\nA key operation in such equivariant networks is the tensor product of representations, an equivariant operation that combines two tensors $\\pmb{x}$ and y with irreps $\\ell_{1},p_{1}$ and $\\ell_{2},p_{2}$ to give an output inhabiting an irrep $\\ell_{\\mathrm{out}},p_{\\mathrm{out}}$ satisfying $|\\ell_{1}-\\ell_{2}|\\leq\\ell_{\\mathrm{out}}\\leq|\\ell_{1}+\\ell_{2}|$ and $p_{\\mathrm{out}}=p_{1}p_{2}$ :  \n\n$$\n(\\mathbf{x}\\otimes\\mathbf{y})_{\\ell_{\\mathrm{out}},m_{\\mathrm{out}}}=\\sum_{m_{1},m_{2}}{\\binom{\\ell_{1}}{m_{1}}}\\ {\\begin{array}{l l l}{\\ell_{2}}&{\\ell_{\\mathrm{out}}}\\\\ {m_{2}}&{m_{\\mathrm{out}}}\\end{array}}\\right)\\mathbf{x}_{\\ell_{1},m_{1}}\\mathbf{y}_{\\ell_{2},m_{2}}\n$$  \n\nwhere $\\left(\\begin{array}{c c c}{\\ell_{1}}&{\\ell_{2}}&{\\ell_{\\mathrm{out}}}\\\\ {m_{1}}&{m_{2}}&{m_{\\mathrm{out}}}\\end{array}\\right)$ is the Wigner $3j$ symbol. Two key properties of the tensor product are that it is bilinear (linear in both $\\pmb{x}$ and $\\textbf{y}$ ) and that it combines tensors inhabiting different irreps in a symmetrically valid way. Many simple operations are encompassed by the tensor product, such as for example:  \n\nscalar-scalar multiplication: $(\\ell_{1}=0,p_{1}=1),(\\ell_{2}=0,p_{2}=1)\\to(\\ell_{\\mathrm{out}}=$ $0,p_{\\mathrm{out}}=1)$   \nvector dot product: $(\\ell_{1}=1,p_{1}=-1),(\\ell_{2}=1,p_{2}=-1)\\to(\\ell_{\\mathrm{out}}=0,$ $\\boldsymbol{p_{\\mathrm{out}}}=1)$   \nvector cross product, resulting in a pseudovector: $(\\ell_{1}=1,p_{1}=-1)$ , $(\\ell_{2}=1,p_{2}=-1)\\to(\\ell_{\\mathrm{out}}=1,p_{\\mathrm{out}}=1)$  \n\nThe message function $M_{t}(\\mathbf h_{i}^{t},\\mathbf h_{j}^{t},\\mathbf e_{i j})$ of the NequIP model, for example, uses this tensor product to define a message from atom j to $i$ as a tensor product between equivariant features of the edge $\\ddot{y}$ and the equivariant features of the neighboring node $j$ .  \n\n![](images/e86fa96b5b406c713219f2db7f94f96108ea033e33ece625061d0b50ccfc2500.jpg)  \nFig. 1 | The Allegro network. a shows the Allegro model architecture and $\\boldsymbol{\\mathbf{b}}$ details a tensor product layer. Blue and red arrows represent scalar and tensor information, respectively, $\\otimes$ denotes the tensor product, and $\\circledast$ is concatenation.  \n\n# Atomic cluster expansion  \n\nFinally, parallel to atom-centered message-passing interatomic potentials, the Atomic Cluster Expansion (ACE) has been developed as a unifying framework for various descriptor-based MLIPs12. ACE can also be expressed in terms of the same tensor product operation introduced above, with further details provided in “Methods”.  \n\nIn this work, we present Allegro, an equivariant deep-learning approach that retains the high accuracy of the recently proposed class of equivariant MPNNs15,26,27,37,39,40 while combining it with strict locality and thus the ability to scale to large systems. We demonstrate that Allegro not only obtains state-of-the-art accuracy on a series of different benchmarks but can also be parallelized across devices to access simulations with hundreds of millions of atoms. We further find that Allegro displays a high level of transferability to out-ofdistribution data, significantly outperforming other local MLIPs, in particular including body-ordered approaches. Finally, we show that Allegro can faithfully recover structural and kinetic properties from molecular dynamics simulations of $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ , a complex phosphate electrolyte.  \n\nThe outline of the article is as follows: we first surveyed relevant related work on message-passing interatomic potentials, equivariant neural networks, and the atomic cluster expansion. We then outline the core ideas and design of the Allegro approach, followed by a series of results on standard benchmarks. Finally, we show the results of molecular dynamics simulations on a challenging material, an analysis of the scaling properties of Allegro, and a theoretical analysis of the framework.  \n\n# Energy decomposition  \n\nWe start by decomposing the potential energy of a system into peratom energies $E_{i},$ in line with previous approaches5,6,25:  \n\n$$\nE_{\\mathrm{system}}=\\sum_{i}^{N}\\sigma_{Z_{i}}E_{i}+\\mu_{Z_{i}}\n$$  \n\nwhere $\\sigma_{Z_{i}}$ and $\\mu_{Z_{i}}$ are per-species scale and shift parameters, which may be trainable. Unlike most existing MLIPs, we further decompose the per-atom energy into a sum of pairwise energies, indexed by the central atom and one of its local neighbors  \n\n$$\nE_{i}=\\sum_{j\\in\\mathcal{N}(i)}\\sigma_{Z_{i},Z_{j}}E_{i j}\n$$  \n\nwhere $j$ ranges over the neighbors of atom $i,$ and again one may optionally apply a per-species-pair scaling factor $\\sigma_{Z_{i},Z_{j}}$ . It is important to note that while these pairwise energies are indexed by the atom $i$ and its neighbor ${j,}$ they may depend on all neighboring atoms $k$ belonging to the local environment $\\mathcal{N}(i)$ . Because $E_{i j}$ and $E_{j i}$ contribute to different site energies $E_{i}$ and $E_{j},$ respectively, we choose that they depend only on the environments of the corresponding central atoms. As a result and by design, $E_{i j}\\not=E_{j i}$ . Finally, the force acting on atom i, ${\\vec{F}}_{i},$ is computed using autodifferentiation according to its definition as the negative gradient of the total energy with respect to the position of atom i:  \n\n$$\n\\vec{F}_{i}=-\\nabla_{i}E_{\\mathrm{system}}\n$$  \n\nwhich gives an energy-conserving force field.  \n\n# Results  \n\nIn the following, we describe the proposed method for learning highdimensional potential energy surfaces using strictly local many-body equivariant representations.  \n\n# The Allegro model  \n\nThe Allegro architecture, shown in Fig. 1, is an arbitrarily deep equivariant neural network with $N_{\\mathrm{layer}}{\\geq}1$ layers. The architecture learns representations associated with ordered pairs of neighboring atoms using two latent spaces: an invariant latent space, which consists of scalar $(\\ell=0)$ features, and an equivariant latent space, which processes tensors of arbitrary rank $\\ell\\geq0$ . The two latent spaces interact with each other at every layer. The final pair energy $E_{i j}$ is then computed by a multi-layer perceptron (MLP) acting on the final layer’s scalar features.  \n\nWe use the following notations: $\\scriptstyle{\\vec{r}}_{i}$ : position of the ith atom in the system $\\vec{r}_{i j}$ : relative displacement vector $\\Vec{r_{j}}-\\Vec{r}_{i}$ from i to j $r_{i j}$ : corresponding interatomic distance $\\hat{r}_{i i\\dot{\\cdot}i}$ unit vector of $\\vec{r}_{i j}$ $\\overrightarrow{Y}_{\\ell,p}^{\\upsilon}$ : projection of $\\hat{\\boldsymbol r}_{i j}$ onto the $\\ell$ -th real spherical harmonic which has parity $p=(-1)^{\\ell}$ . We omit the $\\boldsymbol{m}=-\\ell,\\ldots,0,\\ldots\\ell$ index within the representation from the notation for compactness $Z_{i}\\colon$ chemical species of atom i $\\mathsf{M L P}(\\ldots)$ : multi-layer perceptron—a fully connected scalar neural network, possibly with nonlinearities $\\mathbf{x}^{i j,L}$ : invariant scalar latent features of the ordered pair of atoms $\\ddot{y}$ at layer $\\iota$ $\\mathbf{v}_{n,\\ell,p}^{i j,L}$ : equivariant latent features of the ordered pair of atoms $\\ddot{y}$ at layer L. These transform according to a direct sum of irreps indexed by the rotation order $\\ell\\in0,1,\\ldots,\\ell_{\\mathrm{max}}$ and parity $p\\in-1,1$ and thus consist of both scalars $(\\ell=0)$ and higher-order tensors $(\\ell>0)$ . The hyperparameter $\\ell_{\\mathrm{{max}}}$ controls the maximum rotation order to which features in the network are truncated. In Allegro, $n$ denotes the channel index which runs over $0,\\ldots,n_{\\mathrm{equivariant}}-1.$ We omit the $m$ index within each irreducible representation from the notation for compactness.  \n\nTwo-body latent embedding. Before the first tensor product layer, the scalar properties of the pair $\\ddot{y}$ are embedded through a nonlinear MLP to give the initial scalar latent features $\\mathbf{x}^{i j,L=0}$ :  \n\n$$\n\\begin{array}{r}{\\pmb{x}^{i j,L=0}=\\pmb{\\mathrm{MLP}}_{\\mathrm{two-body}}\\left(\\pmb{\\mathrm{1Hot}}(Z_{i})\\parallel\\pmb{\\mathrm{1Hot}}(Z_{j})\\parallel B(r_{i j})\\right)\\cdot\\pmb{u}(r_{i j})}\\end{array}\n$$  \n\nwhere $\\parallel$ denotes concatenation, $\\mathrm{1Hot(\\cdot)}$ is a one-hot encoding of the center and neighbor atom species $Z_{i}$ and $Z_{j},$ and  \n\n4. an equivariant linear layer that mixes channels in the equivariant latent space.  \n\nTensor product: Our goal is to incorporate interactions between the current equivariant state of the center-neighbor pair and other neighbors in the environment, and the most natural operation with which to interact equivariant features is the tensor product. We thus define the updated equivariant features on the pair $\\ddot{y}$ as a weighted sum of the tensor products of the current features with the geometry of the various other neighbor pairs $i k$ in the local environment of $\\dot{t}.$  \n\n$$\n\\mathbf{V}_{n,(\\ell_{1},p_{1},\\ell_{2},p_{2})\\rightarrow(\\ell_{\\mathrm{out}},p_{\\mathrm{out}})}^{i j,L}=\\sum_{k\\in\\mathcal{N}(i)}w_{n,\\ell_{2},p_{2}}^{i k,L}\\left(\\mathbf{V}_{n,\\ell_{1},p_{1}}^{i j,L-1}\\otimes\\overrightarrow{Y}_{\\ell_{2},p_{2}}^{i k}\\right)\n$$  \n\n$$\n=\\sum_{k\\in\\mathcal{N}(i)}\\mathbf{V}_{n,\\ell_{1},p_{1}}^{i j,L-1}\\otimes\\left(w_{n,\\ell_{2},p_{2}}^{i k,L}\\vec{Y}_{\\ell_{2},p_{2}}^{i k}\\right)\n$$  \n\n$$\n=\\mathbf{V}_{n,\\ell_{1},p_{1}}^{i j,L-1}\\otimes\\left(\\sum_{k\\in\\mathcal{N}(i)}w_{n,\\ell_{2},p_{2}}^{i k,L}\\vec{Y}_{\\ell_{2},p_{2}}^{i k}\\right)\n$$  \n\nIn the second and third lines, we exploit the bilinearity of the tensor product in order to express the update in terms of one tensor product, rather than one for each neighbor $k$ which saves significant computational effort. This is a variation on the “density trick”6,41.  \n\nWe note that valid tensor product paths are all those satisfying $|\\ell_{1}-\\ell_{2}|\\leq\\ell_{\\mathrm{out}}\\leq|\\ell_{1}+\\ell_{2}|$ and $p_{\\mathrm{out}}=p_{1}p_{2},$ , so it is possible to have $(\\ell_{1},p_{1})\\neq(\\ell_{2},p_{2})\\neq(\\ell_{\\mathrm{out}},p_{\\mathrm{out}})$ . We additionally enforce $\\ell_{\\mathrm{out}}\\leq\\ell_{\\mathrm{max}}$ . Which tensor product paths to include is a hyperparameter choice. In this work we include all allowable paths but other choices, such as restricting $(\\ell_{\\mathrm{out}},p_{\\mathrm{out}})$ to be among the values of $(\\ell_{1},p_{1})$ , are possible.  \n\n$$\nB(r_{i j})=(B_{1}(r_{i j})\\parallel\\ldots\\parallel B_{N_{\\mathrm{basis}}}(r_{i j})))\n$$  \n\nis the projection of the interatomic distance $r_{i j}$ onto a radial basis. We use the Bessel basis functions with a polynomial envelope function as proposed in ref. 14. The basis is normalized as described in Supplementary Note 1 and plotted in Supplementary Fig. 1. Finally, the function $u(r_{i j}):\\mathbb{R}\\rightarrow\\mathbb{R}$ by which the output of MLPtwo-body is multiplied is the same smooth cutoff envelope function as used in the radial basis function.  \n\nThe initial equivariant features Vinj,‘L, p= 0 are computed as a linear embedding of the spherical harmonic projection of $\\hat{\\boldsymbol r}_{i j}$ :  \n\n$$\n\\mathbf{V}_{n,\\ell,p}^{i j,L=0}=w_{n,\\ell,p}^{i j,L=0}\\vec{Y}_{\\ell,p}^{i j}\n$$  \n\nwhere the channel index is $n{=}0,\\ldots,n_{\\mathrm{equivariant}}{-}1,$ , and where the scalar weights $w_{n,\\ell,p}^{i j,L=0}$ for each center-neighbor pair $\\ddot{y}$ are computed from the initial two-body scalar latent features:  \n\nEnvironment embedding: The second argument to the tensor product, $\\begin{array}{r}{\\sum_{k\\in\\mathcal{N}(i)}w_{n,\\ell_{2},p_{2}}^{i k,L}\\vec{Y}_{\\ell_{2},p_{2}}^{i k},}\\end{array}$ Y!‘2,p2 , is a weighted sum of the spherical harmonic projections of the various neighbor atoms in the local environment. This can be viewed as a weighted spherical harmonic basis projection of the atomic density, similar to the projection onto a spherical-radial basis used in $\\mathsf{A C E}^{12}$ and $\\mathsf{S O A P}^{41}$ . For this reason, we refer to Pk2N ðiÞwink,,‘L2,p2 Y!i‘k2,p2 a s the “embedded environment” of atom i.  \n\nA central difference from the atomic density projections used in descriptor methods, however, is that the weights of the sum are learned. In descriptor approaches such as ACE, the $n$ index runs over a pre-determined set of radial–chemical basis functions, which means that the size of the basis must increase with both the number of species and the desired radial resolution. In Allegro, we instead leverage the previously learned scalar featurization of each center-neighbor pair to further learn  \n\n$$\n\\boldsymbol{w}_{n,\\ell_{2},p_{2}}^{i k,L}=\\mathbf{MLP}_{\\mathrm{embed}}^{L}(\\mathbf{x}^{i k,L-1})_{n,\\ell_{2},p_{2}}\n$$  \n\n$$\n\\begin{array}{r}{\\begin{array}{r}{w_{n,\\ell,p}^{i j,L=0}=\\mathsf{M L P}_{\\mathrm{embed}}^{L=0}(\\mathbf{x}^{i j,L=0})_{n,\\ell,p}.}\\end{array}}\\end{array}\n$$  \n\nLayer architecture. Each Allegro tensor product layer consists of four components:  \n\n1. an MLP that generates weights to embed the central atom’s environment   \n2. an equivariant tensor product using those weights   \n3. an MLP to update the scalar latent space with scalar information resulting from the tensor product  \n\nwhich yields an embedded environment with a fixed, chosen number of channels nequivariant. It is important to note that $w_{n,\\ell_{2},p_{2}}^{i k,L}$ is learned as a function of the existing scalar latent representation of the centerneighbor pair $i k$ from previous layers. At later layers, this can contain significantly more information about the environment of i than a twobody radial basis. We typically choose $\\mathsf{M L P_{e m b e d}}$ to be a simple onelayer linear projection of the scalar latent space.  \n\nLatent MLP: Following the tensor product defined in Eq. (11), the scalar outputs of the tensor product are reintroduced into the scalar  \n\nTable 1 | MAE on the revised MD-17 dataset for energies and force components, in units of [meV] and [meV/Å], respectively   \n\n\n<html><body><table><tr><td>Molecule</td><td></td><td>FCHL1913, 43</td><td>UNiTE26</td><td>GAP6</td><td>ANI- pretrained48,49</td><td>ANI- random48.49</td><td>ACE12</td><td>GemNet- (T/Q)%</td><td>NequlP (l=3)15</td><td>Allegro</td></tr><tr><td>Aspirin</td><td>Energy</td><td>6.2</td><td>2.4</td><td>17.7</td><td>16.6</td><td>25.4</td><td>6.1</td><td></td><td>2.3</td><td>2.3</td></tr><tr><td></td><td>Forces</td><td>20.9</td><td>7.6</td><td>44.9</td><td>40.6</td><td>75.0</td><td>17.9</td><td>9.5</td><td>8.2</td><td>7.3</td></tr><tr><td>Azobenzene</td><td>Energy</td><td>2.8</td><td>1.1</td><td>8.5</td><td>15.9</td><td>19.0</td><td>3.6</td><td>二</td><td>0.7</td><td>1.2</td></tr><tr><td></td><td>Forces</td><td>10.8</td><td>4.2</td><td>24.5</td><td>35.4</td><td>52.1</td><td>10.9</td><td>_</td><td>2.9</td><td>2.6</td></tr><tr><td>Benzene</td><td>Energy</td><td>0.3</td><td>0.07</td><td>0.75</td><td>3.3</td><td>3.4</td><td>0.04</td><td></td><td>0.04</td><td>0.3</td></tr><tr><td></td><td>Forces</td><td>2.6</td><td>0.73</td><td>6.0</td><td>10.0</td><td>17.5</td><td>0.5</td><td>0.5</td><td>0.3</td><td>0.2</td></tr><tr><td>Ethanol</td><td>Energy</td><td>0.9</td><td>0.62</td><td>3.5</td><td>2.5</td><td>7.7</td><td>1.2</td><td>-</td><td>0.4</td><td>0.4</td></tr><tr><td></td><td>Forces</td><td>6.2</td><td>3.7</td><td>18.1</td><td>13.4</td><td>45.6</td><td>7.3</td><td>3.6</td><td>2.8</td><td>2.1</td></tr><tr><td>Malonaldehyde</td><td>Energy</td><td>1.5</td><td>1.1</td><td>4.8</td><td>4.6</td><td>9.4</td><td>1.7</td><td></td><td>0.8</td><td>0.6</td></tr><tr><td></td><td>Forces</td><td>10.2</td><td>6.6</td><td>26.4</td><td>24.5</td><td>52.4</td><td>11.1</td><td>6.6</td><td>5.1</td><td>3.6</td></tr><tr><td>Naphthalene</td><td>Energy</td><td>1.2</td><td>0.46</td><td>3.8</td><td>11.3</td><td>16.0</td><td>0.9</td><td></td><td>0.2</td><td>0.5</td></tr><tr><td></td><td>Forces</td><td>6.5</td><td>2.6</td><td>16.5</td><td>29.2</td><td>52.2</td><td>5.1</td><td>1.9</td><td>1.3</td><td>0.9</td></tr><tr><td>Paracetamol</td><td>Energy</td><td>2.9</td><td>1.9</td><td>8.5</td><td>11.5</td><td>18.2</td><td>4.0</td><td></td><td>1.4</td><td>1.5</td></tr><tr><td></td><td>Forces</td><td>12.2</td><td>7.1</td><td>28.9</td><td>30.4</td><td>63.3</td><td>12.7</td><td>一</td><td>5.9</td><td>4.9</td></tr><tr><td>Salicylic acid</td><td>Energy</td><td>1.8</td><td>0.73</td><td>5.6</td><td>9.2</td><td>13.5</td><td>1.8</td><td></td><td>0.7</td><td>0.9</td></tr><tr><td></td><td>Forces</td><td>9.5</td><td>3.8</td><td>24.7</td><td>29.7</td><td>52.0</td><td>9.3</td><td>5.3</td><td>4.0</td><td>2.9</td></tr><tr><td>Toluene</td><td>Energy</td><td>1.6</td><td>0.45</td><td>4.0</td><td>7.7</td><td>12.6</td><td>1.1</td><td></td><td>0.3</td><td>0.4</td></tr><tr><td></td><td>Forces</td><td>8.8</td><td>2.5</td><td>17.8</td><td>24.3</td><td>52.9</td><td>6.5</td><td>2.2</td><td>1.6</td><td>1.8</td></tr><tr><td>Uracil</td><td>Energy</td><td>0.6</td><td>0.58</td><td>3.0</td><td>5.1</td><td>8.3</td><td>1.1</td><td>-</td><td>0.4</td><td>0.6</td></tr><tr><td></td><td>Forces</td><td>4.2</td><td>3.8</td><td>17.6</td><td>21.4</td><td>44.1</td><td>6.6</td><td>3.8</td><td>3.1</td><td>1.8</td></tr></table></body></html>\n\nResults for GAP, ANI, and ACE as reported in ref. 24. Best results are marked in bold. ANI-pretrained refers to a version of ANI that was pretrained on 8.9 million structures and fine-tuned on the revMD17 dataset, ANI-random refers to a randomly initialized model trained from scratch.  \n\nlatent space as follows:  \n\n$$\n\\mathbf{x}^{i j,L}=\\mathsf{M L P}_{\\mathrm{latent}}^{L}\\left(\\mathbf{x}^{i j,L-1}\\parallel\\bigoplus_{(\\ell_{1},p_{1},\\ell_{2},p_{2})}\\mathbf{V}_{n,(\\ell_{1},p_{1},\\ell_{2},p_{2})\\to(\\ell_{\\mathrm{out}}=0,p_{\\mathrm{out}}=1)}^{i j,L}\\right)\\cdot\\boldsymbol{u}(r_{i j})\n$$  \n\nwhere $\\parallel$ denotes concatenation and $\\circledast$ denotes concatenation over all tensor product paths whose outputs are scalars $(\\ell_{\\mathrm{out}}=0,{p_{\\mathrm{out}}}=1)$ , each of which contributes nequivariant scalars. The function $u(r_{i j}):\\mathbb{R}\\rightarrow\\mathbb{R}$ is again the smooth cutoff envelope from Eq. (7). The purpose of the latent MLP is to compress and integrate information from the tensor product, whatever its dimension, into the fixed dimension invariant latent space. This operation completes the coupling of the scalar and equivariant latent spaces since the scalars taken from the tensor product contain information about non-scalars previously only available to the equivariant latent space.  \n\nMixing equivariant features: Finally, the outputs of various tensor   \nproduct paths with the same irrep $(\\ell_{\\mathrm{out}},p_{\\mathrm{out}})$ are linearly mixed to   \ncgheannernaetles ionudtepxuetdebqyu arsiatnhtefienaptutrefse $\\mathbf{v}_{n,\\ell,p}^{i j,L}$ swhiatdh the same number of $n$  \n\n$$\n\\mathbf{V}_{n,\\ell,p}^{i j,L}=\\sum_{n^{\\prime}}\\begin{array}{l l}{\\displaystyle{w_{n,n^{\\prime},(\\ell_{1},p_{1},\\ell_{2},p_{2})\\to(\\ell,p)}^{L}\\mathbf{V}_{n^{\\prime},(\\ell_{1},p_{1},\\ell_{2},p_{2})\\to(\\ell,p)}^{i j,L}}}\\\\ {\\displaystyle{(\\ell_{1},p_{1},\\ell_{2},p_{2})}}\\end{array}}\\\\ {\\end{array}\n$$  \n\nThe weights $w_{n,n^{\\prime},(\\ell_{1},p_{1},\\ell_{2},p_{2})\\to(\\ell,p)}^{L}$ are learned. This operation compresses the equivariant information from various paths with the same output irrep $(\\ell,p)$ into a single output space regardless of the number of paths.  \n\nWe finally note that an $S E(3)$ -equivariant version of Allegro, which is sometimes useful for computational efficiency, can be constructed identically to the $E(3)$ -equivariant model described here by simply omitting all parity subscripts $p$ .  \n\nResidual update. After each layer, Allegro uses a residual update42 in the scalar latent space that updates the previous scalar features from layer $_{L-1}$ by adding the new features to them (see Supplementary Note 2). The residual update allows the network to easily propagate scalar information from earlier layers forward.  \n\nOutput block. To predict the pair energy $E_{i j},$ we apply a fully connected neural network with output dimension 1 to the latent features output by the final layer:  \n\n$$\nE_{i j}=\\mathsf{M L P}_{\\mathrm{output}}(\\mathbf{x}^{i j,L=N_{\\mathrm{layer}}})\n$$  \n\nFinally, we note that we found normalization, both of the targets and inside the network, to be of high importance. Details are outlined in “Methods”.  \n\n# Dynamics of small molecules  \n\nWe benchmark Allegro’s ability to accurately learn energies and forces of small molecules on the revised MD-17 dataset43, a recomputed version of the original MD-17 dataset10,44,45 that contains ten small, organic molecules at DFT accuracy. As shown in Table 1, Allegro obtains state-of-the-art accuracy in the mean absolute error (MAE) in force components, while NequIP performs better for the energies of some molecules. We note that while an older version of the MD-17 dataset which has widely been used to benchmark MLIPs exists10,44,45, it has been observed to contain noisy labels43 and is thus only of limited use for comparing the accuracy of MLIPs.  \n\n# Transferability to higher temperatures  \n\nFor an interatomic potential to be useful in practice, it is crucial that it be transferable to new configurations that might be visited over the course of a long molecular dynamics simulation. To assess Allegro’s generalization capabilities, we test the transferability to conformations sampled from higher-temperature MD simulations. We use the temperature transferability benchmark introduced in ref. 24: here, a series of data were computed using DFT for the flexible drug-like molecule 3- (benzyloxy)pyridin-2-amine (3BPA) at temperatures 300, 600, and 1200 K. Various state-of-the-art methods were trained on 500 structures from the $T{=}300\\kappa$ dataset and then evaluated on data sampled at all three temperatures. Table 2 shows a comparison of Allegro against existing approaches reported in ref. 24: linear $\\mathsf{A C E}^{12}$ , sGDML10, $\\mathsf{G A P^{6}}$ , a classical force field based on the GAFF functional form46,47 as well as two ANI parametrizations48,49 (ANI-pretrained refers to a version of ANI that was pretrained on 8.9 million structures and fine-tuned on this dataset, while ANI-2x refers to the original parametrization trained on 8.9 million structures, but not fine-tuned on the 3BPA dataset). The equivariant neural networks Allegro and NequIP are observed to generalize significantly better than all other approaches.  \n\nTable 2 | Energy and Force RMSE for the 3BPA temperature transferability dataset, reported in units of [meV] and [meV/Å]   \n\n\n<html><body><table><tr><td></td><td>ACE12</td><td>SGDML10</td><td>GAP6</td><td>FF46,47</td><td>ANI-pretrained48.49</td><td>ANI-2x48, 49</td><td>NequlP15</td><td>Allegro</td></tr><tr><td>Fit to 300 K</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>300 K, E</td><td>7.1</td><td>9.1</td><td>22.8</td><td>60.8</td><td>23.5</td><td>38.6</td><td>3.28 (0.12)</td><td>3.84 (0.10)</td></tr><tr><td>300 K, F</td><td>27.1</td><td>46.2</td><td>87.3</td><td>302.8</td><td>42.8</td><td>84.4</td><td>10.77 (0.28)</td><td>12.98 (0.20)</td></tr><tr><td>600 K, E</td><td>24.0</td><td>484.8</td><td>61.4</td><td>136.8</td><td>37.8</td><td>54.5</td><td>11.16 (0.17)</td><td>12.07 (0.55)</td></tr><tr><td>600 K, F</td><td>64.3</td><td>439.2</td><td>151.9</td><td>407.9</td><td>71.7</td><td>102.8</td><td>26.37 (0.11)</td><td>29.11 (0.27)</td></tr><tr><td>1200 K, E</td><td>85.3</td><td>774.5</td><td>166.8</td><td>325.5</td><td>76.8</td><td>88.8</td><td>38.52 (2.00)</td><td>42.57 (1.79)</td></tr><tr><td>1200 K, F</td><td>187.0</td><td>711.1</td><td>305.5</td><td>670.9</td><td>129.6</td><td>139.6</td><td>76.18 (1.36)</td><td>82.96 (2.17)</td></tr></table></body></html>\n\nAll models were trained on ${\\cal T}=300$ K. Results for all models except for NequIP and Allegro from ref. 24. Best results are marked in bold. For NequIP and Allegro, we report the mean over three different seeds as well as the sample standard deviation in parentheses.  \n\n<html><body><table><tr><td colspan=\"2\">Table 3 | Comparison of models onthe QM9 dataset, mea- sured by the MAE in units of [meV]</td></tr><tr><td>Model</td><td></td></tr><tr><td>U。 Schnet25 14</td><td>U H G</td></tr><tr><td>DimeNet++77</td><td>19 14 14</td></tr><tr><td>6.3</td><td>6.3 6.5 7.6</td></tr><tr><td>Cormorant23 22</td><td>21 21 20</td></tr><tr><td>LieConv78 19</td><td>19 24 22</td></tr><tr><td>L1Net79 13.5</td><td>13.8 14.4 14.0</td></tr><tr><td>SphereNet80 6.3</td><td>7.3 6.4 8.0</td></tr><tr><td>EGNN40 11</td><td>12 12 12</td></tr><tr><td>ET38 6.2</td><td>6.3 6.5 7.6</td></tr><tr><td>NoisyNodes81 7.3</td><td>7.6 7.4 8.3</td></tr><tr><td>PaiNN27 5.9</td><td>5.7 6.0 7.4</td></tr><tr><td>Allegro, 1 layer 5.7 (0.3)</td><td>5.3 5.3 6.6</td></tr><tr><td>Allegro, 3 layers 4.7 (0.2)</td><td>4.4 4.4 5.7</td></tr></table></body></html>\n\nAllegro outperforms all existing atom-centered message-passing and transformer-based models, in particular even with a single layer. Best methods are shown in bold.  \n\n# Quantum-chemical properties of small molecules  \n\nNext, we assess Allegro’s ability to accurately model properties of small molecules across chemical space using the popular QM9 dataset50. The QM9 dataset contains molecular properties computed with DFT of approximately 134k minimum-energy structures with chemical elements (C, H, O, N, F) that contain up to 9 heavy atoms (C, O, N, F). We benchmark Allegro on four energy-related targets, in particular: (a) $U_{0},$ - the internal energy of the system at $T\\mathrm{=}0\\mathsf{K}$ , (b) $U,$ , the internal energy at $T=298.15\\mathsf{K},$ (c) $H,$ , the enthalpy at $T=298.15\\mathsf{K},$ and (d) $G,$ the free energy at $\\scriptstyle{T=298.15K}$ . Unlike other experiments in this work, which probed conformational degrees of freedom, we here assess the ability of Allegro to describe properties across compositional degrees of freedom. Table 3 shows a comparison with a series of state-of-the-art methods that also learn the properties described above as a direct mapping from atomic coordinates and species. We find that Allegro outperforms all existing methods. Surprisingly, even an Allegro model with a single tensor product layer obtains higher accuracy than all existing methods based on atom-centered message-passing neural networks and transformers.  \n\n# Li-ion diffusion in a phosphate electrolyte  \n\nIn order to examine Allegro’s ability to describe kinetic properties with MD simulations, we use it to study amorphous structure formation and Li-ion migration in the $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ solid electrolyte. This class of solid-state electrolytes is characterized by the intricate dependence of conductivity on the degree of crystallinity51–54.  \n\nIn particular, the $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ dataset used in this work consists of two parts: a 50 ps ab-initio molecular dynamics (AIMD) simulation in the molten liquid state at $T{=}3000\\upkappa$ , followed by a 50 ps AIMD simulation in the quenched state at $T{=}600\\kappa$ . We train a potential on structures from the liquid and quenched trajectories. The model used here is computationally efficient due to a relatively small number of parameters (9058 weights) and tensor products. In particular, we note that the model used to measure the faithfulness of the kinetics and to measure Allegro’s ability to predict thermodynamic observables is identical to the one used in scaling experiments detailed below. This is crucial for fair assessment of a method that simultaneously scales well and can accurately predict material properties. When evaluated on the test set for the quenched amorphous state, which the simulation is performed on, a MAE in the energies of 1.7 meV/atom was obtained, as well as a MAE in the force components of $73.4\\mathrm{meV}/\\mathring{\\mathbf{A}}$ . We then run a series of ten MD simulations starting from the initial structure of the quenched AIMD simulation, all of length 50 ps at $T{=}600\\kappa$ in the quenched state, in order to examine how well Allegro recovers the structure and kinetics compared to AIMD. To assess the quality of the structure after the phase change, we compare the all-atom radial distribution functions (RDF) and the angular distribution functions (ADF) of the tetrahedral angle $\\mathsf{P}{-}\\mathsf{O}{-}\\mathsf{O}$ (P central atom). We show in Fig. 2 that Allegro can accurately recover both distribution functions. For the aspect of ion transport kinetics, we test how well Allegro can model the Li mean-square-displacement (MSD) in the quenched state. We again find excellent agreement with AIMD, as shown in Fig. 3. The structure of $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ can be seen Fig. 4.  \n\n# Scaling  \n\nMany interesting phenomena in materials science, chemistry, and biology require large numbers of atoms, long timescales, a diversity of chemical elements, or often all three. Scaling to large numbers of atoms requires parallelization across multiple workers, which is difficult in atom-centered MPNNs because the iterative propagation of atomic state information along the atomistic graph increases the size of the receptive field as a function of the number of layers. This is further complicated by the fact that access to energy-conservative force fields requires computing the negative gradient of the predicted energy, which in standard backpropagation algorithms also requires propagating gradient information along the atom graph. Allegro is designed to avoid this issue by strict locality. A given Allegro model scales as:  \n\n![](images/e86524ac37434d0dc159e76a89d1d991ed5e4f6b71cff23429cb7fbeac59606b.jpg)  \nFig. 2 | Structural properties of $\\bf{L i_{3}P O_{4}}$ . Left: radial distribution function, right: angular distribution function of tetrahedral bond angle. All defined as probability density functions. Results from Allegro are shown in red, and those from AIMD are shown in black.  \n\n![](images/5df8bb71aaaec9664463fbd526b1c1607846a7d34ec322acf3bb8f80aec26b62.jpg)  \nFig. 3 | Li dynamics in $\\mathbf{Li_{3}P O_{4}}$ . Comparison of the Li MSD of AIMD vs. Allegro. Results are averaged over 10 runs of Allegro, shading indicates $+/-$ one standard deviation. Results from Allegro are shown in red, and those from AIMD are shown in blue.  \n\n![](images/02b4cf53d2eec8077024d2559c37689896e410a8a4838c47dbc82857d746bcab.jpg)  \nFig. 4 | Structure of $\\mathbf{Li_{3}P O_{4}}$ . The quenched $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ structure at $T{=}600\\kappa$  \n\n$\\mathcal{O}(N)$ in the number of atoms in the system $N_{\\astrosun}$ , in contrast to the $\\mathcal{O}(N^{2})$ scaling of some global descriptor methods such as $\\mathsf{s G D M L}^{10}$ ; $\\mathcal{O}(M)$ in the number of neighbors per atom M, in contrast to the quadratic $\\mathcal{O}(M^{2})$ scaling of some deep-learning approaches such as DimeNet14 or Equivariant Transformers38,55;  \n\n$\\mathcal{O}(1)$ in the number of species $S$ , unlike local descriptors such as SOAP ( S2 ) or ACE ( Sbodyorder\u00041 )12.  \n\nWe note, however, that the per-pair featurization of Allegro has larger memory requirements than if one were to choose the same number of features in a per-atom featurization. In practice, we find this to not be a problem and see that Allegro can be scaled to massive systems by parallelizing over modest computational resources.  \n\nIn particular, in addition to scaling as $\\mathcal{O}(N)$ in the number of atoms, Allegro is strictly local within the chosen cutoff and thus easy to parallelize in large-scale calculations. Recall that Eqs. (5) and (6) define the total energy of a system in Allegro as a sum over scaled pairwise energies $E_{i j}.$ Thus by linearity, the force on atom $a$  \n\n$$\n\\vec{F}_{a}=-\\nabla_{a}E_{\\mathrm{system}}=-\\sum_{i,j}\\nabla_{a}E_{i j},\n$$  \n\nignoring the per-species and per-species-pair scaling coefficients $\\sigma_{Z_{i}}$ and $\\sigma_{Z_{i},Z_{j}}$ for clarity. Because each $E_{i j}$ depends only the atoms in the neighborhood of atom $\\dot{\\iota},-\\nabla_{a}E_{i j}{\\neq}0$ only when $a$ is in the neighborhood of i. Further, for the same reason, pair energy terms $\\boldsymbol{E}_{i j}$ with different central atom indices $i$ are independent. As a result, these groups of terms may be computed independently for each central atom, which facilitates parallelization: the contributions to the force on atom $a$ due to the neighborhoods of various different atoms can be computed in parallel by whichever worker is currently assigned the relevant center’s neighborhood. The final forces are then simple sum reductions over force terms from various parallel workers.  \n\nWe first demonstrate the favorable scaling of Allegro in system size by parallelizing the method across GPUs on a single compute node as well as across multiple GPU nodes. We choose two test systems for the scaling experiments: (a) the quenched state structures of the multicomponent electrolyte $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ and (b) the Ag bulk crystal with a vacancy, simulated at $90\\%$ of the melting temperature. The Ag model used 1000 structures for training and validation, resulting in energy MAE of 0.397 meV/atom and force MAE of $16.8\\mathrm{meV/\\mathring{A}}$ on a test set of 159 structures. Scaling numbers are dependent on a variety of hyperparameter choices, such as network size and radial cutoff, that control the trade-off between evaluation speed and accuracy. For $\\mathsf{L i}_{3}\\mathsf{P O}_{4},$ we explicitly choose these identically to those used in the previous set of experiments in order to demonstrate how well an Allegro potential scales that we demonstrated to give highly accurate prediction of structure and kinetics. Table 4 shows the computational efficiency for varied size and computational resources. We are able to simulate the Ag system with over 100 million atoms on 16 GPU nodes.  \n\nThe parallel nature of the method and its implementation also allows multiple GPUs to be used to increase the speed of the potential calculation for a fixed-size system. Figure 5 shows such strong scaling  \n\nTable 4 | Simulation times obtainable in [ns/day] and time required per atom per step in [microseconds] for varying number of atoms and computational resources   \n\n\n<html><body><table><tr><td>Material</td><td>Number of atoms</td><td>Number of GPUs</td><td>Speed in ns/day</td><td>Microseconds/ (atom ·step)</td></tr><tr><td>LiPO4</td><td>192</td><td>1</td><td>32.391</td><td>27.785</td></tr><tr><td>LiPO4</td><td>421,824</td><td>1</td><td>0.518</td><td>0.552</td></tr><tr><td>LiPO4</td><td>421,824</td><td>2</td><td>1.006</td><td>0.284</td></tr><tr><td>LiPO4</td><td>421,824</td><td>4</td><td>1.994</td><td>0.143</td></tr><tr><td>LiPO4</td><td>421,824</td><td>8</td><td>3.810</td><td>0.075</td></tr><tr><td>LiPO4</td><td>421,824</td><td>16</td><td>6.974</td><td>0.041</td></tr><tr><td>LiPO4</td><td>421,824</td><td>32</td><td>11.530</td><td>0.025</td></tr><tr><td>Li3PO4</td><td>421,824</td><td>64</td><td>15.515</td><td>0.018</td></tr><tr><td>LiPO4</td><td>50,331,648</td><td>128</td><td>0.274</td><td>0.013</td></tr><tr><td>Ag</td><td>71</td><td>1</td><td>90.190</td><td>67.463</td></tr><tr><td>Ag</td><td>1,022,400</td><td>1</td><td>1.461</td><td>0.289</td></tr><tr><td>Ag</td><td>1,022,400</td><td>2</td><td>2.648</td><td>0.160</td></tr><tr><td>Ag</td><td>1,022,400</td><td>4</td><td>5.319</td><td>0.079</td></tr><tr><td>Ag</td><td>1,022,400</td><td>8</td><td>10.180</td><td>0.042</td></tr><tr><td>Ag</td><td>1,022,400</td><td>16</td><td>18.812</td><td>0.022</td></tr><tr><td>Ag</td><td>1,022,400</td><td>32</td><td>28.156</td><td>0.015</td></tr><tr><td>Ag</td><td>1,022,400</td><td>64</td><td>43.438</td><td>0.010</td></tr><tr><td>Ag</td><td>1,022,400</td><td>128</td><td>49.395</td><td>0.009</td></tr><tr><td>Ag</td><td>100,640,512</td><td>128</td><td>1.539</td><td>0.003</td></tr></table></body></html>\n\nTime steps of 2fs and 5fs were used for $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ and Ag, respectively.  \n\n![](images/515cda601bd3e8ff6550af790e20a2e0c65e09864ae2ffc45d979f94f5a16301.jpg)  \nFig. 5 | Scaling results. Strong scaling results on a $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ structure of 421,824 atoms, performed in LAMMPS.  \n\n$$\n\\begin{array}{r l}{\\mathbf{V}_{n_{1}}^{i j,L=1}=}&{\\displaystyle\\sum_{n_{1}^{\\prime}}}\\end{array}w_{n_{1},n_{1}^{\\prime},\\mathrm{path}}^{L=1}\\displaystyle\\sum_{k_{1}\\in\\mathcal{N}(i)}w_{n_{1}^{\\prime}}^{i k_{1},L=1}\\bigg(w_{n_{1}^{\\prime}}^{i j,L=0}\\overrightarrow{\\gamma}^{i j}\\otimes\\overrightarrow{Y}^{i k_{1}}\\bigg)}\\end{array}\n$$  \n\n$$\nk\\in\\mathcal{N}(i)\\\n$$  \n\n$$\n\\begin{array}{r l}{=}&{{}\\displaystyle\\sum_{n_{1}^{\\prime}}}&{{w}_{n_{1},n_{1}^{\\prime},\\mathrm{path}}^{L=1}\\displaystyle\\sum_{k_{1}\\in\\mathcal{N}(i)}{w}_{n_{1}^{\\prime}}^{i k_{1},L=1}{w}_{n_{1}^{\\prime}}^{i j,L=0}\\left(\\overrightarrow{\\gamma}^{i j}\\otimes\\overrightarrow{\\gamma}^{i k_{1}}\\right)}\\end{array}\n$$  \n\nresults on a 421,824 atom $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ structure. The system size was kept constant while varying the number of A100 GPUs.  \n\n# Theoretical analysis  \n\nIn this section, we provide a theoretical analysis of the method by highlighting similarities and differences to the Atomic Cluster Expansion (ACE) framework12. Throughout this section we omit representation indices $\\ell$ and $p$ from the notation for conciseness: every weight or feature that carries $\\ell$ and $p$ indices previously implicitly carries them in this section. Starting from the initial equivariant features for the pair of atoms $\\ddot{y}$ at layer $L=0$  \n\nwhich follows from the bilinearity of the tensor product. The sum over “paths” in this equation indicates the sum over all symmetrically valid combinations of implicit irrep indices on the various tensors present in the equation as written out explicitly in Eq. (16). Repeating this substitution, we can express the equivariant features at layer $L=2$ and reveal a general recursive relationship:  \n\n$$\n{\\bf V}_{n_{0}}^{i j,L=0}=w_{n_{0}}^{i j,L=0}\\vec{Y}^{i j}\n$$  \n\nthe first Allegro layer computes a sum over tensor products between $\\mathbf{V}_{n_{0}}^{i j,L=0}$ and the spherical harmonics projection of all neighbors  \n\n$$\n\\mathbf{V}_{n_{2},\\ell_{2},p_{2}}^{i j,l=2}=\\sum_{n_{2}^{\\prime}}\\mathbf{\\delta}w_{n_{2},n_{2}^{\\prime},\\mathrm{path}}^{L=2}\\sum_{k_{2}\\in\\mathcal{N}(i)}w_{n_{2}^{\\prime}}^{i k_{2},L=2}\\left(\\mathbf{V}_{n_{2}^{\\prime}}^{i j,L=1}\\otimes\\stackrel{\\longrightarrow}{Y}^{i k_{2}}\\right)\n$$  \n\n$$\n=\\sum_{n_{1}^{\\prime},n_{2}^{\\prime}}w_{n_{2},n_{2},\\mathrm{path}}^{L=2}w_{n_{2}^{\\prime},n_{1}^{\\prime},\\mathrm{path}}^{L=1}\\left[\\sum_{k_{2}\\in\\mathcal{N}(i)}\\sum_{k_{1}\\in\\mathcal{N}(i)}w_{n_{2}^{\\prime}}^{i k_{2},L=2}w_{n_{1}^{\\prime}}^{i k_{1},L=1}w_{n_{1}^{\\prime}}^{i j,L=0}\\left(\\stackrel{\\rightarrow}{Y}^{i j}\\otimes\\stackrel{\\rightarrow}{Y}^{i k_{1}}\\otimes\\stackrel{\\rightarrow}{Y}^{i k_{2}}\\right)\\right]\n$$  \n\n$$\n\\mathbf{V}_{n_{\\alpha},\\ell_{L},p_{L}}^{i j,L}=\\sum_{k_{1},\\dots,k_{L}}\\left[\\left(\\prod_{\\alpha\\in1,\\dots,L}w_{n_{\\alpha+1}^{\\prime\\alpha},n_{\\alpha}^{\\prime},\\mathrm{path}}^{L^{-\\alpha}}\\right)\\left(\\prod_{\\alpha\\in0,\\dots,L}w_{n_{\\alpha}^{\\prime\\alpha}}^{i k_{\\alpha},L=\\alpha}\\right)\\left(\\bigotimes_{\\alpha\\in0,\\dots,L}\\vec{Y}^{i k_{\\alpha}}\\right)\\right]\n$$  \n\nwhere $k_{0}=j,n_{L+1}^{\\prime}=n_{L}$ , and $n_{0}^{\\prime}=n_{1}^{\\prime}$ .  \n\nThe ACE descriptor $B_{n_{1}...n_{\\nu}}^{(\\nu)}$ of body order $\\nu+1^{12}$ can also be written as an iterated tensor product, specifically of the projection $A_{n}$ of the local atomic density onto a spherical harmonic and radial–chemical basis. The $n$ index here runs over the $N_{\\mathrm{full-basis}}{=}S\\times N_{\\mathrm{basis}}$ combined radial–chemical basis functions. Starting from this definition we may again use the bilinearity of the tensor product to expand the ACE descriptor:  \n\n$$\nB_{n_{1}...n_{\\nu}}^{(\\nu)}=\\bigotimes_{\\alpha=1,...,\\nu}A_{n_{i}}\n$$  \n\n$$\n=\\bigotimes_{\\alpha=1,...,\\nu}\\left(\\sum_{k_{\\alpha}\\in\\mathcal{N}(i)}R_{n_{\\alpha}}(r_{i k_{\\alpha}},z_{k_{\\alpha}})\\overrightarrow{Y}^{i k_{\\alpha}}\\right)\n$$  \n\n$$\n=\\sum_{k_{1},...,k_{\\nu}}\\left[\\left(\\prod_{\\alpha\\in1,...,\\nu}R_{n_{\\alpha}}(r_{i k_{\\alpha}},z_{k_{\\alpha}})\\right)\\left(\\bigotimes_{\\alpha\\in1,...,\\nu}\\vec{Y}^{i k_{\\alpha}}\\right)\\right]\n$$  \n\nComparing Eqs. (23) and (26) it is immediately evident that an Allegro model with $N_{\\mathrm{layer}}$ layers and an ACE expansion of body order $\\nu+1=N_{\\mathrm{layer}}+2$ share the core equivariant iterated tensor products $\\overrightarrow{Y}^{i j}\\otimes\\overrightarrow{Y}^{i k_{1}}\\otimes...\\otimes\\overrightarrow{Y}^{i k_{N_{\\mathrm{layer}}}}$ Y!ikNlayer . The equivariant Allegro features Vinj,L are analogous—but not equivalent—to the full equivariant ACE basis functions BðL + 1Þ n1 :::nL + 1  \n\nThe comparison of these expansions of the two models emphasizes, as discussed earlier in the scaling section, that the ACE basis functions carry a full set of $n_{\\alpha}$ indices (which label radial–chemical twobody basis functions), the number of which increases at each iteration, while the Allegro features do not exhibit this increase as a function of the number of layers. This difference is the root of the contrast between the $\\mathcal{O}(N_{\\mathrm{full-basis}}^{\\nu})$ scaling of ACE in the size of the radial–chemical basis $N_{\\mathrm{full-basis}}$ and the $\\mathcal{O}(1)$ of Allegro. Allegro achieves this more favorable scaling through the learnable channel mixing weights.  \n\nA key difference between Allegro and ACE, made clear here, is their differing construction of the scalar pairwise weights. In ACE, the scalar weights carrying $i k_{\\alpha}$ indices are the radial–chemical basis functions $R$ , which are two-body functions of the distance between atoms i and $k_{\\alpha}$ and their chemistry. These correspond in Allegro to the environment embedding weights $w_{i k_{\\alpha},n}^{L},$ , which—critically—are functions of all the lower-order equivarianαt features $\\mathbf{v}_{n}^{i j,L^{\\prime}<L}$ : the environment embedding weights at layer $\\iota$ are a function of the scalar features from layer $_{L-1}$ (Eq. (14)) which are a function of the equivariant features from layer $L-2$ (Eq. (15)) and so on. As a result, the “pairwise” weights have a hierarchical structure and depend on all previous weights:  \n\n$$\nw_{i x,n}^{L}=f(\\mathbf{V}^{i x,L-1})\n$$  \n\n$$\n=f\\Big(\\{w_{i x^{\\prime},n^{\\prime}}^{L^{\\prime}}\\mathrm{forall}n^{\\prime},L^{\\prime}<L,x^{\\prime}\\in\\mathcal{N}(i)\\}\\Big)\n$$  \n\nwhere $f$ contains details irrelevant to conveying the existence of the dependence. We hypothesize that this hierarchical nature is in part of why Allegro performs so much better than the ACE model and is a key difference to ACE and its variants, such as NICE. We finally note that the expanded features of Eq. (23)—and thus the final features of any Allegro model—are of finite body order if the environment embedding weights $w_{i k_{\\alpha}n}^{L}$ are themselves of finite body order. This condition holds if the latent and embedding MLPs are linear. If any of these MLPs contain nonlinearities whose Taylor expansions are infinite, the body order of the environment embedding weights, and thus of the entire model, becomes infinite. Nonlinearities in the two-body MLP are not relevant to the body order and correspond to the use of a nonlinear radial basis in methods such as ACE. Allegro models whose only nonlinearities lie in the two-body embedding MLP are still highly accurate and such a model was used in the experiments on the 3BPA dataset described above.  \n\n# Discussion  \n\nA new type of deep-learning interatomic potential is introduced that combines high prediction accuracy on energies and forces, enabled by its equivariant architecture, with the ability to scale to large system sizes, due to the strict locality of its geometric representations. The Allegro method surpasses the state-of-the-art set by atom-centered message-passing neural network models for interatomic interactions in terms of combined accuracy and scalability. This makes it possible to predict structural and kinetic properties from molecular dynamics simulations of complex systems of millions of atoms at nearly firstprinciples fidelity.  \n\nOur findings enable the study of molecular and materials system with equivariant neural networks that were previously inaccessible and raise broad questions about the optimal choice of representation and learning algorithm for machine learning on molecules and materials. We note that the Allegro method naturally offers a trade-off between accuracy and computational speed, while still offering efficient parallel scalability. Models of higher accuracy can be obtained by choosing networks with higher capacity (including larger numbers of features and more layers), but we also found a small, fast model to work sufficiently well to capture complex structural and kinetic properties in our example applications. It would be of great value to the community to conduct a detailed analysis of this accuracy-speed trade-off across different machine learning interatomic potentials and materials.  \n\nThe correspondences between the Allegro architecture and the atomic cluster expansion (ACE) formalism also raise questions about how and why Allegro is able to outperform the systematic ACE basis expansion. We speculate that our method’s performance is due in part to the learned dependence of the environment embedding weights at each layer on the full scalar latent features from all previous layers. This dependence may allow the importance of an atom to higher bodyorder interactions to be learned as a function of lower body-order descriptions of its environment. It stands in stark contrast to ACE, where the importance of any higher body-order interaction is learned separately from lower body-order descriptions of the local structure. We believe further efforts to understand this correspondence are a promising direction for future work. Similarly, we believe a systematic study of the completeness of the prescribed architecture will be of high interest.  \n\nAnother important goal for future work is to obtain a better understanding of when explicit long-range terms are required in machine learning interatomic potentials, how to optimally incorporate them with local models, and to what extent message-passing interatomic potentials may or may not implicitly capture these interactions. For example, it would be interesting to combine the Allegro potential with an explicit long-range energy term. In particular, the strict locality of the Allegro model naturally facilitates separation of the energy into a short-range term and a physically motivated long-range term.  \n\n# Methods  \n\n# Software  \n\nAll experiments were run with the Allegro code available at https:// github.com/mir-group/allegro under git commit a5128c2a8635076 2215dad6bd8bb42875ebb06cb. In addition, we used the NequIP code available at https://github.com/mir-group/nequip with version 0.5.3, git commit eb6f9bca7b36162abf69ebb017049599b4ddb09c, as well as e3nn with version $0.4.4^{56}$ , PyTorch with version $1.10.0^{57}.$ , and Python with version 3.9.7. The LAMMPS experiments were run with the LAMMPS code available at https://github.com/lammps/lammps.git under git commit 9b989b186026c6fe9da354c79cc9b4e152ab03af with the pair_allegro code available at https://github.com/ mir-group/pair_allegro, git commit 0161a8a8e2fe0849165- de9eeae3fbb987b294079. The VESTA software was used to generate Fig. $4^{58}$ . Matplotlib was used for plotting results59.  \n\n# Reference training sets  \n\nrevised MD-17. The revised MD-17 dataset consists of ten small organic molecules, for which 100,000 structures were computed at DFT (PBE/ def2-SVP) accuracy using a very tight SCF convergence and very dense DFT integration grid43. The structures were recomputed from the original MD-17 dataset10,44,45. The data can be obtained at https:// figshare.com/articles/dataset/Revised_MD17_dataset_rMD17_/ 12672038. We use 950 structures for training, 50 structures for validation (both sampled randomly), and evaluate the test error on all remaining structures.  \n\n3BPA. The 3BPA dataset consists of 500 training structures at $T{=}300\\kappa$ , and test data at 300 K, $600\\mathsf{K},$ and $1200\\mathsf{K}$ , of dataset size of 1669, 2138, and 2139 structures, respectively. The data were computed using Density Functioal Theory with the ωB97X exchange-correlation functional and the 6-31G(d) basis set. For details, we refer the reader $\\scriptstyle\\mathbf{to}^{24}$ . The dataset was downloaded from https://pubs.acs.org/doi/full/ 10.1021/acs.jctc.1c00647.  \n\nQM9. The QM9 data consist of 133,885 structures with up to 9 heavy elements and consisting of species H, C, N, O, F in relaxed geometries.  \n\nStructures are provided together with a series of properties computed at the DFT/B3LYP/6-31G(2df,p) level of theory. The dataset was downloaded from https://figshare.com/collections/Quantum_ chemistry_structures_and_properties_of_134_kilo_molecules/978904. In line with previous work, we excluded the 3054 structures that failed the geometry consistency check, resulting in 130,831 total structures, of which we use 110,000 for training, 10,000 for validation and evaluate the test error on all remaining structures. Training was performed in units of [eV].  \n\n$\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ . The $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ structure consists of 192 atoms. The reference dataset was obtained from two AIMD simulations both of 50 ps duration, performed in the Vienna Ab-Initio Simulation Package $(\\mathsf{V A S P})^{60-62}$ using a generalized gradient PBE functional63, projector augmented wave pseudopotentials64, a plane-wave cutoff of $400\\mathrm{eV}$ and a Γ-point reciprocal-space mesh. The integration was performed with a time step of 2 fs in the NVT ensemble using a Nosé–Hoover thermostat. The first 50 ps of the simulation were performed at $T{=}3000\\mathsf{K}$ in the molten phase, followed by an instant quench to $T{=}600\\kappa$ and a second 50 ps simulation at $T{=}600\\kappa$ . The two trajectories were combined and the training set of 10,000 structures as well as the validation set of 1000 were sampled randomly from the combined dataset of 50,000 structures.  \n\nAg. The Ag system is created from a bulk face-centered-cubic structure with a vacancy, consisting of 71 atoms. The data were sampled from AIMD simulations at ${\\cal T}=1111\\mathsf{K}$ ( $90\\%$ of the melting temperature of Ag) with Gamma-point $\\mathbf{k}$ -sampling as computed in VASP using the PBE exchange-correlation functional60–62. Frames were then extracted at least 25 fs apart, to limit correlation within the trajectory, and each frame was recalculated with converged DFT parameters. For these calculations, the Brillouin zone was sampled using a $(2\\times2\\times3)$ Gammacentered k-point grid, and the electron density at the Fermi-level was approximated using Methfessel–Paxton smearing65 with a sigma value of 0.05. A cutoff energy of 520 eV was employed, and each calculation was non-spin-polarized.  \n\n# Molecular dynamics simulations  \n\nMolecular Dynamics simulations were performed in LAMMPS66 using the pair style pair_allegro implemented in the Allegro interface, available at https://github.com/mir-group/pair_allegro. We run the $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ production and timing simulations under an NVT ensemble at $T{=}600\\mathsf{K},$ , using a time step of 2 fs, a Nosé-Hoover thermostat and a temperature damping parameter of 40 time steps. The Ag timing simulations are run also in NVT, at a temperature of $T{=}300\\kappa$ using a time step of 5 fs, a Nosé-Hoover thermostat and a temperature damping parameter of 40 time steps. The larger systems are created by replicating the original structures of 192 and 71 atoms of $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ and Ag, respectively. We compute the RDF and ADFs for $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ with a maximum distance of $10\\mathring{\\mathsf{A}}$ (RDF) and $2.5\\mathring{\\mathbf{A}}$ (ADFs). We start the simulation from the first frame of the AIMD quench simulation. RDF and ADF for Allegro were averaged over ten runs with different initial velocities, the first 10 ps of the 50 ps simulation were discarded in the RDF/ADF analysis to account for equilibration.  \n\n# Training details  \n\nModels were trained on a NVIDIA V100 GPU in single-GPU training.  \n\nrevMD-17 and 3BPA. The revised MD-17 models were trained with a total budget of 1000 structures, split into 950 for training and 50 for validation. The 3BPA model was trained with a total budget of 500 structures, split into 450 for training and 50 for validation. The dataset was re-shuffled after each epoch. We use three layers, 128 features for both even and odd irreps and a $\\ell_{\\mathrm{max}}=3$ . The 2-body latent MLP consists of four hidden layers of dimensions [128, 256, 512, 1024], using SiLU nonlinearities on the outputs of the hidden layers67. The later latent MLPs consist of three hidden layers of dimensionality [1024, 1024, 1024] using SiLU nonlinearities for revMD-17 and no nonlinearities for 3BPA. The embedding weight projection was implemented as a single matrix multiplication without a hidden layer or a nonlinearity. The final edge energy MLP has one hidden layer of dimension 128 and again no nonlinearity. All four MLPs were initialized according to a uniform distribution of unit variance. We used a radial cutoff of $7.0\\mathring{\\mathbf{A}}$ for all molecules in the revMD-17 dataset, except for naphthalene, for which a cutoff of $9.0\\mathring{\\mathbf{A}}$ was used, and a cutoff of $5.0\\mathring{\\mathbf{A}}$ for the 3BPA dataset. We have also included an ablation study on the cutoff radius for the large naphthalene molecule which can be found in Supplementary Table 1. We use a basis of eight non-trainable Bessel functions for the basis encoding with the polynomial envelope function using $\\scriptstyle p=6$ for revMD-17 and $p{=}2$ for 3BPA. We found it particularly important to use a low exponent $p$ in the polynomial envelope function for the 3BPA experiments. We hypothesize that this is due to the fact that a lower exponent provides a stronger decay with increasing interatomic distance (see Supplementary Fig. 1), thereby inducing a stronger inductive bias that atoms $j$ further away from a central atom i should have smaller pair energies $E_{i j}$ and thus contribute less to atom $i\\gamma_{s}$ site energy $E_{i}.$ RevMD-17 models were trained using a joint loss function of energies and forces:  \n\n$$\n\\mathcal{L}=\\frac{\\lambda_{E}}{B}\\sum_{b}^{B}\\left(\\hat{E}_{b}-E_{b}\\right)^{2}+\\frac{\\lambda_{F}}{3B N}\\sum_{i=1}^{B N}\\sum_{\\alpha=1}^{3}\\lvert|-\\frac{\\partial\\hat{E}}{\\partial r_{i,\\alpha}}-F_{i,\\alpha}\\rvert|^{2}\n$$  \n\nwhere $B,N,E_{b},\\hat{E}_{b},F_{i,\\alpha}$ denote the batch size, number of atoms, batch of true energies, batch of predicted energies, and the force component on atom $i$ in spatial direction $\\alpha$ , respectively and $\\lambda_{E},\\lambda_{F}$ are energy and force weights. Following previous works, for the revMD-17 data the force weight was set to 1000 and the weight on the total potential energies was set to 1. For the 3BPA molecules, as in ref. 68, we used a per-atom MSE term that divides the energy term by $N_{a t o m s}^{2}$ because (a) the potential energy is a global size-extensive property, and (b) we use a MSE loss function:  \n\n$$\n\\mathcal{L}=\\frac{\\lambda_{E}}{B}\\sum_{b}^{B}\\left(\\frac{\\hat{E}_{b}-E_{b}}{N}\\right)^{2}+\\frac{\\lambda_{F}}{3B N}\\sum_{i=1}^{B N}\\sum_{\\alpha=1}^{3}\\lVert-\\frac{\\partial\\hat{E}}{\\partial r_{i,\\alpha}}-F_{i,\\alpha}\\rVert^{2}\n$$  \n\nAfter this normalization, both the energy and the force term receive a weight of 1. Models were trained with the Adam optimizer69 in PyTorch57, with default parameters of $\\beta_{1}=0.9$ , $\\beta_{2}=0.999$ , and $\\epsilon=10^{-8}$ without weight decay. We used a learning rate of 0.002 and a batch size of 5. The learning rate was reduced using an on-plateau scheduler based on the validation loss with a patience of 100 and a decay factor of 0.8. We use an exponential moving average with weight 0.99 to evaluate on the validation set as well as for the final model. Training was stopped when one of the following conditions was reached: (a) a maximum training time of 7 days, (b) a maximum number of epochs of 100,000, (c) no improvement in the validation loss for 1000 epochs, (d) the learning rate dropped lower than 1e-6. We note that such long wall times are usually not required and highly accurate models can typically be obtained within a matter of hours or even minutes. All models were trained with float32 precision.  \n\n3BPA, NequIP. The NequIP models on the 3BPA dataset were trained with a total budget of 500 molecules, split into 450 for training and 50 for validation. The dataset was re-shuffled after each epoch. We use 5 layers, 64 features for both even and odd irreps and a $\\ell_{\\mathrm{max}}=3$ . We use a radial network of three layers with 64 hidden neurons and SiLU nonlinearities. We further use equivariant, SiLU-based gate nonlinearities as outlined in ref. 15, where even and odd scalars are not gated, but operated on directly by SiLU and tanh nonlinearities, respectively. We used a radial cutoff of $5.0\\mathring{\\mathbf{A}}$ and a non-trainable Bessel basis of size 8 for the basis encoding with a polynomial envelope function using $p{=}2$ . Again, a low $p$ value was found to be important. We use again a peratom MSE loss function in which both the energy and the force term receive a weight of 1. Models were trained with Adam with the AMSGrad variant in the PyTorch implementation57,69–71, with default parameters of $\\beta_{1}=0.9$ , $\\beta_{2}=0.999$ , and $\\epsilon=10^{-8}$ without weight decay. We used a learning rate of 0.01 and a batch size of 5. The learning rate was reduced using an on-plateau scheduler based on the validation loss with a patience of 50 and a decay factor of 0.8. We use an exponential moving average with weight 0.99 to evaluate on the validation set as well as for the final model. Training was stopped when one of the following conditions was reached: (a) a maximum training time of 7 days, (b) a maximum number of epochs of 100,000, (c) no improvement in the validation loss for 1000 epochs, (d) the learning rate dropped lower than 1e-6. We note that such long wall times are usually not required and highly accurate models can typically be obtained within a matter of hours or even minutes. All models were trained with float32 precision. We use a per-atom shift $\\mu_{Z_{i}}$ via the average per-atom potential energy over all training frames and a peratom scale $\\sigma_{Z_{i}}$ as the root-mean-square of the components of the forces over the training set.  \n\n$\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ . The $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ model was trained with a total budget of 11,000 structures, split into 10,000 for training and 1000 for validation. The dataset was re-shuffled after each epoch. We use one layer, 1 feature of even parity and $\\ell_{\\mathrm{max}}=1$ . The 2-body latent MLP consists of 2 hidden layers of dimensions [32, 64], using SiLU nonlinearities67. The later latent MLP consist of 1 hidden layer of dimensionality [64], also using a SiLU nonlinearity. The embedding weight projection was implemented as a single matrix multiplication without a hidden layer or a nonlinearity. The final edge energy MLP has one hidden layer of dimension 32 and again no nonlinearity. All four MLPs were initialized according to a uniform distribution of unit variance. We used a radial cutoff of $4.{\\overset{\\cdot}{0}}{\\overset{\\cdot}{\\mathbf{A}}}$ and a basis of eight non-trainable Bessel functions for the basis encoding with the polynomial envelope function using $p=48$ . The model was trained using a joint loss function of energies and forces. We use again the per-atom MSE as describe above and a weighting of 1 for the force term and 1 for the per-atom MSE term. The model was trained with the Adam optimizer69 in PyTorch57, with default parameters of $\\beta_{1}=0.9$ , $\\beta_{2}=0.999$ , and $\\epsilon=10^{-8}$ without weight decay. We used a learning rate of 0.001 and a batch size of 1. The learning rate was reduced using an on-plateau scheduler based on the validation loss with a patience of 25 and a decay factor of 0.5. We use an exponential moving average with weight 0.99 to evaluate on the validation set as well as for the final model. Training was stopped when one of the following conditions was reached: (a) a maximum training time of 7 days, (b) a maximum number of epochs of 100,000, (c) no improvement in the validation loss for 1000 epochs, (d) the learning rate dropped lower than 1e-5. The model was trained with float32 precision.  \n\nAg. The Ag model was trained with a total budget of 1000 structures, split into 950 for training and 50 for validation, and evaluated on a separate test set of 159 structures. The dataset was re-shuffled after each epoch. We use 1 layer, 1 feature of even parity and $\\ell_{\\mathrm{max}}=1$ . The 2-body latent MLP consists of 2 hidden layers of dimensions [16, 32], using SiLU nonlinearities67. The later latent MLP consists of 1 hidden layer of dimensionality [32], also using a SiLU nonlinearity. The embedding weight projection was implemented as a single matrix multiplication without a hidden layer or a nonlinearity. The final edge energy MLP has one hidden layer of dimension 32 and again no nonlinearity. All four MLPs were initialized according to a uniform distribution. We used a radial cutoff of $4.0\\mathring\\mathrm{A}$ and a basis of eight nontrainable Bessel functions for the basis encoding with the polynomial envelope function using $p=48$ . The model was trained using a joint loss function of energies and forces. We use again the per-atom MSE as describe above and a weighting of 1 for the force term and 1 for the peratom MSE term. The model was trained with the Adam optimizer69 in PyTorch57, with default parameters of $\\beta_{1}=0.9$ , $\\beta_{2}=0.999$ , and $\\epsilon=10^{-8}$ without weight decay. We used a learning rate of 0.001 and a batch size of 1. The learning rate was reduced using an on-plateau scheduler based on the validation loss with patience of 25 and a decay factor of 0.5. We use an exponential moving average with weight 0.99 to evaluate on the validation set as well as for the final model. The model was trained for a total of approximately 5 h with float32 precision.  \n\nQM9. We used 110,000 molecular structures for training, 10,000 for validation, and evaluated the test error on all remaining structures, in line with previous approaches9,27. We note that Cormorant and EGNN are trained on 100,000 structures, L1Net is trained on 109,000 structures while NoisyNodes is trained on 114,000 structures. To give an estimate of the variability of training as a function of random seed, we report for the $U_{0}$ target the mean and sample standard deviation across three different random seeds, resulting in different samples of training set as well as different weight initialization. We report two models, one with three layers and $\\ell_{\\mathrm{max}}=2$ and another one with 1 layer and $\\ell_{\\mathrm{max}}=3$ , both with 256 features for both even and odd irreps. The 1-layer and 3-layer networks have 7,375,237 and 17,926,533 parameters, respectively. The 2-body latent MLP consists of four hidden layers of dimensions [128, 256, 512, 1024], using SiLU nonlinearities67. The later latent MLPs consist of three hidden layers of dimensionality [1024, 1024, 1024], also using SiLU nonlinearities. The embedding weight projection was implemented as a single matrix multiplication without a hidden layer or a nonlinearity. The final edge energy MLP has one hidden layer of dimension 128 and again no nonlinearity. All four MLPs were initialized according to a uniform distribution. We used a radial cutoff of $10.0\\mathring{\\mathsf{A}}$ . We use a basis of 8 non-trainable Bessel functions for the basic encoding with the polynomial envelope function using $p=6$ . Models were trained using a MSE loss on the energy with the Adam optimizer69 in PyTorch57, with default parameters of $\\beta_{1}=0.9$ , $\\beta_{2}=0.999$ , and $\\epsilon=10^{-8}$ without weight decay. In addition, we use gradient clipping by norm with a maximum norm of 100. The dataset was re-shuffled after each epoch. We used a learning rate of 0.001 and a batch size of 16. The learning rate was reduced using an on-plateau scheduler based on the validation MAE of the energy with a patience of 25 and a decay factor of 0.8. We use an exponential moving average with weight 0.999 to evaluate on the validation set as well as for the final model. Training was stopped when one of the following conditions was reached: (a) a maximum training time of approximately 14 days, (b) a maximum number of epochs of 100,000, (c) no improvement in the validation loss for 1000 epochs, (d) the learning rate dropped lower than 1e-5. All models were trained with float32 precision. Again, we note that such long wall times are not required to obtain highly accurate models. We subtract the sum of the reference atomic energies and then apply the linear fitting procedure described above using every 100th reference label in the training set.  \n\n# Scaling experiments  \n\nScalability across devices is achieved by implementing an Allegro extension to the LAMMPS molecular dynamics code66. The local nature of the Allegro model is compatible with the spatial decomposition approach used in LAMMPS and thus all communication between MPI ranks is handled by existing LAMMPS functionality. The Allegro extension simply transforms the LAMMPS neighbor lists into the format required by the Allegro PyTorch model and stores the resulting forces and energies in the LAMMPS data structures. These operations are performed on the GPU and use the Kokkos performance portability library72 to entirely avoid expensive CPU work or CPU-GPU data transfer. The scaling experiments were performed on NVIDIA DGX A100s on the ThetaGPU cluster at the Argonne Leadership Computing  \n\nFacility, where each node contains 8 GPUs and a total of 320 GB of GPU memory. For the $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ simulation, we use a time step of 2 fs, identical to the reference AIMD simulations, float32 precision, and a temperature of $T{=}600\\kappa$ on the quenched structure, identical to the production simulations used in the quench simulation. For Ag, we use a time step of 5 fs, a temperature of $T{=}300\\kappa$ and again float32 precision. Simulations were performed for 1000 time steps after initial warm-up.  \n\n# Atom-density representations  \n\nThe Atomic Cluster Expansion (ACE) is a systematic scheme for representing local atomic environments in a body-ordered expansion. The coefficients of the expansion of a particular atomic environment serve as an invariant description of that environment. To expand a local atomic environment, the local atomic density is first projected onto a combination of radial basis functions $R$ and spherical harmonic angular basis functions $\\vec{\\gamma}$ :  \n\n$$\nA_{z n\\ell}=\\sum_{j\\in\\mathcal{N}(i)\\mathsf{S.t.}z_{j}=z}R_{n\\ell}(\\pmb{r}_{i j})\\overrightarrow{Y}_{\\ell}^{m}(\\hat{\\pmb{r}}_{i j})\n$$  \n\nwhere $z$ runs over all atom species in the system, $z_{j}$ is the species of atom j, $\\mathcal{N}(\\dot{\\iota})$ is the set of all atoms within the cutoff distance of atom $i,$ also known as its “neighborhood”, and the $n$ index runs over the radial basis functions. The $m$ index on $A$ is implicit. The basis projection of body order $\\nu+1$ is then defined as:  \n\n$$\nB_{z_{1},n_{1}}^{(\\nu=1)}=A_{z_{1}n_{1}\\ell_{1}=0}\n$$  \n\n$$\n\\begin{array}{c}{{B_{z_{1},z_{2},n_{1},n_{2}}^{(\\nu=2)}=A_{z_{1}n_{1}\\ell_{1}}\\otimes A_{z_{2}n_{2}\\ell_{2}}}}\\\\ {{\\ell_{1},\\ell_{2}}}\\end{array}\n$$  \n\n$$\n\\begin{array}{c}{{B_{z_{1}...z_{\\nu},n_{1}...n_{\\nu}}^{(\\nu)}=\\bigotimes_{\\alpha=1,...,\\nu}A_{z_{\\alpha}n_{\\alpha}\\ell_{\\alpha}}.}}\\\\ {{\\ell_{1}...\\ell_{\\nu}}}\\end{array}\n$$  \n\nOnly tensor products outputting scalars—which are invariant, like the final target total energy—are retained here. For example, in Eq. (33), only tensor products combining basis functions inhabiting the same rotation order $\\ell_{1}=\\ell_{2}$ can produce scalar outputs. The final energy is then fit as a linear model over all the scalars $B$ up to some chosen maximum body order $\\nu+1.$  \n\nIt is apparent from Eq. (35) that a core bottleneck in the Atomic Cluster Expansion is the polynomial scaling of the computational cost of evaluating the $B$ terms with respect to the total number of two-body radial–chemical basis functions Nfull-basis as the body order $\\nu{+}1$ increases: $\\mathcal{O}(N_{\\mathrm{full-basis}}^{\\nu})$ . In the basic ACE descriptor given above, $N_{\\mathrm{full-basis}}{=}N_{\\mathrm{basis}}\\times S$ is the number of radial basis functions times the number of species. Species embeddings have been proposed for ACE to remove the direct dependence on $S^{73}$ . It retains, however, the $\\mathcal{O}(N_{\\mathrm{full-basis}}^{\\nu})$ scaling in the dimension of the embedded basis $N_{\\mathrm{full}}$ -basis. NequIP and some other existing equivariant neural networks avert this unfavorable scaling by only computing tensor products of a more limited set of combinations of input tensors. The NICE framework74 is an idea closely related to ACE that aims to solve the problem of increasing numbers of features by selecting only certain features at each iteration based on principal component analysis.  \n\n# Normalization  \n\nInternal normalization. The normalization of neural networks’ internal features is known to be of great importance to training. In this work we follow the normalization scheme of the e3nn framework75, in which the initial weight distributions and normalization constants are chosen so that all components of the network produce outputs that element-wise have approximately zero mean and unit variance. In particular, all sums over multiple features are normalized by dividing by the square root of the number of terms in the sum, which follows from the simplifying assumption that the terms are uncorrelated and thus that their variances add. Two consequences of this scheme that merit explicit mention are the normalization of the embedded environment and atomic energy. Both the embedded environment (Eq. (4)) and atomic energy (Eq. (6)) are sums over all neighbors of a central atom. Thus we divide both by $\\sqrt{\\langle|\\mathcal{N}(\\dot{t})|\\rangle}$ where $\\langle\\lvert\\mathcal{N}(i)\\rvert\\rangle$ is the average number of neighbors over all environments in the entire training dataset.  \n\nNormalization of targets. We found the normalization of the targets, or equivalently the choice of final scale and shift parameters for the network’s predictions (see Eq. (5)), to be of high importance. For systems of fixed chemical composition, our default initialization is the following: $\\mu_{Z}$ is set for all species $Z$ to the average per-atom potential energy over all training frames $\\left<\\frac{E_{\\mathrm{config}}}{N}\\right>$ ; $\\sigma_{Z}$ is set for all species $Z$ to the root-mean-square of the components of the forces on all atoms in the training dataset. This scheme ensures size extensivity of the potential energy, which is required if one wants to evaluate the potential on systems of different size than what it was trained on. We note that the widely used normalization scheme of subtracting the mean total potential energy across the training set violates size extensivity.  \n\nFor systems with varying chemical composition, we found it helpful to normalize the targets using a linear pre-fitting scheme that explicitly takes into account the varying chemical compositions: $\\mu_{Z}$ is computed by $[N_{\\mathrm{config},Z}]^{-1}[E_{\\mathrm{config}}].$ , where $[N_{\\mathrm{config},Z}]$ is a matrix containing the number of atoms of each species in the reference structures, and $[E_{\\mathrm{config}}]$ is a vector of reference energies. Details of the normalization calculations and the comparison between different schemes can be found in ref. 68.  \n\n# Reporting summary  \n\nFurther information on research design is available in the Nature Portfolio Reporting Summary linked to this article.  \n\n# Data availability  \n\nThe $\\mathsf{L i}_{3}\\mathsf{P O}_{4}$ and Ag data generated in this study have been deposited in the MaterialsCloud database at https://archive.materialscloud.org/ record/2022.128. The revMD-17, 3BPA, and QM9 datasets are publicly available (see “Methods”).  \n\n# Code availability  \n\nAn open-source software implementation of Allegro is available at https://github.com/mir-group/allegro together with an implementation of the LAMMPS software interface at https://github.com/mirgroup/pair_allegro.  \n\n# References  \n\n1. Richards, W. D. et al. Design and synthesis of the superionic conductor na 10 snp 2 s 12. Nat. Commun. 7, 1–8 (2016).   \n2. Lindorff-Larsen, K., Piana, S., Dror, R. O. & Shaw, D. 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Simple gnn regularisation for 3d molecular property prediction and beyond. In International Conference on Learning Representations (2021).  \n\n# Acknowledgements  \n\nAuthors thank Dr. Nicola Molinari for the helpful discussions. This work was supported primarily by the US Department of Energy. S.B., A.J., and B.K. were supported by DOE Office of Basic Energy Sciences Award No. DE-SC0022199. L.S. and B.K. were supported by the Integrated Mesoscale Architectures for Sustainable Catalysis (IMASC), an Energy Frontier Research Center, Award No. DE-SC0012573. B.K. acknowledges partial support from NSF through the Harvard University Materials Research Science and Engineering Center Grant No. DMR-2011754 and Bosch Research. A.M. is supported by U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Computational Science Graduate Fellowship under Award Number(s) DESC0021110. C.J.O. is supported by the National Science Foundation Graduate Research Fellowship Program, Grant No. DGE1745303. Work at Bosch Research by M.K. was partially supported by ARPA-E Award No. DE-AR0000775 and used resources of the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract DEAC05-00OR22725. The authors acknowledge computing resources provided by the Harvard University FAS Division of Science Research Computing Group and by the Texas Advanced Computing Center (TACC) at The University of Texas at Austin under allocation DMR20013. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357.  \n\n# Author contributions  \n\nS.B. and A.M. jointly conceived the Allegro model architecture, derived the theoretical analysis of the model, and wrote the first version of the manuscript. A.M. implemented the software and contributed to running experiments. S.B. originally proposed to work on an architecture that can capture many-body information without atom-centered message passing, conducted the experiments, and contributed to the software implementation. A.J. wrote the LAMMPS interface, including parallelization across devices. L.S. proposed the linear fitting for the per-species initialization and implemented it. C.J.O. generated the Ag data. M.K. generated the AIMD dataset of $\\mathsf{L i}_{3}\\mathsf{P O}_{4},$ wrote software for the analysis of MD results, and contributed to the analysis of results on this system. B.K. supervised and guided the project from conception to design of experiments, implementation, theory, as well as analysis of data. All authors contributed to the manuscript.  \n\n# Competing interests  \n\nThe authors declare no competing interests.  \n\nAdditional information   \nSupplementary information The online version contains supplementary material available at   \nhttps://doi.org/10.1038/s41467-023-36329-y.  \n\nCorrespondence and requests for materials should be addressed to Simon Batzner or Boris Kozinsky.  \n\nPeer review information Nature Communications thanks Huziel Sauceda, Claudio Zeni and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.  \n\nReprints and permissions information is available at http://www.nature.com/reprints  \n\nPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\nOpen Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.  \n\n$\\circledcirc$ The Author(s) 2023  ",
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    "id": "10.1038_s41586-020-1994-5",
    "DOI": "10.1038/s41586-020-1994-5",
    "DOI Link": "http://dx.doi.org/10.1038/s41586-020-1994-5",
    "Relative Dir Path": "mds/10.1038_s41586-020-1994-5",
    "Article Title": "Closed-loop optimization of fast-charging protocols for batteries with machine learning",
    "Authors": "Attia, PM; Grover, A; Jin, N; Severson, KA; Markov, TM; Liao, YH; Chen, MH; Cheong, B; Perkins, N; Yang, Z; Herring, PK; Aykol, M; Harris, SJ; Braatz, RD; Ermon, S; Chueh, WC",
    "Source Title": "NATURE",
    "Abstract": "Simultaneously optimizing many design parameters in time-consuming experiments causes bottlenecks in a broad range of scientific and engineering disciplines(1,2). One such example is process and control optimization for lithium-ion batteries during materials selection, cell manufacturing and operation. A typical objective is to maximize battery lifetime; however, conducting even a single experiment to evaluate lifetime can take months to years(3-5). Furthermore, both large parameter spaces and high sampling variability(3,6,7) necessitate a large number of experiments. Hence, the key challenge is to reduce both the number and the duration of the experiments required. Here we develop and demonstrate a machine learning methodology to efficiently optimize a parameter space specifying the current and voltage profiles of six-step, ten-minute fast-charging protocols for maximizing battery cycle life, which can alleviate range anxiety for electric-vehicle users(8,9). We combine two key elements to reduce the optimization cost: an early-prediction model(5), which reduces the time per experiment by predicting the final cycle life using data from the first few cycles, and a Bayesian optimization algorithm(10,11), which reduces the number of experiments by balancing exploration and exploitation to efficiently probe the parameter space of charging protocols. Using this methodology, we rapidly identify high-cycle-life charging protocols among 224 candidates in 16 days (compared with over 500 days using exhaustive search without early prediction), and subsequently validate the accuracy and efficiency of our optimization approach. Our closed-loop methodology automatically incorporates feedback from past experiments to inform future decisions and can be generalized to other applications in battery design and, more broadly, other scientific domains that involve time-intensive experiments and multi-dimensional design spaces.",
    "Times Cited, WoS Core": 538,
    "Times Cited, All Databases": 577,
    "Publication Year": 2020,
    "Research Areas": "Science & Technology - Other Topics",
    "UT (Unique WOS ID)": "WOS:000543762000001",
    "Markdown": "# Article  \n\n# Closed-loop optimization of fast-charging protocols for batteries with machine learning  \n\nhttps://doi.org/10.1038/s41586-020-1994-5  \n\nReceived: 6 August 2019  \n\nAccepted: 19 December 2019  \n\nPublished online: 19 February 2020 Check for updates  \n\nPeter M. Attia1,7, Aditya Grover2,7, Norman Jin1, Kristen A. Severson3, Todor M. Markov2, Yang-Hung Liao1, Michael H. Chen1, Bryan Cheong1,2, Nicholas Perkins1, Zi Yang1, Patrick K. Herring4, Muratahan Aykol4, Stephen J. Harris1,5, Richard D. Braatz3 ✉, Stefano Ermon2 ✉ & William C. Chueh1,6 ✉  \n\nSimultaneously optimizing many design parameters in time-consuming experiments causes bottlenecks in a broad range of scientific and engineering disciplines1,2. One such example is process and control optimization for lithium-ion batteries during materials selection, cell manufacturing and operation. A typical objective is to maximize battery lifetime; however, conducting even a single experiment to evaluate lifetime can take months to years3–5. Furthermore, both large parameter spaces and high sampling variability3,6,7 necessitate a large number of experiments. Hence, the key challenge is to reduce both the number and the duration of the experiments required. Here we develop and demonstrate a machine learning methodology  to efficiently optimize a parameter space specifying the current and voltage profiles of six-step, ten-minute fast-charging protocols for maximizing battery cycle life, which can alleviate range anxiety for electric-vehicle users8,9. We combine two key elements to reduce the optimization cost: an early-prediction model5, which reduces the time per experiment by predicting the final cycle life using data from the first few cycles, and a Bayesian optimization algorithm10,11, which reduces the number of experiments by balancing exploration and exploitation to efficiently probe the parameter space of charging protocols. Using this methodology, we rapidly identify high-cycle-life charging protocols among 224 candidates in 16 days (compared with over 500 days using exhaustive search without early prediction), and subsequently validate the accuracy and efficiency of our optimization approach. Our closed-loop methodology automatically incorporates feedback from past experiments to inform future decisions and can be generalized to other applications in battery design and, more broadly, other scientific domains that involve time-intensive experiments and multi-dimensional design spaces.  \n\nOptimal experimental design (OED) approaches are widely used to reduce the cost of experimental optimization. These approaches often involve a closed-loop pipeline where feedback from completed experiments informs subsequent experimental decisions, balancing the competing demands of exploration—that is, testing regions of the experimental parameter space with high uncertainty—and exploitation—that is, testing promising regions based on the results of the completed experiments. Adaptive OED algorithms have been successfully applied to physical science domains, such as materials science1,2,12–14, chemistry15,16, biology17 and drug discovery18, as well as to computer science domains, such as hyperparameter optimization for machine learning19,20. However, while a closed-loop approach is designed to minimize the number of experiments required for optimizing across a multi-dimensional parameter space, the time (and cost) per experiment may remain high, as is the case for lithium-ion batteries. Therefore, an OED approach should account for both the number of experiments and the cost per experiment. Multi-fidelity optimization approaches have been developed to learn from both inexpensive, noisy signals and expensive, accurate signals. For example, in hyperparameter optimization for machine learning algorithms, several low-fidelity signals for predicting the final performance of an algorithmic configuration (for example, extrapolated learning curves19,20, rapid testing on a subset of the full training dataset21) are used in tandem with more complete configuration evaluations22,23. For lithium-ion batteries, classical  \n\n# Article  \n\n![](images/b9d25a890ae53043c9b16c527b371fa098f78249747470b0c387fec852e30591.jpg)  \nFig. 1 | Schematic of our CLO system. First, batteries are tested. The cycling data from the first 100 cycles (specifically, electrochemical measurements such as voltage and capacity) are used as input for an early outcome prediction of cycle life. These cycle life predictions from a machine learning (ML) model are subsequently sent to a BO algorithm, which recommends the next protocols to test by balancing the competing demands of exploration (testing protocols with high uncertainty in estimated cycle life) and exploitation   \n(testing protocols with high estimated cycle life). This process iterates until the testing budget is exhausted. In this approach, early prediction reduces the number of cycles required per tested battery, while optimal experimental design reduces the number of experiments required. A small training dataset of batteries cycled to failure is used both to train the early outcome predictor and to set BO hyperparameters. In future work, design of battery materials and processes could also be integrated into this closed-loop system.  \n\nmethods such as factorial design that use predetermined heuristics to select experiments have been applied24–26, but the design and use of low-fidelity signals is challenging and unexplored. These previously considered approaches do not discover and exploit the patterns present in the parameter space for efficient optimization, nor do they address the issue of time per experiment.  \n\nIn this work, we develop a closed-loop optimization (CLO) system with early outcome prediction for efficient optimization over large parameter spaces with expensive experiments and high sampling variability. We employ this system to experimentally optimize fastcharging protocols for lithium-ion batteries; reducing charging times to approach gasoline refuelling time is critical to reduce range anxiety for electric vehicles8,9 but often comes at the expense of battery lifetime. Specifically, we optimize over a parameter space consisting of 224 unique six-step, ten-minute fast-charging protocols (that is, how current and voltage are controlled during charging) to find charging protocols with high cycle life (defined as the battery capacity falling to $80\\%$ of its nominal value). Our system uses two key elements to reduce the optimization cost (Extended Data Fig. 1). First, we reduce the time per experiment by using machine learning to predict the outcome of the experiment based on data from early cycles, well before the batteries reach the end of life5. Second, we reduce the number of experiments by using a Bayesian optimization (BO) algorithm to balance the exploration–exploitation tradeoff in choosing the next round of experiments10,11. Testing a single battery to failure under our fast-charging conditions requires approximately 40 days, meaning that when 48 experiments are performed in parallel, assessing all 224 charging protocols with triplicate measurements takes approximately 560 days. Here, using CLO with early outcome prediction, only 16 days were required to confidently identify protocols with high cycle lives (48 parallel experiments). In a subsequent validation study, we find that CLO ranks these protocols by lifetime accurately (Kendall rank correlation coefficient, 0.83) and efficiently (15 times less time than a baseline ‘brute-force’ approach that uses random search without early prediction). Furthermore, we find that the charging protocols identified as optimal by CLO with early prediction outperform existing fast-charging protocols designed to avoid lithium plating (a common fast-charging degradation mode), the approach suggested by conventional battery wisdom4,8,9,26. This work highlights the utility of combining CLO with inexpensive early outcome predictors to accelerate scientific discovery.  \n\nCLO with early outcome prediction is depicted schematically in Fig. 1. The system consists of three components: parallel battery cycling, an early predictor for cycle life and a BO algorithm. At each sequential round, we iterate over these three components. The first component is a multi-channel battery cycler; the cycler used in this work tests 48 batteries simultaneously. Before starting CLO, the charging protocols for the first round of 48 batteries are chosen at random (without replacement) from the complete set of 224 unique multi-step protocols (Methods). Each battery undergoes repeated charging and discharging for 100 cycles (about 4 days; average predicted cycle life 905 cycles), beyond which the experiments are terminated.  \n\nThese cycling data are then fed as input to the early outcome predictor, which estimates the final cycle lives of the batteries given data from the first 100 cycles. The early predictor is a linear model trained via elastic net regression27 on features extracted from the charging data of the first 100 cycles (Supplementary Table 1), similar to that presented in Severson et al.5. Predictive features include transformations of both differences between voltage curves and discharge capacity fade trends. To train the early predictor, we require a training dataset of batteries cycled to failure. Here, we used a pre-existing dataset of 41 batteries cycled to failure (cross-validation root-mean-square error, 80.4 cycles; see Methods and Supplementary Discussion 1). Whereas obtaining this dataset itself requires running full cycling experiments for a small training set of batteries (the cost we are trying to offset), this one-time cost could be avoided if pretrained predictors or previously collected datasets are available. If unavailable, we pay an upfront cost in collecting this dataset; this dataset could also be used for warm-starting the BO algorithm. The size of the dataset collected should best tradeoff the upfront cost in acquiring the dataset to train an accurate model with the anticipated reduction in experimentation requirements for CLO.  \n\nFinally, these predicted cycle lives from early-cycle data are fed into the BO algorithm (Methods and Supplementary Discussion 2), which recommends the next round of 48 charging protocols that best balance the exploration–exploitation tradeoff. This algorithm (Methods and Supplementary Discussion 2) builds on the prior work of Hoffman et al.10 and Grover et al.11. The algorithm maintains an estimate of both the average cycle life and the uncertainty bounds for each protocol; these estimates are initially equal for all protocols and are refined as additional data are collected. Crucially, to reduce the total optimization cost, our algorithm performs these updates using estimates from the early outcome predictor instead of using the actual cycle lives. The mean and uncertainty estimates for the cycle lives are obtained via a Gaussian process (Methods), which has a smoothing effect and allows for updating the cycle life estimates of untested protocols with the predictions from related protocols. The closed-loop process repeats until the optimization budget, in our case 192 batteries tested (100 cycles each), is exhausted.  \n\n![](images/52e2ebbd06b58469921ab0dda55b3ebd80dd818325678b16075ce759ce79d277.jpg)  \nFig. 2 | Structure of our six-step, ten-minute fast-charging protocols. Currents are defined as dimensionless C rates; here, 1C is 1.1 A, or the current required to fully (dis)charge the nominal capacity (1.1 A h) in 1 h. a, Current versus SOC for an example charging protocol, 7.0C–4.8C–5.2C–3.45C (bold lines). Each charging protocol is defined by five constant current (CC) steps followed by one constant voltage (CV) step. The last two steps (CC5 and CV1) are identical for all charging protocols. We optimize over the first four constant-current steps, denoted CC1, CC2, CC3 and CC4. Each of these steps comprises a $20\\%$ SOC window, such that CC1 ranges from $0\\%$ to $20\\%$ SOC, CC2   \nranges from $20\\%$ to $40\\%500$ , and so on. CC4 is constrained by specifying that all protocols charge in the same total time $(10\\mathrm{{min})}$ from $0\\%$ to $80\\%500$ . Thus, our parameter space consists of unique combinations of the three free parameters CC1, CC2 and CC3. For each step, we specify a range of acceptable values; the upper limit is monotonically decreasing with increasing SOC to avoid the upper cutoff potential (3.6 V for all steps). b, CC4 (colour scale) as a function of CC1, CC2 and CC3 (on the x, y and z axes, respectively). Each point represents a unique charging protocol.  \n\nOur objective is to find the charging protocol which maximizes the expected battery cycle life for a fixed charging time (ten minutes) and state-of-charge (SOC) range (0 to $80\\%$ ). The design space of our 224 sixstep extreme fast-charging protocols is presented in Fig. 2a. Multi-step charging protocols, in which a series of different constant-current steps are applied within a single charge, are considered advantageous over single-step charging for maximizing cycle life during fast charging4,8, though the optimal combination remains unclear. As shown in Fig. 2b, each protocol is specified by three independent parameters (CC1, CC2 and CC3); each parameter is a current applied over a fixed SOC range $(0{-}20\\%,20{-}40\\%$ and $40\\text{\\textperthousand}$ , respectively). A fourth parameter, CC4, is dependent on CC1, CC2, CC3 and the charging time. Given constraints on the current values (Methods), a total of 224 charging protocols are permitted. We test commercial lithium iron phosphate (LFP)/graphite cylindrical batteries (A123 Systems) in a convective environmental chamber ( $30^{\\circ}\\mathbf{C}$ ambient temperature). A maximum voltage of 3.6 V is imposed. These batteries are designed to fast-charge in 17 min (rate testing data are presented in Extended Data Fig. 2). The cycle life decreases dramatically with faster charging time4,5, motivating this optimization. Since the LFP positive electrode is generally considered to be stable4,5, we select this battery chemistry to isolate the effects of extreme fast charging on graphite, which is universally employed in lithium-ion batteries.  \n\nIn all, we ran four CLO rounds sequentially, consisting of 185 batteries in total (excluding seven batteries; see Methods). Using early prediction, each CLO round requires four days to complete 100 cycles, resulting in a total testing time of sixteen days—a major reduction from the 560 days required to test each charging protocol to failure three times. Figure 3 presents the predictions and selected protocols (Fig. 3a), as well as the evolution of cycle life estimates over the parameter space as the optimization progresses (Fig. 3a). Initially, the estimated cycle lives for all protocols are equal. After two rounds, the overall structure of the parameter space (that is, the dependence of cycle life on charging protocol parameters CC1, CC2 and CC3) emerges, and a prominent region with high cycle life protocols has been identified. The confidence of CLO in this high-performing region is further improved from round 2 to round 4, but overall the cycle life estimates do not change substantially (Extended Data Fig. 3). By learning and exploiting the structure of the parameter space, we avoid evaluating charging protocols with low estimated cycle life and concentrate more resources on the highperforming region (Extended Data Figs. 3–5). Specifically, 117 of 224 protocols are never tested (Fig. 3c); we spend $67\\%$ of the batteries testing $21\\%$ of the protocols (0.83 batteries per protocol on average). CLO repeatedly tests several protocols with high estimated cycle life to decrease uncertainties due to manufacturing variability and the error introduced by early outcome prediction. The uncertainty is expressed as the prediction intervals of the posterior predictive distribution over cycle life (Extended Data Figs. 3g, 4, 5).  \n\nTo the best of our knowledge, this work presents the largest known map of cycle life as a function of charging conditions (Extended Data Fig. 5). This dataset can be used to validate physics-based models of battery degradation. Most fast-charging protocols proposed in the battery literature suggest that current steps decreasing monotonically as a function of SOC are optimal to avoid lithium plating on graphite, a well-accepted degradation mode during fast charging4,8,9,26. In contrast, the protocols identified as optimal by CLO (for example, Fig. 3d) are generally similar to single-step constant-current charging (that is, $\\mathbf{CC1}\\approx\\mathbf{CC}2\\approx\\mathbf{CC}3\\approx\\mathbf{CC}4)$ . Specifically, of the 75 protocols with the highest estimated cycle lives, only ten are monotonically decreasing (that is, $\\mathbf{CC}_{i}{\\geq}\\mathbf{CC}_{i+1}$ for all i) and two are strictly decreasing (that is, $\\mathbf{CC}_{i}>$ $\\mathbf{CC}_{i+1})$ . We speculate that minimizing parasitic reactions caused by heat generation may be the operative optimization strategy for these cells, as opposed to minimizing the propensity for lithium plating (Supplementary Discussion 3). While the optimal protocol for a new scenario would depend on the selected charge time, SOC window, temperature control conditions and battery chemistry, this unexpected result highlights the need for data-driven approaches for optimizing fast charging.  \n\n# Article  \n\n![](images/57adaf43064c50291962c705101f4a157cdf383cc528afb5d4d74e7c893f9ea5.jpg)  \nFig. 3 | Results of closed-loop experiments. a, Early cycle life predictions per round. The tested charging protocols and the resulting predictions are plotted for rounds 1–4. Each point represents a charging protocol, defined by CC1, CC2 and CC3 (the x, y and z axes, respectively). The colour scale represents cycle life predictions from the early outcome prediction model. The charging protocols in the first round of testing are randomly selected. As the BO algorithm shifts from exploration to exploitation, the charging protocols selected for testing by the closed loop in subsequent rounds fall primarily into the high-performing region. b, Evolution of the parameter space per round. The colour scale represents cycle life, as estimated by the BO algorithm. The initial cycle life   \nestimates are equivalent for all protocols; as more predictions are generated, the BO algorithm updates its cycle life estimates. The CLO-estimated mean cycle lives after four rounds for all fast-charging protocols in the parameter space are also presented in Extended Data Fig. 5 and Supplementary Table 3. c, Distribution of the number of repetitions for each charging protocol (excluding failed batteries). Only 46 of 224 protocols $(21\\%)$ are tested multiple times. d, Current versus SOC for the top three fast-charging protocols, as estimated by CLO. CC1–CC4 are displayed in the legend. All three protocols have relatively uniform charging (that is, $\\mathbf{CC1}\\approx\\mathbf{CC}2\\approx\\mathbf{CC}3\\approx\\mathbf{CC}4;$ .  \n\nWe validate the performance of CLO with early prediction on a subset of nine extreme fast-charging protocols. For each of these protocols, we cycle five batteries each to failure and use the sample average of the final cycle lives as an estimate of the true lifetimes. We use this validation study to (1) confirm that CLO is able to correctly rank protocols based on cycle life, (2) compare the cycle lives of protocols recommended by CLO to protocols inspired by the battery literature and (3) compare the performance of CLO to baseline ablation approaches for experimental design. The charging protocols used in validation, some of which are inspired by existing battery fast-charging literature (see Methods), span the range of estimated cycle lives (Extended Data Fig. 6 and Extended Data Table 1). We adjust the voltage limits and charging times of these literature protocols to match our protocols, while maintaining similar current ratios as a function of SOC. Whereas the literature protocols used in these validation experiments are generally designed for batteries with high-voltage positive electrode chemistries, fast-charging optimization strategies generally focus on the graphitic negative electrode4,8. For these nine protocols, we validate the ‘CLO-estimated’ cycle lives against the sample average of the five final cycle lives.  \n\n![](images/73f69396a8d137dd4c4a7a88dbcde7d660aca35299297c1351566be6795e6ad4.jpg)  \nFig. 4 | Results of validation experiment. a, Discharge capacity versus cycle number for all batteries in the validation experiment. The nine validation protocols include the top three protocols as estimated by CLO (‘CLO top 3’), four protocols inspired by the battery literature39–44 (‘Literature-inspired’) and two protocols selected to obtain a representative sampling from the distribution of CLO-estimated cycle lives among the validation protocols (‘Other’). b, Comparison of early-predicted cycle lives from validation to closed-loop estimates, averaged on a protocol basis. Each ten-minute charging protocol is tested with five batteries. Error bars represent the $95\\%$ confidence intervals. c, Observed versus early-predicted cycle life for the validation experiment. Although our early predictor tends to overestimate cycle life,   \nprobably owing to calendar ageing effects (Supplementary Discussion 4), the trend is correctly captured (Pearson correlation coefficient $r{=}0.86$ . d, Final cycle lives from validation, sorted by CLO ranking. The length of each bar and the annotations represents the mean final cycle life from validation per protocol. Error bars represent the $95\\%$ confidence intervals. e, Ablation study of various optimization approaches using the protocols and data in the validation set (Methods). Error bars represent the $95\\%$ confidence intervals $(n=2,000,$ ). With contributions from both early prediction and Bayesian optimization, CLO can rapidly identify high-performing charging protocols. The gains from Bayesian optimization are larger when resources are constrained (Extended Data Fig. 8).  \n\nThe validation results are presented in Fig. 4. The discharge capacity fade curves (Fig. 4a) exhibit the nonlinear decay typical of fast charg$\\mathrm{ing}^{5,7}$ . If we apply our early-prediction model to the batteries in the validation experiment, these early predictions (averaged over each protocol) match the CLO-estimated mean cycle lives well (Pearson correlation coefficient $r=0.93$ ; Fig. 4b). This result validates the performance of the BO component of CLO in particular, since the CLOestimated cycle lives were inferred from early predictions. However, our early-prediction model exhibits some bias (Fig. 4c), probably owing to calendar ageing effects from different battery storage times28 (Supplementary Table 2 and Supplementary Discussion 4). Despite this bias in our predictive model, we generally capture the ranking well (Kendall rank correlation coefficient, 0.83; Fig. 4d and Extended Data Fig. 7). At the same time, we note that the final cycle lives for the top-ranked protocols are similar. Furthermore, the optimal protocols identified by CLO outperform protocols inspired by previously published fastcharging protocols (895 versus 728 cycles on average; Extended Data Fig. 6 and Extended Data Table 1). This result suggests that the efficiency of our approach does not come at the expense of accuracy.  \n\nOur method greatly reduces the optimization time required compared to baseline optimization approaches (Fig. 4e). For instance, a procedure that does not use early outcome prediction and simply selects protocols randomly to test begins to saturate at a competitive performance level after about 7,700 battery-hours of testing. To achieve a similar level of performance, CLO with both early outcome prediction and the BO algorithm requires only 500 battery-hours of testing. For this small-scale validation experiment, we observe that the early-prediction component of CLO greatly reduces the time per experiment. Here, random selection is equivalent to a pure exploration strategy and can achieve a performance similar to the BO-based approaches for smaller experimental budgets. In later stages, random selection is eventually outperformed by BO-based approaches, which exploit the structure across the protocols and focus on reducing the uncertainty in the promising regions of the parameter space. Although these results are specific to this validation study, we observe similar or larger gains in simulations when fewer batteries or fewer parallel experiments (relative to the size of the parameter space) are available (Extended Data Fig. 8). The relative gains from BO over random selection are largest with minimal resources.  \n\nFinally, we compare our early predictor with other low-fidelity predictors proposed in state-of-the-art multi-fidelity optimization algorithms  \n\n# Article  \n\nin the literature19,20, and find that our approach outperforms these algorithms (Supplementary Discussion 2 and Supplementary Table 4). The generic early-prediction models in these previous works fit composites of parametric functions to the capacity fade curves, while our model uses additional features recorded at every cycle (for example, voltage). This result highlights the value of designing predictive models for the target application in multi-fidelity optimization.  \n\nIn summary, we have successfully accelerated the optimization of extreme fast charging for lithium-ion batteries using CLO with early outcome prediction. This method could extend to other fast-charging design spaces, such as pulsed26,28 and constant-power8 charging, as well as to other objectives, such as slower charging and calendar ageing. Additionally, this work opens up new applications for battery optimization, such as formation29, adaptive cycling30 and parameter estimation for battery management system models31. Furthermore, provided that a suitable early outcome predictor exists, this method could also be applied to optimize other aspects of battery development, such as electrode materials and electrolyte chemistries. 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Phys. 148, 241733 (2018).  \n\nPublisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.  \n\n# Methods  \n\n# Experimental  \n\nCommercial high-power lithium iron phosphate (LFP)/graphite A123 APR18650M1A cylindrical cells were used in this work (packing date 2015-09-26, lot number EL1508007-R). These cells have a nominal capacity of 1.1 A h and a nominal voltage of 3.3 V. All currents are defined in units of C rate; here, 1C is 1.1 A, or the current required to fully (dis) charge the nominal capacity (1.1 A h) in 1 h. The manufacturer’s recommended fast-charging protocol is 3.6C (3.96 A) CC-CV. The rate capability of these cells is shown in Extended Data Fig. 2. The graphite and LFP electrodes are $40\\upmu\\mathrm{m}$ thick and ${80\\upmu\\mathrm{m}}$ thick, respectively, as quantified via X-ray tomography (Zeiss Xradia 520 Versa).  \n\nThe cells were cycled with various charging protocols but identically discharged. Cells were charged with one of 224 candidate six-step, tenminute charging protocols from $0\\%$ to $80\\%$ SOC, as detailed below. After a five-second rest, all cells then charged from $80\\%$ to $100\\%$ SOC with a 1C CC-CV charging step to $3.6\\mathsf{V}$ and a current cutoff of $\\mathbf{C}/20$ . After another five-second rest, all cells subsequently discharged with a CC-CV discharge at 4C to $2.0\\upnu$ and a current cutoff of C/20. The cells rested for another five seconds before the subsequent charging step started. The lower and upper cutoff voltages were 2.0 V and $3.6\\mathsf{V}.$ respectively, as recommended by the manufacturer. In this work, cycle life is defined as the number of cycles until the discharge capacity falls below $80\\%$ of the nominal capacity.  \n\nAll cells were tested in cylindrical fixtures with 4-point contacts on a 48-channel Arbin Laboratory Battery Testing battery cycler placed in an environmental chamber (Amerex Instruments) at $30^{\\circ}\\mathrm{C}$ . The cycler calibration was validated before the state of the experiment.  \n\nIn the closed-loop experiment, four experiments did not reach 100 cycles owing to contact issues either at the start or partially through the experiment. These experiments were run on channels 17 and 27 in round 1 (oed_0) and channels 4 and 5 in round 2 (oed_1). Additionally, in each round, one protocol per round that should have been selected (that is, with a top-48 upper bound) was not selected and replaced with the protocol with the 49th-highest upper bound owing to a processing error (Extended Data Fig. 4), but this error is not expected to have a large effect. Additional experimental issues are documented in the notes of the data repository.  \n\n# Charging protocol and parameter space design  \n\nCells were charged with one of 224 different four-step charging protocols. Each of the first four steps is a single constant-current step applied over a $20\\%$ SOC range; thus, the 224 charging protocols represent different combinations of current steps within the $0\\%$ to $80\\%$ SOC range. We can define the charging time from $0\\%$ to $80\\%$ SOC by:  \n\n$$\nt_{0-80\\%}={\\frac{0.2}{\\mathrm{CC1}}}+{\\frac{0.2}{\\mathrm{CC2}}}+{\\frac{0.2}{\\mathrm{CC3}}}+{\\frac{0.2}{\\mathrm{CC4}}}\n$$  \n\nIn all protocols considered here, we constrain $t_{0\\cdot80\\%}$ to be 10 min. We now write CC4 as a function of the first three charging steps, as:  \n\n$$\n\\mathsf C{C4}=\\frac{0.2}{\\frac{10}{60}-\\left(\\frac{0.2}{\\mathsf C\\mathsf C1}+\\frac{0.2}{\\mathsf C\\mathsf C2}+\\frac{0.2}{\\mathsf C\\mathsf C3}\\right)}\n$$  \n\nThus, each protocol can be uniquely defined by CC1, CC2 and CC3.  \n\nEach independent parameter can take on one of the following discrete values: 3.6C, 4.0C, 4.4C, 4.8C, 5.2C and 5.6C. Furthermore, CC1 can take on values of 6.0C, 7.0C and 8.0C, and CC2 can take on values of 6.0C and 7.0C. CC4 is not allowed to exceed 4.81C. The maximum allowable current for each parameter decreases with increasing SOC to avoid reaching the upper cutoff voltage of 3.6 V. With these constraints, a total of 224 charging protocols are permitted.  \n\nFor a consistent protocol nomenclature, we define each fast-charging protocol as CC1-CC2-CC3-CC4. For example, the charging protocol with the highest CLO-estimated mean cycle life is written 4.8C-5.2C5.2C-4.160C.  \n\n# Early outcome predictor  \n\nThe early outcome predictor for cycle life is similar to that presented in Severson et al.5. This linear model predicts the final $\\mathbf{log}_{10}$ cycle life (number of cycles to reach $80\\%$ of nominal capacity, or 0.88 A h) using features from the first 100 cycles. The training set is identical to the one used in Severson et al.5 and consists of 41 batteries. The linear model takes the form:  \n\n$$\n\\widehat{y_{i}}=\\widehat{\\mathbf{w}}^{\\mathrm{T}}\\mathbf{x}_{i}\n$$  \n\nHere $\\widehat{y_{i}}$ is the predicted cycle life for battery $i,\\pmb{x}_{i}$ is a $p$ -dimensional feature vector for battery i and $\\widehat{\\mathbf{w}}$ is a $p$ -dimensional model coefficient vector. Features are $z$ -scored (mean-subtracted and normalized by the standard deviation) to the training set before model evaluation.  \n\nRegularization, or simultaneous feature selection and model fitting, was performed using the elastic net27. Regularization penalizes overly complex fits to improve both generalizability and interpretability. Specifically, the coefficient vector $\\widehat{\\mathbf{w}}$ is found via the following expression:  \n\n$$\n\\widehat{w}=\\mathrm{argmin}_{w}\\bigg[\\|y-X w\\|_{2}^{2}+\\lambda(\\frac{1-\\alpha}{2}\\|w\\|_{2}^{2}+\\alpha\\|w\\|_{1})\\bigg]\n$$  \n\nHere λ and $\\alpha$ are hyperparameters; λ is a non-negative scalar and $\\alpha$ is a scalar between 0 and 1. The first term minimizes the squared loss, and the second term performs both continuous shrinkage and automatic feature selection. During model development, we apply fourfold crossvalidation and Monte Carlo sampling with the training set to optimize the values of the hyperparameters λ and $\\alpha$ .  \n\nAs in Severson et al.5, the available features were based on the difference between discharge voltage curves of cycles 100 and 10, or trends in the discharge capacity. The five selected features, their corresponding weights and the $z$ -scored values are presented in Supplementary Table 1. The training (cross-validated) error was 80.4 cycles $(10.2\\%)$ ; the test error on a test set from Severson et al.5 was 122 cycles $(12.6\\%)$ .  \n\nThe early predictor automatically flags predictions as anomalous if the $95\\%$ prediction interval exceeds 2,000 cycles. The two-tailed $95\\%$ prediction interval is computed by:  \n\n$$\n95\\%\\mathsf{P l}=2t_{(a/2,n-p)}\\times\\mathsf{R M S E}\\sqrt{1+x_{i}^{\\top}(X^{\\top}X)^{-1}x_{i}}\n$$  \n\nwhere $t$ is the Student’s $t$ value, $\\alpha$ is the significance level (0.05 for $95\\%$ confidence), $n$ is the number of samples, $p$ is the number of features, RMSE is the root-mean-square error of the training set (in units of cycles), $x_{i}$ is the vector of selected features for battery i and $\\chi$ is the matrix of selected features for all observations in the training set.  \n\nIn the closed-loop experiment, three tests returned predictions with a prediction interval outside of the threshold; these anomalous predictions were excluded. These tests were run on channel 27 in round 1 (oed_0), channel 12 in round 3 (oed_2) and channel 6 in round 4 (oed_3). Furthermore, in the validation experiment, one test returned a prediction with a prediction interval outside of the threshold (channel 12; 3.6C-6.0C-5.6C-4.755C), although the final cycle life was reasonable.  \n\nWe note that the predictions from this model exhibited systematic bias for the cells in the validation experiments, which we attribute to the increased calendar ageing of these cells relative to the training set (Supplementary Table 2 and Supplementary Discussion 4).  \n\n# Article  \n\n# Bayesian optimization algorithm  \n\nTo perform optimal experimental design, we consider the setting of best-arm identification using multi-armed bandits. Here each arm is a charging protocol and the goal is to identify the best arm, or equivalently the charging protocol with the highest expected cycle life. Many variants of the problem have been studied in prior work33–35; our algorithm builds on the approaches of Hoffman et al.10 and Grover et al.11. We consider further modifications in Supplementary Discussion 2.  \n\nIn particular, we assume a Bayesian regression setting, where there exists an unknown set of parameters $(\\theta\\in R^{d})$ that relate a charging protocol $x$ to its cycle life (a scalar) via a Gaussian likelihood function. Here, $x$ denotes the CC1, CC2, CC3 configurations of a charging protocol, which is projected onto a $d$ -dimensional feature vector $\\varphi(x)$ . We set $d=224$ , and the feature representations $\\varphi(x)$ are obtained by approximating a radial-basis function kernel, $K(x_{i},x_{j})=\\exp(\\gamma\\vert\\vert x_{i}-x_{j}\\vert\\vert_{2}^{2})$ , using Nystroem’s method. Here, $x_{i}$ and $x_{j}$ are the CC1, CC2 and CC3 configurtions for two arbitrary charging protocols and the inverse of the kernel bandwidth, $\\gamma{>}0$ is treated as a hyperparameter.  \n\nThe Gaussian likelihood function relates a charging protocol to its cycle life distribution. For a protocol $x_{i}$ , the mean of this likelihood function is given as $\\theta^{\\mathrm{{r}}}\\varphi(x)$ . The variance of this likelihood function is the sum of two uncertainty terms, both of which we assume to be homoskedastic (that is, uniform across all protocols). The first term is the empirical variance averaged across the repeated runs of individual protocols present in the training dataset (same as that used for training the early predictor). This accounts for variability due to exogenous factors such as manufacturing. Second, since we do not wait for an experiment to complete, the likelihood variance additionally needs to accommodate an additional uncertainty term due to the early outcome prediction component of the pipeline. We do so by computing the residual variance of the early predictions on the held-out portion of the dataset and set the aforementioned uncertainty term to be the maximum of the residual variances. We assume that the two sources of uncertainty are independent, and hence the overall variance of the likelihood distribution is given by the sum of the squares of both variance terms described above.  \n\nTo perform inference over the unknown parameters $\\theta$ and subsequent predictions of cycle lives, we employ a Gaussian process. In a Gaussian process, the prior over $\\theta$ is assumed to be isotropic Gaussian; such a prior is conjugate to the Gaussian likelihood, and as a consequence the Gaussian posterior can be obtained in closed-form via the Bayes rule. This posterior is used to define a Gaussian predictive distribution over the cycle life for any given charging protocol with mean $\\mu$ and variance $\\sigma^{2}$ .  \n\nFinally, to select a charging protocol, we optimize an acquisition function based on upper confidence bounds. The acquisition function selects protocols where the noisy predictive distribution over cycle life has high mean $\\mu$ (to encourage exploitation) and high variance $\\sigma^{2}$ (to encourage exploration). The mean and upper and lower confidence bounds for any arm i is given by $\\mu_{k,i}\\pm\\beta_{k}\\sigma_{k,i}$ at round $k$ , such that the relative weighting of the two terms is controlled by the exploration tradeoff hyperparameter, $\\beta>0$ . The exploration tradeoff hyperparameter at round $k,\\beta_{k},$ is decayed multiplicatively at every round of the closed loop by another hyperparameter, $\\varepsilon\\in(0,1]$ , as given by ${\\beta_{k}}\\mathrm{{=}}{\\beta_{0}}{\\varepsilon^{k}}$ .  \n\n# BO hyperparameter optimization  \n\nThe BO algorithm relies on eight hyperparameters, each of which can be categorized as either a resource hyperparameter, a parameter space hyperparameter or an algorithm hyperparameter. We note that the BO algorithm runs in the fixed-budget setting; here, the budget refers to the number of iterations of the closed loop we run, excluding validation experiments. We describe each category of hyperparameters below; the values of each hyperparameter are tabulated in Supplementary Table 5.  \n\nResource hyperparameters are specified by the available testing resources. The ‘batch size’ represents the number of parallel tests. We set a batch size of 48 given our 48-channel battery cycler. The ‘budget’ represents the number of batches tested during CLO. The budget excludes batches used to develop the early predictor and validation batches. The budget is typically constrained by either the available testing time or the number of cells. In this case, we set a budget of 4, yielding a cell budget of 192 cells and a time budget of 16 days (4 days per batch of 48 cells tested for 100 cycles).  \n\nParameter space hyperparameters are specified by the optimization problem. Here, we use the same data available from the training set of the early predictor to estimate these parameters, despite a different charging protocol structure. The ‘standardization mean’ represents the estimated mean cycle life across all protocols. The ‘standardization standard deviation’ represents the estimated standard deviation of cycle life across all protocols; in other words, this parameter represents the range of cycle lives in the parameter space. The ‘likelihood standard deviation’ represents the estimated standard deviation of a single protocol tested multiple times, which is a measure of the sampling error; this sampling error includes both the intrinsic variability and the prediction error.  \n\nAlgorithm hyperparameters control the performance of the Bayesian optimization algorithm. γ is the kernel bandwidth, which controls the interaction strength between neighbouring protocols in the parameter space. High γ favours under-smoothing of the parameter space, that is, the protocols have weak relationships with their neighbours. $\\beta_{0}$ represents the initial value of $\\boldsymbol{\\beta}$ , the exploration tradeoff hyperparameter; $\\beta$ controls the balance of exploration versus exploitation. $\\mathsf{H i g h}\\beta_{0}$ favours exploration over exploitation. ε represents the decay constant of beta per round; as the experiment progresses, $\\varepsilon$ shifts towards stronger exploitation (given by $\\beta_{k}=\\beta_{0}\\varepsilon^{k}$ , where $\\beta_{k}$ represents the exploration constant at round $k$ , 0-indexed). High ε favours a rapid transition from exploration to exploitation.  \n\nThe algorithm hyperparameters were estimated by creating a physics-based simulator based on the range of cycle lives obtained in the preliminary batch, testing all hyperparameter combinations on the simulator, and selecting the hyperparameter combination with the best performance (that is, that which most consistently obtains the true cycle life). These results are visualized in Extended Data Fig. 9; we note that the performance of BO is relatively insensitive to the selected combination of algorithm hyperparameters, meaning sufficiently high performance can be achieved even with suboptimal algorithm hyperparameters. Other approaches, such as using the early-predictor training dataset, are also possible for optimization of the algorithm hyperparameters (Supplementary Discussion 1).  \n\n# Physics-based simulator  \n\nWe used a physics-based simulator for hyperparameter optimization; this simulator allows us to estimate the shape and range of cycle lives in the parameter space, although the simulator is not designed to be an accurate representation of battery degradation during fast charging. This finite element simulator was originally designed to estimate the heat generation during charging in an 18650 cylindrical battery by approximating the battery as a long cylinder, which simplifies to a onedimensional radial heat transfer problem. The equations and thermal properties were sourced from Drake et al.36 and Çengel and Boles37. The output from these simulations is a matrix of temperature as a function of both radial position and time. We use total solid-electrolyte interphase (SEI) growth as a proxy for degradation. First, we estimate the temperature dependence of SEI growth from the C/10 series of figure 7 from Smith et al.38 (Supplementary Table 6). Simultaneously, we compute the expected temperature profiles in the battery as a function of charging protocol with respect to time and position. We then approximate the kinetics of SEI growth with an Arrhenius equation, such that SEI growth increases with increasing temperature. SEI growth (in arbitrary units) is calculated for each temperature element in the position-time array via:  \n\n$$\n\\scriptstyle D=\\sum_{r}\\sum_{t}\\exp\\left(-{\\frac{E_{\\mathrm{a}}}{k_{\\mathrm{B}}T}}\\right)\n$$  \n\nwhere $D$ is the degradation parameter, $E_{\\mathrm{a}}$ is the effective activation energy for SEI growth (Supplementary Table 6) and $k_{\\mathrm{{B}}}$ is Boltzmann’s constant. The cycle life is then calculated from the degradation parameter using the range of expected cycle lives (as estimated from the early-predictor training dataset):  \n\n$$\n\\mathrm{Cycle\\life\\=\\500+}C/D\n$$  \n\nwhere $c$ is a constant $(5\\times10^{-11})$ ) that scales $D$ to reasonable values of cycle life.  \n\n# Validation experiments  \n\nAfter the closed-loop experiment completed, we selected nine protocols to test to failure (five batteries per charging protocol). This experiment allowed us to (1) evaluate the performance of the closed loop by comparing the CLO-estimated mean cycle lives to the mean cycle life of multiple batteries tested to failure for multiple protocols, (2) compare the protocols with the highest CLO-estimated mean cycle lives to conventional fast-charging protocol design principles from the battery literature, and (3) generate a small dataset with which we can evaluate the performance of the closed loop relative to baseline optimization approaches.  \n\nThe selected protocols are displayed in Extended Data Fig. 6 and Extended Data Table 1. Of our nine fast-charging protocols, three were the top three CLO-estimated protocols; four were based on approximations of multi-step fast-charging protocols in the battery literature (see Extended Data Table 1); and two were selected to obtain a representative sampling from the distribution of CLO-estimated cycle lives. The four protocols based on approximations of multi-step fast-charging protocols in the battery literature were obtained by determining the current ratios between various steps and translating those ratios to our ten-minute fast-charging space. The voltage limits were consistent with our charging protocols, that is, 2.0 V and 3.6 V.  \n\nFive batteries per charging protocol were tested to obtain a reasonable estimate of the true cycle lives. In this experiment, one test returned a prediction with a prediction interval outside of the threshold (channel 12; 3.6C-6.0C-5.6C-4.755C) and was excluded. A comparison of the three different methods for cycle life results (CLO, early predictions from validation, and final measurements from validation) are presented in Extended Data Fig. 7.  \n\n# Validation ablation study  \n\nFor the ablation study using the charging protocols and data from the validation experiments, we systematically compared the full closedloop system against three other ablation baselines which use (1) only early prediction (no BO exploration–exploitation, purely random exploration), (2) only BO exploration–exploitation (no early prediction), (3) purely random exploration without any early prediction. As highlighted earlier, since the final cycle lives for the protocols in the validation study have a noticeable bias that can be explained by calendar ageing (Supplementary Discussion 4), we perform a simple additive bias correction for each of the final cycle lives beforehand to suppress any undesirable influence of this bias in interpreting the results.  \n\nWe run the four ablation baselines for a varying number of sequential rounds. Since our validation space is relatively small (nine charging protocols, five batteries tested per protocol in our validation dataset), we run only one battery per round (that is, we assume a one-channel battery cycler). The baselines that use BO exploration–exploitation additionally require hyperparameters to be specified before beginning the experiment, as described in the Methods section ‘BO hyperparameter optimization’. The best hyperparameters are chosen separately for each round based on the performance obtained on the physics-based simulator, averaged over 100 random seeds.  \n\nWhen an ablation baseline queries for the cycle life of a given charging protocol, the returned value corresponds to one of the five runs in our validation dataset, chosen via random sampling with replacement (that is, bootstrapped). The experimental time cost of this query is equal to 100 cycles for ablation baselines that use early prediction and equals the full cycle life otherwise. Finally, to account for the randomness at the beginning of the experiment (that is, round 0 when every ablation baseline randomly selects a protocol), we report the performance of each ablation baseline averaged over a sequence of 2,000 randomly initialized experiments. To specify the y-axis of Fig. 4e, we assume that each full cycle (charging, discharging, resting) requires one hour of experimental testing.  \n\n# Overpotential analysis  \n\nTo determine the dependence of overpotential on current and SOC during charging (Extended Data Fig. 2e–f), we perform a pseudo-galvanostatic intermittent titration technique experiment on two minimally cycled batteries and two degraded batteries ( $80\\%$ of nominal capacity remaining). We probe currents ranging from 3.6C to 8C at $20\\%$ , $40\\%$ , $60\\%$ and $80\\%$ SOC, mirroring the current and SOC values used in charging protocol design. In this experiment, we start at an initial SOC $20\\%$ lower than the target, for example, we start at $0\\%$ SOC to probe $20\\%$ SOC. We then charge at a given current rate, for example, 3.6C, until we reach $20\\%$ SOC. The cell rests for 1 h, and then the cell discharges at 1C back to $0\\%500$ . We repeat this sequence for all current values, after which we charge the cell at 1C to the next initial SOC, for example, $20\\%$ SOC to probe $40\\%$ SOC, and repeat for each SOC of interest.  \n\nTo compute the overpotential, we compare the voltage at the start and end of the 1-h rest periods. Nearly all of the potential drop occurs immediately $(<100\\mathrm{m}\\mathsf{s})$ after the start of the rest period. Given the linear trends observed (implying ohmic-limited rate capability), we then perform a linear fit on each overpotential-current series. In these fits, the slope represents the ohmic resistance.  \n\n# Data availability  \n\nThe datasets used in this study are available at https://data.matr.io/1.  \n\n# Code availability  \n\nThe CLO code, data and figures associated with this manuscript are available at https://github.com/chueh-ermon/battery-fast-chargingoptimization. The data processing and early-prediction code are available at https://github.com/chueh-ermon/BMS-autoanalysis. The charging protocol generation code (automated creation of battery cycler tests) is available at https://github.com/chueh-ermon/automateArbin-schedule-file-creation.  \n\n33.\t Shahriari, B., Swersky, K., Wang, Z., Adams, R. P. & de Freitas, N. Taking the human out of the loop: a review of Bayesian optimization. Proc. IEEE 104, 148–175 (2016).   \n34.\t Audibert, J.-Y., Bubeck, S. & Munos, R. Best arm identification in multi-armed bandits. In Proc. 23rd Conf. on Learning Theory (COLT) 41–53 (2010); http://certis.enpc.fr/\\~audibert/ Mes%20articles/COLT10.pdf.   \n35.\t Srinivas, N., Krause, A., Kakade, S. M. & Seeger, M. W. Information-theoretic regret bounds for Gaussian process optimization in the bandit setting. IEEE Trans. Inf. Theory   \n58, 3250–3265 (2012).   \n36.\t Drake, S. J. et al. Measurement of anisotropic thermophysical properties of cylindrical Li-ion cells. J. Power Sources 252, 298–304 (2014).   \n37.\t Çengel, Y. A. & Boles, M. A. Thermodynamics: An Engineering Approach (McGraw-Hill Education, 2015).   \n38.\t Smith, A. J., Burns, J. C., Zhao, X., Xiong, D. & Dahn, J. R. A high precision coulometry study of the SEI growth in Li/graphite cells. J. Electrochem. Soc. 158, A447–A452 (2011).   \n39.\t Zhang, S. S. The effect of the charging protocol on the cycle life of a Li-ion battery. J. Power Sources 161, 1385–1391 (2006).  \n\n# Article  \n\n40.\t Kim, J. M. et al. Battery charging method and battery pack using the same. US Patent Application US20160226270A1 (2016).   \n41.\t Lee, M.-S., Song, S.-B., Jung, J.-S. & Golovanov, D. Battery charging method and battery pack using the same. US Patent US9917458B2 (2018).   \n42.\t Notten, P. H. L., Op het Veld, J. H. G. & van Beek, J. R. G. Boostcharging Li-ion batteries: a challenging new charging concept. J. Power Sources 145, 89–94 (2005).   \n43.\t Paryani, A. Low temperature charging of Li-ion cells. US Patent US8552693B2 (2013).   \n44.\t Mehta, V. H. & Straubel, J. B. Fast charging with negative ramped current profile. US Patent US8643342B2 (2014).  \n\nAcknowledgements This work was supported by the Toyota Research Institute through the Accelerated Materials Design and Discovery programme. P.M.A. was supported by the Thomas V. Jones Stanford Graduate Fellowship and the National Science Foundation Graduate Research Fellowship under grant number DGE-114747. A.G. was supported by a Microsoft Research PhD Fellowship and a Stanford Data Science Scholarship. N.P. was supported by the SAIC Innovation Center through the Stanford Energy 3.0 industry affiliates programme. S.J.H. was supported by the Assistant Secretary for Energy Efficiency, Vehicle Technologies Office of the US Department of Energy under the Advanced Battery Materials Research Program. X-ray tomography was performed at the Stanford Nano Shared Facilities, supported by the National Science Foundation under award ECCS-1542152. We thank A. Anapolsky, L. Attia, C. Cundy, J.  \n\nHirshman, S. Jorgensen, G. McConohy, J. Song, R. Smith, B. Storey and H. Thaman for discussions.  \n\nAuthor contributions P.M.A., N.J., Y.-H.L., M.H.C., N.P. and W.C.C. conceived and conducted the experiments. A.G., T.M.M., B.C. and S.E. developed the Bayesian optimization algorithm and incorporated early outcome predictions into the closed loop. P.M.A. and K.A.S. performed the early-prediction modelling. P.M.A., Z.Y., P.K.H. and M.A. performed data management. P.M.A., A.G., N.J., S.J.H., S.E. and W.C.C. interpreted the results. All authors edited and reviewed the manuscript. R.D.B., S.E. and W.C.C. supervised the work.  \n\nCompeting interests S.E., W.C.C., A.G., T.M.M., N.P. and P.M.A. have filed a patent application related to this work: US Patent Application No. 16/161,790 (16 October 2018).  \n\n# Additional information  \n\nSupplementary information is available for this paper at https://doi.org/10.1038/s41586-020- 1994-5. Correspondence and requests for materials should be addressed to R.D.B., S.E. or W.C.C. Peer review information Nature thanks Marius Bauer, Matthias Seeger and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Reprints and permissions information is available at http://www.nature.com/reprints.  \n\n![](images/557c8aa9a21d61fbae9f05477748cfaec8746bc3a3731aebc1a3aecfff2a259d.jpg)  \n\nExtended Data Fig. 1 | Illustrations of early outcome predictor and BO components of CLO. a, Illustration of early outcome prediction for two cells (A and B) using data from only the first 100 cycles. Two discharge capacity features are generated: the second-cycle discharge capacity, $Q_{\\mathrm{d},2}^{}$ , and the difference between the maximum and second-cycle discharge capacities, $\\boldsymbol{\\mathrm{max}}(Q_{\\mathrm{d}})-Q_{\\mathrm{d},2}$ . Three voltage features are generated: the logarithm of the minimum, variance and the skewness of the difference in voltage curves between cycles 100 and 10. These five features are combined in a linear model to predict the final cycle life, or the number of cycles until the capacity falls below 0.88 A h. The weights and scalings of each feature are determined by training the model on a training set using the elastic net; the weights and scaling values are presented in Supplementary Table 1. See Severson et al.7 and  \n\nMethods for additional details. b, Illustration of the BO principle. The desired output, cycle life, has a true functional dependence on charging protocol parameters (such as CC1). Here, we show a one-dimensional model (that is, just dependent on one parameter, CC1) for simplicity. By performing Gaussian process regression on the available data, we develop a probabilistic estimate of the true function; our goal is for the estimate to match the true function. The next data point selected is that which maximizes the upper confidence bound (UCB), which is selected by either high uncertainty (exploration) or high predicted value (exploitation). Once this point is selected (right panel), the next point selected is, again, that which maximizes the upper confidence bound.  \n\n![](images/186fc6bd659a7db1145d122afcbc5f37e77c09bbf22e2d735547dcb76ff6e7ed.jpg)  \nExtended Data Fig. 2 | Cell characterization. a, b, Voltage versus capacity during rate testing of A123 18650M1A cylindrical cells under charge (a) and discharge (b). The (dis)charge step not under investigation is cycled at 1C to isolate the rate of each step; for example, the charge rate test is performed with 1-C discharge steps. We note that the discharge rate capability is much higher than that of charge. c, d, Battery surface temperature (‘can temperature’) versus capacity during rate testing under charge (c) and discharge (d). The can temperature is measured via a type T thermocouple secured with thermal   \nepoxy. e, f, Overpotential as a function of SOC and C rate (see Methods section ‘Overpotential analysis’ for details of the measurement) for a minimally cycled cell (e) and an aged cell at $80\\%$ of nominal capacity (f). The trend lines are linear fits of the overpotential as a function of current at fixed SOC (excluding outliers). We note that both of the relationships are linear (indicating that the rate capability is ohmically limited) and that the SOC dependence is weak, particularly for the minimally cycled cell. The initial internal resistance, averaged over two cells and all four SOCs, is $33\\mathsf{m}\\Omega$ .  \n\n![](images/93d90fea3e445b57de4bf4072360d4aa42d3ebcd20e469f73dfc434605963414.jpg)  \nExtended Data Fig. 3 | Additional optimization results. a, b, Mean of the absolute difference in CLO-estimated cycle lives with increasing rounds, expressed as both percentage change (a) and absolute change (b). These changes are relatively small beyond round 2, suggesting that the closed loop can perform well with even smaller time or battery budgets. c, Change in Kendall rank correlation coefficient with increasing rounds. From round 3 to round 4, the ranking of the top protocols shifts, but the cycle lives of these top protocols are similar. d, Distribution of CLO-estimated mean cycle lives after round 4. The mean and standard deviation are 943 cycles and 126 cycles, respectively. e, Correlation between CLO-estimated mean cycle lives and the sum of squared currents, a simplified measure of heat generation $(P=I^{2}R)$ . This relationship suggests that minimizing heat generation, as opposed to avoiding   \nlithium plating, may be the operative optimization strategy for these cells under these conditions. f, Standard deviation $(\\sigma_{4,i})$ versus mean $(\\mu_{4,i})$ of the BO predictive distribution over cycle life after round 4. The standard deviation quantifies the uncertainty in the cycle life estimates and is generally low for protocols estimated to have high mean cycle life, since these protocols are probed more frequently. We start with a relatively wide, flat prior (standard deviation 164) and therefore the uncertainty intervals after four rounds are also wide. g, Mean $\\pm$ standard deviation of the predictive distribution over cycle life after round 4 $(\\mu_{4,i}\\pm\\sigma_{4,i})$ for all charging protocols, sorted by their rank after round 4. The legend indicates the number of repetitions for each protocol (excluding failed batteries).  \n\n![](images/222a20753cccbe00023eed36b07269bd8cbff88a824b66c1dc7c3c90bfaf72db.jpg)  \n\nExtended Data Fig. 4 | See next page for caption.  \n\nExtended Data Fig. 4 | Means and upper/lower confidence bounds $(\\pmb{\\mu}_{k,i}\\pmb{\\beta}_{k}\\pmb{\\sigma}_{k,i})$ on cycle life per round k. Protocol indices on the x-axis are sorted by rank after round 4. The weighted interval around the estimated mean, $\\beta_{k}\\sigma_{k,i}\\mathrm{=}(\\beta_{0}\\varepsilon^{k})\\sigma_{k,i},$ weights the protocol-specific standard deviation at round $k$ , $\\sigma_{k,i}$ (estimated by the Gaussian process model) with the exploration tradeoff hyperparameter at round $k,\\beta_{k}$ . The upper and lower confidence bounds are plotted for all charging protocols before round 1 (a) and after rounds 1 (b), 2 (c), 3 (d) and 4 (e). The predictive distributions for all charging protocols have identical means and standard deviations before the first round of testing. Because the standard deviations are weighted by $\\beta_{k}{=}\\beta_{0}\\varepsilon^{k}$ and $\\scriptstyle{\\varepsilon=0.5}$ , the  \n\nweighted confidence bounds rapidly decrease with increasing round number, favouring exploitation (examination of protocols with high means). The BO algorithm recommends the 48 protocols with the highest upper bounds (red points); the upper bounds are high either due to high uncertainty (exploration) or high means (exploitation). The algorithm rapidly shifts from exploration to exploitation as $\\scriptstyle{\\varepsilon_{k}}$ rapidly shrinks the upper bounds with increasing round index. We note that one protocol per round that should have been selected (that is, with a top-48 upper bound) was not selected owing to a processing error; instead, the protocol with the 49th-highest upper bound was selected.  \n\n![](images/90d8cce5d47ad87a047f9db8ac8c6926b6080f3cbd415daea8acae788e9f9d87.jpg)  \nExtended Data Fig. 5 | Mean and standard deviation of the CLO-estimated $\\mathsf{C C3}=3.6\\mathsf{C}$ , 4.0C, 4.4C, 4.8C, 5.2C, 5.6C and 6.0C, respectively. CC4 is predicted distribution over cycle lives after round 4. In this two-dimensional represented by the contour lines. Note that the protocols with the highest cycle representation, mean estimated cycle life (colour scale) and standard deviation lives generally have the smallest standard deviations, since these protocols of cycle life (marker size) after round 4 are presented as a function of CC1, CC2 have been tested repeatedly. and CC3 (the x axis, y axis and panels a–f, respectively). Panels a–f represent  \n\n![](images/a33d5d83462f16cf5c5f50c8b18c453daff0270a11b9fe21bdb04b7574cfcc80.jpg)  \nExtended Data Fig. 6 | Selected protocols for validation. The three protocols with the highest CLO-estimated mean cycle lives are shown in panels b, c and d. The protocols shown in panels a, f, g and h are approximations of previously proposed battery fast-charging protocols (Extended Data Table 1). The remaining two protocols, shown in panels e and i, were selected to obtain a representative sampling from the entire distribution of CLO-estimated cycle lives. The annotations on each panel represent the cycle lives of each protocol   \nas estimated by CLO (‘CLO’), early outcome prediction from validation (‘Early prediction’), and the final cycle lives from validation (‘Final’). In the annotations, the errors represent the CLO-estimated standard deviation after round 4 $(\\sigma_{k,4})$ for the CLO-estimated cycle lives and the $95\\%$ confidence intervals for the early-predicted and final cycle lives from validation $\\scriptstyle(n=5;n=4$ for the early predictions of 3.6C-6.0C-5.6C-4.755C) (a).  \n\n![](images/d97748526ed42451da7be97b085bf855a44740217d5ee266f676e14a345196d1.jpg)  \n\nExtended Data Fig. 7 | Validation ablation analysis. We perform pairwise comparisons of the cycle lives of the nine validation protocols, as estimated from three sources: closed-loop estimates after four rounds, early predictions from the validation experiment and final cycle lives from the validation experiment. Panels a–c compare closed-loop estimates to early predictions from validation, panels d–f compare final cycle lives from validation to early predictions from validation, and panels g–i compare final cycle lives from validation to closed-loop estimates. The first column (a, d and g) compares cycle lives averaged on a protocol basis; the second column (b, e and h)  \n\ncompares cycle lives on a battery (cell) basis; and the third column (c, f and i) compares the predicted ranking by cycle life via each method. Orange points represent the top three CLO-estimated protocols, blue points represent protocols inspired by the battery literature (Methods), and green points represent protocols selected to sample the distribution of estimated cycle lives. The error bars represent the $95\\%$ confidence intervals $(n=5;n=4$ for the early predictions of 3.6C-6.0C-5.6C-4.755C). The Pearson correlation coefficient and Kendall rank correlation coefficients are displayed for all relevant cycle life and ranking plots, respectively.  \n\n![](images/bfd9bf3550e5661e183903ae566ee71c65475e9d37a050671c9a5a71e38e3830.jpg)  \n\nExtended Data Fig. 8 | Closed-loop performance under resource constraints. Comparison of the closed loop with and without the Bayesian optimization algorithm (that is, with and without the explore/exploit component) as a function of number of channels and number of rounds in the 224-protocol space, using the first-principles simulator as the ground-truth source for cycle lives. Early prediction is not included. Each point represents the mean of 100 simulations; error bars represent the $95\\%$ confidence intervals $(n=100)$ ). Early prediction is not incorporated into these simulations. The complete closed loop (that is, with Bayesian optimization) consistently outperforms the closed loop without Bayesian optimization. Bayesian optimization offers the largest advantage when the number of channels is low relative to the number of protocols.  \n\n# Article  \n\n![](images/d62d259a02f63e1715b3d01d93d1c670641bddc07b134750f3e640d5bf509d58.jpg)  \nExtended Data Fig. 9 | Hyperparameter sensitivity analysis on a cycle life simulator. The true cycle life of the best charging protocol as estimated by CLO, averaged over ten random seeds, is plotted as a function of the initial exploration constant $(\\beta_{0})$ , the exploration decay factor (ε) and the kernel bandwidth $(\\gamma)$ . The values of all other hyperparameters are consistent with the values indicated in the ‘BO hyperparameter optimization’ Methods section and  \n\nin Supplementary Table 5. Overall, CLO achieves acceptable performance over a range of hyperparameter combinations; the highest-cycle-life protocols as estimated by the best and worst hyperparameter combinations differ by only 60 cycles. In our real-world CLO experiment, the selected hyperparameters are $\\beta_{0}=5.0,\\varepsilon=0.5$ and $\\gamma=1_{\\AA}$ ; this combination performed well on a variety of simulated parameter spaces and budgets.  \n\nExtended Data Table 1 | Selected charging protocols for validation   \n\n\n<html><body><table><tr><td>Charging protocol</td><td>CLO-estimated cycle life</td><td>Early-predicted cyl life from</td><td>Final cycle life valfration)</td><td>Source</td></tr><tr><td>3.6C-6.0C-5.6C- 4.755C</td><td>1103 ± 131</td><td>1013 ± 115</td><td>755 ±81</td><td> Zhang39</td></tr><tr><td>4.4C-5.6C-5.2C- 4.252C</td><td>1174 ± 76</td><td>1056 ± 127</td><td>884± 132</td><td>Protocol with third-highest CLO-estimated mean cycle life</td></tr><tr><td>4.8C-5.2C-5.2C- 4.160C</td><td>1185 ± 78</td><td>1047 ± 49</td><td>890 ± 90</td><td>Protocol with highest CLO- estimated mean cycle life</td></tr><tr><td>5.2C-5.2C-4.8C- 4.160C</td><td>1183 ± 86</td><td>1098 ± 134</td><td>912 ±118</td><td>Protocol with second-highest CLO-estimated mean cycle life</td></tr><tr><td>6.0C-5.6C-4.4C- 3.834C</td><td>954 ± 164</td><td>963 ± 26</td><td>880±85</td><td></td></tr><tr><td>7.0C-4.8C-4.8C- 3.652C</td><td>876 ± 183</td><td>964± 43</td><td>870 ± 70</td><td>Samsung patents40,41</td></tr><tr><td>8.0C-4.4C-4.4C- 3.940C</td><td>818± 212</td><td>854 ± 44</td><td>702 ± 51</td><td>Notten et al.42</td></tr><tr><td>8.0C-6.0C-4.8C- 3.000C</td><td>775 ±273</td><td>698 ± 40</td><td>584± 60</td><td>Tesla patents43,44</td></tr><tr><td>8.0C-7.0C-5.2C- 2.680C</td><td>648 ± 174</td><td>580± 68</td><td>496 ± 49</td><td></td></tr></table></body></html>  \n\nThe columns represent the CLO-estimated mean cycle lives of each protocol, early predictions in the validation experiment and the final tested cycle lives. For the CLO-estimated cycle lives, the errors represent the CLO-estimated standard deviation after round 4 $(\\sigma_{k,4})$ . For the early-predicted and final cycle lives from validation, the errors represent $95\\%$ confidence intervals $(n=5;$ but $n=4$ for the early predictions of 3.6C-6.0C-5.6C-4.755C). The two protocols without a source were selected to obtain a representative sampling from the distribution of CLO-estimated cycle lives. Literature fast-charging protocols are from refs. 39–44.  ",
    "Target": null
  },
  {
    "id": "10.1021_jacs.3c05819",
    "DOI": "10.1021/jacs.3c05819",
    "DOI Link": "http://dx.doi.org/10.1021/jacs.3c05819",
    "Relative Dir Path": "mds/10.1021_jacs.3c05819",
    "Article Title": "ChatGPT Chemistry Assistant for Text Mining and the Prediction of MOF Synthesis",
    "Authors": "Zheng, ZL; Zhang, OF; Borgs, C; Chayes, JT; Yaghi, OM",
    "Source Title": "JOURNAL OF THE AMERICAN CHEMICAL SOCIETY",
    "Abstract": "We use prompt engineering to guide ChatGPT in the automationoftext mining of metal-organic framework (MOF) synthesis conditionsfrom diverse formats and styles of the scientific literature. Thiseffectively mitigates ChatGPT's tendency to hallucinate information,an issue that previously made the use of large language models (LLMs)in scientific fields challenging. Our approach involves the developmentof a workflow implementing three different processes for text mining,programmed by ChatGPT itself. All of them enable parsing, searching,filtering, classification, summarization, and data unification withdifferent trade-offs among labor, speed, and accuracy. We deploy thissystem to extract 26 257 distinct synthesis parameters pertainingto approximately 800 MOFs sourced from peer-reviewed research articles.This process incorporates our ChemPrompt Engineering strategy to instructChatGPT in text mining, resulting in impressive precision, recall,and F1 scores of 90-99%. Furthermore, with the data set builtby text mining, we constructed a machine-learning model with over87% accuracy in predicting MOF experimental crystallization outcomesand preliminarily identifying important factors in MOF crystallization.We also developed a reliable data-grounded MOF chatbot to answer questionsabout chemical reactions and synthesis procedures. Given that theprocess of using ChatGPT reliably mines and tabulates diverse MOFsynthesis information in a unified format while using only narrativelanguage requiring no coding expertise, we anticipate that our ChatGPTChemistry Assistant will be very useful across various other chemistrysubdisciplines.",
    "Times Cited, WoS Core": 170,
    "Times Cited, All Databases": 176,
    "Publication Year": 2023,
    "Research Areas": "Chemistry",
    "UT (Unique WOS ID)": "WOS:001052894400001",
    "Markdown": "# ChatGPT Chemistry Assistant for Text Mining and Prediction of MOF Synthesis  \n\nZhiling Zheng,†,‡,§ Oufan Zhang,† Christian Borgs,§,◊ Jennifer T. Chayes, §,◊,††,‡‡,§§ Omar M. Yaghi†,‡,§,∥,\\*  \n\n† Department of Chemistry, University of California, Berkeley, California 94720, United States   \n‡ Kavli Energy Nanoscience Institute, University of California, Berkeley, California 94720, United States   \n§ Bakar Institute of Digital Materials for the Planet, College of Computing, Data Science, and Society, University of California, Berkeley, California 94720, United States   \n◊ Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720, United States   \n†† Department of Mathematics, University of California, Berkeley, California 94720, United States   \n‡‡ Department of Statistics, University of California, Berkeley, California 94720, United States   \n§§ School of Information, University of California, Berkeley, California 94720, United States   \n∥ KACST–UC Berkeley Center of Excellence for Nanomaterials for Clean Energy Applications, King Abdulaziz City for Science and Technology, Riyadh 11442, Saudi Arabia   \nKEYWORDS: ChatGPT, data mining, metal–organic frameworks, synthesis, crystals.  \n\nABSTRACT: We use prompt engineering to guide ChatGPT in the automation of text mining of metal-organic frameworks (MOFs) synthesis conditions from diverse formats and styles of the scientific literature. This effectively mitigates ChatGPT’s tendency to hallucinate information—an issue that previously made the use of Large Language Models (LLMs) in scientific fields challenging. Our approach involves the development of a workflow implementing three different processes for text mining, programmed by ChatGPT itself. All of them enable parsing, searching, filtering, classification, summarization, and data unification with different tradeoffs between labor, speed, and accuracy. We deploy this system to extract 26,257 distinct synthesis parameters pertaining to approximately 800 MOFs sourced from peer-reviewed research articles.  This process incor  \n\nporates our ChemPrompt Engineering strategy to instruct ChatGPT in text mining, resulting in impressive precision, recall, and F1 scores of $90\\mathrm{-}99\\%$ . Furthermore, with the dataset built by text mining, we constructed a machine-learning model with over $87\\%$ accuracy in predicting MOF experimental crystallization outcomes and preliminarily identifying important factors in MOF crystallization. We also developed a reliable data-grounded MOF chatbot to answer questions on chemical reactions and synthesis procedures. Given that the process of using ChatGPT reliably mines and tabulates diverse MOF synthesis information in a unified format, while using only narrative language requiring no coding expertise, we anticipate that our ChatGPT Chemistry Assistant will be very useful across various other chemistry subdisciplines.  \n\n![](images/50d9efbf3a29b65d8edfc58760c594a8660e1409a5c6c0b132d051db3f3452cd.jpg)  \n\n# INTRODUCTION  \n\nThe dream of chemists is to create matter in the hope of advancing human knowledge for the betterment of society.1, 2 As we stand on the precipice of the age of Artificial General Intelligence (AGI), the potential for synergy between AI and chemistry is vast and promising.3, 4 The idea of creating AI-powered chemistry assistants offers unprecedented opportunities to revolutionize the landscape of chemistry research by applying knowledge across various disciplines, efficiently processing laborintensive and time-consuming tasks, such as literature searches, compound screening and data analysis. AI-powered chemistry may ultimately transcend the limits of human cognition.5-8  \n\nIdentifying chemical information for compounds, including ideal synthesis conditions and physical and chemical properties, has been a critical endeavor in chemistry research. The comprehensive summary of chemical information from literature reports, such as publications and patents, and their subsequent storage in an organized database format is the next logical and necessary step toward discovery of materials.9 The challenge lies in efficiently mining the vast amount of available literature to obtain valuable information and insights. Traditionally, specialized natural language processing (NLP) models have been employed to address this issue.10-14 However, these approaches can be labor-intensive and necessitate expertise in coding, computer science, and data science. Furthermore, they are less generalizable, requiring rewriting the program when the target changes. The advent of large language models (LLMs), such as GPT-3, GPT-3.5 and GPT-4, has the potential to fundamentally transform this process and revolutionize the routine of chemistry research in the next decade.9, 15-18  \n\n![](images/1c872d62a3ad0e2f33281e5e0fc90045fbc15100af38f76f558bfec8a8744ae8.jpg)  \nFigure 1. Schematics of ChatGPT Chemistry Assistant workflow having three different processes employing ChatGPT and ChemPrompt for efficient text mining and summarization of MOF synthesis conditions from a diverse set of published research articles. Each process is distinctively labeled with red, blue, and green dots respectively. To illustrate, Process 1 initiates with “Published Research Articles”, proceeds to “Human Preselection”, moves onto the “Synthesis Paragraph”, integrates “ChatGPT with ChemPrompt”, and culminates in “Tabulated Data”. Steps shared among multiple processes are indicated with corresponding color-coded dots. The two-snakes logo of Python is included to indicate the use of the Python programming language, with the logo's credit attributed to the Python Software Foundation (PSF).  \n\nHerein, we demonstrate that LLMs, including ChatGPT based on the GPT-3.5 and GPT-4 model, can act as chemistry assistants to collaborate with human researchers, facilitating text mining and data analysis to accelerate the research process. To harness the power of what we termed as the ChatGPT Chemistry Assistant (CCA), we provide a comprehensive guide on ChatGPT prompt engineering for chemistry-related tasks, making it accessible to researchers regardless of their familiarity with machine learning, thus bridging the gap between chemists and computer scientists. In this report, we present (1) A novel approach to using ChatGPT for text mining the synthesis conditions of metal-organic frameworks (MOFs), which can be easily generalizable to other contexts requiring minimal coding knowledge and operating primarily on verbal instructions. (2) Assessment of ChatGPT's intelligence in literature text mining through accuracy evaluation and its ability for data refinement. (3) Utilization of the chemical synthesis reaction dataset obtained from text mining to train a model capable of predicting reaction results as crystalline powder or single crystals. Furthermore, we demonstrate that the CCA chatbot can be tuned to specialize in answering questions related to MOF synthesis based on literature conditions, with minimal hallucinations. This study underscores the transformative potential of ChatGPT and other LLMs in the realm of chemistry research, offering new avenues for collaboration and accelerating scientific discovery.  \n\n# MATERIALS AND METHODS  \n\nDesign Considerations for ChatGPT-Based Text Mining. In curating research papers for ChatGPT to read and extract information, it is imperative to account for the diversity in MOF synthesis conditions, such as variations in metal sources, linkers, solvents, and equipment, as well as the different writing styles employed. Notably, the absence of a standardized format for reporting MOF synthesis conditions leads to variable reporting templates by research groups and journals. Indeed, by incorporating a broad spectrum of narrative styles, we can examine ChatGPT's robustness in processing information from heterogeneous sources. On the other hand, it is essential to recognize that the challenge of establishing unambiguous criteria to identify MOF compounds in the literature may lead to the inadvertent inclusion of some non-MOF compounds reported in earlier publications that are non-porous inorganic complexes and amorphous coordination polymers (included in some MOF datasets).  As such, maintaining a balance between quality and quantity is vital, and prioritizing the selection of high-quality and well-cited papers, rather than incorporating all associated papers indiscriminately can ensure that the text mining of MOF synthesis conditions yields reliable and accurate data.  \n\nMoreover, papers discussing post-synthetic modifications, catalytic reactions of MOFs, and MOF composites are not directly pertinent to our objective of identifying MOF synthesis conditions. Hence, such papers have been excluded. Another consideration is that MOFs can be synthesized as both microcrystalline powders and single crystals, both of which should be regarded as valid candidates for our dataset. Utilizing the above-mentioned selection criteria, we narrowed our selection to 228 papers from an extensive pool of MOF papers, retrieved from Web of Science, Cambridge Structure Database MOF subset,19 and the CoreMOF database.20, 21 This sample represents a diverse range of MOF synthesis conditions and narrative styles.  \n\nTo enable ChatGPT to process each paper, we devised three different approaches analogous to human paper reading: (1) locating potential sections containing synthesis conditions within the document, (2) confirming the presence of synthesis conditions in the identified sections, and (3) extracting synthesis parameters one by one. For our ChatGPT Chemistry Assistant, these steps are accomplished through filtering, classification, and summarization (Figure 1).  \n\n# ChemPromptEngineering  \n\n# Prompt:  \n\nAnswer the question as truthfully as possible using the provided context.  \n\nPlease summarize the following details in a table: compound name or chemical formula (if the name is not provided), metal source, organic linker(s), solvent(s), reaction temperature, and reaction time. If any information is not provided or you are unsure, use \"N/A\". Please ignore information related to organic linker synthesis, MOF postsynthetic modification or metalation.  \n\nThe table should have 6 columns, all in lowercase:| compound name| metal source|linker solvent|reaction temperature|reaction time|  \n\n# Input:  \n\nIn a 100 mL media bottle were dissolved 1,3,5-benzenetricarboxylic acid (210 mg) and ZrOCl2-8H2O (970 mg) in a solution containing DMF (30 mL) and formic acid $(30m L)$ . The bottle was sealed and heated in a 100 $^{\\circ}C$ isothermal oven for a day. White powder of MOF-808 was collected by centrifugation.  \n\n# Output:  \n\n<html><body><table><tr><td>compound name</td><td>metal source</td><td>linker</td><td>solvent</td><td>reaction temperature</td><td>reaction time</td></tr><tr><td>MOF-808</td><td></td><td>1,3,5- ZrOCl2-8H2O|benzenetricarboxylic acid</td><td>DMF</td><td>100℃</td><td>24h</td></tr></table></body></html>  \n\n![](images/96c369b80185f5746c7a81c589b87732640bf17920530f07ce4e1d5b5aa4d8f3.jpg)  \nFigure 2. Illustration of a carefully designed ChemPrompt (shown on the left), encapsulating all three fundamental principles of ChemPrompt Engineering (shown on the right). The prompt guides ChatGPT to systematically extract and summarize synthesis conditions from a specified section in a research article, organizing the data into a well-structured table.  \n\nIn Process 1, we developed prompts to guide ChatGPT in summarizing text from designated experimental sections contained in those papers. To replace the need for human intervention to obtain synthesis sections, in Process 2, we designed a method for ChatGPT to categorize text inputs as either \"experimental section\" or \"non-experimental section\", enabling it to generate experimental sections for summarization. In Process 3, we further devised a technique to swiftly eliminate irrelevant paper sections, such as references, titles, and acknowledgments, which are unlikely to encompass comprehensive synthesis conditions. This accelerates processing speed for the later classification task. As such, in Process 1, ChatGPT is solely responsible for summarizing and tabulating synthesis conditions and requires one or more paragraphs of experimental text as input, while Process 2 and 3 can be considered as an \"automated paper reading system\". While Process 2 entails a thorough examination of the entire paper to scrutinize each section, the more efficient Process 3 rapidly scans the entire paper, removing the least relevant portions, thereby reducing the number of paragraphs that ChatGPT must meticulously analyze.  \n\nPrompt Engineering. In the realm of chemistry-related tasks, ChatGPT's performance can be significantly enhanced by employing prompt engineering (PE)—a meticulous approach to designing prompts that steer ChatGPT towards generating precise and pertinent information. We propose three fundamental principles in prompt engineering for chemistry-focused applications, denoted as ChemPrompt Engineering:  \n\n(1) Minimizing Hallucination, which entails the formulation of prompts to avoid eliciting fabricated or misleading content from ChatGPT. This is particularly important in the field of chemistry, where the accuracy of information can have significant implications on research outcomes and safety. For instance, when asked to provide synthesis conditions for MOFs without any additional prompt or context, ChatGPT may recognize that MOF-99999 does not exist but will generate fabricated conditions for existing compounds with names like MOF-41, MOF-419, and MOF-519. We should note that with additional prompts followed after the question, it is possible to minimize hallucination and enforce ChatGPT to answer the questions based on its knowledge (Table 1 and Table 2). Furthermore, we demonstrate that with well-designed prompts and context, hallucination occurrences can be minimized (Supporting Information, Section S2.1). We note that this should be the first and foremost principle to follow when designing prompts for ChatGPT to perform in handling text and questions relevant to chemical information.  \n\n(2) Implementing Detailed Instructions, whereby explicit directions are provided in the prompt to assist ChatGPT in understanding the context and desired response format. By incorporating detailed guidance and context into the prompts, we can facilitate a more focused and accurate response from ChatGPT. In chemistry-related tasks, this approach narrows down the potential answer space and reduces the likelihood of irrelevant or ambiguous responses. For example, we can specify not to include any organic linker synthesis conditions and focus solely on MOF synthesis (Supporting Information, Figure S8). In this case, we found that ChatGPT can recognize the features of organic linker synthesis and differentiate them from MOF synthesis. With proper prompts, information from organic linker synthesis will not be included. Additionally, instructions can provide step-by-step guidance, which has proven effective when multiple tasks are included in one prompt (Supporting Information, Section S2.2).  \n\nTable 1. Assessment of hallucination in ChatGPT response without prompt engineering.   \n\n\n<html><body><table><tr><td>Query</td><td>ChatGPT Response (a)</td></tr><tr><td>Which metal is used in the synthesis of MOF-5?</td><td>Zinc (Correct)</td></tr><tr><td>Which metal is used in the synthesis of MOF-519?</td><td>Zirconium (Incorrect)</td></tr><tr><td>What is the linker used in the synthesis of MOF-99999?</td><td>I don't know (Correct)</td></tr><tr><td>What is the linker used in the synthesis of MOF-419?</td><td>Terephthalic acid (Incorrect)</td></tr><tr><td>What is the linker used in the synthesis of ZIF-8?</td><td>2-methylimidazole (Correct)</td></tr></table></body></html>  \n\nTable 2. Improvements in ChatGPT response accuracy utilizing a basic prompt engineering strategy.   \n\n\n<html><body><table><tr><td>Initial Query</td><td>Guided Prompt</td><td>ChatGPT Response (a)</td></tr><tr><td>Which metal is used in the synthesis of MOF-5?</td><td rowspan=\"4\">If you're uncertain, please reply with 'I do not know'.</td><td>Zinc (Correct)</td></tr><tr><td>Which metal is used in the synthesis of MOF-519?</td><td>I don't know (Correct)</td></tr><tr><td>What is the linker used in the synthesis of MOF-99999?</td><td>I don't know (Correct)</td></tr><tr><td>What is the linker used in the synthesis of MOF-419?</td><td>I don't know (Correct)</td></tr><tr><td>What is the linker used in the synthesis of ZIF-8?</td><td></td><td>2-methylimidazole (Correct)</td></tr></table></body></html>\n\na)  Responses are representative answers selected from a series of 100 repeated queries, followed by parenthetical indications of their correctness, which is based on the established facts concerning the respective compounds referenced in the queries.  \n\n(3) Requesting Structured Output, which includes the incorporation of an organized and well-defined response template or instruction to facilitate data extraction. We emphasize that this principle is particularly valuable in the context of chemistry, where data can often be complex and multifaceted. Structured output enables the efficient extraction and interpretation of critical information, which in turn can significantly contribute to the advancement of research and knowledge in the field. Take synthesis condition extraction as an example, without clear instructions on the formatted output, ChatGPT can generate a table, list-like bullet points, or a paragraph, with the order of parameters such as reaction temperature, reaction time, and solvent volume not being uniform, making it challenging for later sorting and storage of the data. This can be easily improved by explicitly asking it to generate a table and providing a fixed header to start with prompt (Supporting Information, Section S2.3). By incorporating these principles, the resulting prompt can ensure that ChatGPT yields accurate and reliable results, ultimately enhancing its utility in tackling complex chemistry-related tasks (Figure 2). We further employ the idea of interactive prompt refinement, in which we start with asking ChatGPT to write a prompt to instruct itself by giving it preliminary descriptions and information (Supporting Information, Figure S15). Through conversation, we add more specific details and considerations to the prompt, testing it with some texts, and once we obtain output, we provide feedback to ChatGPT and ask it to improve the quality of the prompt (Supporting Information, Section S2.4).  \n\nAs there has been almost no literature systematically discussing prompt engineering in Chemistry, and the fact that this field is relatively new, we provide a comprehensive step-by-step ChemPrompt Engineering guide for beginners to start with, including numerous chemistry-related examples in the Supporting Information, Section S2. At present, everyone is at the same starting point, and no one possesses exclusive expertise in this area. It is our hope that this work will stimulate the development of more powerful prompt engineering skills and help every chemist quickly understand the art of ChemPrompt Engineering, thereby advancing the field of chemistry at large.  \n\nProcess 1: Synthesis Conditions Summarization. One revolutionary aspect of ChatGPT is its specialized domain knowledge due to its extensive pre-trained text corpus, which enables an understanding of chemical nomenclature and reaction conditions.18 In contrast to traditional NLP methods, ChatGPT requires no additional training for named entity recognition, and can readily identify inorganic metal sources, organic linkers, solvents, and other compounds within a given experimental text. Another notable feature is ChatGPT's ability to recognize and associate compound abbreviations (e.g., DMF) with their full names (N,N-dimethylformamide) within the context of MOF synthesis (Supporting Information, Figure S5). This capability is crucial as the use of different abbreviations for the same compound can inflate the number of “unique compounds” in the dataset post text mining, leading to redundancy without providing new information. This challenge is difficult to address using traditional NLP methods or packages, as no model can inherently discern that DMF and N,N-dimethylformamide are the same compound without a manually curated dictionary of chemical abbreviations. Although ChatGPT may not cover all abbreviations, its proficiency in identifying and associating the most common ones such as DEF, DI water, EtOH, and $\\mathrm{CH}_{3}\\mathrm{CN}$ with their full names enhances data consistency and reduces redundancy. This, in turn, facilitates data retrieval and analysis, ensuring that different names of the same compound are treated as a single entity with its unique chemical identity and information.  \n\nOur first goal is to develop a ChatGPT-based AI assistant that demonstrates high performance in converting a given experimental section paragraph into a table containing all synthesis parameters (Supporting Information, Figure S22). To design the prompt for this purpose, we incorporate the three principles discussed earlier into ChemPrompt Engineering (Figure 2). The rationale for using tabulation as the output for synthesis condition summarization is that the tabular format simplifies subsequent data sorting, analysis, and storage. In terms of the choice of 11 synthesis parameters, we include those deemed most important and non-negligible for each MOF synthesis. Specifically, these parameters encompass metal sources and quantities, dictating metal centers in the framework and their relative concentrations; the linker and its quantity, which affect connectivity and pore size within the MOF; the modulator and its quantity or volume, which can fine-tune the MOF's structure by impacting the nucleation and growth of the MOF in the reaction; the solvent and its volume, which can influence both the crystallization process and the final MOF structure; and the reaction temperature and duration, which are vital parameters governing the kinetics and thermodynamics of MOF formation in each synthesis. In our prompt, we also account for the fact that some papers may report multiple synthesis conditions for the same compound and instruct ChatGPT to use multiple rows to include each variation. For multiple units of the same synthesis parameters, such as when molarity mass and weight mass are both reported, we encourage ChatGPT to include them in the same cell, separated by a comma, which can be later streamlined depending on the needs. If any information is not provided in the sections, e.g., most MOF reactions may not involve the use of modulators and some papers may not specify the reaction time, we expect ChatGPT to answer \"N/A\" for that parameter. Importantly, to eliminate non-MOF synthesis conditions such as organic linker synthesis, post-synthetic modification, or catalysis reactions, which are not helpful for studying MOF synthesis reactions, we simply add one line of narrative instruction, asking ChatGPT to ignore these types of reactions and focus solely on MOF synthesis parameters. Notably, this natural language-based instruction is highly convenient, requiring no complex and laborious rule-based code to identify unwanted cases and filter them out, and is friendly to researchers without coding experience.  \n\nThe finalized prompts for Process 1 consist of three parts: (i) a request for ChatGPT to summarize and tabulate the reaction conditions, and only use the text or information provided by humans, which adheres to Principle 1 to minimize hallucination; (ii) a specification of the output table's structure, enumerating expectations and handling instructions, which follows Principles 2 and 3 for detailed instructions and structured output requests; and (iii) the context, consisting of MOF synthesis reaction condition paragraphs from experimental sections or supporting information in research articles. Note that parts (i) and (ii) are fixed prompts, while part (iii) is considered as \"input.\" The combined prompt results in a single question-and-answer interaction, allowing ChatGPT to generate a summarization of the given synthesis conditions as output.  \n\nProcess 2: Synthesis Paragraph Classification. The next question to be answered is, “if ChatGPT is given an entire research article, can it correctly locate the sections of experimental sections?” The objective of Process 2 is to accept an entire research paper as input and selectively forward paragraphs containing chemical experiment details to the next assistant for summarization. However, locating the experimental synthesis section within a research paper is a complex task, as simple techniques such as keyword searches often prove insufficient. For instance, the synthesis of MOFs may be embedded within the supporting information or combined with organic linker synthesis. In earlier publications, synthesis information might appear as a footnote. Furthermore, different journals or research groups utilize varying section titles, including \"Experimental,\" \"Methods,\" \"General Methods and Materials,\" \"Experimental methods,\" \"Synthesis and Characterization,\" \"Synthetic Procedures,\" \"Methods Summary,\" and more. Manually enumerating each case is labor-intensive, especially when synthesis paragraphs may be dispersed with non-MOF synthesis, characterization conditions, or instrument details. Even a human might take considerable time to identify the correct section.  \n\nTo address this challenge and enable ChatGPT to accurately discern synthesis details within a lengthy research paper, we draw inspiration from the human process. A chemistry Ph.D. student, when asked to locate the MOF synthesis section in a new research paper, would typically start with the first paragraph and ask themselves if it contains synthesis parameters. They would then draw upon prior knowledge from previously read papers to determine if the section is experimental. This process is repeated paragraph by paragraph until the end of the supporting information is reached, with no guarantee that additional synthesis details will not be encountered later. To train ChatGPT similarly, we prompt it to read paper sections incrementally, focusing on one or two paragraphs at a time. Using a few-shot prompt strategy, we provided ChatGPT with a couple of example cases of both synthesis and non-synthesis paragraphs and asked it to classify the sections it reads as either \"Yes\" (synthesis paragraph) or \"No\" (non-synthesis paragraph). The ChatGPT Chemistry Assistant would then continue processing the research paper section by section, passing only the paragraphs labeled as \"Yes\" to the following assistant for summarization.  \n\nThis few-shot prompt strategy is more convenient than traditional approaches, which require researchers to manually identify and label a large number of paragraphs as \"Synthesis Paragraphs\" and train their models accordingly. In fact, ChatGPT can even perform such classification using a zero-shot prompt strategy with detailed descriptions of what a \"Synthesis Paragraph\" should look like and contain. However, we have found that providing four or five short examples in a few-shot prompt strategy enables ChatGPT to identify the features of synthesis paragraphs more effectively, streamlining the classification process (Supporting Information, Figure S24).  \n\nThe finalized prompt for Process 2 comprises three parts: (i) a request for ChatGPT to determine whether the provided context includes a comprehensive MOF synthesis, answering only with \"Yes\" or \"No\"; (ii) some example contexts labeled as \"Yes\" and other labeled as \"No\"; (iii) the context to be classified, consisting of one or more research article paragraphs. Similar to Process 1's prompt, parts (i) and (ii) are fixed, while part (iii) is replaced with independent sections from the paper to be classified. The entire research article is parsed into sections of 100-500 words, which are iteratively incorporated into the prompt and sent separately to ChatGPT for a \"Yes\" or \"No\" response. Each prompt represents a one-time conversation, and ChatGPT cannot view answers from previous prompts, preventing potential bias in its decision-making for the current prompt.  \n\nProcess 3: Text Embeddings for Search and Filtering. Text embeddings are high-dimensional vector representations of text that capture semantic information, enabling quantification of the relatedness of textual content.22, 23 The distance between these vectors in the embedded space correlates with the semantic similarity between corresponding text strings, with smaller distances indicating greater relatedness.24, 25 While Process 2 can automatically read and summarize papers, it must evaluate every section to identify synthesis paragraphs. To expedite this process, we developed Process 3, which filters sections least likely to contain synthesis parameters using OpenAI embeddings before exposing the article to classification assistant in Process 2. To achieve this, we employed a two-step approach to construct Process 3: first, parsing all papers and converting each segment into embeddings; and second, calculating and ranking the similarity scores of each segment based on their relevance to a predefined prompt encapsulating synthesis parameter.  \n\n![](images/c51a60d13579f3ae061a0dfe4877e544c1c87a746cbeb53972c7cf1fa1a54f2e.jpg)  \nFigure 3. Two-dimensional visualization of 18,248 text segment embeddings, with each point representing a text segment from the research articles selected. Color coding denotes thematic categories: red for “synthesis”, green for “gas sorption”, yellow for “literature reference”, blue for “crystallographic data”, purple for “structural analysis”, orange for “characterization”, and grey for other text segments not emphasized in this study.  \n\nIn particular, we partitioned the 228 research articles into 18,248 individual text segments (Supporting Information, Figure S30−S32). Each segment was converted into a 1536-dimensional text embedding using OpenAI's text-embedding-ada-002, a simple but efficient model for this process (Supporting Information, Figure S33−S35). These vectors were stored for future use. To identify segments  \n\nmost and least likely to contain synthesis parameters, we employed interactive prompt refinement strategy (Supporting Information, Section S2.4), consulting with ChatGPT to optimize the prompt. The prompt used in Process 3, unlike previous prompts, served as a text segment for search and similarity comparison rather than instructing ChatGPT (Supporting Information, Figure S25). Next, the embeddings of all 18,248 text segments were compared with the prompt's embedding, and a relevance score was assigned to each segment based on the cosine similarity between the two embeddings. Highly relevant segments were passed on to classification assistant for further processing, while low similarity segments were filtered out (Figure 1).  \n\nTo evaluate the effectiveness of this approach, we conducted a visual exploration of our embedding data (Figure 3). By reducing the vectors' dimensionality, we observed distinct clusters corresponding to different topics. Notably, we identified distinct clusters related to topics like “gas sorption”, “literature reference”, “characterization”, “structural analysis” and “crystallographic data”, which were separate from the “synthesis” cluster. This observation strongly supports the efficiency of our embedding-based filtering strategy. However, this strategy, while effective at filtering out less relevant text and passing segments of mid to high relevance to the subsequent classification assistant, cannot directly search for synthesis paragraphs to feed to the summarization assistant, thus bypassing the classification assistant. In other words, the searching-to-classifyingto-summarizing pipeline cannot be simplified to a searching-to-summarizing pathway due to the inherent search limitations of the embeddings. As shown in Figure 3, embeddings alone may not accurately identify all relevant “synthesis” sections, particularly when they contain additional information such as characterization and sorption data. The presence of these elements in a synthesis section can reduce its similarity score and its proximity to the center of the “synthesis” cluster. Points between the “synthesis” and “characterization” or “crystallographic data” clusters may not have the highest similarity scores and could be missed. However, by filtering only the lowest scores, mid-relevance points are retained and passed to the classification assistant, which can more accurately classify ambiguous content.  \n\nChatGPT-Assisted Python Code Generation and Data Processing. Rather than relying on singular, time-consuming conversations with web-based ChatGPT to process textual data from a multitude of research articles, OpenAI's GPT-3.5-turbo, which is identical to the one underpinning the ChatGPT product, facilitates a more efficient approach, as it incorporates an Application Programming Interface (API), enabling batch processing of text from an extensive array of articles. This is achieved through iterative context and prompt submissions to ChatGPT, followed by the collection of its responses (Supporting Information, Section S3.4).  \n\nSpecifically, our approach involves having ChatGPT to create Python scripts for parsing academic papers, generating prompts, executing text processing through Processes 1, 2, and 3, and collating the responses into cleaned, tabulated data (Supporting Information, Figures S28−S39). Traditionally, such a process could necessitate substantial coding experience and be time-consuming. However, we leverage the code generation capabilities of ChatGPT to establish Processes 1, 2, and 3 for batch processing using OpenAI’s APIs, namely, gpt-3.5-turbo and text-embedding-ada-002. In essence, researchers only need to express their requirements for each model in natural language - specifying inputs and desired outputs - and ChatGPT will generate the appropriate Python code (Supporting Information, Section S3.5). This code can be copied, pasted, and executed in the relevant environment. Notably, even in the event of an error, ChatGPT, especially when equipped with the GPT-4 model, can assist in code revision. We note that while coding assistance from ChatGPT may not be necessary for those with coding experience, it does provide an accessible platform for individuals lacking such experience to engage in the process. Given the simplicity and straightforwardness of the logic involved in Processes 1, 2, and 3, ChatGPT-generated Python code exhibits minimal errors and significantly accelerates the programming process.  \n\n![](images/a05a94670a0445bac9c81df6d9f011a5c281c3c63bbd486613a989097d7ec440.jpg)  \nFigure 4. Schematic representation of the diverse data unification tasks managed either directly by ChatGPT or through Python code written by ChatGPT. The figure distinguishes between simpler tasks handled directly by ChatGPT, such as standardizing chemical notation, and converting time and temperature units in reactions. More complex tasks, such as matching linker abbreviations to their full names, converting these to SMILES codes, classifying product morphology, and calculating metal amounts, are accomplished via Python code generated by ChatGPT. The Python logo displayed is credited to PSF.  \n\nChatGPT also aids in entity resolution post text mining (Figure 4). This step involves standardizing data formats including units, notation, and compound representations. For each task, we designed a specific prompt for ChatGPT to handle data directly or a specialized Python code generated by ChatGPT. More details on designing prompts to handle different synthesis parameters are available in a cookbook style in Supporting Information, Section S4. In simpler cases, ChatGPT can directly handle conversions such as time and reaction temperature. For complex calculations, we take advantage of ChatGPT in generating Python code. For instance, to calculate the molar mass of each metal source, ChatGPT can generate the appropriate Python code based on the given compound formulas. For harmonizing notation of compound pairs or mixtures, ChatGPT can standardize different notations to a unified format, facilitating subsequent data processing.  \n\nTo standardize compound representations, we employ the Simplified Molecular Input Line Entry System (SMILES). We faced challenges with some synthesis procedures, where only abbreviations were provided. To overcome this, we designed prompts for ChatGPT to search for the full names of given abbreviations. We then created a dictionary linking each unique PubChem Compound identification number (CID) or Chemical Abstracts Service (CAS) number to multiple full names and abbreviations and generated the corresponding SMILES code. We note that for complicated linkers or those with missing full names, inappropriate nomenclature or non-existent CID or CAS numbers,26-33 manual intervention was occasionally necessary to generate SMILES codes for such chemicals (Supporting Information, Figure S50−S54). However, most straightforward cases were handled efficiently by ChatGPT's generated Python code. As a result, we achieved uniformly formatted data, ready for subsequent evaluation and utilization.  \n\n# RESULTS AND DISCUSSION  \n\nEvaluation of Text Mining Performance. We began our performance analysis by first evaluating the execution time consumption for each process (Figure 5a). As previously outlined, the ChatGPT assistant in Process 1 exclusively accepts preselected experimental sections for summarization. Consequently, Process 1 requires human intervention for the identification and extraction of the synthesis section from a paper to operate autonomously. As illustrated in Figure 5a, this process can vary in duration based on the length and structure of the document and its supporting information file. In our study, the complete selection procedure spanned 12 hours for 228 papers, averaging around 2.5 minutes per paper. This period must be considered as the requisite time for Process 1's execution. For summarization tasks, ChatGPT Chemistry Assistant demonstrated an impressive performance, taking an average of 13 seconds per paper. This is noteworthy considering that certain papers in the dataset contained more than 20 MOF compounds, and human summarization in the traditional way without AI might consume a significantly larger duration. By accelerating the summarization process, we alleviate the burden of repetitive work and free up valuable time for researchers.  \n\nIn contrast, Process 2 operates in a fully automated manner, integrating the classification and result-passing processes to the next assistant for summarization. There is no doubt that it outperforms the manual identification and summarization combination of Process 1 in terms of speed due to ChatGPT's superior text processing capabilities. Lastly, Process 3, as anticipated, is the fastest due to the incorporation of section filtering powered by embedding, reducing the classification tasks, and subsequently enhancing the speed. The efficiency of Process 3 can be further optimized by storing the embeddings locally as a CSV file during the first reading of a paper, which reduces the processing time by 15-20 seconds $(28\\%-37\\%$ faster) in subsequent readings. This provides a convenient solution in scenarios necessitating repeated readings for comparison or extraction of diverse information.  \n\nTo evaluate the accuracy of the three processes in text mining, instead of sampling, we conducted a comprehensive analysis of the entire result dataset. In particular, we manually wrote down the ground truth for all 11 parameters for approximately 800 compounds reported in all papers across the three processes, which was used to judge the text mining output. This involved the grading of nearly 26,000 synthesis parameters by us. Each synthesis parameter was assigned one of three labels: True Positive (TP, correct identification of synthesis parameters by ChatGPT), False Positive (FP, incorrect assignment of a compound to the wrong synthesis parameter or extraction of irrelevant information), and False Negative (FN, failure of ChatGPT to extract some synthesis parameters). Notably, a special rule for assigning labels on modulators, most of which were anticipated to be acid and base, was introduced to accommodate the neutral solvents in a mixed solvent system, due to the inherent challenges in distinguishing between co-solvents and modulators. For instance, in a $\\mathrm{)}\\mathrm{MF}\\mathrm{:}\\mathrm{H}_{2}0=10{:}1$ solution, the role of $\\mathrm{H}_{2}0$ becomes ambiguous. In such situations, we labeled the result as a TP if $\\mathrm{H}_{2}0$ was considered either as a solvent or modulator. However, we labeled it as FP or FN if it appeared or was absent in both solvent and modulator columns. Nevertheless, acids and bases were still classified as modulators, and if labeled as solvents, they were graded as FP.  \n\nThe distribution of TP labels counted for each of the 11 synthesis parameters across all papers is presented in Figure 5b. It should be noted that not all MOF synthesis conditions necessitate reporting of all 11 parameters; for instance, some syntheses do not involve modulators, and in such cases, we asked ChatGPT to assign an $\"\\mathrm{N}/\\mathrm{A}\"$ to the corresponding column and its amount. Subsequently, we computed the precision, recall, and F1 scores for each parameter across all three processes, illustrated in Figure 5c and d. All processes demonstrated commendable performance in identifying compound names, metal source names, linker names, modulator names, and solvent names. However, they encountered difficulties in accurately determining the quantities or volumes of the chemicals involved. Meanwhile, parameters like reaction temperature and reaction time, which usually have fixed patterns (e.g., units such as $^{\\circ}\\mathrm{C}$ , hours), were accurately identified by all processes, resulting in high recall, precision, and F1 scores. The lowest scores were associated with the recall of solvent volumes. This is because ChatGPT often captured only one volume in mixed solvent systems instead of multiple volumes. Moreover, in some literatures, the stock solution was used for dissolving metals and linkers, and in principle these volumes should be added to the total volume and unfortunately, ChatGPT lacked the ability to report the volume for each portion in these cases.  \n\nNevertheless, it should be noted that our instructions did not intend for ChatGPT to perform arithmetic operations in these cases, as the mathematical reasoning of the large languages models is limited, and the diminishment of the recall scores is unavoidable. In other instances, only one exemplary synthesis condition for MOF was reported, and then for similar MOFs, the paper would only state \"following similar procedures\". In such cases, while occasionally ChatGPT could duplicate conditions, most of the time it recognized solvents, reaction temperature, and reaction time as $\"\\mathrm{N}/\\mathrm{A}\"$ , which was graded as a FN, thus reducing the recall scores across all processes.  \n\nDespite these irregularities, which were primarily attributable to informal synthesis reporting styles, the precision, recall, and F1 scores for all three processes remained impressively high, with less than $9.8\\%$ of NP and 0 cases of hallucination detected by human evaluators. We further calculated the average and standard deviation of each process on precision, recall, and F1 scores, respectively, as shown in Figure 5c. By considering and averaging precision, recall, and F1 scores across the 11 parameters, given their equal importance in evaluating overall performance of the process, we found that all three processes achieved impressive precision $(>95\\%)$ , recall $(>90\\%)$ , and F1 scores $(>92\\%)$ .  \n\n![](images/09058b8473201b41d3c0f48ce995d699773ebba7f70d45a51ef7ea995fc78663.jpg)  \nFigure 5. Multifaceted performance analysis of ChatGPT-based text mining processes. (a) Comparison of the average execution time required by each process to read and process a single paper, highlighting their relative efficiency. (b) Distribution of true positive counts for each of the 11 synthesis parameters, derived from the cumulative results of Processes 1, 2, and 3 based on a total of 2387 synthesis conditions. Despite minor discrepancies, the counts are closely aligned, demonstrating the assistants' proficiency in effectively extracting the selected parameters. (c) Aggregate average precision, recall, and F1 scores for each process, indicating their overall accuracy and reliability. Standard deviations are represented by grey error bars in the chart. (d) Heatmap illustrating the detailed percentage precision, recall, and F1 scores for each synthesis parameter across the three processes, providing a nuanced understanding of the ChatGPT-based assistants’ performance in accurately identifying specific synthesis parameters.  \n\nThe performance metrics of Process 1 substantiated our hypothesis that ChatGPT excels in summarization tasks. Upon comparing the performance of Processes 2 and 3 — both of which are fully automated paper-reading systems capable of generating datasets from PDFs with a single click — we observed that Process 2, by meticulously examining every paragraph across all papers, ensures high precision and recall by circumventing the omission of any synthesis paragraphs or extraction of incorrect data from irrelevant sections. Conversely, while Process 3's accuracy is marginally lower than that of Process 2, it provides a significant reduction in processing time, thus enabling faster paper reading while maintaining acceptable accuracy, courtesy of its useful filtration process.  \n\nTo the best of our knowledge, these scores surpass most of other models in text mining in the MOF-related domain.11, 13, 14, 34, 35 Notably, the entire workflow, established via code and programs generated from ChatGPT, can be assembled by one or two researchers with only basic coding proficiency in a period as brief as a week, whilst maintaining remarkable performance. The successful establishment of this innovative ChatGPT Chemistry Assistant workflow including the ChemPrompt  \n\nEngineering system, which harnesses AI for processing chemistry-related tasks, promises to significantly streamline scientific research. It liberates researchers from routine laborious work, enabling them to concentrate on more focused and innovative tasks. Consequently, we anticipate that this approach will catalyze potentially revolutionary shifts in research practices through the integration of AI-powered tools.  \n\nPrediction Modeling of MOF Synthesis Outcomes. Given the large quantity of synthesis conditions obtained through our ChatGPT-based text mining programs, our aim is to utilize this data to investigate, comprehend, and predict the crystallization conditions of a material of interest. Specifically, our goal was to determine the crystalline state based on synthesis conditions - we seek to discern which synthesis conditions will yield MOFs in the form of single crystals, and which conditions are likely to yield non-single crystal forms of MOFs, such as microcrystalline powder or solids.  \n\nWith this objective in mind, we identified the need for a label signifying the crystalline state of the resulting MOF for each synthesis condition, thereby forming a target variable for prediction. Fortunately, nearly all research papers in the MOF field consistently include the description of crystal morphological characteristics such as the color and shape of as-synthesized MOFs (e.g. yellow needle crystals, red solid, sky-blue powdered product). This facilitated in re-running our processes with the same synthesis paragraphs as input and modifying the prompt to instruct ChatGPT to extract the description of reaction products, summarizing and categorizing them (Supporting Information, Figure S23 and Figure S47). The final label for each condition will either be Single-Crystal (SC) or Polycrystalline (P), and our objective is to construct a machine learning model capable of accurately predicting whether a given condition will yield SC or P. Furthermore, we recognized that the crystallization process is intrinsically linked with the synthesis method (e.g., vapor diffusion, solvothermal, conventional, microwaveassisted method). Thus, we incorporated an additional synthesis variable, \"Synthesis Method\", to categorize each synthesis condition into four distinct groups. Extracting the reaction type variable for each synthesis condition can be achieved using the same input but a different few-shot prompt to guide our ChatGPT-based assistants for classification and summarization, subsequently merging this data with the existing dataset. This process parallels the method for obtaining MOF crystalline state outcomes, and both processes can be unified in a single prompt. Moreover, as the name of the MOF is a user-defined term and does not influence the synthesis result, we have excluded this variable for the purposes of prediction modeling.  \n\nAfter unifying and organizing the data to incorporate 11 synthesis parameter variables and 1 synthesis outcome target variable, we designed respective descriptors for each synthesis parameter capable of robustly representing the diversity and complexity in the synthesis conditions and facilitating the transformation of these variables into features suitable for machine learning algorithms. A total of six sets of chemical descriptors were formulated for the metal node(s), linker(s), modulator(s), solvent(s), their respective molar ratios, and the reaction condition(s) - aligning with the extracted synthesis parameters (Supporting Information, Section S5).36-40 These MOF-tailored, hierarchical descriptors have been previously shown to perform well in various prediction tasks.13, 41 To distill the most pertinent features and streamline the model, a recursive feature elimination (REF) with 5-fold cross-validation was performed on $80\\%$ of the total data. The rest was preserved as a held out set unseen during the learning process for independent evaluation (Figure 6a). This down-selection process reduced the number of descriptors from 70 to 33, thereby preserving comparative model performance on the held out set while removing the non-informative features that can lead to overfitting (Supporting Information Section S5).  \n\nSubsequently, we constructed a machine learning model to train for synthesis conditions to predict if a given synthesis condition can yield single crystals. A binary classifier was trained based on a random forest model (Supporting Information, Section S5). The random forest (RF) is an ensemble of decision trees, whose independent predictions are max voted in the classification case to arrive at the more precise prediction.42 In our study, we trained an RF classifier to predict crystalline states from synthesis parameters, given its ability to work with both continuous and categorical data, its advantage in ranking important features towards prediction, its robustness against noisy data,43  and its demonstrated efficacy in various chemistry applications such as chemical property estimation,44-47 spectroscopic analysis,48-51 and material characterization and discovery.52  \n\nThe dimension-reduced data was randomly divided into different training sizes; for each train test split, optimal hyperparameters, in particular, number of tree estimators and minimum samples required for leaf split, were determined with 5-fold cross validation of the training set. Model performance was gauged in terms of class weighted accuracy, precision, recall, and F1 score over 10 runs on the held out set and test set (Figure 6b and Supporting Information, Figure S64). The model converged to an average accuracy of $87\\%$ and an F1 score of $92\\%$ on the held out set, indicating a reasonable performance in the presence of the imbalanced classification challenge.  \n\nFollowing the creation of the predictive model, our objective was to apply this model for descriptor analysis to illuminate the factors impacting MOF crystalline outcomes. This aids in discerning which features in the synthesis protocol are more crucial in determining whether a synthesis condition will yield MOF single crystals. Although the random forest model is not inherently interpretable, we probed the relative importance of descriptors used in building the model. One potential measure of a descriptor's importance is the percent decrease in the model's accuracy score when values for that descriptor are randomly shuffled and the model is retrained. We found that among the descriptors involved, the top ten most influential descriptors are key in predicting MOF crystallization outcomes (Figure 6c). In fact, these descriptors broadly align with the chemical intuition and our understanding on MOF crystal growth.53, 54 For example, the descriptors related to stoichiometry of the MOF synthesis, namely the “modulator to metal ratio\", \"solvent to metal ratio”, and \"linker to metal ratio\",  take precedence in the ranking. These descriptors reflect the vital role of precise stoichiometric control in MOF crystal formation, and they directly impact the crystallization process, playing critical roles in determining the quality and morphology of the MOF crystals.  \n\n![](images/0a3f3a1709b3433c0ecc32827084e4928d66a15053f09f2d6c54add5eae87b79.jpg)  \nFigure 6. Performance of the classification models in predicting the crystalline state of MOFs from synthesis. (a) Learning curves of the classifier model with 1σ standard deviation error bars. (b) Model performance evaluation through the F1 Score, Precision, Recall, and Area Under the Curve metrics. The training set fraction was in ratio to the data excluding the held out set. (c) The ten most significant descriptors of the trained random forest model, determined by accuracy score increase. (d) Six examples of MOFs, MOF520, MOF-74, ZIF-8, Al-fum, CAU-32, and MOF-808, along with their synthesis conditions derived from the literature.55-60 Circle positions on the bar represent the likelihood of resulting in single-crystal or polycrystalline states predicted by the model. The model's predictions for these six examples aligned with actual experimental results.  \n\nFollowing closely is the descriptor \"time\", and it highlights the significant role of reaction duration in the crystallization process. Additionally, the \"metal valence\" descriptor emphasizes the key role of the nature and reactivity of the metal ions used in MOF synthesis. The valence directly influences the secondary building units (SBUs) and the final crystalline state of the MOF. In the meantime, descriptors related to the molecular and the linker can impact the kinetics of the synthesis, influencing the orderliness of crystal growth. Together, this result provides a greater understanding of the crucial factors affecting the crystallization of MOFs and will aid in the design and optimization of synthesis conditions for the targeted preparation of single-crystal or polycrystalline MOFs (Figure 6d).  \n\nInterrogating the Synthesis Dataset via a Chatbot. Having utilized text mining techniques to construct a comprehensive MOF Synthesis Dataset, our aim was to leverage this resource to its fullest potential. To enhance data accessibility and aid in the interpretation of its intricate contents, we embarked on a journey to convert this dataset into an interactive and userfriendly dialogue system, which effectively converts the dataset to dialogue. The resulting chatbot is part of the umbrella concept of ChatGPT Chemistry Assistant thus serving as a reliable and fact-based assistant in chemistry, proficient in addressing a broad spectrum of queries pertaining to chemical reactions, in particular MOF synthesis. Unlike typical and more general web-based ChatGPT provided by OpenAI, which may suffer from limitations such as the inability to access the most recent data and a propensity for hallucinatory errors. This chatbot is grounded firmly in the factual data contained within the MOF synthesis dataset from text mining and is engineered to ensure that responses during conversations are based on accurate information and synthesis conditions derived from text mining the literature (Supporting Information, Section S6).  \n\nIn particular, to construct the chemistry chatbot, our initial step was the creation of distinct entries corresponding to each MOF we identified from the text mining, which encompasses a comprehensive array of synthesis parameters, such as the reaction time, temperature, metal, and linker, among others, using the dataset we have. Recognizing the value of bibliographic context, we compiled a list of paper information, such as authors, DOI, and publication years, collated from Web of Science, into each section (Supporting Information, Table S3). Subsequently, we generated embeddings for each of these information cards of different compounds, thereby constructing an embedding dataset (Figure 7). When a user asks a question, if it is the first query, the system first navigates to the embedding dataset to locate the most relevant information card using the question's embedding, which is based on a similarity score calculation and is similar to the foundation of Process 3 in text mining. The information of the highest-ranking entry is then dispatched to the prompt engineering module of MOF chatbot, guiding it to construct responses centered solely around the given synthesis information.  \n\nTo mitigate the possibility of hallucination, the chatbot is programmed to refrain from addressing queries that fall outside the scope of the dataset. Instead, it encourages the user to rephrase the question (Supporting Information, Figure S69). It's worth noting that, following the initial query, the chatbot 'memorizes' the conversation context by being presented with the context of prior interactions between user and itself. This includes the synthesis context and paper information identified from the initial query, ensuring that the answers to subsequent queries are also based on factual information from the dataset. Consequently, this strategy guarantees that responses to ensuing queries are contextually accurate, being grounded in the facts outlined in the synthesis dataset and corresponding paper information (Figure 7 and Supporting Information, Figures S71−S74).  \n\nBy virtue of its design, the chatbot addresses the challenge of enhancing data accessibility and interpretation. It accomplishes this by delivering synthesis parameters and procedures in a clear and comprehensible manner. Furthermore, it ensures data integrity and traceability by providing DOI links to the original papers, guiding users directly to the source of information. This functionality proves particularly beneficial for newcomers to the field. By leveraging ChatGPT's general knowledge base, they can receive guided instructions through the synthesis process, even when faced with a procedure in a journal that is ambiguously or vaguely described. In this case, the user can consult ChatGPT to “chat with the paper” for a more precise explanation, thereby simplifying the learning process and facilitating a more efficient understanding of complex synthesis procedures. This capability fosters independent learning and expedites comprehension of intricate synthesis procedures, reinforcing ChatGPT's role as a valuable assistant in the field of chemistry research.  \n\nExploring Adaptability and Versatility in Large Language Models. The adaptability of LLM-based programs, a hallmark feature distinguishing them from traditional NLP programs, lies in their inherent ability to modify search targets or tasks simply by adjusting the input prompt. Whereas traditional NLP models may necessitate a complete overhaul of rules and coding in the event of task modifications, programs powered by ChatGPT and some other LLMs utilize a more intuitive approach. A simple change in narrative language within the prompt can adequately steer the model towards the intended task, obviating the need for elaborate code adjustments.  \n\nHowever, we do recognize limitations within the current workflow, particularly concerning token limitations. Research articles for text mining were parsed into short snippets due to 4096 token limit from GPT-3.5-turbo, since longer research articles can extend to $20,000-40,000$ tokens. This fragmentation may inadvertently result in the undesirable segmentation of synthesis paragraphs or other sections containing pertinent information. To alleviate this, we envision that a large language model that can process higher token memory 61, 62 such as GPT-4-32K (OpenAI),  or Claude-v1 (Anthropic) will be very helpful, since each time it reads the entire paper rather than just sections, which can further increase its accuracy by avoiding undesirable segmentation of the synthesis paragraph or other targeted paragraph containing information. Longer reading capabilities will also have the added benefit of reducing the number of tokens used in repeated questions, thus enhancing processing times. As we continue to refine our workflow, we believe that there are further opportunities for improvement. For instance, parts of the fixed prompt could be more concise to save tokens, and the examples in the few-shot prompt can be further optimized to reduce total tokens. Given that each paper may have around 100 segments, such refinements could dramatically reduce time and costs, particularly for classification and summarization tasks, which must process every section with the same fixed prompt, especially for few-shot instructions.  \n\n![](images/e423b01871241bca1e40a52b87069bd7ef942f0d3e7943c505870f384585b87d.jpg)  \n\nFigure 7. Integrated workflow of the MOF chatbot transforming comprehensive synthesis datasets into contextually accurate dialogue systems and demonstration of conversation with the data-driven chatbot. The process ensures enhanced data accessibility, interpretation, and facilitates independent learning in the field of chemistry research.  \n\nFurthermore, language versatility, a crucial aspect in the realm of text mining, is seamlessly addressed by LLMs. Traditional NLP models, trained in a specific language, often struggle when the task requires processing text data in another language. For example, if the model is trained on English data, it may require substantial adjustments or even a complete rewrite to process text data in Arabic, Chinese, French, German, French, Japanese, Korean and some other languages. However, with LLMs that can handle multiple languages, such as ChatGPT, we showed that researchers just need to slightly alter the instructions or prompts to achieve the goal, without the necessity of substantial code modifications (Supporting Information, Figure S55−S58).  \n\nThe adaptable nature of LLMs can further extend versatility in handling diverse tasks. We demonstrated how prompts can be changed to direct ChatGPT to parse and summarize different types of information from the same pool of research articles. For instance, with minor modification of the prompts, we show that our ChatGPT Chemistry Assistants have the potential to be instructed to summarize diverse information such as thermal stability, BET surface area, $\\mathsf{C O}_{2}$ uptake, crystal parameters, water stability, and even MOF structure or topology (Supporting Information, Section S4). This adaptability was previously a labor-intensive process, requiring experienced specialists to manually collect or establish training sets for text mining each type of information.11, 13, 35, 41, 63-66  \n\nMoreover, the utility of this approach can benefit the broader chemistry domain: it is capable of not only facilitating data mining in research papers addressing MOF synthesis but also extending to all chemistry papers with the accorded modifications. By fine-tuning the prompt, the ChatGPT Chemistry Assistant can effectively extract and tabulate data from diverse fields such as organic synthesis, biochemistry preparations, perovskite preparations, polymer synthesis, and more. This capability underscores the versatility of the ChatGPT-based assistant, not only in terms of subject matter but also in the level of detail it can handle. In the event that key parameters for data extraction are not explicitly defined, ChatGPT can be prompted to suggest parameters based on its trained understanding of the text. This level of adaptability and interactivity is unparalleled in traditional NLP models, highlighting a key advantage of the ChatGPT approach. The shift from a code-intensive approach to a natural language instruction approach democratizes the process of data mining, making it accessible even to those with less coding expertise, makes it an innovative and powerful solution for diverse data mining challenges.  \n\n# CONCLUDING REMARKS  \n\nOur research has successfully demonstrated the potential of LLMs, particularly GPT models, in the domain of chemistry research.  We presented a ChatGPT Chemistry Assistant, which includes three different but connected approaches to text mining with ChemPrompt Engineering: Process 3 is capable of conducting search and filtration, Processes 2 and 3 both classify synthesis paragraphs, and Processes 1, 2 and 3 are capable of summarizing synthesis conditions into structured datasets. Enhanced by three fundamental principles of prompt engineering specific to chemistry text processing, coupled with the interactive prompt refinement strategy, the ChatGPT-based assistant have substantially advanced the extraction and analysis of MOF synthesis literature, with precision, recall, and F1 scores exceeding $90\\%$ .  \n\nWe elucidated two crucial insights from the dataset of synthesis conditions. First, the data can be employed to construct predictive models for reaction outcomes, which shed light into the key experimental factors that influence the MOF crystallization process. Second, it is possible to create a MOF chatbot that can provide accurate answers based on text mining, thereby improving access to the synthesis dataset, and achieving a data-to-dialogue transition. This investigation illustrates the potential for rapid advancement inherent to ChatGPT and other LLMs as a proof-of-concept.  \n\nOn a fundamental level, this study provides guidance on interacting with LLMs to serve as AI assistants for chemists, accelerating research with minimal prerequisite coding expertise and thus bridging the gap between chemistry and the realms of computational and data science more effectively. Through interaction and chatting, the code and design of experiments can be modified, democratizing data mining and enhancing the landscape of scientific research. Our work sets a foundation for further exploration and application of LLMs across various scientific domains, paving the way for a new era of AI-assisted chemistry research.  \n\n# ASSOCIATED CONTENT  \n\nSupporting Information. Detailed instructions and design principles for ChemPrompt Engineering, as well as the specifics of the prompts employed in the ChatGPT Chemistry Assistant for text mining and other chemistry-related tasks. Additional information on the ChatGPT-assisted coding and data processing methods. An extensive explanation of the machine learning models and methods used, as well as the steps involved in setting up the MOF chatbot based on the MOF synthesis condition dataset. This material is available free of charge via the Internet at http://pubs.acs.org.  \n\n# AUTHOR INFORMATION  \n\n# Corresponding Author  \n\nOmar M. Yaghi − Department of Chemistry; Kavli Energy Nanoscience Institute; and Bakar Institute of Digital Materials for the Planet, College of Computing, Data Science, and Society, University of California, Berkeley, California 94720, United States; UC Berkeley–KACST Joint Center of Excellence for Nanomaterials for Clean Energy Applications, King Abdulaziz City for Science and Technology, Riyadh 11442, Saudi Arabia; orcid.org/0000-0002-5611-3325; Email: yaghi@berkeley.edu  \n\n# Other Authors  \n\nZhiling Zheng − Department of Chemistry; Kavli Energy Nanoscience Institute; and Bakar Institute of Digital Materials for the Planet, College of Computing, Data Science, and Society, University of California, Berkeley, California 94720, United States; orcid.org/0000- 0001-6090-2258  \n\nOufan Zhang − Department of Chemistry, University of California, Berkeley, California 94720, United States  \n\nChristian Borgs − Bakar Institute of Digital Materials for the Planet, College of Computing, Data Science, and Society; Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720, United States; orcid.org/0000-0001- 5653-0498  \n\nJennifer T. Chayes − Bakar Institute of Digital Materials for the Planet, College of Computing, Data Science, and Society; Department of Electrical Engineering and Computer Sciences; Department of Mathematics; Department of Statistics; and School of Information, University of California, Berkeley, California 94720, United States; orcid.org/0000-0003-4020-8618  \n\n# ACKNOWLEDGMENTS  \n\nZ.Z. extends special gratitude to Jiayi Weng (OpenAI) for valuable discussions on harnessing the potential of ChatGPT. In addition, Z.Z. acknowledges the inspiring guidance and input from Kefan Dong (Stanford University), Long Lian (University of California, Berkeley), and Yifan Deng (Carnegie Mellon University), all of whom contributed to shaping the study's design and enhancing the performance of ChatGPT. We express our appreciation to Dr. Nakul Rampal from the Yaghi Lab for insightful discussions. Our gratitude is also extended for the financial support received from the Defense Advanced Research Projects Agency (DARPA) under contract HR0011-21-C-0020. O.Z. acknowledges funding and extends thanks for the support provided by the National Institute of Health (NIH) under Grant 5R01GM127627-04. Additionally, Z.Z. thanks for the financial support received through a Kavli ENSI Graduate Student Fellowship and the Bakar Institute of Digital Materials for the Planet (BIDMaP). his work is independently developed by the University of California, Berkeley research team and not affiliated, endorsed, or sponsored by OpenAI.  \n\n# REFERENCES  \n\n1. Yaghi, O. M.;  O'Keeffe, M.;  Ockwig, N. W.;  Chae, H. K.;  Eddaoudi, M.; Kim, J., Reticular synthesis and the design of new materials. Nature 2003, 423 (6941), 705-714.   \n2. Matlin, S. A.;  Mehta, G.;  Hopf, H.; Krief, A., The role of chemistry in inventing a sustainable future. Nat. Chem. 2015, 7 (12), 941-943. 3. 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Yaghi†,‡,§,∥,\\*  \n\n† Department of Chemistry, University of California, Berkeley, California 94720, United States ‡ Kavli Energy Nanoscience Institute, University of California, Berkeley, California 94720, United States   \n§ Bakar Institute of Digital Materials for the Planet, College of Computing, Data Science, and Society, University of California, Berkeley, California 94720, United States   \n◊ Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 94720, United States   \n†† Department of Mathematics, University of California, Berkeley, California 94720, United States ‡‡ Department of Statistics, University of California, Berkeley, California 94720, United States $\\S\\S$ School of Information, University of California, Berkeley, California 94720, United States ∥ KACST–UC Berkeley Center of Excellence for Nanomaterials for Clean Energy Applications, King Abdulaziz City for Science and Technology, Riyadh 11442, Saudi Arabia   \n\\* To whom correspondence should be addressed: yaghi@berkeley.edu  \n\n# Table of Contents  \n\nSection S1. General Information. 3   \nSection S2. Chemistry ChatGPT Prompt Engineering (ChemPrompt Engineering). 4   \nS2.1. Principle 1: Minimizing Hallucination . 4   \nS2.2. Principle 2: Implementing Detailed Instructions .. 11   \nS2.3. Principle 3: Requesting Structured Output .. . 15   \nS2.4. Interactive Prompt Refinement .. 18   \nSection S3. Text Mining with ChatGPT API.. .. 25   \nS3.1. Summarizing Synthesis Conditions with ChatGPT.. .. 25   \nS3.2. Classifying Research Article Sections with ChatGPT . 27   \nS3.3. Filtering Text using OpenAI Embeddings . . 28   \nS3.4. Batch Text Processing with ChatGPT API.. . 29   \nS3.5. Generating Python Code with ChatGPT . . 33   \nSection S4. ChatGPT-Assisted Chemistry Data Processing Cookbook 45   \nSection S5. Prediction Modeling 67   \nSection S6. Dataset to Dialogue: The Creation of a MOF Synthesis Chatbot 74   \nReferences . 84  \n\n# Section S1. General Information  \n\nLarge Language Models  \n\nThree prominent large language models (LLMs) were involved in this study: GPT-3,1 ChatGPT (GPT-3.5), and GPT-4. These models are developed and maintained by OpenAI, and although the comprehensive specifics of their training data and architectural design are proprietary, each model is an instantiation of an autoregressive language model that operates on the transformer architecture.2 For clarity, in this study, we refer to the default GPT-3.5 based chatbot as \"ChatGPT\", whereas we explicitly denote the GPT-4 based chatbot as \"ChatGPT (GPT-4)\" when referenced. Both of these are web-based chatbots and accessible through the OpenAI website chat.openai.com.  \n\n# Application Programming Interface (API)  \n\nTwo LLM APIs were involved in this study: text-embedding-ada-002 and gpt-3.5-turbo. It should be noted that the model gpt-3.5-turbo is essentially the same model as that supporting the default web-based ChatGPT, so we refer to GPT-3.5 API as the ChatGPT API. We note that as of May 2023, access to the GPT-4 API is limited and requires being on a waitlist, and its cost surpasses that of GPT-3.5 significantly. Therefore, our research does not incorporate any usage of the GPT4 API. In this study, we used text-embedding-ada-002 for the Text Mining step in Process 3, whereas gpt-3.5-turbo served the Text Mining steps in Processes 1 and 2.  \n\n# Article Retrieval  \n\nWe obtained 228 papers and their corresponding 225 supporting documents from 32 different journals with the authorization from eight distinct publisher groups: American Chemical Society (ACS), Elsevier, Royal Society of Chemistry (RSC), American Association for the Advancement of Science (AAAS), World Scientific, De Gruyter, Springer Nature, and Wiley. The papers, published between May 2004 and March 2023, were downloaded in PDF format.  \n\n# Prompt Engineering  \n\nThe prompts utilized in this study were developed through a process of interactive prompt refinement. For additional details, please refer to Supporting Information Section S2.  \n\n# Python Code  \n\nThe majority of the code for text mining, data preprocessing, and chatbot operation was generated by ChatGPT and tested on a Jupyter notebook. Comprehensive code and instructions can be found in Supporting Information Sections S3, S4, and S6. Details of code and data can be found at https://github.com/zach-zhiling-zheng/ChatGPT_Chemistry_Assistant.  \n\n# Machine Learning Methods  \n\nThe machine learning model implemented in this study was based on the random forest algorithm, as made available through the scikit-learn library in Python. For additional details, please refer to Supporting Information Section S5.  \n\n# Section S2. Chemistry ChatGPT Prompt Engineering (ChemPrompt Engineering) S2.1. Principle 1: Minimizing Hallucination  \n\n# Human-ChatGPT Interaction  \n\nHuman: What is the synthesis condition of MOF-5?  \n\nChatGPT: MOF-5 (Metal-organic framework 5) is a type of metal-organic framework that consists of zinc ions coordinated with 1,4-benzenedicarboxylic acid (also known as terephthalic acid) linkers to form a three-dimensional porous structure.  \n\nThe synthesis conditions for MOF-5 typically involve the following steps:  \n\n1. Dissolving zinc nitrate hexahydrate $(Z\\mathsf{n}(\\mathsf{N O3})2{\\cdot}6\\mathsf{H}2\\mathsf{O})$ in a solvent such as N,Ndimethylformamide (DMF) or N,N-dimethylacetamide (DMA).   \n2. Adding 1,4-benzenedicarboxylic acid (H2BDC) to the solution and stirring for several hours to form a homogeneous mixture.   \n3. Heating the mixture at a high temperature (e.g. $120-140^{\\circ}{\\mathsf{C}})$ for several hours to promote crystallization.   \n4. Cooling the mixture to room temperature and filtering the resulting solid product.   \n5. Washing the solid product with a suitable solvent (e.g. methanol or DMF) to remove any impurities.   \n6. Drying the product under vacuum to obtain the final MOF-5 material.  \n\nThe precise conditions for MOF-5 synthesis may vary depending on the specific laboratory protocol and the desired properties of the material.  \n\nFi gure S1. An interaction with ChatGPT showcasing its ability to accurately identify the name of MOF-5 and utilize its knowledge to provide a description of the synthesis conditions, including chemicals, steps, and temperature ranges.  \n\n# Human-ChatGPT Interaction  \n\nHuman: What is the synthesis condition of MOF-99999?  \n\nChatGPT: I'm sorry, but MOF-99999 is not a known or registered MOF. In fact, the highest numbered MOF as of my knowledge cutoff in September 2021 was MOF-901, which was reported in a paper by Zhang et al. in 2017. Without knowing the specific MOF you are referring to, I cannot provide information on its synthesis conditions. If you have any additional information or clarification on the MOF you are interested in, I may be able to help you further.  \n\nHuman: What is the synthesis condition of MOF-419?  \n\n$$\n\\mathrm{~---~---~---~---~---~---~---~---~---~}-\\mathrm{~---~---~---~}.\n$$  \n\nChatGPT: MOF-419 is a metal-organic framework (MOF) that is synthesized under solvothermal conditions. Specifically, MOF-419 is typically synthesized by mixing a solution of the metal precursor (e.g. zinc nitrate hexahydrate) and the organic ligand (e.g. 2,5-dihydroxyterephthalic acid) in a solvent (e.g. N,N-dimethylformamide) in a sealed reaction vessel. The mixture is then heated under autogenous pressure at a specific temperature (e.g. $120^{\\circ}\\mathsf{C})$ for a certain duration (e.g. 24 hours) to allow for the formation of the MOF crystals. After the reaction, the resulting MOF-419 crystals can be isolated and purified through various methods such as centrifugation, filtration, or solvent exchange.  \n\nFigure S2. Two examples of dialogues with ChatGPT that demonstrate its limitations due to hallucinations, where it provides incorrect information about the synthesis conditions of MOFs. In th  first example, ChatGPT recognizes that MOF-99999 does not exist but inaccurately describes MOF-901. In the second example, ChatGPT provides incorrect synthesis conditions for MOF$419^{3}$ , which was not known at the time of its training.  \n\n# Human-ChatGPT Interaction  \n\nHuman: What is the synthesis condition of MOF-99999?  \n\nChatGPT (GPT-4): As of my knowledge cutoff date in September 2021, MOF-99999 is not a known or documented metal-organic framework (MOF). It is possible that the designation MOF-99999 has been assigned to a new material since then, but I do not have any information on its synthesis or properties.  \n\nHuman: What is the synthesis condition of MOF-419?  \n\nChatGPT (GPT-4): As an AI language model, I don't have real-time access to the latest research or data. However, based on the information available up until September 2021, MOF-419 isn't a wellknown or extensively studied metal-organic framework (MOF). It's possible that MOF-419 may have been developed or studied more recently, but I cannot provide any specific synthesis conditions for it.  \n\nFigure S3. A comparison of responses from GPT-4, illustrating its improved handling of hallucination issues compared to the previous GPT-3.5 model. In the case of MOF-99999, GPT-4 correctly identifies that it does not have any information on the compound. However, it still cannot pro vide useful information on MOF-419, as this compound was published in March 2023, beyond GPT-4's knowledge cutoff date of September 2021.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context. If any information is not provided or you are unsure, use \"N/A\".  \n\nQ: What is the synthesis condition of MOF-99999?  \n\nContext: Section S3: Synthesis of MOF-901   \nBenzene-1,4-dialdehyde (10.0 mg, 0.0750 mmol) was dissolved in 1 mL of methanol and then dropped into a mixture of 4-aminobenzoic acid (16.0 mg, 0.117 mmol) and titanium   \nisopropoxide (9.0 µL, 0.030 mmol). To increase the solubility, an additional 2 mL of methanol was added to the mixture. The solution was then introduced to a $75\\times1.5$ cm Pyrex tube. The tube was flash frozen at 77 K using a liquid nitrogen bath, evacuated for 10 min in order to remove oxygen, and then flamed seal to maintain a vacuum during the reaction process. The reaction mixture was left at $125^{\\circ}C$ for $3d.$ .  \n\nChatGPT: N/A  \n\nHuman: Answer the question as truthfully as possible using the provided context. If any information is not provided or you are unsure, use \"N/A\".  \n\nQ: What is the synthesis condition of MOF-419?  \n\nContext: Synthesis of MOF-419. In a 20 mL scintillation vial, H2BDH (98 mg, 0.5 mmol) was dispersed in DMF (10 mL). The resulting mixture was heated for 30 min in a preheated oven at $100^{\\circ}C$ to yield a clear solution. Yttrium (III) nitrate hexahydrate (191 mg, 0.5 mmol) was dissolved in DMF (2 mL) and added to the H2BDH solution. The combined solution was sonicated for 10 min and formic acid $(4m L,$ 100 mmol) was added. The vial was placed in an oven at $100^{\\circ}C$ for 90 min.  \n\nChatGPT: The synthesis condition of MOF-419 involves heating a mixture of H2BDH and DMF at 100 $^{\\circ}C$ for 30 minutes, followed by adding a solution of Yttrium (III) nitrate hexahydrate in DMF, sonication for 10 minutes, addition of formic acid, and heating at $100^{\\circ}\\mathsf C$ for 90 minutes. However, the exact synthesis condition of MOF-419 may involve additional steps or conditions that are not mentioned in the provided context.  \n\nFigure S4. Dialogue demonstrating the performance of ChatGPT when provided with context3, 4 fo answering questions on MOF synthesis conditions. In the first example, ChatGPT correctly id ntifies that the synthesis condition of MOF-99999 is not available based on the context provided. In the second example, ChatGPT successfully extracts the synthesis conditions of MOF419 from the most recent information provided in the “context”, showcasing its ability to adapt and utilize context-driven information.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context. If any information is not provided or you are unsure, use \"N/A\".  \n\nQ: What is the organic linker used in this reaction? What is the metal source in this reaction?  \n\nContext: Synthesis of MOF-419. In a 20 mL scintillation vial, H2BDH (98 mg, 0.5 mmol) was dispersed in DMF (10 mL). The resulting mixture was heated for 30 min in a preheated oven at $100^{\\circ}C$ to yield $a$ clear solution. Yttrium (III) nitrate hexahydrate (191 mg, 0.5 mmol) was dissolved in DMF $(2m L)$ and added to the H2BDH solution. The combined solution was sonicated for 10 min and formic acid $(4m L,$ 100 mmol) was added. The vial was placed in an oven at $100^{\\circ}C$ for 90 min.  \n\nChatGPT: The organic linker used in this reaction is H2BDH (no information is provided about its full name). The metal source in this reaction is Yttrium (III) nitrate hexahydrate.  \n\nHuman: Can you write the chemical formula of the metal source?  \n\nChatGPT: Yes, the chemical formula of Yttrium (III) nitrate hexahydrate is Y(NO3)3·6H2O.  \n\nHuman: What solvent was used?  \n\nChatGPT: The solvent used in this reaction is DMF (N,N-Dimethylformamide).  \n\nHuman: What is the reaction time and temperature?  \n\nChatGPT: The reaction time is 90 minutes and the reaction temperature is $100^{\\circ}\\mathsf C$ .  \n\nHuman: Can you summarize the synthesis conditions into a table?  \n\nChatGPT: Yes, here is a summary of the synthesis conditions:  \n\n<html><body><table><tr><td>Compound/Parameter</td><td>Amount/Value</td></tr><tr><td>H2BDH</td><td>98 mg, 0.5 mmol</td></tr><tr><td>DMF</td><td>12 mL</td></tr><tr><td>Yttrium (ll) nitrate hexahydrate</td><td>191 mg, 0.5 mmol</td></tr><tr><td>Formic acid</td><td>4 mL, 100 mmol</td></tr><tr><td>Reaction time</td><td>90 minutes</td></tr><tr><td>Reaction temperature</td><td>100 ℃</td></tr></table></body></html>  \n\nFigure S5. An interaction with ChatGPT showcasing its ability to extract and summarize relevant information from the provided context for MOF-419 synthesis.3 The AI model identifies the organic linker, metal source, solvent, and reaction conditions, and then presents a summarized table of the synthesis parameters, demonstrating its effectiveness in processing and presenting complex information.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context. If any information is not provided or you are unsure, use \"N/A\".  \n\nQ: Can you summarize the synthesis conditions into a table?  \n\nContext:  \n\nMOF-802. H2PZDC $(0.27g,$ 1.5 mmol) and ZrOCl2 · 8H2O (0.40 g, 1. 3 mmol) in a solvent mixture of DMF/formic acid (50 mL/ 35mL) were placed in a 125 mL screw capped glass jar, which was heated at $130^{\\circ}C$ for three days.  \n\nMOF-808. H3BTC (0.11 g, 0.50 mmol) and ZrOCl2 ·8H2O (0.1 6 g, 0.50 mmol) in a solvent mixture of DMF/formic acid (20 mL/ 20mL) were placed in a 60 mL screw capped glass jar, which was heated at 1 $00^{\\circ}C$ for seven days.  \n\nDUT-67. DUT-67 was prepared according to published procedures with slight modification (S4). H2TDC (0. 069 g, 0.40 mmol) and ZrOCl2 · 8H2O (0.19 g, 0.60 mmol) in a solvent mixture of DMF/formic acid (20 mL/ 11 mL) were placed in a 60 mL screw capped glass jar, which was heated at $130^{\\circ}C$ for three days.  \n\nChatGPT: Yes, a table can be made to summarize the synthesis conditions of the MOFs. The table can include the following information:  \n\n<html><body><table><tr><td>MOF Type</td><td>Ligand</td><td>Metal Source</td><td> Solvent Mixture</td><td>Solvent Volume</td><td>Temperature</td><td>Duration</td></tr><tr><td>MOF-802</td><td>H2PZDC</td><td>ZrOCI2-8H20</td><td>DMF/formic acid</td><td>50 mL/35 mL</td><td>130 ℃</td><td>3 days</td></tr><tr><td>MOF-808</td><td>H3BTC</td><td>ZrOCI2-8H20</td><td>DMF/formic acid</td><td>20mL/20mL</td><td>100 ℃</td><td>7 days</td></tr><tr><td>DUT-67</td><td>H2TDC</td><td>ZrOCI2-8H20</td><td>DMF/formic acid</td><td>20 mL/11 mL</td><td>130 ℃</td><td>3 days</td></tr></table></body></html>  \n\nFigure S6. An interaction with ChatGPT where it successfully extracts and organizes synthesis co nditions for three different MOFs (MOF-802, MOF-808, and DUT-67) from the provided context5. The AI model presents the information in a concise and easily readable table format, demonstrating its ability to process and structure complex information from multiple sources. Note that ChatGPT can efficiently extract and summarize information from multiple synthesis conditions in a single step, without the need for separate conversations or step-by-step extraction.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context. If any information is not provided or you are unsure, use \"N/A\".  \n\nQ: Can you summarize the synthesis conditions into a table?  \n\nContext:  \n\nOptimized Synthesis of MOF-521-H Single Crystals. In a 4 mL scintillation vial, 3.0 mL DMF was added to dissolve Al(NO3)3·9H2O (18.0 mg, 0.048 mmol) and H3BTB (15.8 mg, 0.036 mmol). After adding 0.24 mL formic acid and 30 µL deionized water to the solution, the vial was capped and placed in the preheated $140^{\\circ}C$ for 2 days, and colorless needle-shape crystals were obtained. The crystals were washed with MeOH $(3\\times20m L)$ for 3 days before activation and characterization. Full activation of the MOF was conducted under dynamic vacuum (\\~10-3 mbar) at ${100^{\\circ}C}$ for 12 hours, yielding pure and desolvated product.  \n\nChatGPT: Based on the provided context, here is a summarized table of the synthesis conditions for MOF-521-H single crystals:  \n\n<html><body><table><tr><td>Compound/Reagent</td><td>Amount/Volume</td></tr><tr><td>AI(NO3)3-9H20</td><td>18.0 mg (0.048 mmol)</td></tr><tr><td>H3BTB</td><td>15.8 mg (0.036 mmol)</td></tr><tr><td>DMF</td><td>3.0 mL</td></tr><tr><td>Formic Acid</td><td>0.24 mL</td></tr><tr><td>Deionized Water</td><td>30 μL</td></tr><tr><td>Preheating Temperature</td><td>140 ℃</td></tr><tr><td>Preheating Duration</td><td>2 days</td></tr></table></body></html>  \n\nFigure S7. An interaction with ChatGPT wherein the model successfully delineates and organizes the synthesis conditions for MOF-521, an unpublished compound as of May 2023, from the provided context. It should be noted that prior to this instance, the synthesis description for this compound was inaccessible to ChatGPT as the compound is from an unpublished manuscript. The AI model exhibits its proficiency in processing and structuring intricate information by presenting it in a concise, reader-friendly table format. Please note that the structured output instructions have not been incorporated in the prompt at this stage, leading to potential variances in formatting elements (such as column headers and layout) when compared to outputs illustrated in Figure S5 and Figure S6. Solutions to these discrepancies will be thoroughly explored in the ensuing Sections S2.2 and S2.3.  \n\n# S2.2. Principle 2: Implementing Detailed Instructions  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context. If any information is not provided or you are unsure, use \"N/A\".  \n\nPlease focus on extracting experimental conditions from only the MOF synthesis and ignore information related to organic linker synthesis.  \n\nQ: What is the synthesis condition of the following compound?  \n\nContext:  \n\n![](images/6b6c8f1fb507fe3f2f1f54be33630470c5008c992c1748b256469bb5ab63fba3.jpg)  \nFigure S8. Engagement of ChatGPT in a context-specific organic synthesis discussion6, adhering to the explicit instruction to exclude organic linker synthesis, thereby minimizing ambiguity. ChatGPT's response illustrates its capability to discern the relevance of organic linker synthesis. Note that accompanying reaction schematic is not a component of the input but is provided to enhance reader comprehension.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context.  \n\nPlease summarize the following details in a table: compound name, metal source, metal amount, organic linker(s), linker amount, reaction temperature, and reaction time. If any information is not provided or you are unsure, use \"N/A\".  \n\nIf multiple conditions are provided for the same compound, use multiple rows to represent them. If multiple units or components are provided for the same factor (e.g.  g and mol for the weight, multiple linker or metals, multiple temperature and reaction time, mixed solvents, etc.), include them in the same cell and separate by comma.  \n\nContext:  \n\nSolvothermal synthesis of MOF-LA2-1: In a 4-mL scintillation vial, the linker H2PZVDC (91.0 mg, 0.5 mmol, 1 equiv.) was dissolved in N,Ndimethylformamide (DMF) (0.6 mL) upon sonication. An aqueous solution of AlCl3·6H2O (2.4 mL, 0.2 M, 1 equiv.) was added dropwise, and the resulting mixture was heated in a $120^{\\circ}C$ oven for 24 hours.  \n\n![](images/aa9870ddc8ff88ab454a6d8c3705feb75c9b590a11de6c7a59b50b4fc9144020.jpg)  \n\nGreen synthesis of MOF-LA2-1:   \nIn a 50-mL round-bottom flask, the linker H2PZVDC (364 mg, 2 mmol, 1 equiv.) and NaOH (160 mg, 4 mmol, 2 equiv.) were dissolved in deionized water ( $70m L)$ upon sonication. An aqueous solution of AlCl3·6H2O (6 mL, 0.33 M, 1 equiv.) was added dropwise for 10 minutes, and the reaction mixture was heated to $120^{\\circ}C$ and refluxed for 2 hours.  \n\n![](images/42a743487c2347760383e3bae8a818a827374dce648d02ed20d569e78f9b4a79.jpg)  \n\nChatGPT:   \n\n\n<html><body><table><tr><td>Compound Name</td><td>Metal Source</td><td>Metal Amount</td><td>Organic Linker(s)</td><td>Linker Amount</td><td>Reaction Temperature</td><td>Reaction Time</td></tr><tr><td>MOF-LA2-1</td><td>AICI3·6H20</td><td>2.4 mL, 0.2 M, 1 equiv.</td><td>H2PZVDC</td><td>91.0 mg, 0.5 mmol, 1 equiv.</td><td>120 ℃</td><td>24 hours</td></tr><tr><td>MOF-LA2-1</td><td>AICI3-6H20</td><td>6 mL, 0.33 M, 1 equiv.</td><td>H2PZVDC</td><td>364 mg, 2 mmol, 1 equiv.</td><td>120 ℃</td><td>2 hours </td></tr></table></body></html>  \n\nFigure S9. An illustration of an interaction with ChatGPT, demonstrating its capability to generate multiple synthesis paragraphs with varying parameters.6 This figure emphasizes the essential role of prompt engineering in guiding ChatGPT to manage specific scenarios effectively. It is evident that ChatGPT can comprehend detailed instructions and yield a well-organized output table. Note that included reaction schematic is not part of the input but is provided solely for the convenience of the reader.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context.  \n\nQuestion: Describe the solvent(s) in the synthesis of MOF-2.  \n\nContext:  \n\nSynthesis of MOF-2: Slow vapor diffusion at room temperature of triethylamine $(0.05~m L)$ and toluene (5 mL) into a DMF solution (2 mL) containing a mixture of Zn(NO3)2 6H2O (0.073 g, 0.246 mmol) and the acid form of BDC (0.040 g, 0.241 mmol) diluted with toluene $(8~m L)$ yields colorless prism-shaped crystals that were formulated as Zn(BDC)(DMF)(H2O).  \n\n# ChatGPT:  \n\nThe solvent used in the synthesis of MOF-2 is a mixture of DMF $(2~\\mathrm{mL},$ ) and toluene $(8~\\mathrm{mL})$ , into which triethylamine $(0.05~\\mathrm{mL}$ ) and toluene $(5~\\mathrm{mL})$ were slowly vapor diffused at room temperature.  \n\n# Human:  \n\nTask: Summarize the synthesis condition of this MOF into a table.   \nThe table should have 6 columns, all in lowercase:   \n| compound name | metal source | metal amount | linker | linker amount | reaction temperature |  \n\nChatGPT:   \n\n\n<html><body><table><tr><td>compound name</td><td>metal source</td><td>metal amount</td><td>linker</td><td>linker amount</td><td>reaction temperature</td></tr><tr><td>MOF-2</td><td>Zn(NO3)2 6H20</td><td>0.073 g (0.246 mmol)</td><td>BDC (acid form)</td><td>0.040 g (0.241 mmol)</td><td>Room temperature</td></tr></table></body></html>  \n\nFigure S10. A schematic representation of interaction with ChatGPT, highlighting the effective use of delimiters, such as $\\\"$ and ':', in prompt engineering for chemistry-related tasks. The application of these delimiters in separating input and output sections enables the AI model to process complex information, including chemical formulas, reactions, and equations, more efficiently and accurately.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Answer the question as truthfully as possible using the provided context.  \n\nTask:  \n\n1. Create a new table that includes only the MOF synthesis temperature and reaction time.   \n2. Convert temperatures to $^{\\circ}C$ .   \n3. Convert reaction times to hours.  \n\n# Context:  \n\n<html><body><table><tr><td>MOF</td><td>Ligand</td><td>Metal Source</td><td>Solvent Mixture</td><td>Solvent Volume</td><td>Temperature</td><td>Duration</td></tr><tr><td>MOF-808</td><td>H3BTC</td><td>ZrOCI2-8H20</td><td>DMF/formic acid</td><td>20 mL/20 mL</td><td>100 °℃</td><td>7 days</td></tr><tr><td>MFM-300(Cr)</td><td>H4BPTC</td><td>CrCI3-6H20</td><td>deionisedwater</td><td>100 mL</td><td>483K</td><td>3 days</td></tr><tr><td>MIL-101</td><td>H2BDC</td><td>Cr(NO3)3·9H2O</td><td>water</td><td>75 mL</td><td>493K</td><td>9 h</td></tr></table></body></html>  \n\nChatGPT: Based on the provided context, a new table including only the MOF synthesis temperature and reaction time can be created as follows:  \n\n<html><body><table><tr><td>MOF</td><td>Temperature (°C)</td><td>Reaction Time (hours)</td></tr><tr><td>MOF-808</td><td>100</td><td>168</td></tr><tr><td>MFM-300(Cr)</td><td>210</td><td>72</td></tr><tr><td>MIL-101</td><td>220</td><td>9</td></tr></table></body></html>  \n\n1. To convert the temperatures to ${}^{\\circ}{\\mathsf{C}},$ the following conversions can be made:  \n\n$$\n\\begin{array}{c}{{483\\mathrm{K}=210^{\\circ}{\\mathsf C}}}\\\\ {{493\\mathrm{K}=220^{\\circ}{\\mathsf C}}}\\end{array}\n$$  \n\n2. To convert the reaction times to hours, the following conversions can be made:  \n\n7 days $=168$ hours   \n3 days $=72$ hours   \n9 hours (already in hours)  \n\nFigure S11. An illustration of an interaction with ChatGPT, underscoring the importance of prompt engineering for guiding the AI model through a multi-step task within a single conversation. This approach optimizes ChatGPT's performance and provides detailed insights into its problem-solving process for each step.  \n\n# S2.3. Principle 3: Requesting Structured Output  \n\n# Human-ChatGPT Interaction  \n\nHuman: Task: Please summarize the following details in a table: compound, metal source, metal amount, organic linker(s), linker amount, solvent(s), solvent volume(s), reaction temperature, and reaction time. If any information is not provided or you are unsure, use \"N/A\".  \n\nOutput table should have 9 columns: | Compound name | Metal source | Metal amount | Linker | Linker amount | Solvent | Solvent volume | Reaction temperature | Reaction time |  \n\nContext:  \n\nMOF-950. A solvent mixture of H3BTAC (0.033 g, 0.11 mmol) and Zn(NO3)2·6H2O (0.25 g, 0.84 mmol) in 20 mL DMF was placed in a 20 mL screw-capped scintillation vial, which was heated at $85~^{\\circ}C$ for ${3d}.$  \n\nMOF-905. A solvent mixture of H3BTAC (0.045 g, 0.16 mmol), H2BDC (0.048 g, 0.29 mmol) and Zn(NO3)2·6H2O (0.26 g, 0.87 mmol) in 18 mL DMF and 1.8 mL ethanol was placed in a 20 mL screwcapped scintillation vial, which was heated at85 $^{\\circ}C$ for 1 d.  \n\nMOF-905-Me2. A solvent mixture of H3BTAC (0.045 g, 0.16 mmol), H2BDC-Me2 (0.067 g, 0.34 mmol) and Zn(NO3)2·6H2O (0.26 g, 0.87 mmol) in 18 mL DMF and 1.8 mL ethanol was placed in a 20 mL screw-capped scintillation vial, which was heated at $85~^{\\circ}C$ for 1 d.  \n\nMOF-905-Naph. A solvent mixture of H3BTAC (0.045 g, 0.16 mmol), H2NDC (0.074 g, 0.34 mmol) and Zn(NO3)2·6H2O (0.26 g, 0.87 mmol) in 18 mL DMF and 1.8 mL ethanol was placed in a 20 mL screwcapped scintillation vial, which was heated at $85~^{\\circ}C$ for 1 d.  \n\nMOF-905-NO2. A solvent mixture of H3BTAC (0.045 g, 0.16 mmol), H2BDC-NO2 (0.062 g, 0.29 mmol) and Zn(NO3)2·6H2O (0.26 g, 0.87 mmol) in 18 mL DMF and 1.8 mL ethanol was placed in a 20 mL screw-capped scintillation vial, which was heated at $85~^{\\circ}C$ for $36\\:h.$ .  \n\n# ChatGPT:  \n\n<html><body><table><tr><td>Compound name</td><td>Metal source</td><td>Metal amount</td><td>Linker</td><td>Linker amount</td><td>Solvent</td><td>Solvent volume</td><td>Reaction temperature</td><td>Reaction time</td></tr><tr><td>MOF-950</td><td>Zn(NO3)2·6H20</td><td>0.84 mmol</td><td>H3BTAC</td><td>0.11 mmol</td><td>DMF</td><td>20 mL</td><td>85°℃</td><td>3d</td></tr><tr><td>MOF-905</td><td>Zn(NO3)2·6H20</td><td>0.87 mmol</td><td>H3BTAC, H2BDC</td><td>0.16 mmol, 0.29mmol</td><td>DMF, ethanol</td><td>18 mL, 1.8 mL</td><td>85 °℃</td><td>1d</td></tr><tr><td>MOF-905- Me2</td><td>Zn(NO3)2·6H20</td><td>0.87 mmol</td><td>H3BTAC, H2BDC-Me2</td><td>0.16 mmol, 0.34mmol</td><td>DMF, ethanol</td><td>18 mL, 1.8 mL</td><td>85 °℃</td><td>1d</td></tr><tr><td>MOF-905- Naph</td><td>Zn(NO3)2·6H20</td><td>0.87 mmol</td><td>H3BTAC, H2NDC</td><td>0.16 mmol, 0.34mmol</td><td>DMF, ethanol</td><td>18 mL, 1.8 mL</td><td>85 °℃</td><td>1 d</td></tr><tr><td>MOF-905- NO2</td><td>Zn(NO3)2·6H20</td><td>0.87 mmol</td><td>H3BTAC, H2BDC-NO2</td><td>0.16 mmol, 0.29 mmol</td><td>DMF, ethanol</td><td>18 mL, 1.8 mL</td><td>85 ℃</td><td>36 h</td></tr></table></body></html>  \n\nFigure S12. An illustration of interaction with ChatGPT, highlighting the necessity for structured output requests. This example showcases a prompt that clarifies tasks and provides headers, thus enabling ChatGPT to understand that the task is 'tabulation'.7 The specified headers subsequently guide the AI model to generate a table in the desired format.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Task: Given the following text describing the synthesis of MOFs, please extract the MOF compound name and reaction product morphology. Present the information in a table format with a header and a new row for each compound. If the compound name, or reaction results is not mentioned in the text, use $\"\\mathsf{N}/\\mathsf{A}^{\\prime\\prime}$ .  Header: | MOF Compound | Reaction Results |  \n\nExample 1:   \nInput: Synthesis of Zn(2-mbim)2: the mixture of Zn(CH3CO2)2·2(H2O), 2-methylbenzimdazole in DMF was sealed in a 20 mL vial and heated to 120 oC for 2 days, and then cooled to room-temperature. The pale yellow polyhedral crystals were obtained.   \nOutput:   \n| MOF Compound | Reaction Results |   \n| Zn(2-mbim)2 | pale yellow polyhedral crystals| Example 2:   \nInput: The synthesis of HKUST-1 and MOF-199 was performed in a 20 mL Teflon-lined autoclave. Blue octahedral crystals were obtained for HKUST-1.   \nOutput:   \n| MOF Compound | Reaction Results |   \n| HKUST-1 | Blue octahedral crystals |   \n| MOF-199 | N/A  | Example 3:   \nInput: A mixture of H2PylDC (0.2 mol) and NaOH (0.6 mol) was dissolved and heated to ${120^{\\circ}C}.$ After refluxing for 6 hours, the white crystalline powder was collected.   \nOutput:   \n| MOF Compound | Reaction Results |   \n| N/A | white crystalline powder |  \n\nPlease provide the extracted information for the following text: ZIF-21: A solid mixture of cobalt(II) nitrate hexahydrate Co(NO3)2·6H2O and purine was dissolved in DMF in a 20 ml vial. To the solution, 2.0M dimethylamine solution in methanol was added. The vial was tightly capped and heated at $85~\\circ C$ in an isothermal oven for $24h$ to yield purple octrahedral crystals.  \n\n# ChatGPT:  \n\n| MOF Compound | Reaction Results | | ZIF-21 | purple octahedral crystals |  \n\nFigure S13. An illustration of interaction with ChatGPT using few-shot prompts. In contrast to the zero-shot approach, ChatGPT is presented with several example contexts along with their corresponding correct responses. This process aids the model in identifying and extracting specific parameters such as synthesis vessel type and crystal class, tasks that would otherwise be challenging to accomplish without the provision of examples.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Task: Answer the question as truthfully as possible using the provided context. Please summarize the following details in a table: compound name or chemical formula (if the name is not provided), metal source, metal amount, organic linker(s), linker amount, modulator, modulator amount or volume, solvent(s), solvent volume(s), reaction temperature, and reaction time. If any information is not provided or you are unsure, use \"N/A\".  \n\nThe table should have 11 columns, all in lowercase:| compound name | metal source | metal amount | linker | linker amount | modulator | modulator amount or volume | solvent | solvent volume | reaction temperature | reaction time |  \n\nContext:  \n\nOptimized Synthesis of MOF-521-H Single Crystals. In a 4 mL scintillation vial, 3.0 mL DMF was added to dissolve Al(NO3)3·9H2O (18.0 mg, 0.048 mmol) and H3BTB (15.8 mg, 0.036 mmol). After adding 0.24 mL formic acid and 30 µL deionized water to the solution, the vial was capped and placed in the preheated $140^{\\circ}C$ for 2 days, and colorless needle-shape crystals were obtained. The crystals were washed with MeOH $(3\\times20m L)$ for 3 days before activation and characterization. Full activation of the MOF was conducted under dynamic vacuum (\\~10-3 mbar) at ${100^{\\circ}C}$ for 12 hours, yielding pure and desolvated product.  \n\n# ChatGPT:  \n\n<html><body><table><tr><td>comamund</td><td> mural </td><td> ametant</td><td>linker</td><td>alinkert</td><td>modulator</td><td></td><td> solvent</td><td> voluemnt</td><td>tereactiure</td><td>reaction</td></tr><tr><td>MOF-521- H</td><td>Al(NO3)3- 9H20</td><td>18.0 mg</td><td>H3BTB</td><td>15.8 mg</td><td>Formic acid</td><td>0.24 mL</td><td>DMF</td><td>3.0 mL</td><td>140 °℃</td><td>2 days</td></tr></table></body></html>  \n\nFigure S14. An illustration of interaction with ChatGPT, demonstrating how specific instructions can direct the AI model to generate a table in a desired format. In contrast to the prompt displayed in Figure S12, the number and nomenclature of headers can be effortlessly altered through natural language commands, thereby modifying the output. It should be noted that the text utilized in this example originates from an unpublished manuscript, which was not previously introduced to ChatGPT. The structured output directive effectively guides the AI model to handle various synthesis parameters derived from text sources, encompassing both published manuscripts and unpublished works.  \n\n# S2.4. Interactive Prompt Refinement  \n\n# Human-ChatGPT Interaction  \n\nHuman: I am going to write a prompt to instruct ChatGPT to help me extract the MOF linker and its amount in the synthesis from the text I provide. Could you please begin by asking me a few questions to gather the necessary information to include in the prompt? Do you understand?  \n\nChatGPT (GPT-4): Yes, I understand. I'll ask you a few questions to gather the necessary information for creating the prompt.  \n\n1. What specific text or document are you planning to provide for ChatGPT to extract the MOF linker and its amount from? Is it a research article, a patent, or some other type of document?   \n2. Are there any specific types or classes of MOF linkers that you want ChatGPT to focus on?   \n3. Do you want ChatGPT to provide the information in a particular format or style, such as a table, bullet points, or a summary?   \n4. Are there any additional details or context you'd like ChatGPT to consider when extracting the information?  \n\n# Human:  \n\n1. This will be a section from a research article.   \n2. No, you should extract the linker as long as it is involved in the reaction to form the MOF.   \n3. Yes, please make it a table and ensure that the table has a fixed header for each output.   \n4. If multiple linkers are presented, please separate them by semicolons within the same cell. Do   \nthe same for their amount and ensure that the order is consistent.  \n\nChatGPT (GPT-4): Thank you for providing the necessary information. Based on your responses, I'll create a prompt for you to instruct ChatGPT to help you extract MOF linker and its amount in the synthesis from a section of a research article. Here's the prompt:  \n\n\"Please help me extract the MOF linker and its amount used in the synthesis from the following text taken from a research article. Present the information in a table format with fixed headers: 'MOF Linker' and 'Amount'. If multiple linkers are present, separate them using a semicolon (;) in the same cell. Do the same for their corresponding amounts, ensuring the order is consistent between linkers and amounts.\"  \n\nFigure S15. A conversation with ChatGPT (GPT-4) for the interactive prompt refinement process, highlighting the nitial engagement with GPT-4 to develop a guiding prompt, followed by iterative discussions and refinements to generate a testable prompt for subsequent evaluation with another in ance of ChatGPT (GPT-4) model.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Please help me extract the MOF linker and its amount used in the synthesis from the following text taken from a research article. Present the information in a table format with fixed headers: 'MOF Linker' and 'Amount'. If multiple linkers are present, separate them using a semicolon (;) in the same cell. Do the same for their corresponding amounts, ensuring the order is consistent between linkers and amounts  \n\n# Context:  \n\nZIF-20: Zn(Pur)2·(DMF)0.75(H2O)1.5. A solid mixture of zinc nitrate tetrahydrate Zn(NO3)2·4H2O (65 mg, 0. 25 mmol, EM Science) and purine (150 mg, 1.25 mmol, Fluka or Sigma) was dissolved in 5 ml DMF (Fisher) in a 20 ml vial to obtain a slightly turbid solution. The vial was tightly capped and heated at $65~\\circ C$ in an isothermal oven for 3 days to yield pale yellow octahedral crystals.  \n\nChatGPT:   \n\n\n<html><body><table><tr><td>MOFLinker</td><td>Amount</td></tr><tr><td>Purine</td><td>1.25 mmol</td></tr><tr><td>Zn(NO3)2·4H20</td><td>0.25 mmol</td></tr></table></body></html>  \n\nFigure S16. A conversation illustrating the evaluation of a preliminarily devised prompt. The process involves integrating the prompt within a given context and assessing the outcomes. It was observed that the initial prompt did not account for potential metal sources, leading to erroneous classification of these sources as linkers in the synthetic process.8  \n\n# Human-ChatGPT Interaction  \n\n---Continued from previous conversation shown in Figure S15---  \n\nChatGPT (GPT-4):  \\*\\*previous prompt\\*\\*  \n\nHuman: When I use this prompt, I sometimes find that the metal source, such as \"Zn(NO3)2·4H2O\", is mistakenly counted as a MOF linker. In addition, there is no header. Could you please revise the prompt to fix this issue?  \n\nChatGPT (GPT-4): I apologize for the confusion. I have revised the prompt to explicitly mention that metal sources should not be considered as MOF linkers. Here's the updated prompt:  \n\n\"Please help me extract the organic MOF linker and its amount used in the synthesis from the following text taken from a research article. Exclude any metal sources or inorganic components, such as Co(NO3)2·4H2O, AlCl3, and HfOCl2, focusing only on the organic linkers. Present the information in a table format with fixed headers: 'MOF Linker' and 'Amount'. If multiple linkers are present, separate them using a semicolon (;) in the same cell. Do the same for their corresponding amounts, ensuring the order is consistent between linkers and amounts.\"  \n\nFigure S17. A continued conversation following on from Figure S15 with ChatGPT (GPT-4), di cussing the evaluation outcomes of a preliminary prompt, providing feedback on identified in ccuracies, and suggesting areas of potential improvement. ChatGPT responds by generating a re ned prompt, thereby addressing the initially identified issues.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Please help me extract the organic MOF linker and its amount used in the synthesis from the following text taken from a research article. Exclude any metal sources or inorganic components, such as Co(NO3)2·4H2O, AlCl3, and HfOCl2, focusing only on the organic linkers. Present the information in a table format with fixed headers: 'MOF Linker' and 'Amount'. If multiple linkers are present, separate them using a semicolon (;) in the same cell. Do the same for their corresponding amounts, ensuring the order is consistent between linkers and amounts.  \n\n# Context:  \n\nZIF-20: Zn(Pur)2·(DMF)0.75(H2O)1.5. A solid mixture of zinc nitrate tetrahydrate Zn(NO3)2·4H2O (65 mg, 0. 25 mmol, EM Science) and purine (150 mg, 1.25 mmol, Fluka or Sigma) was dissolved in 5 ml DMF (Fisher) in a 20 ml vial to obtain a slightly turbid solution. The vial was tightly capped and heated at $65~\\circ C$ in an isothermal oven for 3 days to yield pale yellow octahedral crystals.  \n\nChatGPT:   \n\n\n<html><body><table><tr><td>MOF Linker</td><td>Amount</td></tr><tr><td>purine</td><td>150 mg (1.25 mmol)</td></tr></table></body></html>  \n\nFigure S18. A new conversation representing the ongoing evaluation of the updated prompt from Figure S17. The previously highlighted issue regarding the metal source was rectified, but a new challenge, specifically related to the formatting of the linker amount, surfaced during the process.  \n\n# Human-ChatGPT Interaction  \n\n---Continued from previous conversation shown in Figure S17---  \n\nChatGPT (GPT-4): \\*\\*revised prompt\\*\\*  \n\nHuman: Upon examination, it has come to my attention that the weight of the linker is sometimes expressed as \"150 mg (1.25 mmol),\" which presents both molar (mol) and mass (g) units. To ensure consistency, I kindly request that when indicating the amount of the linker, the use of mol units is preferred over g units. Moreover, it is important to display only one unit of measurement, either mol or g, when presenting the amount. Please note that the terms g and mol encompass all units of measurement, such as mg, g, kg, μg, mmol, mol, and $\\upmu\\mathrm{mol}$ . Thank you.  \n\nChatGPT (GPT-4): Thank you for the clarification. I have updated the prompt to prioritize the mole unit over the mass unit when extracting the amount of the MOF linker. Here's the revised prompt: \"Please help me extract the organic MOF linker and its amount used in the synthesis from the following text taken from a research article. Exclude any metal sources or inorganic components, such as Co(NO3)2·4H2O, AlCl3, and HfOCl2, focusing only on the organic linkers. When extracting the amount, prioritize the mole unit (e.g., mmol, mol, μmol) over the mass unit (e.g., mg, g, kg, μg) and display only one of them. Present the information in a table format with fixed headers: 'MOF Linker' and 'Amount'. If multiple linkers are present, separate them using a semicolon (;) in the same cell. Do the same for their  \n\nFigure S19. An ongoing dialogue with ChatGPT (GPT-4), detailing observed issues related to fo matting and initiating a request for further refinement.  \n\n# Human-ChatGPT Interaction  \n\nHuman: Please help me extract the organic MOF linker and its amount used in the synthesis from the following text taken from a research article. Exclude any metal sources or inorganic components, such as Co(NO3)2·4H2O, AlCl3, and HfOCl2, focusing only on the organic linkers. When extracting the amount, prioritize the mole unit (e.g., mmol, mol, μmol) over the mass unit (e.g., mg, g, kg, μg) and display only one of them. Present the information in a table format with fixed headers: 'MOF Linker' and 'Amount'. If multiple linkers are present, separate them using a semicolon (;) in the same cell. Do the same for their corresponding amounts, ensuring the order is consistent between linkers and amounts.  \n\n# Context:  \n\nZIF-20: Zn(Pur)2·(DMF)0.75(H2O)1.5. A solid mixture of zinc nitrate tetrahydrate Zn(NO3)2·4H2O (65 mg, 0. 25 mmol, EM Science) and purine (150 mg, 1.25 mmol, Fluka or Sigma) was dissolved in 5 ml DMF (Fisher) in a 20 ml vial to obtain a slightly turbid solution. The vial was tightly capped and heated at $65~\\circ C$ in an isothermal oven for 3 days to yield pale yellow octahedral crystals.  \n\nChatGPT:   \n\n\n<html><body><table><tr><td>MOF Linker</td><td>Amount</td></tr><tr><td>purine</td><td>1.25 mmol</td></tr></table></body></html>  \n\nFigure S20. A conversation with ChatGPT, examining the outcomes of the further refined prompt shown in Figure S19. All previously identified issues were found to be effectively addressed.  \n\n# Human-ChatGPT Interaction  \n\nHuman: \\*\\*revised prompt shown in Figure $S19^{\\star\\star}$   \nContext:   \nMOF-905. A solvent mixture of H3BTAC (0.045 g, 0.16 mmol), H2BDC (0.048 g, 0.29 mmol) and Zn(NO3)2·6H2O (0.26 g, 0.87 mmol) in 18 mL DMF and 1.8 mL ethanol was placed in a 20 mL screwcapped scintillation vial, which was heated at85 $^{\\circ}C$ for 1 d.  \n\nChatGPT:   \n\n\n<html><body><table><tr><td>MOF Linker</td><td>Amount</td></tr><tr><td>H3BTAC; H2BDC</td><td>0.16 mmol; 0.29 mmol</td></tr></table></body></html>  \n\nHuman: \\*\\*revised prompt shown in Figure S19\\*\\*  \n\nContext:  \n\nSolvothermal synthesis of MOF-LA2-1: In a 4-mL scintillation vial, the linker H2PZVDC (91.0 mg, 0.5 mmol, 1 equiv.) was dissolved in N,Ndimethylformamide (DMF) (0.6 mL) upon sonication. An aqueous solution of AlCl3·6H2O (2.4 mL, 0.2 M, 1 equiv.) was added dropwise, and the resulting mixture was heated in a $120^{\\circ}C$ oven for 24 hours.  \n\nChatGPT:  \n\n<html><body><table><tr><td>MOF Linker</td><td>Amount</td></tr><tr><td>H2PZVDC</td><td>0.5 mmol</td></tr></table></body></html>  \n\nHuman: \\*\\*revised prompt shown in Figure S19\\*\\*  \n\nContext:  \n\n[Al(C6H2O4S)(OH)] ∙ xH2O, CAU-23. 4.30 g H2TDC (25 mmol) was mixed with $2.0\\ g$ (50 mmol) sodium hydroxide in 100 mL distilled water until a clear solution of Na2TDC was obtained. After adding 18.75 mL of aqueous aluminum chloride solution (1 mol/L, 18.75 mmol), and $12.5\\:m L$ of aqueous sodium aluminate solution (0.5 mol/L, 6.25 mmol), the slurry was stirred under reflux conditions for 6 h,and then filtered off and dried at $100^{\\circ}C$ for 4 h.  \n\n# ChatGPT:  \n\n<html><body><table><tr><td>MOF Linker</td><td>Amount</td></tr><tr><td>H2TDC</td><td>25 mmol</td></tr></table></body></html>  \n\nFigure S21. An extended discussion with ChatGPT, continually testing the refined prompt across various scenarios. The consistent outcomes highlight the robustness of the finalized prompt. The interactive prompt engineering process is deemed completed once user satisfaction is achieved. If new concerns or issues arise, the process loops back to further refine the prompt through continued dialogue.  \n\n# Section S3. Text Mining with ChatGPT API  \n\n# S3.1. Summarizing Synthesis Conditions with ChatGPT  \n\n# ChatGPT Processing Demo  \n\nPrompt: Answer the question as truthfully as possible using the provided context. Please summarize the following details in a table: compound name or chemical formula (if the name is not provided), metal source, metal amount, organic linker(s), linker amount, modulator, modulator amount or volume, solvent(s), solvent volume(s), reaction temperature, and reaction time. If any information is not provided or you are unsure, use \"N/A\".  \n\nPlease focus on extracting experimental conditions from only the MOF synthesis and ignore information related to organic linker synthesis, MOF postsynthetic modification or metalation, high throughput (HT) experiment details, or catalysis reactions.  \n\nIf multiple conditions are provided for the same compound, use multiple rows to represent them. If multiple units or components are provided for the same factor (e.g.  g and mol for the weight, multiple linker or metals, multiple temperature and reaction time, mixed solvents, etc.), include them in the same cell and separate by comma.  \n\nThe table should have 11 columns, all in lowercase:| compound name | metal source | metal amount | linker | linker amount | modulator | modulator amount or volume | solvent | solvent volume | reaction temperature | reaction time |  \n\nInput:  \n\nSynthesis of MOF-808. In a 100 mL media bottle were dissolved 1,3,5-benzenetricarboxylic acid (210 mg) and ZrOCl2·8H2O (970 mg) in a solution containing DMF $(30m L)$ and formic acid $(30m L)$ . The bottle was sealed and heated in a $100^{\\circ}C$ isothermal oven for a day. White powder was collected by centrifugation (8000 rpm, 3 min), washed with DMF three times $(60\\:m L\\times3)$ over a 24 h period, and then washed with acetone three times $(60m L\\times3)$ over a $24h$ period.  \n\n# Output:  \n\n<html><body><table><tr><td rowspan=\"2\">compound</td><td rowspan=\"2\">mural </td><td rowspan=\"2\">metunt</td><td rowspan=\"2\">linker</td><td rowspan=\"2\">linkeunt</td><td rowspan=\"2\">modulator</td><td rowspan=\"2\">modulator amount orsolventolumtteature volume</td><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td rowspan=\"2\"></td><td rowspan=\"2\">reaction</td></tr><tr><td></td></tr><tr><td>MOF-808</td><td>ZrOCl2-8H20</td><td>970 mg</td><td>1,3,5- benzenetricarboxylic 210 mg acid</td><td></td><td>formic acid</td><td>30 mL</td><td>DMF</td><td>30 mL</td><td>100 ℃</td><td>24 h </td></tr></table></body></html>  \n\nFigure S22. An illustration of prompt components, designed to guide ChatGPT in summarizing synthesis conditions, with reference to the three fundamental principles of chemistry prompt engineering. This illustration includes an example input and output table for context. It should be noted that this is a demonstration and various prompt versions could be constructed based on specific research needs.  \n\n# ChatGPT Processing Demo  \n\nPrompt: Given the following text describing the synthesis of MOFs, please extract the MOF compound name, synthesis equipment/reaction vessel, reaction type, and reaction product morphology. Present the information in a table format with a header and a new row for each compound. If the compound name, equipment, or reaction results are not mentioned in the text, use \"N/A.\" Determine the reaction type based on the following keywords and prioritize them in this order: Diffusion, Microwave, Conventional, Solvothermal.  \n\nExample 1:   \nInput: Synthesis of Zn(2-mbim)2:the mixture of Zn(CH3CO2)2·2(H2O), 2-methylbenzimdazole in DMF was sealed in a 20 mL vial and heated to 120 oC for 2 days, and then cooled to room-temperature. The pale yellow polyhedral crystals were obtained. Output:   \nMOF Compound | Equipment | Reaction Type | Reaction Results   \nZn(2-mbim)2 | 20 mL vial | Solvothermal | pale yellow polyhedral crystals   \nExample 2:   \nInput: The synthesis of HKUST-1 and MOF-199 was performed in a 20 mL Teflon-lined autoclave. Blue octahedral crystals were obtained for HKUST-1, while MOF-199 yielded a green crystalline powder.   \nOutput:   \nMOF Compound | Equipment | Reaction Type | Reaction Results   \nHKUST-1 | 20 mL Teflon-lined autoclave | Solvothermal | Blue octahedral crystals   \nMOF-199 | 20 mL Teflon-lined autoclave | Solvothermal | Green crystalline powder   \nExample 3:   \nInput: MOF-313. In a 1 L glass round bottom flask, a mixture of H2PylDC (0.2 mol) and NaOH (0.6 mol) was dissolved. The resulting solution was stirred for 10 minutes until all the solids were completely dissolved. Afterward, the reaction mixture was heated to ${120^{\\circ}C}.$ After refluxing for 6 hours, the white crystalline powder was collected.   \nOutput:   \nMOF Compound | Equipment | Reaction Type | Reaction Results   \nMOF-313 | 1 L glass round bottom flask | Conventional | white crystalline powder   \nExample 4:   \nInput: H3BTB and Bi(NO3)3·5H2O were mixed into 30 ml microwave glass reaction vessel. The reaction mixture homogenized and heated to $120^{\\circ}C$ for 20 min in a microwave synthesizer. The mixture was stirred with a magnetic stirring bar during the reaction. The yellow precipitated of CAU-7 was filtered, washed with methanol.   \nOutput:   \nMOF Compound | Equipment | Reaction Type | Reaction Results   \nCAU-7 | 30 ml microwave glass reaction vessel | Microwave | yellow precipitated  \n\nPlease provide the extracted information for the following text:  \n\nInput:  \n\nSynthesis of ZIF-1001: A mixture of Zn(NO3)2·4H2O (1.2 mmol), HbTZ (1.68 mmol), HbIM (0.75 mmol) was dissolved in DMF $72m L)$ under ultrasound. The solution was capped in a 20-ml glass vial and heated at $100^{\\circ}C$ for 48 h. White prism-shaped crystals were collected and washed with DMF $(6\\times10$ ml). (Yield: $30\\%$ based on Zn).  \n\n# Output:  \n\n<html><body><table><tr><td>MOF Compound</td><td>Equipment</td><td>Reaction Type</td><td>Reaction Results</td></tr><tr><td>ZIF-1001</td><td>20-ml glass vial</td><td>Solvothermal</td><td>White prism-shaped crystals</td></tr></table></body></html>  \n\nFigure S23. An illustration of prompt components, designed to guide ChatGPT in summarizing reaction equipment and reaction results using the few-shot prompt strategy. This illustration includes an example input and output table for context.  \n\n# S3.2. Classifying Research Article Sections with ChatGPT  \n\n# ChatGPT Processing Demo  \n\nPrompt: Determine whether the provided context includes a comprehensive MOF synthesis with explicit reactant quantities or solvent volumes, and answer with either Yes or No.  \n\nContext: In a 4-mL scintillation vial, the linker H2PZVDC (91.0 mg, 0.5 mmol, 1 equiv.) was dissolved in N,N-dimethylformamide (DMF) (0.6 mL) upon sonication. An aqueous solution of AlCl3·6H2O $(2.4~m L,$ 0.2 M) was added dropwise, and the resulting mixture was heated in a $120^{\\circ}C$ oven for 24 hours. Question: Does the section contain a comprehensive MOF synthesis with explicit reactant quantities or solvent volumes?  \n\nAnswer: Yes.  \n\nContext: A 0.150 M solution of imidazole in DMF and a 0.075M solution of Zn(NO3)2·4H2O in DMF were used as stock solutions, and heated in an $85~^{\\circ}C$ isothermal oven for 3 days.   \nQuestion: Does the section contain a comprehensive MOF synthesis with explicit reactant quantities or solvent volumes?   \nAnswer: Yes. Context: Solvothermal reactions of Co(NO3)·6H2O, Hatz, and L1/L2 in a 2:2:1 molar ratio in DMF solvent at $180^{\\circ}C$ for $24h$ yielded two crystalline products, 1 and 2, respectively.   \nQuestion: Does the section contain a comprehensive MOF synthesis with explicit reactant quantities or solvent volumes?   \nAnswer: No. Context: A $22.9\\%$ weight loss was observed from 115 to $350^{\\circ}C,$ which corresponds to the loss of one DEF molecule per formula unit (calcd: $23.5\\%$ .   \nQuestion: Does the section contain a comprehensive MOF synthesis with explicit reactant quantities or solvent volumes?   \nAnswer: No. Input #1:   \nIn a 100 mL media bottle were dissolved 1,3,5-benzenetricarboxylic acid (210 mg) and ZrOCl2·8H2O (970 mg) in a solution containing DMF (30 mL) and formic acid (30 mL).  \n\nOutput #1: Yes.  \n\nInput #2: Single crystal X-ray analyses were performed at room temperature on a Siemens SMART platform diffractometer outfitted with an APEX II area detector and monochromatized Mo Kα radiation.  \n\nOutput #2: No.  \n\nFigure S24. An illustration of use of few-shot prompts to guide ChatGPT in classifying synthesis paragraphs. Two example inputs and their respective outputs are provided for clarification.  \n\n# S3.3. Filtering Text using OpenAI Embeddings  \n\n# Embedding Demo  \n\nPrompt Embedding (ada-002): Identify the experimental section or synthesis method. This section should cover essential information such as the compound name (e.g., MOF-5, ZIF-1, Cu(Bpdc), compound 1, etc.), metal source (e.g., ZrCl4, CuCl2, AlCl3, zinc nitrate, iron acetate, etc.), organic linker (e.g., terephthalate acid, H2BDC, H2PZDC, H4Por, etc.), amount (e.g., 25mg, 1.02g, 100mmol, 0.2mol, etc.), solvent (e.g., N,N Dimethylformamide, DMF, DCM, DEF, NMP, water, EtOH, etc.), solvent volume (e.g., $12\\mathsf{m L}$ , 100mL, 1L, 0.1mL, etc.), reaction temperature (e.g., $120^{\\circ}\\mathsf C,$ 293K, 100C, room temperature, reflux, etc.), and reaction time (e.g., 120h, 1 day, 1d, 1h, 0.5h, 30min, a week, etc.).  \n\n$$\n\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\n$$  \n\nInput #1:  \n\nThe synthesis of $Z r$ -DTDC was performed by adding ZrCl 4 (0.466 g), H2DTDC (1.025 g), and hydrochloric acid $(0.33m l)$ into DMF $(12m l)$ at room temperature. The slurry was transferred to $20m l$ Teflon-lined steel autoclave . The autoclave was placed in an oven with $2^{\\circ}C,$ /min hearting up to $220^{\\circ}C,$ then held at $220^{\\circ}C$ for $76~h$ .  \n\nOutput #1: 0.8771 (Yes)  \n\nInput #2:   \nFurukawa, Nakeun Ko, Yong Bok Go, Naoki Aratani, Sang Beom Choi, Eunwoo Choi, A. Özgür Yazaydin, Randall Q. Snurr, Michael O’Keeffe, Jaheon Ki.  \n\nOutput #2: 0.6651 (No)  \n\nInput #3: Thus, a fixed bed was packed with activated ZIF-204 and subsequently subjected to a binary gas mixture containing CO 2 $35\\%,$ v/v) and CH 4 $65\\%,$ v/v) at room temperature. It is worthwhile to note that the composition of this binary gas mixture was chosen to simulate the typical volumetric percentage of CO 2 and CH4 found in biogas sources produced from the decomposition of organic matter.  \n\nOutput #3: 0.7904 (No)  \n\n$$\n\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_}\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\n$$  \n\nFigure S25. An illustration of the prompt to be converted into embeddings to facilitate the search and filter process for synthesis paragraphs. This is achieved by determining the semantic similarity between the context and the prompt, assessed via cosine similarity scores ranging between 0 and 1. A high score (denoted as \"Yes\" in the output) implies the paragraph is relevant and retained, while a low score (denoted as \"No\") leads to the exclusion of the paragraph. As with previous figures, multiple prompt versions could be used, and the details are largely dependent on the specific requirements of the study.  \n\n# S3.4. Batch Text Processing with ChatGPT API  \n\nIn the preceding sections, we have demonstrated how fixed prompts with context derived from research articles can guide ChatGPT in performing non-dialogue text processing tasks such as summarization and classification.  \n\nThe ChatGPT API offers a notable advantage over web-based interactions with the model. This advantage lies in its ability to iterate over multiple text inputs using a programming construct like a 'for' loop. This approach enables concurrent processing of a multitude of requests, thus enhancing efficiency for large-scale tasks.  \n\n# ChatGPT API  \n\n# import openai  \n\n![](images/e1a1dd1657c76b17f7d9814b8fbbd492327bee75621d9ad2b246ed84a3288c0c.jpg)  \nFigure S26. An illustration of Python code utilizing the ChatGPT API to ask the question on the synthesis condition of MOF-5 and displaying the subsequent output. This output aligns closely with responses generated via the web-based ChatGPT interface.  \n\n#  \n\n![](images/f35aa0be9b5c7e2bf29b69fb2f8faa325826180cde6c23aff79d49cafd4f3a7e.jpg)  \n\nFigure S27. Python code demonstrating the iterative question-asking functionality made possible through the use of a 'for' loop. The ChatGPT API enables efficient batch processing of text, allowing a pre-compiled series of inputs to be queried rapidly, which otherwise is a process significantly more time-consuming on the web-based ChatGPT due to manual input requirements.  \n\n# ChatGPT API  \n\n![](images/1dcd8d25a630c63fbe460d8cbf1459e0df57e82a5ff1986e020523f9914f1abe.jpg)  \nengage the ChatGPT API. Preloaded paragraphs serve as the context, with the Python code assembling the prompts and context into a series of inputs. Once the model generates an answer, it is collected, and the process moves on to the next question automatically. The responses from the API are string-based; therefore, to represent answers in a tabular format, \"|\" and \"-\" symbols are used. To maintain clarity and brevity, full prompts and context details are not shown here. Please refer to Figure S22 for complete prompt information.  \n\n# ChatGPT API  \n\ncontexts $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ [  \n\n![](images/71acad397eba4511877d0c349debcd5d5ff6e83e255982f70a310131ac5ed56d.jpg)  \n\nFigure S29. Illustration of performing batch synthesis paragraph classification using a 'for' loop in Python to engage the ChatGPT API. As with Process 1, preloaded paragraphs provide context, and the Python code prepares the prompts and context as inputs. Answers are collected automatically and the process continues to the subsequent question. In this example, ChatGPT has been directed to respond with \"Yes\" or \"No\" for classification purposes, following the instructions provided in the fixed prompt. To ensure clarity and brevity, full prompts and context details are omitted here. Please refer to Figure S23 for complete prompt information.  \n\n# S3.5. Generating Python Code with ChatGPT  \n\nPaper Parsing  \n\nAs the first step towards enabling ChatGPT to autonomously parse and read academic papers, it is necessary to convert the PDF files into text strings, which can subsequently be incorporated into the prompt as context and processed using the ChatGPT API. This conversion can be achieved by employing a Python function, as illustrated in the following figures.  \n\n# ChatGPT Code Synthesis  \n\nHuman: Write a Python code to extract text from multiple research article PDF files. For each page in each file, divide the text into four sections and record their page and section numbers (1-4). Count the tokens for each section and construct a dataframe with the following columns: 'file name', 'page number', section number', 'content', and 'tokens'. The input will be a list of PDF file names as strings, and the output should be a dataframe.  \n\n# ChatGPT (GPT-4):  \n\nimport PyPDF2   \nimport pandas as pd   \nfrom nltk.tokenize import word_tokenize   \nimport nltk   \nnltk.download('punkt') # Download the Punkt tokenizer for tokenizing  \n\n![](images/97f06ae66d75983d8830684d4169c5a96eeaebe84048b8c1755b06116d0b0edf.jpg)  \nFigure S30. Demonstration of the interaction with ChatGPT (GPT-4) to generate a Python function capable of parsing academic papers. The function extracts text from multiple PDF files, dividing each page into four sections and recording their respective page and section numbers.  \n\nIt should be noted that while the majority of the time ChatGPT generates code with fluent and seemingly correct syntax, the code might not always execute as expected. In such instances, ChatGPT can be interactively tasked with debugging by providing the error messages received when executing the code.  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous conversation shown in Figure S30---  \n\nHuman: DeprecationError: reader.getNumPages is deprecated and was removed in PyPDF2 3.0.0. Use len(reader.pages) instead.  \n\n# ChatGPT (GPT-4):  \n\nApologies for the confusion, and thank you for pointing out the change.  \n\nYou're right, in the recent versions of PyPDF2, getNumPages has been replaced with len(reader.pages). Here's the corrected code:  \n\n![](images/5544a8ce7ee513b2a8f6227d52f8642c51d58876276949ab7a66f7ad5cafc5e4.jpg)  \n\nHuman: DeprecationError: reader.getPage(pageNumber) is deprecated and was removed in PyPDF2 3.0.0. Use reader.pages[page_number] instead.  \n\n# ChatGPT (GPT-4):  \n\nApologies for the confusion, and thank you for the correction. Here's the updated code:  \n\nFigure S31. Illustration of the debugging process, where human simply copies and pastes the error message and ChatGPT (GPT-4) assists in revising code that had issues during execution. Please note that for brevity, only the revised sections of the code are shown.  \n\nFurthermore, users can direct the code implementation using narrative language to articulate the request. This feature allows the user to specify the use of targeted models or functions.  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous conversation shown in Figure S31---   \nHuman: Revise the code to use OpenAI’s tiktoken \"cl100k_base\" to count for tokens   \nChatGPT (GPT-4):   \nTo count tokens using the OpenAI's tiktoken library, you first need to install it via pip:   \npip install tiktoken   \nHere's how you can modify the function to use tiktoken:   \nimport PyPDF2   \nimport pandas as pd   \nimport tiktoken   \ndef count_tokens(text): encoding $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ tiktoken.get_encoding(\"cl100k_base\") num_tokens $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ len(encoding.encode(text)) return num_tokens   \ndef get_txt_from_pdf(pdf_files): data $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ [] # Create an empty list to store the data for pdf in pdf_files: $\\#$ Iterate over the PDF with open(pdf, 'rb') as pdf_content: # Fetch the PDF content from the pdf pdf_reader $\\mathbf{\\Psi}=\\mathbf{\\Psi}$ PyPDF2.PdfReader(pdf_content) # Create a PDF reader object for page_num in range(len(pdf_reader.pages)): # Iterate over all the pages page $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ pdf_reader.pages[page_num] # Extract the text from the current page page_text $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ page.extract_text() words $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ page_text.split() # Split the page text into individual words page_text_join $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ ' '.join(words) # Join the words page_len $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ len(page_text_join) div_len $\\mathbf{\\Psi}=\\mathbf{\\Psi}$ page_len // 4 # Divide the page into 4 parts page_parts $\\mathbf{\\Psi}=\\mathbf{\\Psi}$ [page_text_join[i\\*div_len:(i+1)\\*div_len] for i in range(4)] for i, page_part in enumerate(page_parts): # Append the data to the list data.append({ 'file name': pdf, 'page number': page_num + 1, 'page section': i+1, 'content': page_part, 'tokens': count_tokens(page_part) }) return pd.DataFrame(data)  \n\n# Text Encoding  \n\nUpon obtaining text segments from a research paper and storing them in a DataFrame, the execution of Process 3 requires OpenAI embeddings of each segments. This can be achieved by instructing ChatGPT about the DataFrame's contents and providing it with an exemplary code snippet derived from OpenAI's official documentation, which illustrates how to engage the OpenAI model for embedding conversion.  \n\n# ChatGPT Code Synthesis  \n\nHuman: Task: Write a function called add_embedding  \n\nYou need to implement a Python function called add_embedding that takes a dataframe as input and returns a new dataframe with an additional column called 'embedding'. The 'embedding' column will contain the embeddings generated from the strings in the 'content' column.  \n\nTo generate the embeddings, you can use the following code:  \n\nresponse $\\mathbf{\\tau}=\\mathbf{\\tau}$ openai.Embedding.create(input $\\c=$ \"Your text string goes here\", model $\\c=$ \"text-embedding-ada-002\") embeddings $\\mathbf{\\tau}=\\mathbf{\\tau}$ response['data'][0]['embedding']  \n\n![](images/a227d3b1958c804f146196d8de1a47dd6311135a02ff2bbd0f68c1762cd28351.jpg)  \nFigure S33. Demonstration of GPT-4's auto-generation of a function to append embeddings to all research paper segments in a DataFrame. This process is enabled by describing the Dataframe's contents and demonstrating the use of an example code from OpenAI's official documentation. Please note that the placeholder 'your-api-key' should be replaced with the user's actual OpenAI API key.  \n\nTo circumvent the need for recurring embedding conversions for the same paper pool, the embeddings can be stored locally for subsequent reloading. This process can be facilitated by informing ChatGPT about the requirement and requesting it to generate an additional code that allows for the conversion between the DataFrame and locally saved CSV files storing the embeddings. Moreover, the function should be adapted to ensure that if the DataFrame already contains embeddings, no further actions are taken.  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous conversation shown in Figure S33---  \n\n![](images/3dfc75782b9d1d95228d6dab556fa7ebdc79828561bddabe4ef96600794d3f7a.jpg)  \nFigure S34. Depiction of code modifications and implementations by ChatGPT (GPT-4) to append functions for the storage and loading of embeddings.  \n\nNonetheless, the manner in which ChatGPT stores and loads embeddings is rudimentary, leading to type inconsistency issues when attempting to utilize reloaded embeddings from the CSV files. This arises from the storage of embeddings as text in the CSV files, and upon reloading, ChatGPT, unaware of the embeddings' type, simply reloads the CSV files. This issue can be readily resolved by informing ChatGPT about the accurate form of embeddings and requesting it to revise the function accordingly.  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous conversation shown in Figure S34---  \n\nHuman: I noticed that df['embedding'] is a string instead of a list of floats after loading it from a CSV.   \nPlease modify the code to rectify this issue.  \n\n# ChatGPT (GPT-4):  \n\nYou're right. When data is saved to a CSV file, the list of floats in the 'embedding' column is converted into a string representation of the list. When this data is read back into a DataFrame, the 'embedding' column consists of strings that need to be converted back into lists of floats.  \n\nWe can achieve this by modifying the load_df_from_csv function to include conversion of the 'embedding' column back to lists of floats using the ast.literal_eval function.  \n\nHere's the modified code:  \n\nAfter having a dataframe with segmented paper content and their respective embeddings, we can proceed to develop Process 3. The primary objective of Process 3 lies in search and filtering operations, as illustrated in the conversation below.  \n\n# ChatGPT Code Synthesis  \n\nHuman: Your task is to write Python code to construct a function called Model_3. The function takes a dataframe called df as input, which contains columns named 'content' and 'embeddings'. Your job is to first convert a fixed prompt, which is a text string used for search purposes, into OpenAI embeddings using the following code template:  \n\nresponse $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ openai.Embedding.create(input $\\c=$ \"Your text string goes here\", model $\\c=$ \"text-embedding-ada-002\") embeddings $\\mathbf{\\Psi}=\\mathbf{\\Psi}$ response['data'][0]['embedding']  \n\nNext, you need to compare the converted embedding of the fixed prompt with all embeddings in the dataframe. Calculate the cosine similarity for each comparison and select the top 10 neighbors. Drop the remaining rows to obtain the filtered dataframe. Finally, you should pass the filtered dataframe to a new function called Model_2 for further processing. To calculate the cosine similarity and select the top neighbors, you can write two additional functions: add_similarity and select_top_neighbors.  \n\n![](images/fa6322b37342324b7a99242245ddcfd046e721903ce35c166c36580364dbb4aa.jpg)  \nFigure S36. Illustration of Process 3 configuration, guided by code generated by ChatGPT (GPT4). For information regarding the fixed prompt, please refer to Figure S25.  \n\nThe fine tuning of the functions can be achieved by providing more detailed instructions to ChatGPT (GPT-4) and requesting its revision. For instance, we aimed to filter less relevant content for each paper, therefore necessitating the selection of the top 10 sections based on each unique file name.  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous conversation shown in Figure S36---  \n\nHuman: There is another column called 'file name'. Please modify your code accordingly to select the top neighbors for each different file name.  \n\nChatGPT (GPT-4): If you want to select the top neighbors for each different file name, you can modify the select_top_neighbors function to group by the 'file name' column before applying the sorting and selection operation. Here is the updated code:  \n\nf select_top_neighbors(df): df $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ df.sort_values('similarity', ascending $\\mathbf{\\Psi}=$ False) top_neighbors $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ df.head(10) return top_neighbors  \n\nUpon setting up Process 3, we can shift our attention to the configuration of Process 2. The approach mirrors the workflow of Process 3: the user provides a detailed description of the function's name, input, and expected output to ChatGPT (GPT-4). It's crucial to inform ChatGPT (GPT-4) about the standard Python code needed to call the ChatGPT API, which can be obtained from OpenAI’s official documentation.  \n\n# ChatGPT Code Synthesis  \n\nHuman: Your task involves writing Python code to construct a function named Model_2. This function should accept a dataframe, denoted as 'df', which includes a column labeled 'content'. Start by constructing a prompt that combines \"given few-shot prompt\" with 'content', followed by:  \n\n\"\"\"Question: Does the section contain a comprehensive MOF synthesis with explicit reactant quantities or solvent volumes? Answer: \"\"\"  \n\nThen, you will pass this prompt to the ChatGPT API using the following code:  \n\nresponse $\\mathbf{\\tau}=\\mathbf{\\tau}$ openai.ChatCompletion.create(mode $|=$ \"gpt-3.5-turbo\",messages $\\c=$ [{\"role\": \"user\", \"content\": \"<your_prompt $>$ \"}]) Answer $\\mathbf{\\tau}=\\mathbf{\\tau}$ response.choices[0].message.content  \n\nYou will assign the first choice of the response content to a variable, 'Answer' The answers will then be appended to the dataframe as a new column, named 'classification'.   \nNext, add a mask to remove rows with the value of 'No' in the 'classification' column. Finally, pass the resultant dataframe to a function named model_1 for further processing.  \n\nChatGPT (GPT-4): import openai import pandas as pd  \n\n![](images/5b6147fbac2d2a794c95ee9b541bba718a0a5859e9901b17819e88b30d736af6.jpg)  \nFigure S38. Depiction of Process 2 setup, aided by code generated by ChatGPT (GPT-4). For details concerning the fixed few-shot prompt, please refer to Figure S24.  \n\nSimilarly, Process 1 can be constructed following the same approach. We note that two sources can provide the input dataframe for Process 1. The first source is the dataframe generated by Process 2 after classification, which contains only the text segments classified as synthesis paragraphs, each having a few hundred tokens. The second source is a dataframe composed of longer, human-selected synthesis paragraphs for each paper, with lengths varying from a few hundred to many thousand tokens. Given the 4096-token limitation, we must consider the total token count of synthesis paragraph context plus the fixed token. Hence, we established a cutoff at 3000 tokens to segment the input content.  \n\n# ChatGPT Code Synthesis  \n\nHuman: Your task is to write Python code to construct a function named Model_1. This function should accept a dataframe, df, which contains a column named 'content'. Initially, apply the count_tokens() function to determine the number of tokens for each row in 'content'. If the token count exceeds 3000 max_tokens, split the content at each 3000-token boundary. Then, for each string in the 'content' column, concatenate \"Context: \" $^+$ 'content' $^+$ \"Given tubulation prompt\". Following this, pass the resulting prompt to the ChatGPT API using the code:  \n\nresponse $\\mathbf{\\tau}=\\mathbf{\\tau}$ openai.ChatCompletion.create(model=\"gpt-3.5-turbo\",messages $\\mathbf{\\sigma}=\\mathbf{\\sigma}$ [{\"role\": \"user\", \"content\": \"<your_prompt>\"}]) Answer $\\mathbf{\\tau}=\\mathbf{\\tau}$ response.choices[0].message.content Assign the first choice of the response content to a variable, 'Answer'. Append the 'Answer' to the dataframe as a new column, named 'summarized'. Finally, return the updated dataframe.  \n\n# ChatGPT (GPT-4):  \n\n![](images/d559bd237de4a3a3ae1c0a5dccf893abaf695a963ce84357fbabb0b4ba5cc88c.jpg)  \nFigure S39. Demonstration of Process 1 establishment, supported by code produced by ChatGPT (GPT-4). For specifics of the fixed prompt, please refer to Figure S22 and Figure S22.  \n\n# Text Parsing  \n\nAs previously illustrated in Figure S26, responses from the ChatGPT API present tabular information as text strings filled with '|' symbols, rendering them unreadable. To address this, we enlisted ChatGPT to write a Python function to extract this data, restructure it into a clear, readable format, and convert it into a dataframe. This was achieved by providing ChatGPT with an exemplary output from the 'summarized' column in Process 1, which stores the API responses, and instructs it to reassemble the table into a dataframe and compile all the synthesis parameters.  \n\n# ChatGPT Code Synthesis  \n\nHuman: Your task is to write Python code to construct a function named 'tabulate_condition'. This function should accept a dataframe, 'df', which contains a column named 'summarized'. This column consists of text strings of tabulated data that could take the following format:  \n\nHeader 1 Header 2 | Header 3 | Header 11 Data 1-1 | Data 1-2 | Data 1-3 Data 1-11 | | Data 2-1 | Data 2-2 | Data 2-3 Data 2-11  \n\nHere, the “……” represent repeating units. Please note that the text strings can have one, two, or more rows of data, as shown in the exemplary input above. In rare cases, they may have no rows of data and consequently, no header. In such instances, no \"|\" symbol will be found in the text string. These text strings should be dropped.  \n\nThe function should iterate over all cells in the 'summarized' column, remove the header, extract the data, merge all the data, and then reinsert a single header. The data will always consist of 11 columns: 'compound name', 'metal source', 'metal amount', 'linker', 'linker amount', 'modulator', 'modulator amount or volume', 'solvent', 'solvent volume', 'reaction temperature', and 'reaction time'. The final dataframe returned by the function will consist of these 11 columns.  \n\n# ChatGPT (GPT-4):  \n\ndef tabulate_condition(df):  \n\n![](images/a2ae78178f871d0e580586152470250ccb6d3c8b88c83c8dd10ec939fb074bdf.jpg)  \nFigure S40. Demonstration of instruction for ChatGPT (GPT-4) to generate a Python function for converting string-based tabular data from the 'summarized' column of a dataframe into a structured Dataframe format.  \n\nMoreover, given the exemplary input already provided, we further directed ChatGPT to generate code to create a sample dataframe. This dataframe was used to call and test the function that was written by ChatGPT, enabling us to evaluate the function's output and ensure its accuracy and effectiveness.  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous conversation shown in Figure S40---  \n\nHuman: Please help me write code to test this function tabulate_condition.  \n\n# ChatGPT (GPT-4):  \n\nSure! Here is an example of how you can test this function:  \n\n$\\#$ Test the function with a sample dataframe sample_data $\\mathbf{\\Psi}=\\mathbf{\\Psi}$ {'summarized': [  \n\n![](images/db5fac0b4a341590795b7489b1c1a4ac2a12ef57a86af4616b96848300e84ee0.jpg)  \n\nsample_df $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ pd.DataFrame(sample_data) result $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ tabulate_condition(sample_df)  \n\nprint(result)  \n\n# Output:  \n\n<html><body><table><tr><td>compound name</td><td>metal source</td><td>metal amount</td><td>linker</td><td>linker amount</td><td>modulator</td><td>aoduat or volume</td><td>solvent</td><td>solvent volume</td><td>reaction temperature</td><td>reaction time</td></tr><tr><td>Data 1-1</td><td>Data 1-2</td><td>Data 1-3</td><td>Data 1-4</td><td>Data 1-5</td><td>Data 1-6</td><td>Data 1-7</td><td>Data 1-8</td><td>Data 1-9</td><td>Data 1-10</td><td>Data 1-11</td></tr><tr><td>Data 5-1</td><td>Data 5-2</td><td>Data 5-3</td><td></td><td>Data 5-4Data 5-5</td><td>Data 5-6</td><td>Data 5-7</td><td>Data 5-8</td><td>Data 5-9</td><td>Data 5-10</td><td>Data 5-11</td></tr></table></body></html>  \n\nFigure S41. Demonstration of testing the function generated by ChatGPT (GPT-4) through instructing ChatGPT to build up a sample dataframe.  \n\n# Section S4. ChatGPT-Assisted Chemistry Data Processing Cookbook  \n\nReaction Time and Temperature Conversion  \n\nTo handle basic entity resolution and data unification tasks, such as unit and format unification, we can either import the data into the ChatGPT API or, more conveniently, copy and paste the data into the web-based ChatGPT platform. Each specific task requires a preconfigured prompt, developed based on the three fundamental principles discussed in Supplementary Information Section S2. The prompts are refined iteratively, as demonstrated in Figures S13−S19, to enhance performance. The final versions of these prompts are displayed in the top box of each figure below. For easier proofreading, we instructed ChatGPT to include the original input in the first column of the output. It should be noted that the examples in this section were executed by ChatGPT based on the GPT-4 model. While both the GPT-4 and GPT-3.5 models perform adequately for such simple conversions, GPT-4 was chosen due to its superior handling of empty cells when processing large data inputs. Based on token limitations, each ChatGPT operation can process between 50 to 100 data points, typically completing in about 10 seconds.  \n\n# ChatGPT Data Processing  \n\nPrompt: Please convert the following reaction times into hours and generate a two-column table. The first column should contain the original text input, and the second column should contain the converted hours without units. The input will consist of various formats like days, weeks, hours, minutes, and seconds, with some entries possibly containing multiple time values. In cases with multiple time values, please use the longest time value. If \"N/A,\" a blank space, or irrelevant information with no indication of the time value is present, treat it as N/A and convert it to the text string \"N/A. Treat \"overnight\" as 12 hours.  \n\nInput:  \n\n# Output:  \n\n■=Ⅱ   \n24h   \n18 hours   \n1 week   \n150 sec   \n20 min   \none day   \novernight   \n3 days   \n\"\"\"  \n\n<html><body><table><tr><td>Original Input</td><td>Converted Hours</td></tr><tr><td>24h</td><td>24</td></tr><tr><td>18 hours</td><td>18</td></tr><tr><td>1 week</td><td>168</td></tr><tr><td>150 sec</td><td>0.0417</td></tr><tr><td>20 min</td><td>0.3333</td></tr><tr><td>one day</td><td>24</td></tr><tr><td>overnight</td><td>12</td></tr><tr><td>3 days</td><td>72</td></tr></table></body></html>  \n\nSimilarly, reaction temperatures can also be standardized using a pre-configured prompt. It's worth noting that ChatGPT could be guided to write Python code, Excel functions, or even use NLP methods for these conversions. However, the most remarkable feature of ChatGPT lies in its ability to accommodate a wide range of situations that would otherwise require extensive rule-based coding. This includes variations in phrases with equivalent meanings such as “overnight,” “for one night,” and “Over the course of a night,” as well as special characters and incorrect representations of units that are easily recognized and understood by ChatGPT with a single sentence instruction.  \n\n# ChatGPT Data Processing  \n\nPrompt: Please convert the given reaction temperatures into numbers without units in $^{\\circ}C$ and generate a two-column table, where the first column shows the original temperature and the second column displays the converted temperature in $^{\\circ}C$ without units. The input may include various units such ${\\sf a s}^{\\circ}{\\sf C},$ $\\mathsf{K},$ and possibly incorrect representations of $^{\\circ}C$ (for example, represented with any special character or alphanumeric character before $\\because C^{\\prime}$ ). If \"N/A,\" a blank space, new line, or irrelevant information with no indication of the temperature value is present, treat it as N/A and convert it to the text string \"N/A\". In cases with multiple temperature values or ramping instructions, use the highest temperature value and disregard ramping time. When encountering temperatures in Kelvin (K), convert them to Celsius $(^{\\circ}\\mathsf{C})$ before adding to the table. Consider \"room temperature\" or \"rt\" or \"ambient\" as $25^{\\circ}C$ . Exclude $\"\"\\mathsf{C}\"$ from the second column of the table. Ensure that the output is formatted as a two-column table.  \n\n# Input:  \n\n# Output:  \n\n<html><body><table><tr><td>Original Temperature</td><td>Converted Temperature (°C)</td></tr><tr><td>120 ℃</td><td>120</td></tr><tr><td>room temperature</td><td>25</td></tr><tr><td>ambient</td><td>25</td></tr><tr><td>383K</td><td>110</td></tr><tr><td>298K</td><td>25</td></tr><tr><td>1208C</td><td>120</td></tr><tr><td>160C</td><td>160</td></tr><tr><td>ramp of 10 ??C per minute to 120 ??C</td><td>120</td></tr><tr><td>160oC</td><td>160</td></tr><tr><td>reflux</td><td>N/A</td></tr><tr><td>reflux at 120 C</td><td>120</td></tr><tr><td>microwave (700W)</td><td>N/A</td></tr><tr><td>100 uC</td><td>100</td></tr><tr><td>85??C</td><td>85</td></tr><tr><td>65C</td><td>65</td></tr></table></body></html>  \n\n![](images/9defeba260d3a20f7f56fa999bc533a84752af283b3769748a049a48c0c9f5ec.jpg)  \nFigure S43. Demonstration of a custom prompt instructing ChatGPT (GPT-4) to automate temperature conversion.  \n\nCompared to unifying reaction temperature and reaction time data, the process of standardizing compound notation often necessitates intricate rule-based codes, making it a time-consuming task. Special characters can further complicate this process. However, the implementation of ChatGPT, equipped with carefully designed prompts, significantly simplifies this task. As depicted in Figure S42, ChatGPT successfully converts a variety of chemical formulae representations into standard notation in a matter of seconds.  \n\n# ChatGPT Data Processing  \n\nPrompt: Given a list of metal salts compound names in various formats, clean, standardize, and unify these names by identifying the correct chemical formula for each compound and converting them into their standard notation (e.g., AlCl3·6H2O, ZnCl4). Then, create a two-column table to display the results, with the left column containing the original text and the right column showing the converted text in the standard format. The input may include compound names written in different ways, such as using common names, abbreviations, or Roman numerals. If \"N/A,\" a new line, or only a metal (cation) is mentioned without any associated anion, treat it as N/A and convert it to the text string \"N/A\".  \n\n# Input:  \n\n# Output:  \n\n\"\"\"   \nZirconium oxynitrate hydrate   \nal2so43 18h2o   \nZn(Ac)2 ·2H2O   \nZnCO3??2Zn(OH)2??H2O   \nZnNO3•6(H2O)   \nstock solution   \niron chloride anhydrous   \nzn(no3)2??4h2o   \nIn(NO3)3??5H2O and   \nZn(NO3)2??4H2O   \nManganese(II) chloride,   \nMagnesium chloride   \ncupric sulfate pentahydrate   \n\"\"\"  \n\n<html><body><table><tr><td>Original Text</td><td>Converted Text</td></tr><tr><td>Zirconium oxynitrate hydrate</td><td>Zr(NO3)4·nH2O</td></tr><tr><td>al2so43 18h2o</td><td>A12(SO4)3·18H20</td></tr><tr><td>Zn(Ac)2 ·2H2O</td><td>Zn(C2H302)2·2H20</td></tr><tr><td>ZnCO3??2Zn(OH)2??H2O</td><td>Zn5(CO3)2(OH)6</td></tr><tr><td>ZnNO3·6(H2O)</td><td>Zn(NO3)2·6H20</td></tr><tr><td>stock solution</td><td>N/A</td></tr><tr><td>iron chloride anhydrous</td><td>FeC13</td></tr><tr><td>zn(no3)2??4h2o</td><td>Zn(NO3)2·4H2O</td></tr><tr><td>In(NO3)3??5H2O and Zn(NO3)2??4H2OZn(NO3)2·4H2O</td><td>In(NO3)3·5H2O,</td></tr><tr><td>Manganese(II) chloride,Magnesium chloride</td><td>MnC12, MgC12</td></tr><tr><td>cupric sulfate pentahydrate</td><td>CuSO4-5H20</td></tr></table></body></html>  \n\nFigure S44. Utilization of ChatGPT (GPT-4) to standardize the notation of metal compounds.  \n\nThe molecular weight of compounds can be calculated in a similar manner. With the standard notation established, it is a matter of extending the prompt to request ChatGPT to generate Python code to perform these calculations.  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous data processing shown in Figure S44 ---  \n\nHuman: Could you write a Python script that calculates the molecular weights of given compounds and adds this information as a new column in a DataFrame? If a compound name includes an undetermined number of water molecules (denoted as 'nH2O'), please exclude this portion during the calculation. Additionally, if multiple compounds are listed in a single entry, please separate them with a semicolon $(^{1};^{1})$ .  \n\n# ChatGPT (GPT-4):  \n\n!pip install mendeleev pandas  \n\n![](images/31975d576107d1a055de16afcbcae78602787a2b2f16edda1e10c1d0a617f349.jpg)  \nFigure S45. Instruction of ChatGPT (GPT-4) to generate Python code to calculate the molecular weight of given compounds. The conversation was continued from the context set up in Figure S44, and ChatGPT was able to know the format of the standard notation.  \n\nAfter executing the ChatGPT-provided code in a Python environment, the resulting DataFrame is presented in Table S1. The code efficiently calculates the molecular weight of the given compounds in standard notation. Subsequently, we instructed ChatGPT to modify the code to create a function, thereby facilitating batch molecular weight calculations for a DataFrame containing standard notations.  \n\nTable S1. Molecular weights of various compounds calculated using the code provided by ChatGPT.   \n\n\n<html><body><table><tr><td>Original Compound</td><td> Standard Notation</td><td>Molar Mass</td></tr><tr><td>Zirconium oxynitrate hydrate</td><td>Zr(NO3)4·nH20</td><td>153.228</td></tr><tr><td>al2so43 18h20</td><td>A12(SO4)3·18H20</td><td>168.034</td></tr><tr><td>Zn(Ac)2 ·2H20</td><td>Zn(C2H302)2·2H20</td><td>142.439</td></tr><tr><td>ZnCO3??2Zn(OH)2??H2O</td><td>Zn5(CO3)2(OH)6</td><td>403.915</td></tr><tr><td>ZnNO3·6(H2O)</td><td>Zn(NO3)2·6H20</td><td>145.399</td></tr><tr><td>stock solution</td><td>N/A</td><td>N/A</td></tr><tr><td>iron chloride anhydrous</td><td>FeC13</td><td>162.195</td></tr><tr><td>zn(no3)2??4h2o</td><td>Zn(NO3)2·4H20</td><td>145.399</td></tr><tr><td>In(NO3)3??5H2O and Zn(NO3)2??4H2O</td><td>In(NO3)3·5H2O, Zn(NO3)2·4H20</td><td>194.837; 145.399</td></tr><tr><td>Manganese(II) chloride, Magnesium chloride</td><td>MnC12, MgCl2</td><td>125.838; 95.205</td></tr><tr><td>cupric sulfate</td><td>CuSO4·5H20</td><td>177.617</td></tr></table></body></html>  \n\n# ChatGPT Code Synthesis  \n\n---Continued from previous conversation shown in Figure S45---  \n\nHuman: I have a DataFrame that includes a 'Standard Notation' column. Could you provide a Python script that uses this DataFrame as input and adds an additional 'Molar Mass' column to it? Each row of the 'Molar Mass' column should correspond to the molecular weight of the compound in the same row of the 'Standard Notation' column.  \n\n![](images/0d972a93082ddcdc3db36e98dcec06eb63114629be353a479b499ca0f3cfd27f.jpg)  \nFigure S46. Instruction of ChatGPT (GPT-4) to modify the Python code for batch processing of molecular weight calculations for compounds listed in a DataFrame.  \n\n# Reaction Outcome Classification  \n\nWe demonstrate here that ChatGPT can assist in classifying synthesis outcomes by crystalline state. As shown in Figure S21, direct synthesis results such as “colorless crystals” “red solid”, and “microcrystalline powder” can be extracted and summarized. Now, we engage ChatGPT to categorize these descriptions, stored in a Dataframe. We note that in literature we select, there are no synthesis condition that yield amorphous, but rather only crystalline MOFs, with difference on whether or not they are single crystals or just microcrystalline powder or solid. Consequently, this MOF crystalline state classification will have two categories, one is “single-crystal” (SC), which includes all cases where the author clearly says that crystals form, and the other is “polycrystalline” (P), where the author mentions solid, powder, or did not mention the single-crystal.  \n\n# ChatGPT Data Processing  \n\nPrompt: Please classify the provided descriptions of materials based on their crystalline structure into either single-crystal (SC) or polycrystalline (P). The descriptions should be parsed and classified as per the following rules: If the description includes words like \"powder\", \"solid\", \"microcrystalline\", \"crystalline product\", or \"N/A\", classify it as \"P\" (polycrystalline). If the description includes words like \"single-crystal\" or \"crystal\", classify it as \"SC\" (single-crystal). Please present the results in a two-column table. The first column should contain the original description, and the second column should display the classification result \"SC\" or \"P\". Ensure that each row in the table represents one description and its corresponding classification.  \n\n# Input:  \n\n\"\"\"  \n\nred solid   \nGreen block crystals   \ncolorless rodlike crystals   \npink precipitate   \nyellow crystalline product   \npurple hexagonal platelike crystals   \nlarge yellow crystalline product   \nColorless rod-shaped crystals   \ncolorless rhombus-shaped block crystals   \nOff-white powdered product   \nTetragonal colorless crystals   \nSky-blue powdered product   \nYellow needle crystals   \nfine powdered MOF   \ndark yellow cubic crystals   \nwhite microcrystalline product   \nNeedlelike colorless crystals   \nⅡⅡⅡ  \n\n# Output:  \n\n<html><body><table><tr><td>Original Description</td><td>Classification</td></tr><tr><td>red solid</td><td>P</td></tr><tr><td>Green block crystals</td><td>SC</td></tr><tr><td>colorless rodlike crystals</td><td>SC</td></tr><tr><td>pink precipitate</td><td>P</td></tr><tr><td>yellow crystalline product</td><td>P</td></tr><tr><td>purple hexagonal platelike crystals</td><td>SC</td></tr><tr><td>large yellow crystalline product</td><td>P</td></tr><tr><td>Colorless rod-shaped crystals</td><td>SC</td></tr><tr><td>colorless rhombus-shaped block crystals</td><td>SC</td></tr><tr><td>Off-white powdered product</td><td>P</td></tr><tr><td>Tetragonal colorless crystals</td><td>SC</td></tr><tr><td>Sky-blue powdered product</td><td>P</td></tr><tr><td>Yellow needle crystals</td><td>SC</td></tr><tr><td>fine powdered MOF</td><td>P</td></tr><tr><td>dark yellow cubic crystals</td><td>SC</td></tr><tr><td>white microcrystalline product</td><td>P</td></tr><tr><td>Needlelike colorless crystals</td><td>SC</td></tr></table></body></html>  \n\n# Unifying Linker Representations  \n\nThe lack of a consistent representation for organic linkers in MOF synthesis often results in data confusion, with full names, abbreviations, and special characters all being used interchangeably. By standardizing linker representations using a format like the SMILES (Simplified Molecular Input Line Entry System) code, we can facilitate better data exchange. However, this standardization process can be challenging and often requires human intervention, especially for cases involving misspelled names or complex linker structures that have not yet been named. Nevertheless, we have developed a method for instructing ChatGPT to write code that handles the majority of relatively simple cases, accelerating the process while reducing manual effort.  \n\nOur approach begins by converting abbreviations to full names. We do this by instructing ChatGPT to implement Process 3, which prompts the system to search for the full name of the linker. The paragraph with the highest similarity score is then presented to ChatGPT, which is then prompted to provide the full name of the abbreviation.  \n\n# Embedding Demo  \n\nPrompt Embedding (ada-002): Provide the full name of linker ({Input_Linker_Abbreviation}) or denoted as {Input_Linker_Abbreviation} in chemicals, abstract, introduction or experimental section.  \n\nInput #1: H2FDC  \n\n# Output #1:  \n\n…The linker 2,5-furanedicarboxylic acid (H2FDC) is considered to be a biorenewable organic building unit 27,28 and hence considered a promising alternative to terephthalic or isophthalic acid…  \n\nInput #2: H3BTCB  \n\n# Output #2:  \n\nSection 2. Ligand Synthesis. The organic linker H3L1 (4,4',4''-((benzene -1,3,5 -tricarbonyl)tris (azanediyl))tribenzoic acid, H3BTCB ) was synthesized according to the literature procedure……  \n\nInput #3: 2-nIM.  \n\n# Output #3:  \n\nExperimental measurements and methods used: General Remarks: 2-nitroimidazole (2-nIM) was purchased from …  \n\nFigure S48. An illustration of how an embedding model (OpenAI ada-002) is employed to derive the full names of linkers from their abbreviations. Please note that the outputs are abridged for clarity.  \n\nOnce the full names of the linkers are obtained, we instruct ChatGPT to fetch the corresponding SMILES code and PubChem ID using an API. Given that a single compound may have multiple synonymous names, we consolidate the names sharing the same PubChem ID and SMILES code to create a dictionary. This cleaned dataset replaces all names and abbreviations recorded in the dictionary with their SMILES codes. The compounds that cannot be fetched are manually reviewed. While it is possible for ChatGPT to provide the SMILES code given the standard full name of a compound, this method is limited to common compounds and is at risk of generating inaccurate information. Hence, we recommend using the API to retrieve this information.  \n\n# ChatGPT Code Synthesis  \n\nHuman: I have a dataframe with two columns: \"full name\" and \"abbreviation\". Please write Python code to create a function. This function will take the dataframe as input and will search for each \"full name\" value (which are strings) using the PubChem API, PubChemPy, to retrieve its PubChem ID and canonical SMILES code. These two pieces of information will then be appended as two additional columns. If the information cannot be found on PubChem, the function should assign the value \"N/A\" to the corresponding cells. The function should then merge the rows based on unique PubChem CID values. In other words, it should create a new dataframe with four columns: \"full name\", \"abbreviation\", \"CID\", and \"SMILES code\". If there are rows with the same CID, they should be merged. The resulting merged row should have full names and abbreviations separated by \";\", with just one \"CID\" and \"SMILES\" code. The function should not merge items with \"N/A\" in their CID column. Finally, the function should return this new dataframe.  \n\n# ChatGPT (GPT-4):  \n\nimport pandas as pd import pubchempy as pcp  \n\n![](images/50e3c9080d7826a691ba85ec6a599d471beee49357b393d0e963c0cbe5d13ae3.jpg)  \nFigure S49. Demonstration of using ChatGPT (GPT-4) to generate code that retrieves CID and SMILES codes from PubChem and merges dataframes based on unique compound identifiers.  \n\n![](images/03ebae059da7a18a165612496d2d1061e86b236fe7fd4fa94ef186285a2d30af.jpg)  \nFigure S50. MOF linkers obtained from text mining and their CID numbers.  \n\n![](images/b6079be84a8f9ded5605e8b28906c98f73ae25c63204b884b4980dddcbabd0eb.jpg)  \nFigure S51. MOF linkers obtained from text mining and their CID numbers.  \n\n![](images/ca943cb065297b6e4ec6f5f48849ae0cbca01182ec2c8b5353ca3683ec22ad2e.jpg)  \nFigure S52. MOF linkers obtained from text mining and their CID numbers.  \n\n![](images/dfb1f54c9b28a345be5de09ca27d52cb9cfdc5b004b3079bd0b3687be5a5697e.jpg)  \nFigure S53. MOF linkers obtained from text mining and their CID numbers.  \n\n![](images/31bc4fe6985b844c4482cadee16f77ea23f0ce77c043d7e4bc4f9cbe0a1a83c1.jpg)  \nFigure S54. MOF linkers obtained from text mining and their CID numbers.  \n\nWe showcase ChatGPT's capability to process research papers written in languages other than English with minor modifications to the prompt. As evidenced, ChatGPT accurately identifies each synthesis parameter and tabulates them in English - a task that proves significantly challenging for traditional English-based NLP methods.9-12  \n\n# ChatGPT Processing Demo  \n\nPrompt: Please provide a truthful response based on the given context. Translate and summarize the following details into an English table: compound name or chemical formula (if the name is not mentioned), metal source, metal quantity, organic linker(s), amount of linker, modulator, volume or quantity of modulator, solvent(s), volume of solvent(s), reaction temperature, and reaction duration. If any of the data is not provided or you are uncertain, please fill in \"N/A\".  \n\nYour attention should be on extracting only the experimental conditions from the MOF synthesis, disregarding information associated with the organic linker synthesis, MOF postsynthetic modifications or metalation, high throughput (HT) experiment details, or catalytic reactions.  \n\nIf there are various conditions mentioned for the same compound, represent them in multiple rows. If there are multiple units or components provided for the same factor (such as g and mol for weight, multiple linkers or metals, varied temperature and reaction duration, mixed solvents, etc.), include them in a single cell, separating each by a comma.  \n\nThe table should have 11 columns, all in lowercase:| compound name | metal source | metal amount | linker | linker amount | modulator | modulator amount or volume | solvent | solvent volume | reaction temperature | reaction time |  \n\nInput:  \n\nSynthese: Die solvothermale Synthese von DUT-9 kann sowohl in DMFals auch in DEF durchgefhrt werden. Die Kristallstruktur von DUT-9 wurde anhand eines in DMF-Lösung gewachsenen Kristalls bestimmt. Weitere analytische Untersuchungen erfolgten mit Material, das aus Synthesen in DEFLösung stammt. In einer typischen Synthese werden 180 mg H3btb (0.41 mmol) und 366 mg Ni(NO3)2·6H2O (1.26 mmol) in ein Pyrex-Röhrchen gegeben und in 10.5 mL DEF gelöst. Die Lösung wird 20 h auf $120^{\\circ}C$ erhitzt. Dieerhaltenen, hellgrünen Kristalle von DUT-9 werden durch Filtration unter Argon von der überstehenden Lösung getrennt, zweimal mit frischem DEF gewaschen und im Argonstrom bei Raumtemperatur getrocknet. Ausbeute: 412 mg $54\\%$ bezogen auf die eingesetzte Menge an H3btb).  \n\n# Output:  \n\n<html><body><table><tr><td>compound name</td><td>metal source</td><td>metal amount</td><td>linker</td><td>linker amount</td><td></td><td>modulatormdulator volume</td><td>solvent solvent</td><td>volume</td><td>reaction temperature</td><td>reaction time</td></tr><tr><td>DUT-9</td><td>Ni(NO3)2·6H2O</td><td>366 mg, 1.26 mmol</td><td>H3btb0.41</td><td>180 mg, mmol</td><td>N/A</td><td>N/A</td><td>DEF</td><td>10.5 mL</td><td>120 ℃</td><td>20 hours</td></tr></table></body></html>\n\nFigure S55. Demonstration of ChatGPT's proficiency in interpreting and summarizing synthesis condition from German text.  \n\n# Prompt:  \n\n---Same prompt as shown in previous demo in Figure S55---  \n\nInput:  \n\n单金属Mn-MOF-74 的制备  \n单金属Mn-MOF-74 的制备参考了已报道的文献[25]。在250 mL 反应瓶中加入MnCl2·4H2O(1 098 mg)和2，5-二羟基对苯二甲酸(DOBDC，333 mg)，再加入150 mL N，N-二甲基甲酰胺(DMF)-乙醇-水(体积比为15:1:1) 混合溶剂，超声20 min 得到澄清溶液后，置入已经预热至 $135^{\\circ}C$ 的烘箱中反应 $24\\:h$ 。从烘箱中取出自然冷却后，倾倒出反应母液，得到棕黑色晶体材料。晶体用DMF 洗涤3 次 $(3\\times20~m L)$ 后，再用甲醇进行溶剂置换 $3d(3\\times3\\times20m L)$ 。甲醇置换过后的Mn-MOF-74 材料先于室温下真空脱溶剂6 h，之后缓慢升温至 $250~^{\\circ}C$ 真空条件 ${\\mathcal F}_{\\cdot}$ 脱溶剂处理 $12h$ 。  \n\n# Output:  \n\n<html><body><table><tr><td>compound</td><td>moutal </td><td>mmtaunt</td><td>linker</td><td>linkeunt</td><td>modulator</td><td>modulator amount or volume</td><td>solvent</td><td>soluent</td><td>reatioraturereaction</td><td></td></tr><tr><td>Mn-MOF- 74</td><td>MnCI2·4H20</td><td>1,098 mg </td><td>2ihydroxytreptaic3 m acid</td><td></td><td>N/A</td><td>N/A</td><td>Dtmanol- water</td><td>150 mL</td><td>135 ℃</td><td>24 hours</td></tr></table></body></html>  \n\nFigure S56. Demonstration of ChatGPT's proficiency in interpreting and summarizing synthesis condition from Chinese text.  \n\n# Prompt:  \n\n---Same prompt as shown in previous demo in Figure S55---  \n\n# Input:  \n\nMOF-5 の合成：テフロンで口を巻いた100 mL のガラス瓶に49 mL のDMF と1 mL  の水を加える。その混合液にZn(NO3)2 · 6H2O (0.45 g, 1.5 mmol) とテレフタル酸H2(bdc) (0.083 g, 0.50 mmol) を溶かす。ガラス瓶を閉め、それを $100^{\\circ}C$ のオーブンで 7 時間加熱する。加熱後、ガラス瓶を室温に冷ます。これ以降の操作は、不活性雰囲気下のグ $\\boldsymbol{\\varpi}$ ーブテントで、無水溶媒を用いて行う。混合物の上澄み液をデカンテーションにより取 $p$ 除き、新しい無水 DMF 50 mL を加えて、固体を 8 時間浸す。この DMF 洗浄を 3 回行ったら、同様 $\\mathit{l}_{-}^{>}$ 無水ジクロロメタン DCM による洗浄を3 回行う。最後のDCM 洗浄の後、無色の結晶をろ過により回収 $L$ 、 $150^{\\circ}C$ で加熱真空引きすることで孔内の溶媒を除去する。  \n\nOutput:   \n\n\n<html><body><table><tr><td>compound</td><td>moural </td><td>mmtunt</td><td>linker</td><td>linkeunt</td><td></td><td>moduiaor </td><td>solvent</td><td>soluemt</td><td>teactierature</td><td>reaction</td></tr><tr><td>MOF-5</td><td>Zn(NO3)2 · 6H20</td><td>1.5 mmol 0.45 g,</td><td>H2(bdc)0.50</td><td>mmol 0.083 g,</td><td>N/A</td><td>N/A</td><td>DMF, water</td><td>49 mL, 1 mL </td><td>100 ℃</td><td>7 hours</td></tr></table></body></html>  \n\nFigure S57. Demonstration of ChatGPT's proficiency in interpreting and summarizing synthesis condition from Japanese text.  \n\n# Prompt:  \n\n---Same prompt as shown in previous demo in Figure S55---  \n\n# Input:  \n\n2.1 금속유기골격체(MOFs):  \nMIL-53(Al), MIL-53-NH2(Al) 합성  \nMIL-53 의합성은Al(NO3)3·9H2O(1.3 g)와H2BDC(0.288 g), 5 mL H2O 를sonication 에30 분정도놓아섞은후autoclave 에넣어, Box furnace 에서분당5 oC 씩220 °C 까지승온한다음220 °C 에서약72 시간반응을진행시켰다. 그후centrifugation 으로수득한후DI water 로2 회, N,N-Dimethylformamide(DMF)로1 회washing 작업을진행하였다. 색이하얀페이스트가얻어지면DMF  \n25 ml 를autoclave 에넣어분당5 °C 씩 $150^{\\circ}C$ 까지승온후150 oC 에서약15 시간동안H2BDC 를충분히 제거 하였다. 이 과정을 통해 하얀색 색을 띄는 파우더 형태 의 MIL-53 을 얻게 되었다.MIL-53-NH2 의합성은AlCl3·6H2O(0.5 g)와H2BDCNH2(0.38 g), 5 mL H2O 를autoclave 에넣어, Boxfurnace 에서분당 $5^{\\circ}C$ 씩 $150^{\\circ}C$ 까지 승온 한 다음 그 온도에 서 약 5 시간 반응을 진행시킨다. 색이노란페이스트가얻어지면DMF 25 ml 를autoclave 에넣어분당 $5^{\\circ}C$ 씩 $150^{\\circ}C$ 까지 승온 후150 °C 에서약15 시간동안H2BDC 를충분히제거하였다. 추가DMF 제거를위해서 $150^{\\circ}C$ 12 시간동안box furnace 에놓아MIL-53-NH2 를얻게되었다.  \n\nOutput:   \n\n\n<html><body><table><tr><td>compound name</td><td>metal source</td><td>metal amount</td><td>linker</td><td>linker amount</td><td>modulator</td><td>modulator volume</td><td>solvent</td><td>solvent volume</td><td>reaction temperature</td><td>reaction time</td></tr><tr><td>MIL-53(AI)</td><td>AI(NO3)3·9H20</td><td>1.3 g </td><td>H2BDC</td><td>0.288 g</td><td>N/A</td><td>N/A</td><td>H2O, DMF</td><td>5 mL, 25 mL</td><td>220 ℃</td><td>72 hours</td></tr><tr><td>MIL-53</td><td>AICI3·6H20</td><td>0.5 g </td><td>H2BDCNH2</td><td>0.38 g</td><td>N/A</td><td>N/A</td><td>2M</td><td>5 mL,25</td><td>220 ℃</td><td>72 hours</td></tr></table></body></html>\n\nFigure S58. Demonstration of ChatGPT's proficiency in interpreting and summarizing synthesis condition from Korean text.  \n\nWhile the primary focus of this paper is to automate the process of reading research papers, identifying and summarizing synthesis procedures using a three-tiered ChatGPT model, the versatility of the large language model extends to diverse tasks. The adaptable nature of ChatGPT enables the search for various information types, such as sorption behaviors, BET surface area, gas uptake, thermal stability, decomposition temperature, and even chemical stability. Additionally, it can extract structural information such as MOF topology and crystal parameters. These tasks would typically require a specialist to manually establish a training set or write specific patterns - an approach that lacks flexibility when the search target changes. By slightly modifying the structure of the search input and the summarization prompt, ChatGPT can efficiently accomplish these tasks.  \n\nFurthermore, we demonstrate the ability to search for specific information from the same pool of papers without the need to process all the text from the papers again, saving significant time. This is achieved by converting all papers into embeddings, which can easily be reloaded. As a demonstration, we design a prompt to search for the decomposition temperature obtained from TGA plots for the compounds reported in the papers, changing the search target from synthesis parameters to decomposition temperature.  \n\n# Embedding Demo  \n\nPrompt Embedding (ada-002): Identify the section discussing thermogravimetric analysis (TGA) and thermal stability. This section typically includes information about weight-loss steps (e.g., $20\\%,$ $30\\%,$ $29.5\\%)$ and a decomposition temperature range (e.g., $450^{\\circ}\\mathsf C,$ $515^{\\circ}{\\mathsf{C}})$ or a plateau.  \n\nArticle Input #1: doi.org/10.1021/jacs.1c04946  \n\n# Search Output #1:  \n\n…Figure S24. TGA plot of the as-synthesized 3W-ROD-2-CH3, $<177^{\\circ}\\mathsf C_{\\iota}$ , loss of free/surface water/ DMF; $>380^{\\circ}C,$ framework degradation…  \n\nArticle Input #2: doi/10.1073/pnas.0602439103  \n\n# Search Output #2:  \n\n…The TGA trace for ZIF-8 showed a gradual weight-loss step of $28.3\\%$ $(25{-}450^{\\circ}C)$ ,corresponding to partial loss of guest species [N,N- dimethylformamide (DMF) and 3H2O; calcd. $35.9\\%$ ], followed by a plateau $(450-550^{\\circ}{\\mathsf{C}})$ . More impressively, the TGA trace for ZIF-11 revealed a sharp weight-loss step of $22.8\\%$ $(25-250^{\\circ}\\mathsf{C})$ , corresponding to the escape of all N,N- diethylformamide (DEF) solvent molecules trapped in the pores (0.9 DEF; calcd. $23.3\\%$ ), despite the fact that DEF is actually much larger than the aperture of ZIF-11 in size. The TGA trace of ZIF-11 also showed a long plateau in the temperature range $250\\mathrm{-}550^{\\circ}\\mathsf C,$ indicating its high thermostability in the absence of guest molecules….  \n\nArticle Input #3: doi.org/10.1021/ja512973b  \n\n# Search Output #3:  \n\n… The results revealed that all the materials are stable at elevated temperature, having decomposition temperatures around $470^{\\circ}\\mathsf C$ (see SI section S8.). The TGA of the solvated samples all show significant weight loss due to guest solvent molecules in the cavity prior to $200^{\\circ}\\mathsf C,$ consistent with the highly porous nature of these materials. ……  \n\nFigure S59. An illustration of ChatGPT's adaptability, where the prompt is modified to target TGA data. The text parsed from the research article is not displayed for clarity, and the DOI of the paper is provided instead. The output displays one of the top similarity results, with the remainder omitted for brevity.  \n\nOnce the embedding search yields a selection of paragraphs with embeddings exhibiting high similarity to the prompt's embedding, and these paragraphs potentially contain TGA information, we can adjust the prompt to facilitate summarization of this data.  \n\n# ChatGPT Processing Demo  \n\nPrompt: Answer the question as truthfully as possible using the provided context. Please summarize the following details in a table: compound name and decomposition temperature. If any information is not provided or you are unsure, use \"N/A\".  \n\nThe table should have 2 columns, all in lowercase:| compound name | decomposition temperature |  \n\nInput:  \n\n---Full text from Search Output #1, #2 and #3 in previous Embedding Demo shown in Figure S59---  \n\n# Output:  \n\n<html><body><table><tr><td>compound name</td><td>decomposition temperature</td></tr><tr><td>3w-rod-2-oh</td><td>>375 ℃</td></tr><tr><td>3w-rod-1</td><td>>375 ℃</td></tr><tr><td>3w-rod-2-f</td><td>>371 ℃</td></tr><tr><td>3w-rod-2-ch3</td><td>>380 ℃</td></tr><tr><td>zif-8</td><td>450-550C</td></tr><tr><td>zif-11</td><td>250-550°C</td></tr><tr><td>nu-1101</td><td>~470°C</td></tr><tr><td>nu-1102</td><td>~470C</td></tr></table></body></html>  \n\nFigure S60. A designed prompt to guide ChatGPT in summarizing TGA decomposition temperatures, applying the three principles of chemistry prompt engineering. The figure includes a sample input and output table for context.  \n\nSimilar adaptability can be exhibited by modifying the search and summary modules to extract BET surface area data. By appropriately adjusting the prompt and the summarization request, ChatGPT can effectively extract and summarize critical data from the literature.  \n\n# Embedding Demo  \n\nPrompt Embedding (ada-002): Identify the section discussing nitrogen (N2) sorption, argon sorption, Brunauer-Emmett-Teller (BET) surface area, Langmuir surface area, and porosity. This section typically reports values such as $1000~\\mathrm{m}2/\\mathrm{g},$ , $100\\ c m3/9$ STP, and includes pore diameter or pore size expressed in units of Ångströms (Å).  \n\nArticle Input #1: dx.doi.org/10.1021/ic301961q  \n\n# Search Output #1:  \n\n…The nitrogen sorption experiment clearly yields a type-I-isotherm, proving the microporosity of CAU-8 (Figure 9). The specific surface area according to the Brunauer−Emmett−Teller (BET)-method is $\\mathsf{S B E T}=600\\mathsf{m}2/\\mathsf{g},$ and the observed micropore volume is ${\\mathsf{V M I C}}=0.23~{\\mathsf{c m}}3/{\\mathsf{g}},$ calculated from the amount adsorbed at $\\mathsf{p}/\\mathsf{p}0=0.5$ . The maximum uptake of hydro- gen at $77\\mathsf{K}$ and 1 bar is 1.04 wt $\\%$ . …  \n\nArticle Input #2: dx.doi.org/10.1021/acs.cgd.0c00258  \n\n# Search Output #2:  \n\n… permanent porosity of ZTIF-8 was confirmed by the reversible N2 sorption measurements at $77\\mathsf{K},$ which showed type I adsorption isotherm behavior (Figure 2 a). The Langmuir and BET surface areas were $1981~{\\mathrm m}2/{\\mathrm g}$ and $1430~\\mathrm{m}2/\\mathrm{g}$ for ZTIF-8, respectively. A single data point at relative pressure at 0.98 gives a pore volume of 0.705 for ZTIF-8 by the Horvath −Kawazoe equation. Furthermore, the single component sorption isotherms for CO 2were measured at 1 atm and 273 and $288\\mathsf{K}$ ….  \n\nArticle Input #3: dx.doi.org/10.1021/ja4050828 |  \n\n# Search Output #3:  \n\n… The N2 adsorption isotherm of NU-1000 is best described as type IVc (Figure 3); NU-1000 has a Brunauer−Emmett−Teller (BET) surface area of $2320\\ m2\\ {\\mathfrak{g}}-1$ and a total pore volume of $1.4\\:\\mathsf{c m}3\\:\\mathsf{g}-1$ . The experimentally measured surface area and total pore volumes are in excellent agreement with the theoretical values of $2280~\\mathsf{m}2~\\mathsf{g}-1$ and $1.4\\:\\mathsf{c m}3\\:\\mathsf{g}-1$ obtained from grand canonical Monte Carlo simulations (GCMC) and subsequent BET analysis (Table S3, SI)…  \n\nFigure S61. Illustration of ChatGPT's flexibility in changing the search target to sorption properties. The parsed text from the research article is not displayed for clarity, and the DOI of the paper is provided instead. The output displays one of the top similarity results, with the remainder omitted for brevity.  \n\n# ChatGPT Processing Demo  \n\nPrompt: Answer the question as truthfully as possible using the provided context. Please summarize the following details in a table: compound name, pore volume, and BET surface area. If any information is not provided or you are unsure, use \"N/A\".  \n\nThe table should have 2 columns, all in lowercase:| compound name | pore volume | surface area |  \n\n# Input:  \n\n---Full text from Search Output #1, #2 and #3 in previous Embedding Demo shown in Figure S61---  \n\n# Output:  \n\n<html><body><table><tr><td>compound name</td><td>pore volume</td><td>surface area</td></tr><tr><td>CAU-8</td><td>0.23 cm3/g</td><td>600 m2/g</td></tr><tr><td>ZTIF-8</td><td>0.705 cm3/g</td><td>1430 m2/g</td></tr><tr><td>NU-1000</td><td>1.4 cm3/g</td><td>2320 m2/g</td></tr></table></body></html>  \n\nFigure S62. A prompt designed to instruct ChatGPT to summarize BET surface area and pore volume, with reference to the principles of chemistry prompt engineering. This includes an example input and output table for context, indicating that various prompt versions can be constructed based on specific research requirements.  \n\n# Section S5. Prediction Modeling  \n\nTo train our model, an initial step of data curation was undertaken. Rare cases in which the synthesis condition contained more than three metals or linkers were pruned and dropped. This was done to manage the complexity of the model, as one-hot encoding for such multi-component systems would introduce a large number of additional features, significantly increasing the model's dimensionality. Furthermore, instances with more than three metals or linkers were relatively rare and could act as outliers, potentially disturbing the learning process. After comparing the quality of the text-mined synthesis conditions by different processes, as shown in Figure 5c, we chose the results from Process 1 for training due to the fewest errors presented that could potentially impact the model. Consequently, data curation based on the output from Process 1 resulted in 764 samples that were used for model training.  \n\nSix sets of chemical descriptors were designed in alignment with the extracted synthesis parameters: these pertain to the metal node(s), linker(s), modulator(s), solvent(s), their respective molar ratios, and the reaction condition(s). The metal ions were described by several atomic and chemical properties, including valency, atomic radius13, electron affinity14, ionization potential, and electronegativity15. For the organic linkers, apart from Molecular Quantum Numbers (MQNs) that encode structural features in atomistic, molecular, and topological spaces,16, 17 a set of empirical descriptors were also employed. These were based on counts of defined motifs such as carboxylate and phosphate groups (Figure S48−S52).  \n\nAll solvents and modulators extracted were categorized into eight classes based on the recommendations from ChatGPT, each assigned a number from 1 to 8 and this assignment was made based on ranking the frequency of the compounds within each group (Table S2). These categories were represented by one-hot encodings. Molecular weights were also incorporated as descriptors for the linker(s), modulator(s), and solvent(s) sets. When multiple metals and organic linkers were present in the synthesis, the descriptors were calculated by taking a molar weighted average of the individual components. This approach was also employed to obtain the categorical encoders for multiple solvents and modulators used in combination. Here, the normalized molar fraction was entered into the cell where the corresponding solvent or modulator category was present, while all other entries were zero. In instances where solvents or modulators were absent in the synthesis parameters, arrays of zeros were used.  \n\nThe RF models were trained using Scikit-Learn's RandomForestClassifier implementation for varying train size on $80\\%$ random split of the curated data. We used grid search to determine the optimal hyperparameters for our model, specifically the number of tree estimators and the minimum samples required for a leaf split. Model performance was evaluated using crossvalidation and the metrics used for assessing the model's predictive power included class-weighted accuracy, precision, recall, and F1 score on the test set and the held out set. Feature permutation importance, quantified by the percent decrease in model accuracy by permutating one feature at a time, was used to identify which descriptors were the most influential in predicting the crystalline state outcome of a given synthesis condition.  \n\nTable S2. Classification of solvent and modulator groups.   \n\n\n<html><body><table><tr><td>Solvent and Modulator Class</td><td>Assigned Number for Solvent Class</td><td>Compound Name</td></tr><tr><td>Acids</td><td>8</td><td>acetic anhydride; hydrofluoric acid; hydrochloric acid; tetrafluoroboric acid; formic acid; acetic acid; trifluoroacetic acid; benzoic acid; biphenyl- 4-carboxylic acid; 4-nitrobenzoic acid; 2- fluorobenzoic acid; octanoic acid; nonanoic acid; phosphoric acid; nitric acid; sulfuric acid</td></tr><tr><td>Alcohols</td><td>3</td><td>methanol; ethanol; l-propanol; 2-propanol; ethylene glycol; 2-amino-1-butanol; 3-amino-1- propanol; 1-butanol; 3-methylphenol; phenylmethanol</td></tr><tr><td>Amides, Sulfur- containing, and Cyclic Ethers</td><td>1</td><td>1,4-dioxane; acetone; 1,3-dimethyl-2- imidazolidinone; l-cyclohexyl-2-pyrrolidone; dimethylformamide; diethylformamide; 1-methyl- 2-pyrrolidone; dimethyl sulfoxide; N,N- dimethylacetamide; N-methylformamide; tetrahydrofuran; 2-imidazolidinone</td></tr><tr><td>Amines and Ammonium Compounds</td><td>6</td><td>ammonia; methylamine; dimethylamine; triethylamine; tetrabutylammonium hydroxide; tetramethylammonium bromide; tetraethylammonium hydroxide; ammonium fluoride; 1-ethyl-3-methylimidazolium tetrafluoroborate; 1-ethyl-3-methylimidazolium</td></tr><tr><td>Base</td><td>7</td><td>chloride sodium hydroxide; sodium azide; lithium hydroxide; potassium hydroxide; sodium fluoride</td></tr><tr><td>Heterocyclic Compounds</td><td>5</td><td>2-(1-hydroxyethyl)-1h-benzimidazole; 1,4- diazabicyclo[2.2.2]octane; 4,4'-bipyridine; pyrazine; piperazine; morpholine; pyridine; s- triazine; meso-tetra(n-methyl-4-pyridyl) porphine tetratosylate</td></tr><tr><td>Hydrocarbons and Derivatives</td><td>4</td><td>hexadecyltributylphosphonium bromide; benzene; toluene; chlorobenzene; p-xylene; acetonitrile; dichloromethane</td></tr><tr><td>Water and Derivatives</td><td>2</td><td>water; hydrogen peroxide</td></tr></table></body></html>  \n\nMolar ratios were calculated from the total molar amount in the event of multiple species for each set. For the reaction conditions, four categories were identified: vapor diffusion, solvothermal, conventional, and microwave-assisted reaction. These were classified using ChatGPT (Figure S21). With regards to the crystalline state outcome, if the reaction results contained a description of (single) crystal(s), it was classified as the single-crystal (SC). If it included words like microcrystalline product, powder, solid, or no description of product morphology was given, it was classified as polycrystalline (P).  \n\nThe full descriptor set include the following components: 'temperature', 'time', 'synthesis_method', 'metal_ionenergy', 'metal_affinity', 'metal_radii', 'metal_electronegativity', 'metal_valence', 'n_carboxylate', 'n_N_donnor', 'n_phosphate', 'n_chelating', 'linker_MW', 'solvent_MW', 'modulator_MW', 'linker_metal_ratio', 'solvent_metal_ratio', 'modulator_metal_ratio', 'linker_MQNs1', 'linker_MQNs2', 'linker_MQNs3', 'linker_MQNs4', 'linker_MQNs6', 'linker_MQNs7', 'linker_MQNs8', 'linker_MQNs9', 'linker_MQNs10', 'linker_MQNs11', 'linker_MQNs12', 'linker_MQNs13', 'linker_MQNs14', 'linker_MQNs15', 'linker_MQNs16', 'linker_MQNs17', 'linker_MQNs19', 'linker_MQNs20', 'linker_MQNs21', 'linker_MQNs22', 'linker_MQNs23', 'linker_MQNs24', 'linker_MQNs25', 'linker_MQNs26', 'linker_MQNs27', 'linker_MQNs28', 'linker_MQNs29', 'linker_MQNs30', 'linker_MQNs31', 'linker_MQNs32', 'linker_MQNs34', 'linker_MQNs35', 'linker_MQNs36', 'linker_MQNs37', 'linker_MQNs40', 'linker_MQNs41', 'linker_MQNs42', 'solvent_type1', 'solvent_type2', 'solvent_type3', 'solvent_type4', 'solvent_type5', 'solvent_type6', 'solvent_type7', 'solvent_type8', 'modulator_type1', 'modulator_type2', 'modulator_type3', 'modulator_type5', 'modulator_type6', 'modulator_type7', 'modulator_type8'. In order to extract the most relevant features and to reduce model complexity, a recursive feature elimination (REF) with 5-fold cross validation was performed to yield 26 descriptors from the initial 70 after the down-selection (Figure S61).  \n\n![](images/56a9c55e2182cb1774d7674383d8f6b520abd62133dbfcc77045ea0787495625.jpg)  \nFigure S63. Percent decrease in accuracy by permutating features in descriptors set after REF over 10 runs. The boxes for each descriptor extend from the first to the third quartile, with a green line indicating the median. The whiskers span from the minimum to the maximum values of the data.  \n\n![](images/8a1e2ac6728cccb9fb5b3f40b0476664c5d550d3012474137980fdeefa0f6d21.jpg)  \nFigure S64. Performance of the classification models in predicting the crystalline state of MOFs from synthesis on the train and test set for varying training set ratio to the data excluding the held out set. (a) Learning curves of the classifier model with 1σ standard deviation error bars. (b) Model performance evaluation through the F1 Score, Precision, Recall, and Area Under the Curve metrics.  \n\n![](images/06c470396550173a704fa03357cf1b5b0976136a49054d3f546c1147a338f311.jpg)  \nFigure S65. Frequency analysis of the synthesis condition dataset. In total, 35 unique solvent compounds and 44 unique modulator compounds were identified, and 10 most frequently occurring solvents and modulators from the extracted synthesis parameters were shown. Percent occurrence of solvents were calculated out of 763 experiments with solvent parameters; those of modulators were calculated out of 402 experiments with modulator parameters.  \n\n![](images/1e85fd0b6a54841c3757f9748285a14c4502671c2dc7368ac0b2b26f9f87f2d8.jpg)  \nFigure S66. Frequency analysis of the synthesis condition dataset. 10 most frequently occurring reaction conditions, out of 30 unique reaction temperature and 48 unique reaction time, from the extracted synthesis parameters were shown.  \n\n![](images/e107870aad3e888f790177400a70b7a17569107870db7ba9a118dfbfb7bf29d0.jpg)  \nFigure S67. Frequency analysis of the synthesis condition dataset. 10 most frequently occurring metal elements and linkers, out of 29 unique metal and 263 unique linker compounds, from the extracted synthesis parameters are shown.  \n\n# Section S6. Dataset to Dialogue: The Creation of a MOF Synthesis Chatbot  \n\nTo enable an automated chatbot drawing upon our dataset acquired from text mining, we initially reformatted the synthesis parameters for each compound into discrete paragraphs. For each paragraph, we also compiled a list of publication data where the compound was reported, such as authors, DOIs, and publication years, retrieved from Web of Science. This approach facilitated the creation of a synthesis and publication information card for each compound. Subsequently, we developed embeddings for the information cards, which form an embedded dataset (Table S3).  \n\nTable S3. Illustrative information card for MOFs and their respective embeddings   \n\n\n<html><body><table><tr><td colspan=\"2\"></td></tr><tr><td>Synthesis and Paper Information</td><td>Embeddings</td></tr><tr><td>MOF Name: MOF-808 Metal Source: ZrOCl2·8H20 Metal Amount: 0.50 mmol Linker: H3BTC (1,3,5-Benzenetricarboxylic acid, CAS number: 554-95-0) Linker Amount: 0.50 mmol Modulator: formic acid Modulator Amount or Volume: 20 mL Solvent: DMF Solvent Volume: 20 mL Reaction Temperature: 100°C Reaction Time: 168 h Reaction Equipment: 60 mL screw capped glass Product Color or Shape: Octahedral colorless crystals Paper DOI: 10.1021/ja500330a Journal: J. Am. Chem. Soc. Publication Year: 2014 Publication Date: MAR 19 Article Title: Water Adsorption in Porous Metal-Organic Frameworks and Related Materials Author Names: Furukawa, Hiroyasu; Gandara, Felipe; Zhang, Yue-Biao; Jiang, Juncong; Queen, Wendy L.; Hudson, Matthew R.; Yaghi, Omar M. MOF Name: ZIF-8</td><td>[0.000997044611722231, -0.021761000156402588, -0.025494899600744247, -0.027127644047141075, -0.006226510275155306, 0.04229075089097023, -0.03372553735971451]</td></tr><tr><td>Metal Source: Zn(NO3)2·4H2O Metal Amount: 0.210 g Linker: H-MeIM (2-methylimidazole, CAS number: 693-98-1) Linker Amount: 0.060 g Modulator: N/A Modulator Amount or Volume: N/A Solvent: DMF Solvent Volume: 18 mL Reaction Temperature: 140°C Reaction Time: 24 h Reaction Equipment: 20-mL vial Product Color or Shape: Colorless polyhedral crystals Paper DOI: 10.1073/pnas.0602439103 Journal: Proc.Natl. Acad. Sci. U. S.A. Publication Year: 2006 Publication Date: JUL 5 Article Title: Exceptional chemical and thermal stability of zeolitic imidazolate frameworks Author Names: Park, Kyo Sung; Ni, Zheng; Cote, Adrien P.; Choi, Jae Yong; Huang, Rudan;</td><td>[-0.0068720560520887375, -0.02060604654252529, -0.03643505275249481, -0.017434848472476006, -0.007826789282262325, 0.05133294314146042, -0.026907961815595627]</td></tr></table></body></html>  \n\nThe system is programmed to navigate to the embedding dataset and locate the most relevant sections based on a user's initial query. This procedure is based on calculating a similarity score between the question and the embeddings and mirrors the foundation of Text Mining Process 3. The highest-ranking entry's information is then dispatched to the ChatGPT Chemistry Assistant's prompt engineering module, which, through the ChatGPT API, crafts responses centered solely around the provided synthesis information. Depending on the user needs, the system can output multiple high similarity scores, such as the top 3 or top 5, provided this does not exceed the token budget (i.e., 4096 tokens for gpt-3.5-turbo).  \n\n# Embedding Demo  \n\nQuery from User: What is the linker used to synthesis MOF-520?  \n\n![](images/af6e7264ae21240bb07ac968922d6e216975581ad055107492888ddbe64bfdc8.jpg)  \nFigure S68. Illustration of embedding user’s initial question to generate context, utilizing the information card of MOF-520 to respond to the query.  \n\nTo establish a chatbot through the ChatGPT API, we followed a similar methodology to that employed in Processes 1, 2, and 3, specifically using ChatGPT to generate the code. The code takes an input prompt from the search output and a fixed prompt to ensure context-based responses. Furthermore, the function should enable access to prior conversations, maintaining a consistent context based on the synthesis information card. The figure below (Figure S69) displays a representative function that our ChatGPT Chemistry Assistant operates on.  \n\n# ChatGPT API  \n\ndef chatbot(question, past_user_messages $\\c=$ None, initial_context=None):  \n\nif past_user_messages is None: past_user_messages $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ []   \npast_user_messages.append(question) $\\#$ Store Synthesis and Paper Information Cards   \ndf_with_emb $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ pd.read_csv(\"xxx.csv\")# Get Information Cards and Embeddings  \n\n![](images/dcd6c2083dd705ff52951a38ee2e75b4a22b91ae00f583d79bda1ed60c0d9970.jpg)  \nFigure S69. Python code snippet demonstrating the utilization of the ChatGPT API for establishing a literature-grounded chatbot.  \n\nWe demonstrate the function's capabilities and its potential for building a robust chatbot application through several dialogues between a human user and the ChatGPT Chemistry Assistant, as depicted below. This foundational function could be integrated into an interactive website or mobile application, offering users real-time dialogues with the chatbot. As a user poses a question through the application interface, a POST request is triggered to the server, prompting the chatbot function with the user's query. The function then computes a pertinent response considering the conversation history, which is then relayed back to the user through the front-end.  \n\n# ChatGPT API  \n\n![](images/56caa12df1f6a7cf378a8ee45c26f423c6f4fe4646d01ddf548753c226f3ec33.jpg)  \nFigure S70. Demonstration of a dialogue with the ChatGPT Chemistry Assistant using Python, showing the potential for conversion into an interactive web or mobile application.  \n\nExpanding upon this, we demonstrate the potential and versatility of the ChatGPT Chemistry Assistant through a series of queries that reflect its ability to make a dataset more accessible, in addition to responding effectively to synthesis inquiries.  \n\nIn the first example (Figure S71), the user sought to understand the difference between two MOFs, ZIF-8 and ZTIF-8. The assistant detailed their distinct characteristics, which includes the metal sources, solvents, co-linkers, and synthesis conditions. When the user expressed interest in synthesizing ZIF-8, the assistant underscored the importance of lab safety and suggested familiarizing oneself with essential lab techniques before attempting the MOF synthesis. In response to the user's request for a detailed synthesis procedure for ZIF-8, the assistant provided a step-by-step process while highlighting necessary safety precautions.  \n\nIn the second example (Figure S72), the assistant showcased its ability to provide information on the synthesis of Zirconium MOFs. It shared two examples of synthesis procedures and explained the concept and role of a modulator. The assistant further elaborated that the outcome might differ if the modulator (acetic acid, in this case) was not included, leading to uncontrolled crystal growth and possibly reducing the yield. When the user inquired about alternative modulators, the assistant explained that the selection of modulators depends on the desired MOF structure and properties, and their choice could significantly impact the outcome.  \n\nThe third example demonstrated how the assistant can effectively handle misinformation (Figure S73). Initially, the user asked about synthesizing MOF-499, a compound the assistant couldn't find information on. However, when the user corrected the typo to MOF-419, the assistant provided a detailed synthesis procedure. The user also sought to understand the post-synthesis steps, and the assistant mentioned common processes such as activation, doping, and functionalization, though specific steps for MOF-419 were not detailed in the reviewed paper.  \n\nThe fourth example highlighted the assistant's capability to provide detailed information about the synthesis of single crystals of MOF-801 (Figure S74). The assistant confirmed that single crystals of MOF-801 could be made and explained the significance of obtaining single crystals for accurate structural characterization. It also explained the potential applications of MOF-801 and the difference in synthesis procedures for single crystal and microcrystalline powder forms. When the user asked about other MOFs reported in the paper, the assistant provided information on the reported zirconium MOFs.  \n\nIn each of these cases, the assistant has shown its utility as an effective tool for helping researchers navigate the complex world of MOF synthesis. By providing contextually aware, focused responses, the assistant makes it easier for researchers to access relevant data and learn more about MOF synthesis, serving as an invaluable tool in the field of chemistry.  \n\n# Human-MOF Chatbot Interaction  \n\nHuman: What is the difference between ZIF-8 and ZTIF-8?  \n\n# MOF Chatbot:  \n\nZIF-8 and ZTIF-8 are different metal-organic frameworks. Although both MOFs share a similar ligand, which is 2- methylimidazole, ZIF-8 is synthesized using $Z n(N O3)2{\\cdot}4120$ as a metal source and DMF as a solvent, while ZTIF-8 is synthesized using Zn(CH3COO)2·2H2O as a metal source and DMF and EtOH as solvents. Additionally, ZTIF-8 uses 5- methyltetrazole as a co-linker, which is not present in ZIF-8. ZIF-8 is also synthesized at a lower temperature for a longer time compared to ZTIF-8. Their different synthesis conditions lead to different crystal structures and properties.  \n\nHuman: I have never conducted any MOF synthesis before, and I want to make ZIF-8. How should I prepare for it?  \n\n# MOF Chatbot:  \n\nMaking MOFs requires knowledge and experience in handling chemicals and laboratory equipment. It also involves following strict safety protocols to avoid any accidents or mishaps. Before attempting to make ZIF-8, it is recommended that you first gain experience in basic laboratory techniques and familiarize yourself with the necessary equipment and safety measures. Additionally, it is essential to carefully read and understand the synthesis procedure from a reliable source, such as a peerreviewed scientific journal article. It is not advised to attempt to make MOFs without proper knowledge, training, and safety precautions.  \n\nHuman: Could you please provide me with a detailed synthesis procedure for this MOF, highlighting the parts that require caution?  \n\n# MOF Chatbot:  \n\nSure! Here's a detailed synthesis procedure for ZIF-8:  \n\n1. Dissolve $Z n(N O3)2{\\cdot}G H20$ in DMF $(20~\\mathrm{mL})$ ) to create a solution.   \n2. Add 2-methylimidazole (0.5 g) to the solution.   \n3. Mix the solution thoroughly.   \n4. Transfer the solution to a Teflon-lined autoclave $(23~\\mathsf{m L})$ .   \n5. Seal the autoclave and heat it at $85^{\\circ}C$ for 24 hours.   \n6. Allow the autoclave to cool to room temperature.   \n7. Remove the resulting white precipitate by centrifugation or filtration.   \n8. Wash the precipitate with solvents such as DMF to remove any impurities.   \n9. Dry the resulting white powder at $60^{\\circ}C$ under vacuum for 12 hours.  \n\nCaution:  \n\n- DMF is toxic and should not be inhaled or ingested. Handle it in a well-ventilated area with protective equipment such as gloves and goggles.   \n- The autoclave should be sealed tightly to prevent leakage or explosion during heating.   \n- The resulting white powder should be handled with care to avoid inhaling or exposing it to moisture, as it can easily absorb water vapor from the air and decompose.  \n\nFigure S71. A conversation with the MOF Chatbot based on gpt-3.5-turbo model.  \n\n# Human-MOF Chatbot Interaction  \n\nHuman: How to make Zirconium MOF?  \n\n# MOF Chatbot:  \n\nThere are different ways to make Zirconium MOFs, and the specific method may depend on various factors such as the linker and modulator used. However, based on the information available from the papers I have reviewed, I can provide two examples:  \n\n1. Zr-CAU-30: Zr-CAU-30 was synthesized using Ni-H8TPPP (linker) and NaF/NaOH (modulators) in H2O solvent at $160^{\\circ}\\mathsf C$ for 3 hours under stirring. ZrOCl2·8H2O (metal source) and Ni-H8TPPP were mixed in H2O solvent, and then NaF and NaOH were added to the mixture. The mixture was heated and stirred until the reaction was completed. The product was obtained as a highly porous metal phosphonate.  \n\n2. Zr-CAU-28: Zr-CAU-28 was synthesized using H2FDC (linker) and acetic acid (modulator) in H2O solvent at $80^{\\circ}C$ for 1 hour under microwave-assisted heating. ZrOCl2·8H2O (metal source) and H2FDC were mixed in H2O solvent, and then acetic acid was added to the mixture. The mixture was heated under microwave irradiation until the reaction was completed. The product was obtained as a white crystalline powder.  \n\nIt's important to note that these are just two examples of Zirconium MOF synthesis, and the specific conditions may vary depending on the desired MOF structure and properties.  \n\nHuman: Could you explain what a modulator is, and if I choose not to add acetic acid during the synthesis, how might it impact the outcome of my results?  \n\n# MOF Chatbot:  \n\nIn MOF synthesis, a modulator is a molecule that can promote or inhibit crystal growth and control the morphology of the MOF crystals. In the synthesis of Zr-CAU-28 as described in the paper, acetic acid was used as a modulator. The addition of acetic acid likely helped to increase the yield of the desired MOF product, as well as promote the formation of smaller and more uniform crystals.  \n\nIf you choose not to add acetic acid during the synthesis of Zr-CAU-28, the outcome of your results may differ. Without a modulator, the crystal growth may be less controlled, leading to the formation of larger or less uniform crystals. Additionally, the yield of the desired MOF product may be lower. It is important to note, however, that the impact of omitting acetic acid may depend on the specific MOF synthesis conditions and may not always have a significant effect on the final product.  \n\nHuman: That's interesting. In the case of synthesizing other MOFs, have you come across any alternative modulators that could potentially be used in place of acetic acid?  \n\n# MOF Chatbot:  \n\nWhile the paper did not mention any other modulators used in the synthesis of Zr-CAU-28, other papers have used different modulators for synthesizing Zr-based MOFs. For example, in the synthesis of Zr-CAU-30, the modulators used were NaF and NaOH. However, the choice of modulator may impact the outcome of the results, so it is recommended to refer to other papers and perform some preliminary experiments to determine the most suitable modulator for the specific synthesis protocol and desired MOF properties.  \n\nFigure S72. A conversation with the MOF Chatbot based on gpt-3.5-turbo model.  \n\n# Human-MOF Chatbot Interaction  \n\nHuman: Can you tell me how to synthesize MOF-499?  \n\n# MOF Chatbot:  \n\nBased on the information available from the MOF papers I have read so far, I cannot provide a reliable answer to this question.   \nPlease provide more information, or the specific paper reference, on the synthesis of MOF-499.  \n\nHuman: Oops, my bad. There was a typo. I actually meant MOF-419.  \n\n# MOF Chatbot:  \n\nTo synthesize MOF-419, you will need to use Y(NO3)3 $(0.5\\:\\mathrm{mmol}$ ) as the metal source, H2BDH (benzene-1,4-dihydroxamic acid, CAS number: 20073-80-7) (0.5 mmol) as the linker, and formic acid ( $100~\\mathrm{{mmol}}$ ) as the modulator. The reaction should be carried out in H2O ( $12\\mathrm{mL})$ ) solvent in a $20~\\mathrm{mL}$ scintillation vial at $100^{\\circ}\\mathsf{C}$ for 12 hours. The product should be obtained as single crystals. If you have any further questions, feel free to ask..  \n\nHuman: Thanks for the explanation. But, how about the post-synthesis steps? What should I do after I get the MOF crystals?  \n\n# MOF Chatbot:  \n\nUnfortunately, the paper I have reviewed did not provide information regarding any post-synthesis steps. However, depending on the intended application of the MOF, various post-synthesis modifications are possible including activation, doping, functionalization, and more. Activating the MOF typically involves removing any solvent or guests trapped within the pores of the MOF through solvent exchange or thermal treatment. Doping and functionalization can be achieved through various methods including deposition, exchange, and covalent attachment. The exact post-synthesis steps required for MOF-419 may depend on the intended application and should be further researched.  \n\nHuman: Could you provide me with the exact title of the paper that discusses MOF-419 so that I can have a check by myself?  \n\n# MOF Chatbot:  \n\nThe paper discussing MOF-419 is titled \"Metal-Organic Frameworks with Rod Yttrium Secondary Building Units\" and was published in the Isr. J. Chem. journal.  \n\nHuman: Thank you. Can you give me a link to get access to the paper?  \n\n# MOF Chatbot:  \n\nUnfortunately, I cannot provide a specific link to access the paper as I do not have information on the journal's access policies. However, you can try searching for the paper by its title \"Metal-Organic Frameworks with Rod Yttrium Secondary Building Units\" in a search engine or a scientific paper database. Some databases that you can use are Google Scholar, PubMed, and ResearchGate.  \n\nFigure S73. A conversation with the MOF Chatbot based on gpt-3.5-turbo model.  \n\n# Human-MOF Chatbot Interaction  \n\nHuman: Is it possible to make single crystals of MOF-801?  \n\nMOF Chatbot: Yes, it is possible to make single crystals of MOF-801. In fact, the paper \"Water Adsorption in Porous MetalOrganic Frameworks and Related Materials\" (DOI: 10.1021/ja500330a) reports the synthesis and characterization of MOF-801 in its single crystal form (MOF-801-SC). The synthesis was performed using ZrOCl2·8H2O as the metal source, fumaric acid as the linker, formic acid as the modulator, DMF as the solvent, and a reaction temperature of $120^{\\circ}\\mathsf C$ for $24\\mathsf{h}$ in a $60~\\mathsf{m L}$ screw capped glass vessel. The product obtained was octahedral colorless single crystals.  \n\nHuman: What is the application of this MOF and why it is important to make it single crystal? Can I make powdered MOF-801?  \n\nMOF Chatbot: MOF-801 has potential applications in water adsorption, dehumidification, and delivery of drinking water in remote areas. It is important to make single crystals of MOF-801 because they allow for more accurate structural characterization, which is important for understanding the properties and behavior of the material. Powdered MOF-801 can also be made and studied for its properties, but single crystals provide more detailed information. Therefore, both single crystal and powdered forms of MOF-801 have their own importance and applications.  \n\nHuman: How do they differ in terms of synthesis procedure?  \n\nMOF Chatbot: Based on the information available from the MOF paper I have read so far, MOF-801 can be synthesized as both a single crystal and a microcrystalline powder form. However, the synthesis procedures are different for each form of MOF-801. The synthesis procedure for MOF-801 single crystal involves the use of ZrOCl2·8H2O as metal source, fumaric acid as linker, formic acid as modulator, DMF as solvent, and a $60~\\mathrm{mL}$ screw capped glass for 24 hours at $120^{\\circ}\\mathsf C$ . On the other hand, MOF-801 microcrystalline powder is synthesized using the same metal source and linker but with different amounts of modulator (benzoic acid) and DMF as solvent.  \n\nHuman: What other MOFs does this paper also report?  \n\nMOF Chatbot: The paper reports on the water adsorption properties of 23 materials, 20 of which are metal-organic frameworks (MOFs). Among the MOFs are 10 zirconium(IV) MOFs, with the following reported for the first time: MOF-801-SC (single crystal form), MOF-802, MOF-805, MOF-806, MOF-808, MOF-812, and MOF-841. MOF-801-P (microcrystalline powder form) was reported earlier and studied for its water adsorption properties. MOF-812 was only made and structurally characterized but not examined for water adsorption because it is a byproduct of MOF-841 synthesis.  \n\nHuman: Thanks. Last question, where can I find this paper?  \n\nMOF Chatbot: You can find the paper on the ACS Publications website using the Digital Object Identifier (DOI) associated with each paper. Here are the DOIs for the papers mentioned in the context:   \n- MOF-801-SC paper: 10.1021/ja500330a   \nAlternatively, you can search for the paper on the ACS Publications website using the journal title I mentioned earlier.  \n\nFigure S74. A conversation with the MOF Chatbot based on gpt-3.5-turbo model.  \n\n# References  \n\n1. Brown, T.;  Mann, B.;  Ryder, N.;  Subbiah, M.;  Kaplan, J. D.;  Dhariwal, P.;  Neelakantan, A.; Shyam, P.;  Sastry, G.; Askell, A., Language models are few-shot learners. NIPS 2020, 33, 1877- 1901.   \n2. Vaswani, A.;  Shazeer, N.;  Parmar, N.;  Uszkoreit, J.;  Jones, L.;  Gomez, A. N.;  Kaiser, Ł.; Polosukhin, I., Attention is all you need. NIPS 2017, 30.   \n3. Zheng, Z.;  Rong, Z.;  Iu‐Fan Chen, O.; Yaghi, O. M., Metal‐Organic Frameworks with Rod Yttrium Secondary Building Units. Isr. J. Chem. 2023, e202300017.   \n4. Nguyen, H. L.;  Gándara, F.;  Furukawa, H.;  Doan, T. L.;  Cordova, K. E.; Yaghi, O. M., A titanium–organic framework as an exemplar of combining the chemistry of metal–and covalent– organic frameworks. J. Am. Chem. Soc. 2016, 138 (13), 4330-4333.   \n5. Furukawa, H.;  Gandara, F.;  Zhang, Y.-B.;  Jiang, J.;  Queen, W. L.;  Hudson, M. R.; Yaghi, O. M., Water adsorption in porous metal–organic frameworks and related materials. J. Am. Chem. Soc. 2014, 136 (11), 4369-4381.   \n6. Hanikel, N.;  Kurandina, D.;  Chheda, S.;  Zheng, Z.;  Rong, Z.;  Neumann, S. E.;  Sauer, J.; Siepmann, J. I.;  Gagliardi, L.; Yaghi, O. M., MOF Linker Extension Strategy for Enhanced Atmospheric Water Harvesting. ACS Cent. Sci. 2023, 9 (3), 551-557.   \n7. Jiang, J.;  Furukawa, H.;  Zhang, Y.-B.; Yaghi, O. M., High methane storage working capacity in metal–organic frameworks with acrylate links. J. Am. Chem. Soc. 2016, 138 (32), 10244-10251. 8. Hayashi, H.;  Côté, A. P.;  Furukawa, H.;  O’Keeffe, M.; Yaghi, O. M., Zeolite A imidazolate frameworks. Nat. Mater. 2007, 6 (7), 501-506.   \n9. Luo, Y.;  Bag, S.;  Zaremba, O.;  Cierpka, A.;  Andreo, J.;  Wuttke, S.;  Friederich, P.; Tsotsalas, M., MOF synthesis prediction enabled by automatic data mining and machine learning. Angew. Chem. Int. Ed. 2022, 61 (19), e202200242.   \n10. Nandy, A.;  Duan, C.; Kulik, H. J., Using machine learning and data mining to leverage community knowledge for the engineering of stable metal–organic frameworks. J. Am. Chem. Soc. 2021, 143 (42), 17535-17547.   \n11. Park, H.;  Kang, Y.;  Choe, W.; Kim, J., Mining Insights on Metal–Organic Framework Synthesis from Scientific Literature Texts. J. Chem. Inf. Model. 2022, 62 (5), 1190-1198. 12. Park, S.;  Kim, B.;  Choi, S.;  Boyd, P. G.;  Smit, B.; Kim, J., Text mining metal–organic framework papers. J. Chem. Inf. Model. 2018, 58 (2), 244-251.   \n13. Shannon, R. D., Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr. A . 1976, 32 (5), 751-767.   \n14. Haynes, W. M., CRC handbook of chemistry and physics. CRC press: 2016.   \n15. Pauling, L., The nature of the chemical bond. IV. The energy of single bonds and the relative electronegativity of atoms. J. Am. Chem. Soc. 1932, 54 (9), 3570-3582.   \n16. Nguyen, K. T.;  Blum, L. C.;  Van Deursen, R.; Reymond, J. L., Classification of organic molecules by molecular quantum numbers. ChemMedChem 2009, 4 (11), 1803-1805.   \n17. Deursen, R. v.;  Blum, L. C.; Reymond, J.-L., A searchable map of PubChem. J. Chem. Inf. Model. 2010, 50 (11), 1924-1934.  ",
    "Target": null
  }
]