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Zhang H, Liu S, You J, Liu C, Zheng S, Lu Z, Wang T, Zheng N, Shao B. Overcoming the barrier of orbital-free density functional theory for molecular systems using deep learning. NATURE COMPUTATIONAL SCIENCE 2024; 4:210-223. [PMID: 38467870 DOI: 10.1038/s43588-024-00605-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2023] [Accepted: 02/07/2024] [Indexed: 03/13/2024]
Abstract
Orbital-free density functional theory (OFDFT) is a quantum chemistry formulation that has a lower cost scaling than the prevailing Kohn-Sham DFT, which is increasingly desired for contemporary molecular research. However, its accuracy is limited by the kinetic energy density functional, which is notoriously hard to approximate for non-periodic molecular systems. Here we propose M-OFDFT, an OFDFT approach capable of solving molecular systems using a deep learning functional model. We build the essential non-locality into the model, which is made affordable by the concise density representation as expansion coefficients under an atomic basis. With techniques to address unconventional learning challenges therein, M-OFDFT achieves a comparable accuracy to Kohn-Sham DFT on a wide range of molecules untouched by OFDFT before. More attractively, M-OFDFT extrapolates well to molecules much larger than those seen in training, which unleashes the appealing scaling of OFDFT for studying large molecules including proteins, representing an advancement of the accuracy-efficiency trade-off frontier in quantum chemistry.
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Affiliation(s)
- He Zhang
- National Key Laboratory of Human-Machine Hybrid Augmented Intelligence, National Engineering Research Center for Visual Information and Applications, and Institute of Artificial Intelligence and Robotics, Xi'an Jiaotong University, Xi'an, China
- Microsoft Research AI4Science, Beijing, China
| | - Siyuan Liu
- Microsoft Research AI4Science, Beijing, China
| | | | - Chang Liu
- Microsoft Research AI4Science, Beijing, China.
| | | | - Ziheng Lu
- Microsoft Research AI4Science, Beijing, China
| | - Tong Wang
- Microsoft Research AI4Science, Beijing, China
| | - Nanning Zheng
- National Key Laboratory of Human-Machine Hybrid Augmented Intelligence, National Engineering Research Center for Visual Information and Applications, and Institute of Artificial Intelligence and Robotics, Xi'an Jiaotong University, Xi'an, China
| | - Bin Shao
- Microsoft Research AI4Science, Beijing, China.
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Mi W, Luo K, Trickey SB, Pavanello M. Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations. Chem Rev 2023; 123:12039-12104. [PMID: 37870767 DOI: 10.1021/acs.chemrev.2c00758] [Citation(s) in RCA: 8] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2023]
Abstract
Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.
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Affiliation(s)
- Wenhui Mi
- Key Laboratory of Material Simulation Methods & Software of Ministry of Education, College of Physics, Jilin University, Changchun 130012, PR China
- State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, PR China
- International Center of Future Science, Jilin University, Changchun 130012, PR China
| | - Kai Luo
- Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, PR China
| | - S B Trickey
- Quantum Theory Project, Department of Physics and Department of Chemistry, University of Florida, Gainesville, Florida 32611, United States
| | - Michele Pavanello
- Department of Physics and Department of Chemistry, Rutgers University, Newark, New Jersey 07102, United States
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Tan CW, Pickard CJ, Witt WC. Automatic differentiation for orbital-free density functional theory. J Chem Phys 2023; 158:124801. [PMID: 37003740 DOI: 10.1063/5.0138429] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/03/2023] Open
Abstract
Differentiable programming has facilitated numerous methodological advances in scientific computing. Physics engines supporting automatic differentiation have simpler code, accelerating the development process and reducing the maintenance burden. Furthermore, fully differentiable simulation tools enable direct evaluation of challenging derivatives-including those directly related to properties measurable by experiment-that are conventionally computed with finite difference methods. Here, we investigate automatic differentiation in the context of orbital-free density functional theory (OFDFT) simulations of materials, introducing PROFESS-AD. Its automatic evaluation of properties derived from first derivatives, including functional potentials, forces, and stresses, facilitates the development and testing of new density functionals, while its direct evaluation of properties requiring higher-order derivatives, such as bulk moduli, elastic constants, and force constants, offers more concise implementations than conventional finite difference methods. For these reasons, PROFESS-AD serves as an excellent prototyping tool and provides new opportunities for OFDFT.
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Affiliation(s)
- Chuin Wei Tan
- Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom
| | - Chris J Pickard
- Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom
| | - William C Witt
- Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 0FS, United Kingdom
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Hao H, Ruiz Pestana L, Qian J, Liu M, Xu Q, Head‐Gordon T. Chemical transformations and transport phenomena at interfaces. WIRES COMPUTATIONAL MOLECULAR SCIENCE 2022. [DOI: 10.1002/wcms.1639] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Affiliation(s)
- Hongxia Hao
- Kenneth S. Pitzer Theory Center and Department of Chemistry University of California Berkeley California USA
- Chemical Sciences Division Lawrence Berkeley National Laboratory Berkeley California USA
| | - Luis Ruiz Pestana
- Department of Civil and Architectural Engineering University of Miami Coral Gables Florida USA
| | - Jin Qian
- Chemical Sciences Division Lawrence Berkeley National Laboratory Berkeley California USA
| | - Meili Liu
- Department of Civil and Architectural Engineering University of Miami Coral Gables Florida USA
| | - Qiang Xu
- Chemical Sciences Division Lawrence Berkeley National Laboratory Berkeley California USA
| | - Teresa Head‐Gordon
- Kenneth S. Pitzer Theory Center and Department of Chemistry University of California Berkeley California USA
- Chemical Sciences Division Lawrence Berkeley National Laboratory Berkeley California USA
- Department of Bioengineering and Chemical and Biomolecular Engineering University of California Berkeley California USA
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Kinetic Energy Density Functionals Based on a Generalized Screened Coulomb Potential: Linear Response and Future Perspectives. COMPUTATION 2022. [DOI: 10.3390/computation10020030] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
Abstract
We consider kinetic energy functionals that depend, beside the usual semilocal quantities (density, gradient, Laplacian of the density), on a generalized Yukawa potential, that is the screened Coulomb potential of the density raised to some power. These functionals, named Yukawa generalized gradient approximations (yGGA), are potentially efficient real-space semilocal methods that include significant non-local effects and can describe different important exact properties of the kinetic energy. In this work, we focus in particular on the linear response behavior for the homogeneous electron gas (HEG). We show that such functionals are able to reproduce the exact Lindhard function behavior with a very good accuracy, outperforming all other semilocal kinetic functionals. These theoretical advances allow us to perform a detailed analysis of a special class of yGGAs, namely the linear yGGA functionals. Thus, we show how the present approach can generalize the yGGA functionals improving the HEG linear behavior and leading to an extended formula for the kinetic functional. Moreover, testing on several jellium cluster model systems allows highlighting advantages and limitations of the linear yGGA functionals and future perspectives for the development of yGGA kinetic functionals.
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Sillaste S, Thompson RB. Molecular Bonding in an Orbital-Free-Related Density Functional Theory. J Phys Chem A 2022; 126:325-332. [PMID: 34994568 DOI: 10.1021/acs.jpca.1c07128] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
A density functional theory based on polymer self-consistent field theory is applied to systems of two atoms in order to show that this approach is capable of predicted molecular bonding. Periodic table elements from hydrogen up to neon are examined and homonuclear diatomic molecules are found to form for H2, N2, O2, and F2, in agreement with known results. The heteronuclear molecules CO and HF, which are known to exist under ambient conditions, are also found to be stable. Bond lengths for most of these molecules agree with experimental results to within less than 8%, with the exception of O2 and F2 which deviate more significantly. The bonding energy for H2 is given and is within 16% of the known value, but fundamental vibrational frequencies do not agree well with experiment. The main approximations of the theory are very simple and include a Fermi-Amaldi correction to the electron-electron interaction to account for self-interactions and a basic expression for the Pauli potential to account for the exclusion principle. The self-consistent equations are solved in terms of basis functions that encode the cylindrical symmetry of diatomic molecules. Since orbitals are not used, the approach is related to orbital-free density functional theory.
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Affiliation(s)
- Spencer Sillaste
- Department of Physics & Astronomy and Waterloo Institute for Nanotechnology, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
| | - Russell B Thompson
- Department of Physics & Astronomy and Waterloo Institute for Nanotechnology, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
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Finzel K. Approximate Analytical Solutions for the Euler Equation for Second-Row Homonuclear Dimers. J Chem Theory Comput 2021; 17:6832-6840. [PMID: 34407616 DOI: 10.1021/acs.jctc.1c00435] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
This work presents a new method how to obtain approximate analytical solutions for the Euler equation for second-row homonuclear dimers. In contrast to the well-known Kohn-Sham method where a system of N nonlinear coupled differential equations must be solved iteratively, orbital-free density functional theory allows to access the minimizing electron density directly via the Euler equation. For simplified models, here, an atom-centered monopole expansion with one free parameter, solutions of the electron density can be obtained analytically by solving the Euler equation at the bond critical point. The procedure is exemplarily carried out for N2, C2, and B2, yielding bound molecules with an internuclear distance of 2.01, 2.43, and 3.07 bohr, respectively.
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Affiliation(s)
- Kati Finzel
- Technische Universität Dresden, Bergstrasse 66c, Dresden 01069, Germany
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Witt WC, Jiang K, Carter EA. Upper bound to the gradient-based kinetic energy density of noninteracting electrons in an external potential. J Chem Phys 2019. [DOI: 10.1063/1.5108896] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- William C. Witt
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, USA
| | - Kaili Jiang
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, USA
| | - Emily A. Carter
- School of Engineering and Applied Science, Princeton University, Princeton, New Jersey 08544-5263, USA
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Finzel K. The first order atomic fragment approach-An orbital-free implementation of density functional theory. J Chem Phys 2019; 151:024109. [PMID: 31301700 DOI: 10.1063/1.5099217] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
An orbital-free implementation of the original Hohenberg-Kohn theorems is presented, making use of the scaling properties from a fictitious Kohn-Sham system, but without reintroducing orbitals. The first order fragment approach does not contain data or parameters that are fitted to the final outcome of the molecular orbital-free calculation and thus represents a parameter-free implementation of orbital-free density functional theory, although it requires the precalculation of atomic data. Consequently, the proposed method is not limited to a specific type of molecule or chemical bonding. The different approximation levels arise from including (first order) or neglecting (zeroth order) the dependency between the potential and the electron density, which in the bifunctional approach are formally treated as independent variables.
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Affiliation(s)
- K Finzel
- Faculty of Chemistry and Food Chemistry, Technische Universität Dresden, Bergstraße 66, 01069 Dresden, Germany
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Thompson RB. An alternative derivation of orbital-free density functional theory. J Chem Phys 2019; 150:204109. [PMID: 31153158 DOI: 10.1063/1.5096405] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Abstract
Polymer self-consistent field theory techniques are used to derive quantum density functional theory without the use of the theorems of density functional theory. Instead, a free energy is obtained from a partition function that is constructed directly from a Hamiltonian so that the results are, in principle, valid at finite temperatures. The main governing equations are found to be a set of modified diffusion equations, and the set of self-consistent equations are essentially identical to those of a ring polymer system. The equations are shown to be equivalent to Kohn-Sham density functional theory and to reduce to classical density functional theory, each under appropriate conditions. The obtained noninteracting kinetic energy functional is, in principle, exact but suffers from the usual orbital-free approximation of the Pauli exclusion principle in addition to the exchange-correlation approximation. The equations are solved using the spectral method of polymer self-consistent field theory, which allows the set of modified diffusion equations to be evaluated for the same computational cost as solving a single diffusion equation. A simple exchange-correlation functional is chosen, together with a shell-structure-based Pauli potential, in order to compare the ensemble average electron densities of several isolated atom systems to known literature results. The agreement is excellent, justifying the alternative formalism and numerical method. Some speculation is provided on considering the timelike parameter in the diffusion equations, which is related to temperature, as having dimensional significance, and thus picturing pointlike quantum particles instead as nonlocal, polymerlike, threads in a higher dimensional thermal-space. A consideration of the double-slit experiment from this point of view is speculated to provide results equivalent to the Copenhagen interpretation. Thus, the present formalism may be considered as a type of "pilot-wave," realist, perspective on density functional theory.
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Affiliation(s)
- Russell B Thompson
- Department of Physics and Astronomy and Waterloo Institute for Nanotechnology, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada
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Constantin LA, Fabiano E, Della Sala F. Performance of Semilocal Kinetic Energy Functionals for Orbital-Free Density Functional Theory. J Chem Theory Comput 2019; 15:3044-3055. [DOI: 10.1021/acs.jctc.9b00183] [Citation(s) in RCA: 24] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Lucian A. Constantin
- Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy
| | - Eduardo Fabiano
- Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy
- Institute for Microelectronics and Microsystems (CNR-IMM), Campus Unisalento, Via Monteroni, 73100 Lecce, Italy
| | - Fabio Della Sala
- Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy
- Institute for Microelectronics and Microsystems (CNR-IMM), Campus Unisalento, Via Monteroni, 73100 Lecce, Italy
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13
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del Rio BG, Chen M, González LE, Carter EA. Orbital-free density functional theory simulation of collective dynamics coupling in liquid Sn. J Chem Phys 2018; 149:094504. [DOI: 10.1063/1.5040697] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Beatriz G. del Rio
- Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, USA
| | - Mohan Chen
- Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA
| | - Luis E. González
- Departamento de Física Teórica, Atómica y Óptica, Universidad de Valladolid, 47011 Valladolid, Spain
| | - Emily A. Carter
- School of Engineering and Applied Science, Princeton University, Princeton, New Jersey 08544-5263, USA
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Constantin LA, Fabiano E, Della Sala F. Semilocal Pauli-Gaussian Kinetic Functionals for Orbital-Free Density Functional Theory Calculations of Solids. J Phys Chem Lett 2018; 9:4385-4390. [PMID: 30019904 DOI: 10.1021/acs.jpclett.8b01926] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Kinetic energy (KE) approximations are key elements in orbital-free density functional theory. To date, the use of nonlocal functionals, possibly employing system-dependent parameters, has been considered mandatory in order to obtain satisfactory accuracy for different solid-state systems, whereas semilocal approximations are generally regarded as unfit to this aim. Here, we show that, instead, properly constructed semilocal approximations, the Pauli-Gaussian (PG) KE functionals, especially at the Laplacian level of theory, can indeed achieve similar accuracy as nonlocal functionals and can be accurate for both metals and semiconductors, without the need for system-dependent parameters.
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Affiliation(s)
- Lucian A Constantin
- Center for Biomolecular Nanotechnologies @UNILE , Istituto Italiano di Tecnologia , Via Barsanti , I-73010 Arnesano , Italy
| | - Eduardo Fabiano
- Center for Biomolecular Nanotechnologies @UNILE , Istituto Italiano di Tecnologia , Via Barsanti , I-73010 Arnesano , Italy
- Institute for Microelectronics and Microsystems (CNR-IMM) , Via Monteroni, Campus Unisalento , 73100 Lecce , Italy
| | - Fabio Della Sala
- Center for Biomolecular Nanotechnologies @UNILE , Istituto Italiano di Tecnologia , Via Barsanti , I-73010 Arnesano , Italy
- Institute for Microelectronics and Microsystems (CNR-IMM) , Via Monteroni, Campus Unisalento , 73100 Lecce , Italy
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Modified Fourth-Order Kinetic Energy Gradient Expansion with Hartree Potential-Dependent Coefficients. J Chem Theory Comput 2017; 13:4228-4239. [DOI: 10.1021/acs.jctc.7b00705] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Dieterich JM, Witt WC, Carter EA. libKEDF: An accelerated library of kinetic energy density functionals. J Comput Chem 2017; 38:1552-1559. [DOI: 10.1002/jcc.24806] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/17/2017] [Revised: 03/22/2017] [Accepted: 03/24/2017] [Indexed: 01/09/2023]
Affiliation(s)
- Johannes M. Dieterich
- Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrinceton New Jersey08544‐5263
| | - William C. Witt
- Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrinceton New Jersey08544‐5263
| | - Emily A. Carter
- School of Engineering and Applied SciencePrinceton UniversityPrinceton New Jersey08544‐5263
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18
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Mi W, Zhang S, Wang Y, Ma Y, Miao M. First-principle optimal local pseudopotentials construction via optimized effective potential method. J Chem Phys 2016; 144:134108. [DOI: 10.1063/1.4944989] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Wenhui Mi
- State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, People’s Republic of China
- Beijing Computational Science Research Center, Beijing 100086, People’s Republic of China
| | - Shoutao Zhang
- State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, People’s Republic of China
- Beijing Computational Science Research Center, Beijing 100086, People’s Republic of China
| | - Yanchao Wang
- State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, People’s Republic of China
| | - Yanming Ma
- State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, People’s Republic of China
| | - Maosheng Miao
- Beijing Computational Science Research Center, Beijing 100086, People’s Republic of China
- Department of Chemistry and Biochemistry, California State University Northridge, Northridge, California 91330, USA
- Materials Research Lab, University of California Santa Barbara, Santa Barbara, California 93110, USA
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