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Gedeon J, Schmidt J, Hodgson MJP, Wetherell J, Benavides-Riveros CL, Marques MAL. Machine learning the derivative discontinuity of density-functional theory. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2022. [DOI: 10.1088/2632-2153/ac3149] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Abstract
Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (a) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (b) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation.
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Kraisler E, Hodgson MJP, Gross EKU. From Kohn-Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential. J Chem Theory Comput 2021; 17:1390-1407. [PMID: 33595312 PMCID: PMC8363072 DOI: 10.1021/acs.jctc.0c01093] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
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Accurately
describing excited states within Kohn–Sham (KS)
density functional theory (DFT), particularly those which induce ionization
and charge transfer, remains a great challenge. Common exchange-correlation
(xc) approximations are unreliable for excited states owing, in part,
to the absence of a derivative discontinuity in the xc energy (Δ),
which relates a many-electron energy difference to the corresponding
KS energy difference. We demonstrate, analytically and numerically,
how the relationship between KS and many-electron energies leads to
the step structures observed in the exact xc potential in four scenarios:
electron addition, molecular dissociation, excitation of a finite
system, and charge transfer. We further show that steps in the potential
can be obtained also with common xc approximations, as simple as the
LDA, when addressed from the ensemble perspective. The article therefore
highlights how capturing the relationship between KS and many-electron
energies with advanced xc approximations is crucial for accurately
calculating excitations, as well as the ground-state density and energy
of systems which consist of distinct subsystems.
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Affiliation(s)
- Eli Kraisler
- Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, 9091401 Jerusalem, Israel
| | - M J P Hodgson
- Department of Physics, Durham University, South Road, Durham DH1 3LE, United Kingdom.,Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
| | - E K U Gross
- Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, 9091401 Jerusalem, Israel
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Carmona-Espíndola J, Gázquez JL, Vela A, Trickey SB. Generalized Gradient Approximation Exchange Energy Functional with Near-Best Semilocal Performance. J Chem Theory Comput 2018; 15:303-310. [PMID: 30481469 DOI: 10.1021/acs.jctc.8b00998] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
We develop and validate a nonempirical generalized gradient approximation (GGA) exchange (X) density functional that performs as well as the SCAN (strongly constrained and appropriately normed) meta-GGA on standard thermochemistry tests. Additionally, the new functional (NCAP, nearly correct asymptotic potential) yields Kohn-Sham eigenvalues that are useful approximations of the density functional theory (DFT) ionization potential theorem values by inclusion of a systematic derivative discontinuity shift of the X potential. NCAP also enables time-dependent DFT (TD-DFT) calculations of good-quality polarizabilities, hyper-polarizabilities, and one-Fermion excited states without modification (calculated or ad hoc) of the long-range behavior of the exchange potential or other patches. NCAP is constructed by reconsidering the imposition of the asymptotic correctness of the X potential (-1/ r) as a constraint. Inclusion of derivative discontinuity and approximate integer self-interaction correction treatments along with first-principles determination of the effective second-order gradient expansion coefficient yields a major advance over our earlier correct asymptotic potential functional [CAP; J. Chem. Phys. 2015 , 142 , 054105 ]. The new functional reduces a spurious bump in the CAP atomic exchange potential and moves it to distances irrelevantly far from the nucleus (outside the tail of essentially all practical basis functions). It therefore has nearly correct atomic exchange-potential behavior out to rather large finite distances r from the nucleus but eventually goes as - c/ r with an estimated value for the constant c of around 0.3, so as to achieve other important properties of exact DFT exchange within the restrictions of the GGA form. We illustrate the results with the Ne atom optimized effective potentials and with standard molecular benchmark test data sets for thermochemical, structural, and response properties.
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Affiliation(s)
- Javier Carmona-Espíndola
- Departamento de Química , CONACYT-Universidad Autónoma Metropolitana-Iztapalapa , Av. San Rafael Atlixco 186 , Ciudad de México 09340 , México
| | - José L Gázquez
- Departamento de Química , Universidad Autónoma Metropolitana-Iztapalapa , Av. San Rafael Atlixco 186 , Ciudad de México 09340 , México
| | - Alberto Vela
- Departamento de Química , Centro de Investigación y de Estudios Avanzados , Av. Instituto Politécnico Nacional 2508 , Ciudad de México 07360 , México
| | - S B Trickey
- Quantum Theory Project, Department of Physics and Deptartment of Chemistry , University of Florida , P.O. Box 118435, Gainesville , Florida 32611-8435 , United States
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Hodgson MJP, Kraisler E, Schild A, Gross EKU. How Interatomic Steps in the Exact Kohn-Sham Potential Relate to Derivative Discontinuities of the Energy. J Phys Chem Lett 2017; 8:5974-5980. [PMID: 29179553 DOI: 10.1021/acs.jpclett.7b02615] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Accurate density functional calculations hinge on reliable approximations to the unknown exchange-correlation (xc) potential. The most popular approximations usually lack features of the exact xc potential that are important for an accurate prediction of the fundamental gap and the distribution of charge in complex systems. Two principal features in this regard are the spatially uniform shift in the potential, as the number of electrons infinitesimally surpasses an integer, and the spatial steps that form, for example, between the atoms of stretched molecules. Although both aforementioned concepts are well known, the exact relationship between them remained unclear. Here we establish this relationship via an analytical derivation. We support our result by numerically solving the many-electron Schrödinger equation to extract the exact Kohn-Sham potential and directly observe its features. Spatial steps in the exact xc potential of a full configuration-interaction (FCI) calculation of a molecule are presented in three dimensions.
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Affiliation(s)
- M J P Hodgson
- Max-Planck-Institut für Mikrostrukturphysik , Weinberg 2, D-06120 Halle, Germany
| | - Eli Kraisler
- Max-Planck-Institut für Mikrostrukturphysik , Weinberg 2, D-06120 Halle, Germany
| | - Axel Schild
- Laboratorium für Physikalische Chemie, ETH Zürich , 8093 Zürich, Switzerland
| | - E K U Gross
- Max-Planck-Institut für Mikrostrukturphysik , Weinberg 2, D-06120 Halle, Germany
- Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem , Jerusalem 91904, Israel
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Andrade X, Strubbe D, De Giovannini U, Larsen AH, Oliveira MJT, Alberdi-Rodriguez J, Varas A, Theophilou I, Helbig N, Verstraete MJ, Stella L, Nogueira F, Aspuru-Guzik A, Castro A, Marques MAL, Rubio A. Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems. Phys Chem Chem Phys 2016; 17:31371-96. [PMID: 25721500 DOI: 10.1039/c5cp00351b] [Citation(s) in RCA: 186] [Impact Index Per Article: 23.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
Real-space grids are a powerful alternative for the simulation of electronic systems. One of the main advantages of the approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. These properties constitute a great advantage at the time of implementing and testing new physical models. Based on our experience with the Octopus code, in this article we discuss how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems. Among these applications are approaches to calculate response properties, modeling of photoemission, optimal control of quantum systems, simulation of plasmonic systems, and the exact solution of the Schrödinger equation for low-dimensionality systems.
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Affiliation(s)
- Xavier Andrade
- Lawrence Livermore National Laboratory, Livermore, CA 94550, USA. and Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, MA 02138, USA
| | - David Strubbe
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Umberto De Giovannini
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Universidad del País Vasco CFM CSIC-UPV/EHU-MPC & DIPC, 20018 Donostia-San Sebastián, Spain
| | - Ask Hjorth Larsen
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Universidad del País Vasco CFM CSIC-UPV/EHU-MPC & DIPC, 20018 Donostia-San Sebastián, Spain
| | - Micael J T Oliveira
- Unité Nanomat, Département de Physique, Université de Liège, Allée du 6 Août 17, B-4000 Liège, Belgium
| | - Joseba Alberdi-Rodriguez
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Universidad del País Vasco CFM CSIC-UPV/EHU-MPC & DIPC, 20018 Donostia-San Sebastián, Spain and Dept. of Computer Architecture and Technology, University of the Basque Country UPV/EHU, M. Lardizabal, 1, 20018 Donostia-San Sebastian, Spain
| | - Alejandro Varas
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Universidad del País Vasco CFM CSIC-UPV/EHU-MPC & DIPC, 20018 Donostia-San Sebastián, Spain
| | - Iris Theophilou
- Peter-Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Nicole Helbig
- Peter-Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Matthieu J Verstraete
- Unité Nanomat, Département de Physique, Université de Liège, Allée du 6 Août 17, B-4000 Liège, Belgium
| | - Lorenzo Stella
- Atomistic Simulation Centre, School of Mathematics and Physics, Queen's University Belfast, University Road, Belfast BT7 1NN, Northern Ireland, UK
| | - Fernando Nogueira
- Center for Computational Physics, University of Coimbra, Rua Larga, 3004-516 Coimbra, Portugal
| | - Alán Aspuru-Guzik
- Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, MA 02138, USA
| | - Alberto Castro
- Institute for Biocomputation and Physics of Complex Systems (BIFI) and Zaragoza Center for Advanced Modeling (ZCAM), University of Zaragoza, E-50009 Zaragoza, Spain and ARAID Foundation, María de Luna 11, Edificio CEEI Aragón, Zaragoza E-50018, Spain
| | - Miguel A L Marques
- Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, Von-Seckendorff-Platz 1, 06120 Halle (Saale), Germany
| | - Angel Rubio
- Nano-Bio Spectroscopy Group and European Theoretical Spectroscopy Facility (ETSF), Universidad del País Vasco CFM CSIC-UPV/EHU-MPC & DIPC, 20018 Donostia-San Sebastián, Spain and Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany
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Kraisler E, Schmidt T, Kümmel S, Kronik L. Effect of ensemble generalization on the highest-occupied Kohn-Sham eigenvalue. J Chem Phys 2015; 143:104105. [DOI: 10.1063/1.4930119] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Eli Kraisler
- Department of Materials and Interfaces, Weizmann Institute of Science, Rehovoth 76100, Israel
| | - Tobias Schmidt
- Theoretical Physics IV, University of Bayreuth, 95440 Bayreuth, Germany
| | - Stephan Kümmel
- Theoretical Physics IV, University of Bayreuth, 95440 Bayreuth, Germany
| | - Leeor Kronik
- Department of Materials and Interfaces, Weizmann Institute of Science, Rehovoth 76100, Israel
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