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Tancogne-Dejean N, Penz M, Laestadius A, Csirik MA, Ruggenthaler M, Rubio A. Exchange energies with forces in density-functional theory. J Chem Phys 2024; 160:024103. [PMID: 38189616 DOI: 10.1063/5.0177346] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2023] [Accepted: 12/12/2023] [Indexed: 01/09/2024] Open
Abstract
We propose exchanging the energy functionals in ground-state density-functional theory with physically equivalent exact force expressions as a new promising route toward approximations to the exchange-correlation potential and energy. In analogy to the usual energy-based procedure, we split the force difference between the interacting and auxiliary Kohn-Sham system into a Hartree, an exchange, and a correlation force. The corresponding scalar potential is obtained by solving a Poisson equation, while an additional transverse part of the force yields a vector potential. These vector potentials obey an exact constraint between the exchange and correlation contribution and can further be related to the atomic shell structure. Numerically, the force-based local-exchange potential and the corresponding exchange energy compare well with the numerically more involved optimized effective potential method. Overall, the force-based method has several benefits when compared to the usual energy-based approach and opens a route toward numerically inexpensive nonlocal and (in the time-dependent case) nonadiabatic approximations.
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Affiliation(s)
- Nicolas Tancogne-Dejean
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science and Department of Physics, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Markus Penz
- Department of Computer Science, Oslo Metropolitan University, 0130 Oslo, Norway
- Basic Research Community for Physics, Innsbruck, Austria
| | - Andre Laestadius
- Department of Computer Science, Oslo Metropolitan University, 0130 Oslo, Norway
- Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, 0315 Oslo, Norway
| | - Mihály A Csirik
- Department of Computer Science, Oslo Metropolitan University, 0130 Oslo, Norway
- Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, 0315 Oslo, Norway
| | - Michael Ruggenthaler
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science and Department of Physics, Luruper Chaussee 149, 22761 Hamburg, Germany
- The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Angel Rubio
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science and Department of Physics, Luruper Chaussee 149, 22761 Hamburg, Germany
- The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA
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2
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Baerends EJ. Chemical potential, derivative discontinuity, fractional electrons, jump of the Kohn-Sham potential, atoms as thermodynamic open systems, and other (mis)conceptions of the density functional theory of electrons in molecules. Phys Chem Chem Phys 2022; 24:12745-12766. [PMID: 35593143 DOI: 10.1039/d2cp01585d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Many references exist in the density functional theory (DFT) literature to the chemical potential of the electrons in an atom or a molecule. The origin of this notion has been the identification of the Lagrange multiplier μ = ∂E/∂N in the Euler-Lagrange variational equation for the ground state density as the chemical potential of the electrons. We first discuss why the Lagrange multiplier in this case is an arbitrary constant and therefore cannot be a physical characteristic of an atom or molecule. The switching of the energy derivative ("chemical potential") from -I to -A when the electron number crosses the integer, called integer discontinuity or derivative discontinuity, is not physical but only occurs when the nonphysical noninteger electron systems and the corresponding energy and derivative ∂E/∂N are chosen in a specific discontinuous way. The question is discussed whether in fact the thermodynamical concept of a chemical potential can be defined for the electrons in such few-electron systems as atoms and molecules. The conclusion is that such systems lack important characteristics of thermodynamic systems and do not afford the definition of a chemical potential. They also cannot be considered as analogues of the open systems of thermodynamics that can exchange particles with an environment (a particle bath or other members of a Gibbsian ensemble). Thermodynamical (statistical mechanical) concepts like chemical potential, open systems, grand canonical ensemble etc. are not applicable to a few electron system like an atom or molecule. A number of topics in DFT are critically reviewed in light of these findings: jumps in the Kohn-Sham potential when crossing an integer number of electrons, the band gap problem, the deviation-from-straight-lines error, and the role of ensembles in DFT.
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Lacombe L, Maitra NT. Developing new and understanding old approximations in TDDFT. Faraday Discuss 2020; 224:382-401. [PMID: 32926040 DOI: 10.1039/d0fd00049c] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
When a system has evolved far from a ground-state, the adiabatic approximations commonly used in time-dependent density functional theory calculations completely fail in some applications, while giving qualitatively good predictions in others, and sometimes even quantitative predictions. It is not clearly understood why this is so, and developing practical approximations going beyond the adiabatic approximation remains a challenge. This paper explores three different lines of investigation. First, an expression for the exact time-dependent exchange-correlation potential suggests that the accuracy of an adiabatic approximation is intimately related to the deviation between the natural orbital occupation numbers of the physical system and those of the Kohn-Sham system, and we explore this on some exactly-solvable model systems. The exact expression further suggests a path to go beyond the adiabatic approximations, and in the second part we discuss a newly proposed class of memory-dependent approximations developed in this way. Finally, we derive a new expression for the exact exchange-correlation potential from a coupling-constant path integration.
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Affiliation(s)
- Lionel Lacombe
- Department of Physics, Rutgers University, Newark, New Jersey 07102, USA.
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Brütting M, Trepl T, de Las Heras D, Schmidt M. Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics. Molecules 2019; 24:molecules24203660. [PMID: 31614514 PMCID: PMC6832129 DOI: 10.3390/molecules24203660] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2019] [Revised: 09/15/2019] [Accepted: 09/30/2019] [Indexed: 11/30/2022] Open
Abstract
We apply the formally exact quantum power functional framework (J. Chem. Phys.2015, 143, 174108) to a one-dimensional Hooke’s helium model atom. The physical dynamics are described on the one-body level beyond the density-based adiabatic approximation. We show that gradients of both the microscopic velocity and acceleration field are required to correctly describe the effects due to interparticle interactions. We validate the proposed analytical forms of the superadiabatic force and transport contributions by comparison to one-body data from exact numerical solution of the Schrödinger equation. Superadiabatic contributions beyond the adiabatic approximation are important in the dynamics and they include effective dissipation.
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Affiliation(s)
- Moritz Brütting
- Theoretische Physik II, Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth, Germany
| | - Thomas Trepl
- Theoretische Physik II, Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth, Germany
| | - Daniel de Las Heras
- Theoretische Physik II, Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth, Germany
| | - Matthias Schmidt
- Theoretische Physik II, Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth, Germany.
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Lacombe L, Maitra NT. Density-Matrix Coupled Time-Dependent Exchange-Correlation Functional Approximations. J Chem Theory Comput 2019; 15:1672-1678. [DOI: 10.1021/acs.jctc.8b01159] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Lionel Lacombe
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, United States
| | - Neepa T. Maitra
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, United States
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Dittmann N, Splettstoesser J, Helbig N. Nonadiabatic Dynamics in Single-Electron Tunneling Devices with Time-Dependent Density-Functional Theory. PHYSICAL REVIEW LETTERS 2018; 120:157701. [PMID: 29756889 DOI: 10.1103/physrevlett.120.157701] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2017] [Revised: 11/28/2017] [Indexed: 06/08/2023]
Abstract
We simulate the dynamics of a single-electron source, modeled as a quantum dot with on-site Coulomb interaction and tunnel coupling to an adjacent lead in time-dependent density-functional theory. Based on this system, we develop a time-nonlocal exchange-correlation potential by exploiting analogies with quantum-transport theory. The time nonlocality manifests itself in a dynamical potential step. We explicitly link the time evolution of the dynamical step to physical relaxation timescales of the electron dynamics. Finally, we discuss prospects for simulations of larger mesoscopic systems.
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Affiliation(s)
- Niklas Dittmann
- Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, SE-41298 Göteborg, Sweden
- Institute for Theory of Statistical Physics, RWTH Aachen, 52056 Aachen, Germany
- Peter-Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany
| | - Janine Splettstoesser
- Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, SE-41298 Göteborg, Sweden
| | - Nicole Helbig
- Peter-Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany
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Fuks JI, Lacombe L, Nielsen SEB, Maitra NT. Exploring non-adiabatic approximations to the exchange–correlation functional of TDDFT. Phys Chem Chem Phys 2018; 20:26145-26160. [DOI: 10.1039/c8cp03957g] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Decomposition of the exact time-dependent exchange–correlation potential offers a new starting point to build approximations with memory.
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Affiliation(s)
- Johanna I. Fuks
- Departamento de Física and IFIBA
- FCEN
- Universidad de Buenos Aires
- Ciudad Universitaria
- C1428EHA Ciudad de Buenos Aires
| | - Lionel Lacombe
- Department of Physics and Astronomy
- Hunter College and the Graduate Center of the City University of New York
- New York
- USA
| | - Søren E. B. Nielsen
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science
- 22761 Hamburg
- Germany
| | - Neepa T. Maitra
- Department of Physics and Astronomy
- Hunter College and the Graduate Center of the City University of New York
- New York
- USA
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8
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Suzuki Y, Lacombe L, Watanabe K, Maitra NT. Exact Time-Dependent Exchange-Correlation Potential in Electron Scattering Processes. PHYSICAL REVIEW LETTERS 2017; 119:263401. [PMID: 29328727 DOI: 10.1103/physrevlett.119.263401] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2017] [Indexed: 06/07/2023]
Abstract
We identify peak and valley structures in the exact exchange-correlation potential of time-dependent density functional theory that are crucial for time-resolved electron scattering in a model one-dimensional system. These structures are completely missed by adiabatic approximations that, consequently, significantly underestimate the scattering probability. A recently proposed nonadiabatic approximation is shown to correctly capture the approach of the electron to the target when the initial Kohn-Sham state is chosen judiciously, and it is more accurate than standard adiabatic functionals but ultimately fails to accurately capture reflection. These results may explain the underestimation of scattering probabilities in some recent studies on molecules and surfaces.
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Affiliation(s)
- Yasumitsu Suzuki
- Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
| | - Lionel Lacombe
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Kazuyuki Watanabe
- Department of Physics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
| | - Neepa T Maitra
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
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Maitra NT. Charge transfer in time-dependent density functional theory. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:423001. [PMID: 28766507 DOI: 10.1088/1361-648x/aa836e] [Citation(s) in RCA: 58] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Charge transfer plays a crucial role in many processes of interest in physics, chemistry, and bio-chemistry. In many applications the size of the systems involved calls for time-dependent density functional theory (TDDFT) to be used in their computational modeling, due to its unprecedented balance between accuracy and efficiency. However, although exact in principle, in practise approximations must be made for the exchange-correlation functional in this theory, and the standard functional approximations perform poorly for excitations which have a long-range charge-transfer component. Intense progress has been made in developing more sophisticated functionals for this problem, which we review. We point out an essential difference between the properties of the exchange-correlation kernel needed for an accurate description of charge-transfer between open-shell fragments and between closed-shell fragments. We then turn to charge-transfer dynamics, which, in contrast to the excitation problem, is a highly non-equilibrium, non-perturbative, process involving a transfer of one full electron in space. This turns out to be a much more challenging problem for TDDFT functionals. We describe dynamical step and peak features in the exact functional evolving over time, that are missing in the functionals currently used. The latter underestimate the amount of charge transferred and manifest a spurious shift in the charge transfer resonance position. We discuss some explicit examples.
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Affiliation(s)
- Neepa T Maitra
- Department of Physics and Astronomy, Hunter College and the Physics Program at the Graduate Center of the City University of New York, 695 Park Avenue, NY 10065, United States of America
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Luo K, Fuks JI, Maitra NT. Studies of spuriously shifting resonances in time-dependent density functional theory. J Chem Phys 2016; 145:044101. [DOI: 10.1063/1.4955447] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Kai Luo
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
- Department of Chemistry and Chemical Biology, Cornell University, 259 East Ave., Ithaca, New York 14850, USA
| | - Johanna I. Fuks
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Neepa T. Maitra
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
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11
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Maitra NT. Perspective: Fundamental aspects of time-dependent density functional theory. J Chem Phys 2016; 144:220901. [DOI: 10.1063/1.4953039] [Citation(s) in RCA: 228] [Impact Index Per Article: 28.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Neepa T. Maitra
- Department of Physics and Astronomy, Hunter College and the Physics Program at the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
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12
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Fuks JI, Nielsen SEB, Ruggenthaler M, Maitra NT. Time-dependent density functional theory beyond Kohn–Sham Slater determinants. Phys Chem Chem Phys 2016; 18:20976-85. [DOI: 10.1039/c6cp00722h] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022]
Abstract
Different choices of initial Kohn Sham wavefunction shape the time-dependent exchange–correlation potential.
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Affiliation(s)
- Johanna I. Fuks
- Department of Physics and Astronomy
- Hunter College and the Graduate Center of the City University of New York
- New York
- USA
| | - Søren E. B. Nielsen
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science & Department of Physics
- 22761 Hamburg
- Germany
- Institut für Theoretische Physik
- Universität Innsbruck
| | - Michael Ruggenthaler
- Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science & Department of Physics
- 22761 Hamburg
- Germany
- Institut für Theoretische Physik
- Universität Innsbruck
| | - Neepa T. Maitra
- Department of Physics and Astronomy
- Hunter College and the Graduate Center of the City University of New York
- New York
- USA
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13
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Wang J, Baerends EJ. Real-space representation of electron correlation in π-conjugated systems. J Chem Phys 2015; 142:204311. [DOI: 10.1063/1.4921725] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jian Wang
- School of Science, Huzhou University, Zhejiang 10083, China
| | - Evert Jan Baerends
- Afdeling Theoretische Chemie, FEW, Vrije Universiteit, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands
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14
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Fuks JI, Luo K, Sandoval ED, Maitra NT. Time-resolved spectroscopy in time-dependent density functional theory: an exact condition. PHYSICAL REVIEW LETTERS 2015; 114:183002. [PMID: 26000998 PMCID: PMC6135656 DOI: 10.1103/physrevlett.114.183002] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/23/2014] [Indexed: 05/17/2023]
Abstract
A fundamental property of a quantum system driven by an external field is that when the field is turned off the positions of its response frequencies are independent of the time at which the field is turned off. We show that this leads to an exact condition for the exchange-correlation potential of time-dependent density functional theory. The Kohn-Sham potential typically continues to evolve after the field is turned off, which leads to time dependence in the response frequencies of the Kohn-Sham response function. The exchange-correlation kernel must cancel out this time dependence. The condition is typically violated by approximations currently in use, as we demonstrate by several examples, which has severe consequences for their predictions of time-resolved spectroscopy.
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Affiliation(s)
- Johanna I Fuks
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Kai Luo
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Ernesto D Sandoval
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
| | - Neepa T Maitra
- Department of Physics and Astronomy, Hunter College and the Graduate Center of the City University of New York, 695 Park Avenue, New York, New York 10065, USA
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