1
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Tarantino W, Ullrich CA. A reformulation of time-dependent Kohn-Sham theory in terms of the second time derivative of the density. J Chem Phys 2021; 154:204112. [PMID: 34241164 DOI: 10.1063/5.0039962] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The Kohn-Sham approach to time-dependent density-functional theory (TDDFT) can be formulated, in principle, exactly by invoking the force-balance equation for the density, which leads to an explicit expression for the exchange-correlation potential as an implicit density functional. It is shown that this suggests a reformulation of TDDFT in terms of the second time derivative of the density, rather than the density itself. The result is a time-local Kohn-Sham scheme of second-order in time whose causal structure is more transparent than that of the usual Kohn-Sham formalism. The scheme can be used to construct new approximations at the exchange-only level and beyond, and it offers a straightforward definition of the exact adiabatic approximation.
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Affiliation(s)
- Walter Tarantino
- Dipartimento di Fisica, Università degli Studi di Cagliari, Cittadella Universitaria, I-09042 Monserrato, Cagliari, Italy
| | - Carsten A Ullrich
- Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
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2
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Hofmann F, Kümmel S. Molecular excitations from meta-generalized gradient approximations in the Kohn-Sham scheme. J Chem Phys 2020; 153:114106. [PMID: 32962375 DOI: 10.1063/5.0023657] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/03/2023] Open
Abstract
Meta-Generalized Gradient Approximations (meta-GGAs) can, in principle, include spatial and temporal nonlocality in time-dependent density functional theory at a much lower computational cost than functionals that use exact exchange. We here test whether a meta-GGA that has recently been developed with a focus on capturing nonlocal response properties and the particle number discontinuity can realize such features in practice. To this end, we extended the frequency-dependent Sternheimer formalism to the meta-GGA case. Using the Krieger-Li-Iafrate (KLI) approximation, we calculate the optical response for the selected paradigm molecular systems and compare the meta-GGA Kohn-Sham response to the one found with exact exchange and conventional (semi-)local functionals. We find that the new meta-GGA captures important properties of the nonlocal exchange response. The KLI approximation, however, emerges as a limiting factor in the evaluation of charge-transfer excitations.
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Affiliation(s)
- Fabian Hofmann
- Theoretical Physics IV, University of Bayreuth, D-95440 Bayreuth, Germany
| | - Stephan Kümmel
- Theoretical Physics IV, University of Bayreuth, D-95440 Bayreuth, Germany
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3
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Pathak H, Sato T, Ishikawa KL. Study of laser-driven multielectron dynamics of Ne atom using time-dependent optimised second-order many-body perturbation theory. Mol Phys 2020. [DOI: 10.1080/00268976.2020.1813910] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022]
Affiliation(s)
- Himadri Pathak
- Department of Nuclear Engineering and Management, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
| | - Takeshi Sato
- Department of Nuclear Engineering and Management, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
- Photon Science Center, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
- Research Institute for Photon Science and Laser Technology, The University of Tokyo, Tokyo, Japan
| | - Kenichi L. Ishikawa
- Department of Nuclear Engineering and Management, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
- Photon Science Center, Graduate School of Engineering, The University of Tokyo, Tokyo, Japan
- Research Institute for Photon Science and Laser Technology, The University of Tokyo, Tokyo, Japan
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4
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Li X, Govind N, Isborn C, DePrince AE, Lopata K. Real-Time Time-Dependent Electronic Structure Theory. Chem Rev 2020; 120:9951-9993. [DOI: 10.1021/acs.chemrev.0c00223] [Citation(s) in RCA: 64] [Impact Index Per Article: 16.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Xiaosong Li
- Department of Chemistry, University of Washington, Seattle, Washington 98195-1700, United States
| | - Niranjan Govind
- Physical and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 99354, United States
| | - Christine Isborn
- Department of Chemistry and Chemical Biology, University of California, Merced, California 95343, United States
| | - A. Eugene DePrince
- Department of Chemistry and Biochemistry, Florida State University, Tallahassee, Florida 32306-4390, United States
| | - Kenneth Lopata
- Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, United States
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5
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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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6
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Nazarov VU. Many-Body Quantum Dynamics by the Reduced Density Matrix Based on Time-Dependent Density-Functional Theory. PHYSICAL REVIEW LETTERS 2019; 123:095302. [PMID: 31524458 DOI: 10.1103/physrevlett.123.095302] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2019] [Indexed: 06/10/2023]
Abstract
We evaluate the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method of Görling and Levy, from which the expansion of the many-body density matrix in powers of the coupling constant λ naturally arises. We then find the reduced density matrix ρ_{λ}(r,r^{'},t), which, by construction, has the λ independent diagonal elements ρ_{λ}(r,r,t)=n(r,t), n(r,t) being the particle density. The off-diagonal elements of ρ_{λ}(r,r^{'},t) contribute importantly to the processes unaccessible via the density, directly or by the use of the known TDDFT functionals. Of those, we consider the momentum-resolved photoemission, doing this to the first order in λ, i.e., on the level of the exact exchange theory. In illustrative calculations of photoemission from the quasi-2D electron gas and isolated atoms, we find quantitatively strong and conceptually far-reaching differences with the independent-particle Fermi's golden rule formula.
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Affiliation(s)
- Vladimir U Nazarov
- Research Center for Applied Sciences, Academia Sinica, Taipei 11529, Taiwan
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7
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Yu W, Gao CZ, Jiang T, Zou Y, Wang JG, Wu Y, Wei B. A theoretical study of Ar 8+-acetylene collisions at 1.2 MeV: Ionization and dissociation dynamics. J Chem Phys 2019; 150:124304. [PMID: 30927893 DOI: 10.1063/1.5082729] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We theoretically study Ar8+-induced dissociation of C2H2 molecule at 1.2 MeV using the time-dependent density-functional theory non-adiabatically coupled to nuclear dynamics. We find that molecular dissociation depends strongly on the ionization at the initial stage and the collision configuration. A detailed analysis shows a correspondence between the charge state of [C2H2]q+ and the final fragments. A remarkable impact parameter effect provides deep insights of bond breakup and electronic transport. We analyze two typical sequential dissociation channels reported in experiments by tracking structural and electronic dynamics in real time. Our results provide better understanding of experiments. Moreover, the comparison between various exchange-correlation functionals reveals that electrons' correlation and self-interaction do not significantly impact the initial ionization and fragment distribution in the present study.
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Affiliation(s)
- W Yu
- Department of Nuclear Science and Technology, Institute of Modern Physics, Fudan University, Shanghai 200433, China
| | - C-Z Gao
- Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
| | - T Jiang
- Department of Nuclear Science and Technology, Institute of Modern Physics, Fudan University, Shanghai 200433, China
| | - Y Zou
- Department of Nuclear Science and Technology, Institute of Modern Physics, Fudan University, Shanghai 200433, China
| | - J-G Wang
- Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
| | - Y Wu
- Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
| | - B Wei
- Department of Nuclear Science and Technology, Institute of Modern Physics, Fudan University, Shanghai 200433, China
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8
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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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9
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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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10
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Liao SL, Ho TS, Rabitz H, Chu SI. Time-Local Equation for the Exact Optimized Effective Potential in Time-Dependent Density Functional Theory. PHYSICAL REVIEW LETTERS 2017; 118:243001. [PMID: 28665665 DOI: 10.1103/physrevlett.118.243001] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2015] [Indexed: 06/07/2023]
Abstract
A long-standing challenge in the time-dependent density functional theory is to efficiently solve the exact time-dependent optimized effective potential (TDOEP) integral equation derived from orbital-dependent functionals, especially for the study of nonadiabatic dynamics in time-dependent external fields. In this Letter, we formulate a completely equivalent time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham and effective memory orbitals. The time-local formulation is numerically implemented, with the incorporation of exponential memory loss to address the unaccounted for correlation component in the exact-exchange-only functional, to enable the study of the many-electron dynamics of a one-dimensional hydrogen chain. It is shown that the long time behavior of the electric dipole converges correctly and the zero-force theorem is fulfilled in the current implementation.
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Affiliation(s)
- Sheng-Lun Liao
- Center for Quantum Science and Engineering and Department of Physics, National Taiwan University, Taipei 10617, Taiwan
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - Tak-San Ho
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - Herschel Rabitz
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
| | - Shih-I Chu
- Center for Quantum Science and Engineering and Department of Physics, National Taiwan University, Taipei 10617, Taiwan
- Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, USA
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11
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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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12
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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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13
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Smith CE, Odoh SO, Ghosh S, Gagliardi L, Cramer CJ, Frisbie CD. Length-Dependent Nanotransport and Charge Hopping Bottlenecks in Long Thiophene-Containing π-Conjugated Molecular Wires. J Am Chem Soc 2015; 137:15732-41. [DOI: 10.1021/jacs.5b07400] [Citation(s) in RCA: 69] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
Affiliation(s)
- Christopher E. Smith
- Department of Chemistry and ‡Department of
Chemical Engineering and Materials
Science, §Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Samuel O. Odoh
- Department of Chemistry and ‡Department of
Chemical Engineering and Materials
Science, §Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Soumen Ghosh
- Department of Chemistry and ‡Department of
Chemical Engineering and Materials
Science, §Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Laura Gagliardi
- Department of Chemistry and ‡Department of
Chemical Engineering and Materials
Science, §Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - Christopher J. Cramer
- Department of Chemistry and ‡Department of
Chemical Engineering and Materials
Science, §Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
| | - C. Daniel Frisbie
- Department of Chemistry and ‡Department of
Chemical Engineering and Materials
Science, §Chemical Theory Center, and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, United States
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14
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Pellegrini C, Flick J, Tokatly IV, Appel H, Rubio A. Optimized Effective Potential for Quantum Electrodynamical Time-Dependent Density Functional Theory. PHYSICAL REVIEW LETTERS 2015; 115:093001. [PMID: 26371646 DOI: 10.1103/physrevlett.115.093001] [Citation(s) in RCA: 52] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2014] [Indexed: 05/20/2023]
Abstract
We propose an orbital exchange-correlation functional for applying time-dependent density functional theory to many-electron systems coupled to cavity photons. The time nonlocal equation for the electron-photon optimized effective potential (OEP) is derived. In the static limit our OEP energy functional reduces to the Lamb shift of the ground state energy. We test the new approximation in the Rabi model. It is shown that the OEP (i) reproduces quantitatively the exact ground-state energy from the weak to the deep strong coupling regime and (ii) accurately captures the dynamics entering the ultrastrong coupling regime. The present formalism opens the path to a first-principles description of correlated electron-photon systems, bridging the gap between electronic structure methods and quantum optics for real material applications.
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Affiliation(s)
- Camilla Pellegrini
- Departamento de Fisica de Materiales, Nano-bio Spectroscopy Group and ETSF Scientific Development Centre, Universidad del Pais Vasco UPV/EHU, E-20018 San Sebastian, Spain
| | - Johannes Flick
- Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
| | - Ilya V Tokatly
- Departamento de Fisica de Materiales, Nano-bio Spectroscopy Group and ETSF Scientific Development Centre, Universidad del Pais Vasco UPV/EHU, E-20018 San Sebastian, Spain
- IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
| | - Heiko Appel
- Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Angel Rubio
- Departamento de Fisica de Materiales, Nano-bio Spectroscopy Group and ETSF Scientific Development Centre, Universidad del Pais Vasco UPV/EHU, E-20018 San Sebastian, Spain
- Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany
- Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
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15
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van Meer R, Gritsenko OV, Baerends EJ. Physical Meaning of Virtual Kohn–Sham Orbitals and Orbital Energies: An Ideal Basis for the Description of Molecular Excitations. J Chem Theory Comput 2014; 10:4432-41. [DOI: 10.1021/ct500727c] [Citation(s) in RCA: 170] [Impact Index Per Article: 17.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022]
Affiliation(s)
- R. van Meer
- Faculty
of Exact Sciences,
Theoretical Chemistry, VU University, 1081 HV Amsterdam, The Netherlands
| | - O. V. Gritsenko
- Faculty
of Exact Sciences,
Theoretical Chemistry, VU University, 1081 HV Amsterdam, The Netherlands
| | - E. J. Baerends
- Faculty
of Exact Sciences,
Theoretical Chemistry, VU University, 1081 HV Amsterdam, The Netherlands
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16
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Ronca E, Angeli C, Belpassi L, De Angelis F, Tarantelli F, Pastore M. Density Relaxation in Time-Dependent Density Functional Theory: Combining Relaxed Density Natural Orbitals and Multireference Perturbation Theories for an Improved Description of Excited States. J Chem Theory Comput 2014; 10:4014-24. [DOI: 10.1021/ct5004675] [Citation(s) in RCA: 33] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Enrico Ronca
- Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123, Perugia, Italy
- Dipartimento
di Chimica, Biologia e Biotecnologie, Università degli Studi di Perugia, via Elce di Sotto 8, I-06123, Perugia, Italy
| | - Celestino Angeli
- Dipartimento
di Scienze Chimiche e Farmaceutiche, Università degli Studi di Ferrara, via Borsari 46, I-44100, Ferrara, Italy
| | - Leonardo Belpassi
- Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123, Perugia, Italy
| | - Filippo De Angelis
- Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123, Perugia, Italy
| | - Francesco Tarantelli
- Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123, Perugia, Italy
- Dipartimento
di Chimica, Biologia e Biotecnologie, Università degli Studi di Perugia, via Elce di Sotto 8, I-06123, Perugia, Italy
| | - Mariachiara Pastore
- Istituto CNR di Scienze e Tecnologie Molecolari, via Elce di Sotto 8, I-06123, Perugia, Italy
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17
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Huang C, Libisch F, Peng Q, Carter EA. Time-dependent potential-functional embedding theory. J Chem Phys 2014; 140:124113. [DOI: 10.1063/1.4869538] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023] Open
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18
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Agnihotri N. Computational studies of charge transfer in organic solar photovoltaic cells: A review. JOURNAL OF PHOTOCHEMISTRY AND PHOTOBIOLOGY C-PHOTOCHEMISTRY REVIEWS 2014. [DOI: 10.1016/j.jphotochemrev.2013.10.004] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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19
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Ringholm M, Jonsson D, Ruud K. A general, recursive, and open-ended response code. J Comput Chem 2014; 35:622-33. [DOI: 10.1002/jcc.23533] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2013] [Revised: 12/09/2013] [Accepted: 12/12/2013] [Indexed: 12/27/2022]
Affiliation(s)
- Magnus Ringholm
- Department of Chemistry; Centre for Theoretical and Computational Chemistry (CTCC), University of Tromsø-The Arctic University of Norway; N-9037 Tromsø Norway
| | - Dan Jonsson
- Department of Chemistry; Centre for Theoretical and Computational Chemistry (CTCC), University of Tromsø-The Arctic University of Norway; N-9037 Tromsø Norway
- High-Performance Computing Group; University of Tromsø-The Arctic University of Norway; N-9037 Tromsø Norway
| | - Kenneth Ruud
- Department of Chemistry; Centre for Theoretical and Computational Chemistry (CTCC), University of Tromsø-The Arctic University of Norway; N-9037 Tromsø Norway
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20
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Chu SI, Telnov DA. Generalized Floquet Formulation of Time-Dependent Density Functional Theory for Multiphoton Processes in Intense Laser Fields. J CHIN CHEM SOC-TAIP 2013. [DOI: 10.1002/jccs.200200107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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21
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Tokatly IV. Time-dependent density functional theory for many-electron systems interacting with cavity photons. PHYSICAL REVIEW LETTERS 2013; 110:233001. [PMID: 25167487 DOI: 10.1103/physrevlett.110.233001] [Citation(s) in RCA: 68] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2013] [Indexed: 05/20/2023]
Abstract
Time-dependent (current) density functional theory for many-electron systems strongly coupled to quantized electromagnetic modes of a microcavity is proposed. It is shown that the electron-photon wave function is a unique functional of the electronic (current) density and the expectation values of photonic coordinates. The Kohn-Sham system is constructed, which allows us to calculate the above basic variables by solving self-consistent equations for noninteracting particles. We suggest possible approximations for the exchange-correlation potentials and discuss implications of this approach for the theory of open quantum systems. In particular we show that it naturally leads to time-dependent density functional theory for systems coupled to the Caldeira-Leggett bath.
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Affiliation(s)
- I V Tokatly
- Nano-bio Spectroscopy group and ETSF Scientific Development Centre, Departamento de Física de Materiales, Universidad del País Vasco UPV/EHU, E-20018 San Sebastían, Spain and IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
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22
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Gao Y, Neuhauser D. Communication: Dynamical embedding: Correct quantum response from coupling TDDFT for a small cluster with classical near-field electrodynamics for an extended region. J Chem Phys 2013; 138:181105. [DOI: 10.1063/1.4804544] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023] Open
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23
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Hofmann D, Kümmel S. Self-interaction correction in a real-time Kohn-Sham scheme: Access to difficult excitations in time-dependent density functional theory. J Chem Phys 2012; 137:064117. [DOI: 10.1063/1.4742763] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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24
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Hofmann D, Körzdörfer T, Kümmel S. Kohn-Sham self-interaction correction in real time. PHYSICAL REVIEW LETTERS 2012; 108:146401. [PMID: 22540810 DOI: 10.1103/physrevlett.108.146401] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2011] [Indexed: 05/31/2023]
Abstract
We present a solution scheme for the time-dependent Kohn-Sham self-interaction correction. Based on the generalized optimized effective potential approach, the multiplicative Kohn-Sham potential is constructed in real time and real space for the self-interaction corrected local density approximation. Excitations of different character, including charge-transfer excitations that had been regarded as prime examples for the failure of standard time-dependent density functionals, are described correctly by this approach. We analyze the time-dependent exchange-correlation potential and density, revealing features that are decisive for the correct description of the response.
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Affiliation(s)
- D Hofmann
- Theoretical Physics IV, University of Bayreuth, D-95440 Bayreuth, Germany
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25
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DOLTSINIS NIKOSL, MARX DOMINIK. FIRST PRINCIPLES MOLECULAR DYNAMICS INVOLVING EXCITED STATES AND NONADIABATIC TRANSITIONS. JOURNAL OF THEORETICAL & COMPUTATIONAL CHEMISTRY 2012. [DOI: 10.1142/s0219633602000257] [Citation(s) in RCA: 126] [Impact Index Per Article: 10.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Extensions of traditional molecular dynamics to excited electronic states and non-Born–Oppenheimer dynamics are reviewed focusing on applicability to chemical reactions of large molecules, possibly in condensed phases. The latter imposes restrictions on both the level of accuracy of the underlying electronic structure theory and the treatment of nonadiabaticity. This review, therefore, exclusively deals with ab initio "on the fly" molecular dynamics methods. For the same reason, mainly mixed quantum-classical approaches to nonadiabatic dynamics are considered.
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Affiliation(s)
- NIKOS L. DOLTSINIS
- Lehrstuhl für Theoretische Chemie, Ruhr–Universität Bochum, 44780 Bochum, Germany
| | - DOMINIK MARX
- Lehrstuhl für Theoretische Chemie, Ruhr–Universität Bochum, 44780 Bochum, Germany
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26
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Abstract
The classic density-functional theory (DFT) formalism introduced by Hohenberg, Kohn, and Sham in the mid-1960s is based on the idea that the complicated N-electron wave function can be replaced with the mathematically simpler 1-electron charge density in electronic structure calculations of the ground stationary state. As such, ordinary DFT cannot treat time-dependent (TD) problems nor describe excited electronic states. In 1984, Runge and Gross proved a theorem making TD-DFT formally exact. Information about electronic excited states may be obtained from this theory through the linear response (LR) theory formalism. Beginning in the mid-1990s, LR-TD-DFT became increasingly popular for calculating absorption and other spectra of medium- and large-sized molecules. Its ease of use and relatively good accuracy has now brought LR-TD-DFT to the forefront for this type of application. As the number and the diversity of applications of TD-DFT have grown, so too has our understanding of the strengths and weaknesses of the approximate functionals commonly used for TD-DFT. The objective of this article is to continue where a previous review of TD-DFT in Volume 55 of the Annual Review of Physical Chemistry left off and highlight some of the problems and solutions from the point of view of applied physical chemistry. Because doubly-excited states have a particularly important role to play in bond dissociation and formation in both thermal and photochemistry, particular emphasis is placed on the problem of going beyond or around the TD-DFT adiabatic approximation, which limits TD-DFT calculations to nominally singly-excited states.
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Affiliation(s)
- M E Casida
- Laboratoire de Chimie Théorique, Département de Chimie Moléculaire, Institut de Chimie Moléculaire de Grenoble, Université Joseph Fourier, France.
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27
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Srebro M, Autschbach J. Tuned Range-Separated Time-Dependent Density Functional Theory Applied to Optical Rotation. J Chem Theory Comput 2011; 8:245-56. [DOI: 10.1021/ct200764g] [Citation(s) in RCA: 96] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Monika Srebro
- Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States
- Department of Theoretical Chemistry, Faculty of Chemistry, Jagiellonian University, R. Ingardena 3, 30-060 Krakow, Poland
| | - Jochen Autschbach
- Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States
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28
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Strong-field ionization of Li and Be: a time-dependent density functional theory with self-interaction correction. Chem Phys 2011. [DOI: 10.1016/j.chemphys.2011.03.023] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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29
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Lehtovaara L, Havu V, Puska M. All-electron time-dependent density functional theory with finite elements: Time-propagation approach. J Chem Phys 2011; 135:154104. [DOI: 10.1063/1.3651239] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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30
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Kowalczyk T, Yost SR, Voorhis TV. Assessment of the ΔSCF density functional theory approach for electronic excitations in organic dyes. J Chem Phys 2011; 134:054128. [DOI: 10.1063/1.3530801] [Citation(s) in RCA: 133] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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31
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Abstract
Abstract
The currently available methods for the computation of structures and their properties are reviewed. After a brief introduction into some common technical aspects, the capabilities and limitations of the most commonly used approaches are discussed. Examples are given to show the state of the art in Computational “Crystallography”, and possible future developments are outlined
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32
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Ködderitzsch D, Ebert H, Engel E, Akai H. Self-interaction Free Relativistic Spin-density Functional Theory. Z PHYS CHEM 2010. [DOI: 10.1524/zpch.2010.6115] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Abstract
Abstract
We review the progress in the recent development of the relativistic optimized effective potential (ROEP) method within spin-density functional theory. The ROEP-equations for spin-polarized systems are derived and their application to open-shell atoms using the exact exchange approximation for the exchange correlation-functional is presented. Further, we expand the ROEP framework to treat extended systems within the KKR-multiple scattering formalism. We illustrate the application of the theory to open-shell free atoms and solids.
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Affiliation(s)
| | | | - E. Engel
- J. W. Goethe-Universität Frankfurt, Center for Scientific Computing, Frankfurt, Deutschland
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33
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Affiliation(s)
- Roi Baer
- Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904 Israel;
| | - Ester Livshits
- Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904 Israel;
| | - Ulrike Salzner
- Department of Chemistry, Bilkent University, 06800 Bilkent, Ankara, Turkey
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34
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Messud J, Wang Z, Dinh P, Reinhard PG, Suraud E. Polarizabilities as a test of localized approximations to the self-interaction correction. Chem Phys Lett 2009. [DOI: 10.1016/j.cplett.2009.08.027] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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35
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Elliott P, Furche F, Burke K. Excited States from Time-Dependent Density Functional Theory. REVIEWS IN COMPUTATIONAL CHEMISTRY 2009. [DOI: 10.1002/9780470399545.ch3] [Citation(s) in RCA: 118] [Impact Index Per Article: 7.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
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36
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Vieira D, Capelle K, Ullrich CA. Physical signatures of discontinuities of the time-dependent exchange–correlation potential. Phys Chem Chem Phys 2009; 11:4647-54. [DOI: 10.1039/b902613d] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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37
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Thorvaldsen AJ, Ruud K, Kristensen K, Jørgensen P, Coriani S. A density matrix-based quasienergy formulation of the Kohn–Sham density functional response theory using perturbation- and time-dependent basis sets. J Chem Phys 2008; 129:214108. [DOI: 10.1063/1.2996351] [Citation(s) in RCA: 95] [Impact Index Per Article: 5.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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38
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Verdozzi C. Time-dependent density-functional theory and strongly correlated systems: insight from numerical studies. PHYSICAL REVIEW LETTERS 2008; 101:166401. [PMID: 18999689 DOI: 10.1103/physrevlett.101.166401] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2007] [Indexed: 05/27/2023]
Abstract
We illustrate the scope of time-dependent density-functional theory for strongly correlated (lattice) models out of equilibrium. Using the exact many-body time evolution, we reverse engineer the exact exchange correlation (xc) potential v_(xc) for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation to v_(xc) for the 1D Hubbard model and compare it to exact results, to gain insight about approximate xc potentials. Finally, we provide some remarks on the v-representability for the 1D Hubbard model.
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Affiliation(s)
- Claudio Verdozzi
- Division of Mathematical Physics and European Theoretical Spectroscopy Facility (ETSF), Lund University, Lund, Sweden
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39
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Messud J, Dinh PM, Reinhard PG, Suraud E. Time-dependent density-functional theory with a self-interaction correction. PHYSICAL REVIEW LETTERS 2008; 101:096404. [PMID: 18851631 DOI: 10.1103/physrevlett.101.096404] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2007] [Indexed: 05/26/2023]
Abstract
We discuss an implementation of the self-interaction correction for the local-density approximation to time-dependent density-functional theory. A variational formulation is given, taking care of the necessary constraints. A manageable and transparent propagation scheme using two sets of wave functions is proposed and applied to laser excitation with subsequent ionization of a dimer molecule.
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Affiliation(s)
- J Messud
- Laboratoire de Physique Théorique, IRSAMC, CNRS, Université de Toulouse, F-31062 Toulouse, France
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40
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Kjærgaard T, Jørgensen P, Olsen J, Coriani S, Helgaker T. Hartree-Fock and Kohn-Sham time-dependent response theory in a second-quantization atomic-orbital formalism suitable for linear scaling. J Chem Phys 2008; 129:054106. [DOI: 10.1063/1.2961039] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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41
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42
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Wijewardane HO, Ullrich CA. Real-time electron dynamics with exact-exchange time-dependent density-functional theory. PHYSICAL REVIEW LETTERS 2008; 100:056404. [PMID: 18352401 DOI: 10.1103/physrevlett.100.056404] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2007] [Indexed: 05/26/2023]
Abstract
The exact exchange potential in time-dependent density-functional theory is defined as an orbital functional through the time-dependent optimized effective potential (TDOEP) method. We numerically solve the TDOEP integral equation for the real-time nonlinear intersubband electron dynamics in a semiconductor quantum well with two occupied subbands. It is found that memory effects become significant in the vicinity of intersubband resonances.
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Affiliation(s)
- H O Wijewardane
- Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
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43
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Burke K, Werschnik J, Gross EKU. Time-dependent density functional theory: past, present, and future. J Chem Phys 2007; 123:62206. [PMID: 16122292 DOI: 10.1063/1.1904586] [Citation(s) in RCA: 505] [Impact Index Per Article: 29.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/05/2023] Open
Abstract
Time-dependent density functional theory (TDDFT) is presently enjoying enormous popularity in quantum chemistry, as a useful tool for extracting electronic excited state energies. This article discusses how TDDFT is much broader in scope, and yields predictions for many more properties. We discuss some of the challenges involved in making accurate predictions for these properties.
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Affiliation(s)
- Kieron Burke
- Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 08854, USA
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44
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Ullrich CA. Time-dependent density-functional theory beyond the adiabatic approximation: Insights from a two-electron model system. J Chem Phys 2006; 125:234108. [PMID: 17190548 DOI: 10.1063/1.2406069] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Most applications of time-dependent density-functional theory (TDDFT) use the adiabatic local-density approximation (ALDA) for the dynamical exchange-correlation potential V(xc)(r,t). An exact (i.e., nonadiabatic) extension of the ground-state LDA into the dynamical regime leads to a V(xc)(r,t) with a memory, which causes the electron dynamics to become dissipative. To illustrate and explain this nonadiabatic behavior, this paper studies the dynamics of two interacting electrons on a two-dimensional quantum strip of finite size, comparing TDDFT within and beyond the ALDA with numerical solutions of the two-electron time-dependent Schrodinger equation. It is shown explicitly how dissipation arises through multiple particle-hole excitations, and how the nonadiabatic extension of the ALDA fails for finite systems but becomes correct in the thermodynamic limit.
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Affiliation(s)
- C A Ullrich
- Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
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45
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Chu X, Dalgarno A. Linear response time-dependent density functional theory for van der Waals coefficients. J Chem Phys 2006; 121:4083-8. [PMID: 15332953 DOI: 10.1063/1.1779576] [Citation(s) in RCA: 113] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A linear response time-dependent density functional theory is described and used to calculate the dynamic polarizabilities and van der Waals C(6) coefficients of complex atom pairs. We present values of C(6) for dimers of main group atoms and the first row of transition metal atoms.
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Affiliation(s)
- X Chu
- ITAMP Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
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46
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Leeuwen NEDASAR. Nonequilibrium Green function theory for excitation and transport in atoms and molecules. ACTA ACUST UNITED AC 2006. [DOI: 10.1088/1742-6596/35/1/030] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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47
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Durbeej B, Eriksson LA. Protein-bound chromophores astaxanthin and phytochromobilin: excited state quantum chemical studies. Phys Chem Chem Phys 2006; 8:4053-71. [PMID: 17028694 DOI: 10.1039/b605682b] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We present an overview of excited state quantum chemical calculations aimed at elucidating controversial issues regarding the photochemistry of the protein-bound chromophores astaxanthin and phytochromobilin. In particular, we show how the application of time-dependent density functional theory and other single-reference quantum chemical excited state methods have contributed to shed new light on the origin of the >0.5 eV bathochromic shift of the electronic absorption by the carotenoid astaxanthin in the protein macromolecular complex crustacyanin, and the mechanism for C15-Z,syn --> C15-E,anti isomerization of the tetrapyrrole phytochromobilin that underlies the photoactivation of the plant photoreceptor phytochrome. Within the approximation that exciton coupling is neglected, the calculations on astaxanthin provide support for the notion that the bathochromic shift, which is responsible for the slate-blue coloration of lobster shell, is due to polarization rather than a conformational change of the chromophore in the protein-bound state. Furthermore, the polarization is attributed to a hydrogen-bonded protonated histidine residue. The calculations on phytochromobilin, in turn, suggest that a stepwise C15-Z,syn --> C15-E,syn (photochemical), C15-E,syn --> C15-E,anti (thermal) mechanism is much more favorable than a concerted, fully photochemical mechanism, and that neutral forms of the chromophore are much less likely to photoisomerize than the parent, protonated form. Accordingly, the calculations indirectly support the view that the photoactivation of phytochrome does not involve a proton transfer from the chromophore to the surrounding protein.
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Affiliation(s)
- Bo Durbeej
- Department of Chemistry, University of Siena, Via Aldo Moro 2, I-53100, Siena, Italy.
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48
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Generalized functional theory of interacting coupled Liouvillean Quantum Fields of condensed matter. Top Curr Chem (Cham) 2005. [DOI: 10.1007/bfb0016644] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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49
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50
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Mundt M, Kümmel S. Derivative discontinuities in time-dependent density-functional theory. PHYSICAL REVIEW LETTERS 2005; 95:203004. [PMID: 16384054 DOI: 10.1103/physrevlett.95.203004] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/15/2005] [Indexed: 05/05/2023]
Abstract
Based on the Runge-Gross theorem for ensembles we investigate the influence of particle-number-changes on the exchange-correlation potential in time-dependent density-functional theory. We show that the potential changes discontinuously when the particle number crosses an integer value. Real-time simulations of an atomic ionization process demonstrate that this discontinuity appears naturally in the theory of the time-dependent optimized effective potential. The importance of such a discontinuity for physical processes, even such ones in which the particle number is a constant, is discussed.
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Affiliation(s)
- Michael Mundt
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany
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