1
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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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Hellgren M, Baguet L. Strengths and limitations of the adiabatic exact-exchange kernel for total energy calculations. J Chem Phys 2023; 158:2889488. [PMID: 37158324 DOI: 10.1063/5.0146423] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2023] [Accepted: 04/24/2023] [Indexed: 05/10/2023] Open
Abstract
We investigate the adiabatic approximation to the exact-exchange kernel for calculating correlation energies within the adiabatic-connection fluctuation-dissipation framework of time-dependent density functional theory. A numerical study is performed on a set of systems having bonds of different character (H2 and N2 molecules, H-chain, H2-dimer, solid-Ar, and the H2O-dimer). We find that the adiabatic kernel can be sufficient in strongly bound covalent systems, yielding similar bond lengths and binding energies. However, for non-covalent systems, the adiabatic kernel introduces significant errors around equilibrium geometry, systematically overestimating the interaction energy. The origin of this behavior is investigated by studying a model dimer composed of one-dimensional, closed-shell atoms, interacting via soft-Coulomb potentials. The kernel is shown to exhibit a strong frequency dependence at small to intermediate atomic separation that affects both the low-energy spectrum and the exchange-correlation hole obtained from the corresponding diagonal of the two-particle density matrix.
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Affiliation(s)
- Maria Hellgren
- Sorbonne Université, MNHN, UMR CNRS 7590, IMPMC, 4 place Jussieu, 75005 Paris, France
| | - Lucas Baguet
- CEA, DAM, DIF, F-91297 Arpajon, France
- Université Paris-Saclay, CEA, Laboratoire Matière en Conditions Extrêmes, 91680 Bruyères-le-Châtel, France
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3
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Ranka K, Isborn CM. Size-dependent errors in real-time electron density propagation. J Chem Phys 2023; 158:2887545. [PMID: 37125706 DOI: 10.1063/5.0142515] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2023] [Accepted: 04/14/2023] [Indexed: 05/02/2023] Open
Abstract
Real-time (RT) electron density propagation with time-dependent density functional theory (TDDFT) or Hartree-Fock (TDHF) is one of the most popular methods to model the charge transfer in molecules and materials. However, both RT-TDHF and RT-TDDFT within the adiabatic approximation are known to produce inaccurate evolution of the electron density away from the ground state in model systems, leading to large errors in charge transfer and erroneous shifting of peaks in absorption spectra. Given the poor performance of these methods with small model systems and the widespread use of the methods with larger molecular and material systems, here we bridge the gap in our understanding of these methods and examine the size-dependence of errors in RT density propagation. We analyze the performance of RT density propagation for systems of increasing size during the application of a continuous resonant field to induce Rabi-like oscillations, during charge-transfer dynamics, and for peak shifting in simulated absorption spectra. We find that the errors in the electron dynamics are indeed size dependent for these phenomena, with the largest system producing the results most aligned with those expected from linear response theory. The results suggest that although the RT-TDHF and RT-TDDFT methods may produce severe errors for model systems, the errors in charge transfer and resonantly driven electron dynamics may be much less significant for more realistic, large-scale molecules and materials.
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Affiliation(s)
- Karnamohit Ranka
- Chemistry and Biochemistry, University of California Merced, Merced, California 95343, USA
| | - Christine M Isborn
- Chemistry and Biochemistry, University of California Merced, Merced, California 95343, USA
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4
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Moitra T, Konecny L, Kadek M, Rubio A, Repisky M. Accurate Relativistic Real-Time Time-Dependent Density Functional Theory for Valence and Core Attosecond Transient Absorption Spectroscopy. J Phys Chem Lett 2023; 14:1714-1724. [PMID: 36757216 PMCID: PMC9940299 DOI: 10.1021/acs.jpclett.2c03599] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2022] [Accepted: 02/03/2023] [Indexed: 06/18/2023]
Abstract
First principles theoretical modeling of out-of-equilibrium processes observed in attosecond pump-probe transient absorption spectroscopy (TAS) triggering pure electron dynamics remains a challenging task, especially for heavy elements and/or core excitations containing fingerprints of scalar and spin-orbit relativistic effects. To address this, we formulate a methodology for simulating TAS within the relativistic real-time, time-dependent density functional theory (RT-TDDFT) framework, for both the valence and core energy regimes. Especially for TAS, full four-component (4c) RT simulations are feasible but computationally demanding. Therefore, in addition to the 4c approach, we also introduce the atomic mean-field exact two-component (amfX2C) Hamiltonian accounting for one- and two-electron picture-change corrections within RT-TDDFT. amfX2C preserves the accuracy of the parent 4c method at a fraction of its computational cost. Finally, we apply the methodology to study valence and near-L2,3-edge TAS processes of experimentally relevant systems and provide additional physical insights using relativistic nonequilibrium response theory.
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Affiliation(s)
- Torsha Moitra
- Hylleraas
Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, 9037 Tromsø, Norway
| | - Lukas Konecny
- Hylleraas
Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, 9037 Tromsø, Norway
- Max
Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Marius Kadek
- Hylleraas
Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, 9037 Tromsø, Norway
- Department
of Physics, Northeastern University, Boston, Massachusetts 02115, United States
- Algorithmiq
Ltd., Kanavakatu 3C, FI-00160 Helsinki, Finland
| | - Angel Rubio
- Max
Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center
for Computational Quantum Physics (CCQ), The Flatiron Institute, 162 Fifth Avenue, New York New York 10010, United States
- Nano-Bio
Spectroscopy Group, Departamento de Física de Materiales, Universidad del País Vasco, 20018 San Sebastian, Spain
| | - Michal Repisky
- Hylleraas
Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, 9037 Tromsø, Norway
- Department
of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University, 84104 Bratislava, Slovakia
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5
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Shepard C, Zhou R, Yost DC, Yao Y, Kanai Y. Simulating electronic excitation and dynamics with real-time propagation approach to TDDFT within plane-wave pseudopotential formulation. J Chem Phys 2021; 155:100901. [PMID: 34525811 DOI: 10.1063/5.0057587] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We give a perspective on simulating electronic excitation and dynamics using the real-time propagation approach to time-dependent density functional theory (RT-TDDFT) in the plane-wave pseudopotential formulation. RT-TDDFT is implemented in various numerical formalisms in recent years, and its practical application often dictates the most appropriate implementation of the theory. We discuss recent developments and challenges, emphasizing numerical aspects of studying real systems. Several applications of RT-TDDFT simulation are discussed to highlight how the approach is used to study interesting electronic excitation and dynamics phenomena in recent years.
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Affiliation(s)
- Christopher Shepard
- Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27514, USA
| | - Ruiyi Zhou
- Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27514, USA
| | - Dillon C Yost
- Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142, USA
| | - Yi Yao
- Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27514, USA
| | - Yosuke Kanai
- Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27514, USA
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6
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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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7
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Omar KA, Hasnaoui K, de la Lande A. First-Principles Simulations of Biological Molecules Subjected to Ionizing Radiation. Annu Rev Phys Chem 2021; 72:445-465. [PMID: 33878897 DOI: 10.1146/annurev-physchem-101419-013639] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
Ionizing rays cause damage to genomes, proteins, and signaling pathways that normally regulate cell activity, with harmful consequences such as accelerated aging, tumors, and cancers but also with beneficial effects in the context of radiotherapies. While the great pace of research in the twentieth century led to the identification of the molecular mechanisms for chemical lesions on the building blocks of biomacromolecules, the last two decades have brought renewed questions, for example, regarding the formation of clustered damage or the rich chemistry involving the secondary electrons produced by radiolysis. Radiation chemistry is now meeting attosecond science, providing extraordinary opportunities to unravel the very first stages of biological matter radiolysis. This review provides an overview of the recent progress made in this direction, focusing mainly on the atto- to femto- to picosecond timescales. We review promising applications of time-dependent density functional theory in this context.
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Affiliation(s)
- Karwan Ali Omar
- Institut de Chimie Physique, CNRS UMR 8000, Université Paris-Saclay, 91405 Orsay, France; .,Department of Chemistry, College of Education, University of Sulaimani, 41005 Kurdistan, Iraq
| | - Karim Hasnaoui
- High Performance Computing User Support Team, Institut du Développement et des Ressources en Informatique Scientifique (IDRIS), 91403 Orsay, France.,Maison de la Simulation, CNRS, Commissariat à l'Energie Atomique et aux Énergies Alternatives (CEA), Université Paris-Saclay, 91191 Gif-sur-Yvette, France
| | - Aurélien de la Lande
- Institut de Chimie Physique, CNRS UMR 8000, Université Paris-Saclay, 91405 Orsay, France;
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8
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Head-Marsden K, Flick J, Ciccarino CJ, Narang P. Quantum Information and Algorithms for Correlated Quantum Matter. Chem Rev 2020; 121:3061-3120. [PMID: 33326218 DOI: 10.1021/acs.chemrev.0c00620] [Citation(s) in RCA: 35] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.
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Affiliation(s)
- Kade Head-Marsden
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Johannes Flick
- Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, United States
| | - Christopher J Ciccarino
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States.,Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States
| | - Prineha Narang
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, United States
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9
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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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10
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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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11
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Lee Y, Yao X, Fischetti MV, Cho K. Real-time ab initio simulation of inelastic electron scattering using the exact, density functional, and alternative approaches. Phys Chem Chem Phys 2020; 22:8616-8624. [DOI: 10.1039/c9cp06376e] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Abstract
Inelastic electron scattering phenomena in chemical/physical/materials interests: electron radiation damage in materials; DNA damaged by electron scattering; electron therapy; electron microscope; electron-beam-induced deposition for nanofabrication.
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Affiliation(s)
- Yeonghun Lee
- Department of Materials Science and Engineering
- University of Texas at Dallas
- Richardson
- USA
| | - Xiaolong Yao
- Department of Materials Science and Engineering
- University of Texas at Dallas
- Richardson
- USA
| | - Massimo V. Fischetti
- Department of Materials Science and Engineering
- University of Texas at Dallas
- Richardson
- USA
| | - Kyeongjae Cho
- Department of Materials Science and Engineering
- University of Texas at Dallas
- Richardson
- USA
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12
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Nenov A, Segatta F, Bruner A, Mukamel S, Garavelli M. X-ray linear and non-linear spectroscopy of the ESCA molecule. J Chem Phys 2019; 151:114110. [DOI: 10.1063/1.5116699] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Affiliation(s)
- Artur Nenov
- Dipartimento di Chimica Industriale “Toso Montanari”, Università degli studi di Bologna, Viale del Risorgimento 4,
40136 Bologna, Italy
| | - Francesco Segatta
- Dipartimento di Chimica Industriale “Toso Montanari”, Università degli studi di Bologna, Viale del Risorgimento 4,
40136 Bologna, Italy
| | - Adam Bruner
- Department of Chemistry and Physics and Astronomy, University of California, Irvine, California 92697,
USA
| | - Shaul Mukamel
- Department of Chemistry and Physics and Astronomy, University of California, Irvine, California 92697,
USA
| | - Marco Garavelli
- Dipartimento di Chimica Industriale “Toso Montanari”, Università degli studi di Bologna, Viale del Risorgimento 4,
40136 Bologna, Italy
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13
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Yost DC, Yao Y, Kanai Y. Propagation of maximally localized Wannier functions in real-time TDDFT. J Chem Phys 2019; 150:194113. [DOI: 10.1063/1.5095631] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
Affiliation(s)
- Dillon C. Yost
- Department of Chemistry, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA
| | - Yi Yao
- Department of Chemistry, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA
| | - Yosuke Kanai
- Department of Chemistry, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, USA
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14
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Yost DC, Kanai Y. Electronic Excitation Dynamics in DNA under Proton and α-Particle Irradiation. J Am Chem Soc 2019; 141:5241-5251. [DOI: 10.1021/jacs.8b12148] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
Affiliation(s)
- Dillon C. Yost
- Department of Chemistry, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, United States
| | - Yosuke Kanai
- Department of Chemistry, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, United States
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15
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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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