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Trushin E, Görling A. Avoiding spin contamination and spatial symmetry breaking by exact-exchange-only optimized-effective-potential methods within the symmetrized Kohn-Sham framework. J Chem Phys 2023; 159:244109. [PMID: 38149736 DOI: 10.1063/5.0171546] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2023] [Accepted: 11/16/2023] [Indexed: 12/28/2023] Open
Abstract
For open-shell atoms and molecules, Kohn-Sham (KS) methods typically resort to spin-polarized approaches that exhibit spin-contamination and often break spatial symmetries. As a result, the KS Hamiltonian operator and the KS orbitals do not exhibit the space and spin symmetry of the physical electron system. The KS formalism can be symmetrized in a rigorous way only in real space, only in spin space, or both in real and spin space. Within such symmetrized KS frameworks, we present exact-exchange-only optimized-effective-potential (OEP) methods that are free of spin contamination and/or spatial symmetry breaking. The effect of symmetrizations on the total energy and its parts and on the exchange potential is analyzed. The presented exact-exchange-only OEP methods may serve as a starting point for high-level symmetrized KS methods based, e.g., on the adiabatic-connection fluctuation-dissipation theorem.
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Affiliation(s)
- Egor Trushin
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany and Erlangen National High Performance Computing Center (NHR@FAU), Martensstr. 1, D-91058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany and Erlangen National High Performance Computing Center (NHR@FAU), Martensstr. 1, D-91058 Erlangen, Germany
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2
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Haasler M, Maier TM, Kaupp M. Toward a correct treatment of core properties with local hybrid functionals. J Comput Chem 2023; 44:2461-2477. [PMID: 37635647 DOI: 10.1002/jcc.27211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2023] [Revised: 08/01/2023] [Accepted: 08/03/2023] [Indexed: 08/29/2023]
Abstract
In local hybrid functionals (LHs), a local mixing function (LMF) determines the position-dependent exact-exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt-LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn-Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high-density and iso-orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K-shell core excitations of second- and third-period systems including accurate state-of-the-art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange-energy densities, as well as by B95c meta-GGA correlation. The resulting LH23pt functional relates to the previous LH20t functional but specifically improves upon the core region.
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Affiliation(s)
- Matthias Haasler
- Technische Universität Berlin, Institute of Chemistry Theoretical Chemistry/Quantum Chemistry, Berlin, Germany
| | - Toni M Maier
- Technische Universität Braunschweig, Institute of Physical and Theoretical Chemistry, Braunschweig, Germany
| | - Martin Kaupp
- Technische Universität Berlin, Institute of Chemistry Theoretical Chemistry/Quantum Chemistry, Berlin, Germany
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3
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Teale AM, Helgaker T, Savin A, Adamo C, Aradi B, Arbuznikov AV, Ayers PW, Baerends EJ, Barone V, Calaminici P, Cancès E, Carter EA, Chattaraj PK, Chermette H, Ciofini I, Crawford TD, De Proft F, Dobson JF, Draxl C, Frauenheim T, Fromager E, Fuentealba P, Gagliardi L, Galli G, Gao J, Geerlings P, Gidopoulos N, Gill PMW, Gori-Giorgi P, Görling A, Gould T, Grimme S, Gritsenko O, Jensen HJA, Johnson ER, Jones RO, Kaupp M, Köster AM, Kronik L, Krylov AI, Kvaal S, Laestadius A, Levy M, Lewin M, Liu S, Loos PF, Maitra NT, Neese F, Perdew JP, Pernal K, Pernot P, Piecuch P, Rebolini E, Reining L, Romaniello P, Ruzsinszky A, Salahub DR, Scheffler M, Schwerdtfeger P, Staroverov VN, Sun J, Tellgren E, Tozer DJ, Trickey SB, Ullrich CA, Vela A, Vignale G, Wesolowski TA, Xu X, Yang W. DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science. Phys Chem Chem Phys 2022; 24:28700-28781. [PMID: 36269074 PMCID: PMC9728646 DOI: 10.1039/d2cp02827a] [Citation(s) in RCA: 54] [Impact Index Per Article: 27.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2022] [Accepted: 08/09/2022] [Indexed: 12/13/2022]
Abstract
In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.
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Affiliation(s)
- Andrew M. Teale
- School of Chemistry, University of Nottingham, University ParkNottinghamNG7 2RDUK
| | - Trygve Helgaker
- Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
| | - Andreas Savin
- Laboratoire de Chimie Théorique, CNRS and Sorbonne University, 4 Place Jussieu, CEDEX 05, 75252 Paris, France.
| | - Carlo Adamo
- PSL University, CNRS, ChimieParisTech-PSL, Institute of Chemistry for Health and Life Sciences, i-CLeHS, 11 rue P. et M. Curie, 75005 Paris, France.
| | - Bálint Aradi
- Bremen Center for Computational Materials Science, University of Bremen, P.O. Box 330440, D-28334 Bremen, Germany.
| | - Alexei V. Arbuznikov
- Technische Universität Berlin, Institut für Chemie, Theoretische Chemie/Quantenchemie, Sekr. C7Straße des 17. Juni 13510623Berlin
| | | | - Evert Jan Baerends
- Department of Chemistry and Pharmaceutical Sciences, Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.
| | - Vincenzo Barone
- Scuola Normale Superiore, Piazza dei Cavalieri 7, 56125 Pisa, Italy.
| | - Patrizia Calaminici
- Departamento de Química, Centro de Investigación y de Estudios Avanzados (Cinvestav), CDMX, 07360, Mexico.
| | - Eric Cancès
- CERMICS, Ecole des Ponts and Inria Paris, 6 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France.
| | - Emily A. Carter
- Department of Mechanical and Aerospace Engineering and the Andlinger Center for Energy and the Environment, Princeton UniversityPrincetonNJ 08544-5263USA
| | | | - Henry Chermette
- Institut Sciences Analytiques, Université Claude Bernard Lyon1, CNRS UMR 5280, 69622 Villeurbanne, France.
| | - Ilaria Ciofini
- PSL University, CNRS, ChimieParisTech-PSL, Institute of Chemistry for Health and Life Sciences, i-CLeHS, 11 rue P. et M. Curie, 75005 Paris, France.
| | - T. Daniel Crawford
- Department of Chemistry, Virginia TechBlacksburgVA 24061USA,Molecular Sciences Software InstituteBlacksburgVA 24060USA
| | - Frank De Proft
- Research Group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, B-1050 Brussels, Belgium.
| | | | - Claudia Draxl
- Institut für Physik and IRIS Adlershof, Humboldt-Universität zu Berlin, 12489 Berlin, Germany. .,Fritz-Haber-Institut der Max-Planck-Gesellschaft, 14195 Berlin, Germany
| | - Thomas Frauenheim
- Bremen Center for Computational Materials Science, University of Bremen, P.O. Box 330440, D-28334 Bremen, Germany. .,Beijing Computational Science Research Center (CSRC), 100193 Beijing, China.,Shenzhen JL Computational Science and Applied Research Institute, 518110 Shenzhen, China
| | - Emmanuel Fromager
- Laboratoire de Chimie Quantique, Institut de Chimie, CNRS/Université de Strasbourg, 4 rue Blaise Pascal, 67000 Strasbourg, France.
| | - Patricio Fuentealba
- Departamento de Física, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile.
| | - Laura Gagliardi
- Department of Chemistry, Pritzker School of Molecular Engineering, The James Franck Institute, and Chicago Center for Theoretical Chemistry, The University of Chicago, Chicago, Illinois 60637, USA.
| | - Giulia Galli
- Pritzker School of Molecular Engineering and Department of Chemistry, The University of Chicago, Chicago, IL, USA.
| | - Jiali Gao
- Institute of Systems and Physical Biology, Shenzhen Bay Laboratory, Shenzhen 518055, China. .,Department of Chemistry, University of Minnesota, Minneapolis, MN 55455, USA
| | - Paul Geerlings
- Research Group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, B-1050 Brussels, Belgium.
| | - Nikitas Gidopoulos
- Department of Physics, Durham University, South Road, Durham DH1 3LE, UK.
| | - Peter M. W. Gill
- School of Chemistry, University of SydneyCamperdown NSW 2006Australia
| | - Paola Gori-Giorgi
- Department of Chemistry and Pharmaceutical Sciences, Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.
| | - Andreas Görling
- Chair of Theoretical Chemistry, University of Erlangen-Nuremberg, Egerlandstrasse 3, 91058 Erlangen, Germany.
| | - Tim Gould
- Qld Micro- and Nanotechnology Centre, Griffith University, Gold Coast, Qld 4222, Australia.
| | - Stefan Grimme
- Mulliken Center for Theoretical Chemistry, University of Bonn, Beringstrasse 4, 53115 Bonn, Germany.
| | - Oleg Gritsenko
- Department of Chemistry and Pharmaceutical Sciences, Amsterdam Institute of Molecular and Life Sciences (AIMMS), Faculty of Science, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.
| | - Hans Jørgen Aagaard Jensen
- Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, DK-5230 Odense M, Denmark.
| | - Erin R. Johnson
- Department of Chemistry, Dalhousie UniversityHalifaxNova ScotiaB3H 4R2Canada
| | - Robert O. Jones
- Peter Grünberg Institut PGI-1, Forschungszentrum Jülich52425 JülichGermany
| | - Martin Kaupp
- Technische Universität Berlin, Institut für Chemie, Theoretische Chemie/Quantenchemie, Sekr. C7, Straße des 17. Juni 135, 10623, Berlin.
| | - Andreas M. Köster
- Departamento de Química, Centro de Investigación y de Estudios Avanzados (Cinvestav)CDMX07360Mexico
| | - Leeor Kronik
- Department of Molecular Chemistry and Materials Science, Weizmann Institute of Science, Rehovoth, 76100, Israel.
| | - Anna I. Krylov
- Department of Chemistry, University of Southern CaliforniaLos AngelesCalifornia 90089USA
| | - Simen Kvaal
- Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
| | - Andre Laestadius
- Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
| | - Mel Levy
- Department of Chemistry, Tulane University, New Orleans, Louisiana, 70118, USA.
| | - Mathieu Lewin
- CNRS & CEREMADE, Université Paris-Dauphine, PSL Research University, Place de Lattre de Tassigny, 75016 Paris, France.
| | - Shubin Liu
- Research Computing Center, University of North Carolina, Chapel Hill, NC 27599-3420, USA. .,Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USA
| | - Pierre-François Loos
- Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France.
| | - Neepa T. Maitra
- Department of Physics, Rutgers University at Newark101 Warren StreetNewarkNJ 07102USA
| | - Frank Neese
- Max Planck Institut für Kohlenforschung, Kaiser Wilhelm Platz 1, D-45470 Mülheim an der Ruhr, Germany.
| | - John P. Perdew
- Departments of Physics and Chemistry, Temple UniversityPhiladelphiaPA 19122USA
| | - Katarzyna Pernal
- Institute of Physics, Lodz University of Technology, ul. Wolczanska 219, 90-924 Lodz, Poland.
| | - Pascal Pernot
- Institut de Chimie Physique, UMR8000, CNRS and Université Paris-Saclay, Bât. 349, Campus d'Orsay, 91405 Orsay, France.
| | - Piotr Piecuch
- Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA. .,Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
| | - Elisa Rebolini
- Institut Laue Langevin, 71 avenue des Martyrs, 38000 Grenoble, France.
| | - Lucia Reining
- Laboratoire des Solides Irradiés, CNRS, CEA/DRF/IRAMIS, École Polytechnique, Institut Polytechnique de Paris, F-91120 Palaiseau, France. .,European Theoretical Spectroscopy Facility
| | - Pina Romaniello
- Laboratoire de Physique Théorique (UMR 5152), Université de Toulouse, CNRS, UPS, France.
| | - Adrienn Ruzsinszky
- Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA.
| | - Dennis R. Salahub
- Department of Chemistry, Department of Physics and Astronomy, CMS – Centre for Molecular Simulation, IQST – Institute for Quantum Science and Technology, Quantum Alberta, University of Calgary2500 University Drive NWCalgaryAlbertaT2N 1N4Canada
| | - Matthias Scheffler
- The NOMAD Laboratory at the FHI of the Max-Planck-Gesellschaft and IRIS-Adlershof of the Humboldt-Universität zu Berlin, Faradayweg 4-6, D-14195, Germany.
| | - Peter Schwerdtfeger
- Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Auckland, 0632 Auckland, New Zealand.
| | - Viktor N. Staroverov
- Department of Chemistry, The University of Western OntarioLondonOntario N6A 5B7Canada
| | - Jianwei Sun
- Department of Physics and Engineering Physics, Tulane University, New Orleans, LA 70118, USA.
| | - Erik Tellgren
- Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
| | - David J. Tozer
- Department of Chemistry, Durham UniversitySouth RoadDurhamDH1 3LEUK
| | - Samuel B. Trickey
- Quantum Theory Project, Deptartment of Physics, University of FloridaGainesvilleFL 32611USA
| | - Carsten A. Ullrich
- Department of Physics and Astronomy, University of MissouriColumbiaMO 65211USA
| | - Alberto Vela
- Departamento de Química, Centro de Investigación y de Estudios Avanzados (Cinvestav), CDMX, 07360, Mexico.
| | - Giovanni Vignale
- Department of Physics, University of Missouri, Columbia, MO 65203, USA.
| | - Tomasz A. Wesolowski
- Department of Physical Chemistry, Université de Genève30 Quai Ernest-Ansermet1211 GenèveSwitzerland
| | - Xin Xu
- Shanghai Key Laboratory of Molecular Catalysis and Innovation Materials, Collaborative Innovation Centre of Chemistry for Energy Materials, MOE Laboratory for Computational Physical Science, Department of Chemistry, Fudan University, Shanghai 200433, China.
| | - Weitao Yang
- Department of Chemistry and Physics, Duke University, Durham, NC 27516, USA.
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4
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Trushin E, Görling A. Numerically stable optimized effective potential method with standard Gaussian basis sets. J Chem Phys 2021; 155:054109. [PMID: 34364359 DOI: 10.1063/5.0056431] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
Abstract
We present a numerically stable optimized effective potential (OEP) method based on Gaussian basis sets. The key point of the approach is a sequence of preprocessing steps of the auxiliary basis set used to represent exchange or correlation potentials, the Kohn-Sham (KS) response function, and the right-hand side of the OEP equation in conjunction with a representation of exchange or correlation potentials via exchange or correlation charge densities whose electrostatic potentials generate the potentials. Due to the preprocessing, standard Gaussian basis sets from basis set libraries can be used in OEP calculations. As examples, we present numerical stable computational setups based on aux-cc-pwCVXZ basis sets with X = T, Q, 5 for the orbitals and aux-cc-pVDZ/mp2fit and aux-cc-pVTZ/mp2fit auxiliary basis sets and use them to calculate KS exchange potentials with the exact exchange-only KS method for various atoms and molecules. The resulting exchange potentials not only are numerically stable and physically reasonable but also show convergence with increasing quality of the orbital basis sets. The effect of incorporating exact conditions that the KS exchange potential has to obey is discussed. Moreover, it is briefly demonstrated that the presented approach not only works for KS exchange potentials but equally well for correlation potentials within the direct random phase approximation. Besides for OEP methods, the introduced preprocessing of auxiliary basis sets should also be beneficial in procedures to calculate back effective KS potentials from given electron densities.
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Affiliation(s)
- Egor Trushin
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany and Erlangen National High Performance Computing Center (NHR@FAU), Martensstr. 1, D-91058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany and Erlangen National High Performance Computing Center (NHR@FAU), Martensstr. 1, D-91058 Erlangen, Germany
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5
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Ikabata Y, Nakai H. Picture-change correction in relativistic density functional theory. Phys Chem Chem Phys 2021; 23:15458-15474. [PMID: 34278401 DOI: 10.1039/d1cp01773j] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Relativistic quantum chemical calculations are performed based on one of two physical pictures, namely the Dirac picture and the Schrödinger picture. With regard to the latter, the so-called picture-change effect (PCE) and picture-change correction (PCC) have been studied. The PCE, which is the change in the expectation value associated with the transformation, is not commonly a minor effect. The electron density, which is given by the expectation value of the density operator, is a fundamental variable in relativistic density functional theory (RDFT). Thus, performing the PCC in RDFT calculations is essential not only in terms of numerical agreement with the Dirac picture, but also from the viewpoint of fundamental theory. This paper explains theories and numerical studies of PCE and PCC in RDFT after overviewing those in properties, which involves the authors' works on the development of RDFT in the Schrödinger picture and relativistic exchange-correlation functionals based on picture-change-corrected variables.
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Affiliation(s)
- Yasuhiro Ikabata
- Waseda Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan.
| | - Hiromi Nakai
- Waseda Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan. and Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan and Elements Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University, Katsura, Kyoto 615-8520, Japan
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6
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Paquier J, Giner E, Toulouse J. Relativistic short-range exchange energy functionals beyond the local-density approximation. J Chem Phys 2020; 152:214106. [DOI: 10.1063/5.0004926] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Julien Paquier
- Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France
| | - Emmanuel Giner
- Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France
| | - Julien Toulouse
- Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France
- Institut Universitaire de France, F-75005 Paris, France
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7
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Maier TM, Ikabata Y, Nakai H. Relativistic local hybrid functionals and their impact on 1s core orbital energies. J Chem Phys 2020; 152:214103. [DOI: 10.1063/5.0010400] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Toni M. Maier
- Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
- Technische Universität Berlin, Institut für Chemie, Theoretische Chemie/Quantenchemie, Sekr. C7, Straße des 17. Juni 135, D-10623 Berlin, Germany
| | - Yasuhiro Ikabata
- Waseda Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
| | - Hiromi Nakai
- Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
- Waseda Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
- Elements Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University, Katsura, Kyoto 615-8520, Japan
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8
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Paquier J, Toulouse J. Four-component relativistic range-separated density-functional theory: Short-range exchange local-density approximation. J Chem Phys 2018; 149:174110. [DOI: 10.1063/1.5049773] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
Affiliation(s)
- Julien Paquier
- Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France
| | - Julien Toulouse
- Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France
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9
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Abstract
The foundations, formalisms, technicalities, and practicalities of relativistic time-dependent density functional theories (R-TD-DFT) for spinor excited states of molecular systems containing heavy elements are critically reviewed.
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Affiliation(s)
- Wenjian Liu
- Beijing National Center for Molecular Sciences
- Institute of Theoretical and Computational Chemistry
- College of Chemistry and Molecular Engineering
- Peking University
- Beijing 100871
| | - Yunlong Xiao
- Beijing National Center for Molecular Sciences
- Institute of Theoretical and Computational Chemistry
- College of Chemistry and Molecular Engineering
- Peking University
- Beijing 100871
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10
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Zabrodin E, Bravina L, Bleibel J. Basic features of proton-proton interactions at ultra-relativistic energies and RFT-based quark-gluon string model. EPJ WEB OF CONFERENCES 2017. [DOI: 10.1051/epjconf/201716401005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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11
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Liu W, Wang F, Li L. The Beijing Density Functional (BDF) Program Package: Methodologies and Applications. JOURNAL OF THEORETICAL & COMPUTATIONAL CHEMISTRY 2011. [DOI: 10.1142/s0219633603000471] [Citation(s) in RCA: 115] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The Beijing Density Functional (BDF) program package is such a code that can perform nonrelativistic, one-, two-, and four-component relativistic density functional calculations on medium-sized molecular systems with various functionals in most compact and yet sufficient basis set expansions. The mergence of different approaches in a single code facilitates direct and systematic comparisons between different Hamiltonians, since they share all the same numerical and technical issues. In this account, the methodologies adopted in the code will be discussed in great detail and some applications of the code will be briefly presented.
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Affiliation(s)
- Wenjian Liu
- Institute of Theoretical and Computational Chemistry and State Key, Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, P. R. China
| | - Fan Wang
- Institute of Theoretical and Computational Chemistry and State Key, Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, P. R. China
| | - Lemin Li
- Institute of Theoretical and Computational Chemistry and State Key, Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, P. R. China
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13
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Belpassi L, Storchi L, Quiney HM, Tarantelli F. Recent advances and perspectives in four-component Dirac–Kohn–Sham calculations. Phys Chem Chem Phys 2011; 13:12368-94. [DOI: 10.1039/c1cp20569b] [Citation(s) in RCA: 55] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023]
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14
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Abstract
Relativistic effects in atomic and molecular propertiesWe present an overview of basic principles and methods of the relativistic quantum chemistry. Practical aspects of different methods will be discussed stressing their capability of providing accurate predictions of molecular properties, particularly in species containing a heavy metal element. We will present a series of examples showing the importance of relativistic effects in a variety of molecular properties including electron affinities, ionization potentials, reaction and dissociation energies, electric, spectroscopic and other properties. It is possible to recognize a link between these properties and behaviour of materials in some cases. Particular attention is paid to relativistic calculations of the nuclear quadrupole moments for which accurate theoretical electric field gradient is combined with data from the microwave spectra. Important aspect of the present paper is understanding of trends in electronically related atoms throughout the Mendeleev Periodic Table rather than focusing on highly accurate numbers. We will show that relativistic effects represent an unavoidable instrument for explaining some unexpected properties of heavy metal containing compounds. We will also discuss an interplay between the many-electron correlation and relativistic effects.
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15
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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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16
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Heßelmann A, Görling A. Random phase approximation correlation energies with exact Kohn–Sham exchange. Mol Phys 2010. [DOI: 10.1080/00268970903476662] [Citation(s) in RCA: 93] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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17
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18
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van Wüllen C. Relativistic Density Functional Theory. CHALLENGES AND ADVANCES IN COMPUTATIONAL CHEMISTRY AND PHYSICS 2010. [DOI: 10.1007/978-1-4020-9975-5_5] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
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19
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Ködderitzsch D, Ebert H, Akai H, Engel E. Relativistic optimized effective potential method-application to alkali metals. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2009; 21:064208. [PMID: 21715911 DOI: 10.1088/0953-8984/21/6/064208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
We present a relativistic formulation of the optimized effective potential method (ROEP) and its implementation within the Korringa-Kohn-Rostoker multiple scattering formalism. The scheme is an all-electron approach, treating core and band states formally on the same footing. We use exact exchange (EXX) as an approximation to the exchange correlation functional. Numerical four-component wavefunctions for the description of core and valence electrons and the corresponding ingredients of the ROEP integral equation are employed. The exact exchange expression for the valence states is reformulated in terms of the electronic Green's function that in turn is evaluated by making use of multiple scattering formalism. We present and discuss the application of the formalism to non-magnetic alkali metals.
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Affiliation(s)
- D Ködderitzsch
- Ludwig-Maximilians-Universität München, Department Chemie und Biochemie, Physikalische Chemie, Butenandtstraße 11, D-81377 München, Germany
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20
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Heßelmann A, Görling A. Comparison between optimized effective potential and Kohn–Sham methods. Chem Phys Lett 2008. [DOI: 10.1016/j.cplett.2008.02.042] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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21
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Hesselmann A, Götz AW, Della Sala F, Görling A. Numerically stable optimized effective potential method with balanced Gaussian basis sets. J Chem Phys 2007; 127:054102. [PMID: 17688329 DOI: 10.1063/1.2751159] [Citation(s) in RCA: 95] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A solution to the long-standing problem of developing numerically stable optimized effective potential (OEP) methods based on Gaussian basis sets is presented by introducing an approach consisting of an exact exchange OEP method with an accompanying construction and balancing scheme for the involved auxiliary and orbital Gaussian basis sets that is numerically stable and that properly represents an exact exchange Kohn-Sham method. The method is a purely analytical method that does not require any numerical grid, scales like Hartree-Fock or B3LYP procedures, is straightforward to implement, and is easily generalized to take into account orbital-dependent density functionals other than the exact exchange considered in this work. Thus, the presented OEP approach opens the way to the development and application of novel orbital-dependent exchange-correlation functionals. It is shown that adequately taking into account the continuum part of the Kohn-Sham orbital spectrum is crucial for numerically stable Gaussian basis set OEP methods. Moreover, it is mandatory to employ orbital basis sets that are converged with respect to the used auxiliary basis representing the exchange potential. OEP calculations in the past often did not meet the latter requirement and therefore may have led to erroneously low total energies.
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Affiliation(s)
- Andreas Hesselmann
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstrasse 3, D-91058 Erlangen, Germany
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22
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Witek HA, Köhler C, Frauenheim T, Morokuma K, Elstner M. Relativistic parametrization of the self-consistent-charge density-functional tight-binding method. 1. Atomic wave functions and energies. J Phys Chem A 2007; 111:5712-9. [PMID: 17567112 DOI: 10.1021/jp070786o] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
A detailed treatment of a confined relativistic atom, needed as an initial step for the parametrization of the self-consistent-charge density-functional tight-binding method, is presented and discussed. The required one-component quantities, i.e., orbital energies, orbital wave functions, and Hubbard parameters, are obtained by weighted averaging of the corresponding numbers determined for the atomic spinors. The wave function and density confinement is achieved by introducing the Woods-Saxon potential in the atomic four-component Dirac-Kohn-Sham problem. The effect of the additional confining potential on energy eigenvalues and the shape of atomic wave functions and densities is discussed and numerical examples are presented for the valence spinors of carbon, germanium, and lead.
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Affiliation(s)
- Henryk A Witek
- Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, Hsinchu 30010, Taiwan.
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23
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Matveev AV, Majumder S, Rösch N. Efficient treatment of the Hartree interaction in the relativistic Kohn-Sham problem. J Chem Phys 2005; 123:164104. [PMID: 16268678 DOI: 10.1063/1.2079907] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We elaborate the two-component Douglas-Kroll reduction of the Dirac-Kohn-Sham problem of relativistic density-functional theory as introduced by Matveev and Rosch [J. Chem. Phys. 118, 3997 (2003)]. That method retains corrections to the Coulomb self-interaction (or Hartree) term of the energy functional that are due to the picture change. Using analytic expressions for the matrix elements, one is able to abandon the resolution of the identity approach for a crucial step of the relativistic transformation. Thus, a major source of uncertainties of the method is eliminated because basis sets no longer have to be extended by functions of higher angular momentum, previously required to ensure kinetic balance. This approach also relies on the electron charge-density fitting scheme via an auxiliary basis set. An efficient approximate implementation results if one restricts the relativistic transformation to the spherically symmetric atom-centered auxiliary functions. It provides accurate results while simplifying greatly the expressions for the matrix elements of the relativistically transformed operators and significantly reducing the computational effort. We demonstrate the performance of the method for the fine structure of one-electron levels of the Hg atom, the g-tensor shifts of NO2, and the properties of the diatomic molecules Bi2, Pb2, PbO, and TlH.
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Affiliation(s)
- Alexei V Matveev
- Department Chemie, Theoretische Chemie, Technische Universität München, 85747 Garching, Germany
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25
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Ackad E, Horbatsch M. Numerical solution of the Dirac equation by a mapped Fourier grid method. ACTA ACUST UNITED AC 2005. [DOI: 10.1088/0305-4470/38/14/007] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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26
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Gao J, Liu W, Song B, Liu C. Time-dependent four-component relativistic density functional theory for excitation energies. J Chem Phys 2004; 121:6658-66. [PMID: 15473721 DOI: 10.1063/1.1788655] [Citation(s) in RCA: 72] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
Time-dependent four-component relativistic density functional theory within the linear response regime is developed for calculating excitation energies of heavy element containing systems. Since spin is no longer a good quantum number in this context, we resort to time-reversal adapted Kramers basis when deriving the coupled Dirac-Kohn-Sham equation. The particular implementation of the formalism into the Beijing density functional program package utilizes the multipolar expansion of the induced density to facilitate the construction of the induced Coulomb potential. As the first application, pilot calculations on the valence excitation energies and fine structures of the rare gas (Ne to Rn) and Group 12 (Zn to Hg) atoms are reported. To the best of our knowledge, it is the first time to be able to account for spin-orbit coupling within time-dependent density functional theory for excitation energies.
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Affiliation(s)
- Jun Gao
- Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, People's Republic of China
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27
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Rösch N, Matveev A, Nasluzov VA, Neyman KM, Moskaleva L, Krüger S. Quantum chemistry with the Douglas-Kroll-Hess approach to relativistic density functional theory: Efficient methods for molecules and materials. THEORETICAL AND COMPUTATIONAL CHEMISTRY 2004. [DOI: 10.1016/s1380-7323(04)80038-4] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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28
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Wang F, Li L. Numerical examination of performance of some exchange-correlation functionals for molecules containing heavy elements. J Comput Chem 2004; 25:669-77. [PMID: 14978710 DOI: 10.1002/jcc.10421] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Abstract
The performance of 17 exchange-correlation functionals for molecules containing heavy elements are numerically examined through four-component relativistic density DFT calculations. The examined functionals show the similar accuracy as they do for the molecules containing light elements only except for bond lengths. LDA and OP86 produce good results for bond lengths and frequencies but bad bond energies. Different functionals do not show much different performance for bond energies except LDA. BP86 and GP86 produce results with average accuracy while LYP does not perform well. Although encouraging results are obtained with functional B97GGA-1, other heavily parameterized and meta-GGA functionals do not produce impressive results.
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Affiliation(s)
- Fan Wang
- State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry, and Molecular Engineering, Peking University, Beijing 100871, People's Republic of China
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29
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Majumder S, Matveev AV, Rösch N. Spin–orbit interaction in the Douglas–Kroll approach to relativistic density functional theory: the screened nuclear potential approximation for molecules. Chem Phys Lett 2003. [DOI: 10.1016/j.cplett.2003.10.072] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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30
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Orbital-Dependent Functionals for the Exchange-Correlation Energy: A Third Generation of Density Functionals. ACTA ACUST UNITED AC 2003. [DOI: 10.1007/3-540-37072-2_2] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/04/2023]
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31
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Matveev A, Rösch N. The electron–electron interaction in the Douglas–Kroll–Hess approach to the Dirac–Kohn–Sham problem. J Chem Phys 2003. [DOI: 10.1063/1.1540615] [Citation(s) in RCA: 38] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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32
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33
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Relativistic Density Functional Theory. ACTA ACUST UNITED AC 2003. [DOI: 10.1007/978-94-017-0105-1_11] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register]
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34
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Quiney HM, Belanzoni P. Relativistic density functional theory using Gaussian basis sets. J Chem Phys 2002. [DOI: 10.1063/1.1502245] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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35
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Abstract
The theoretical and technical foundations are presented for the efficient relativistic electronic structure theories to treat heavy-atomic molecular systems. This review contains two surveys of four-component and two-component quasi-relativistic approaches. First, we review our highly efficient computational scheme for four-component relativistic ab initio molecular orbital (MO) methods over generally contracted spherical harmonic Gaussian-type spinors (GTSs). Illustrative calculations, which are performed with a new four-component relativistic ab initio molecular orbital program package REL4D, clearly show the efficiency of our computational scheme by the Dirac-Hartree-Fock (DHF) and Dirac-Hartree-Fock (DKS) methods. Next, in the two-component quasi-relativistic framework, two relativistic Hamiltonians, RESC and higher order Douglas-Kroll (DK) Hamiltonians, are introduced, and several illustrative calculations are shown. Numerical results for several systems show that good accuracy can be obtained with our third-order DK (DK3) Hamiltonian.
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Affiliation(s)
- Takahito Nakajima
- Department of Applied Chemistry, Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan
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36
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Van Wüllen C. Spin densities in two-component relativistic density functional calculations: noncollinear versus collinear approach. J Comput Chem 2002; 23:779-85. [PMID: 12012354 DOI: 10.1002/jcc.10043] [Citation(s) in RCA: 135] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
With present day exchange-correlation functionals, accurate results in nonrelativistic open shell density functional calculations can only be obtained if one uses functionals that do not only depend on the electron density but also on the spin density. We consider the common case where such functionals are applied in relativistic density functional calculations. In scalar-relativistic calculations, the spin density can be defined conventionally, but if spin-orbit coupling is taken into account, spin is no longer a good quantum number and it is not clear what the "spin density" is. In many applications, a fixed quantization axis is used to define the spin density ("collinear approach"), but one can also use the length of the local spin magnetization vector without any reference to an external axis ("noncollinear approach"). These two possibilities are compared in this work both by formal analysis and numerical experiments. It is shown that the (nonrelativistic) exchange-correlation functional should be invariant with respect to rotations in spin space, and this only holds for the noncollinear approach. Total energies of open shell species are higher in the collinear approach because less exchange energy is assigned to a given Kohn-Sham reference function. More importantly, the collinear approach breaks rotational symmetry, that is, in molecular calculations one may find different energies for different orientations of the molecule. Data for the first ionization potentials of Tl, Pb, element 113, and element 114, and for the orientation dependence of the total energy of I+2 and PbF indicate that the error introduced by the collinear approximation is approximately 0.1 eV for valence ionization potentials, but can be much larger if highly ionized open shell states are considered. Rotational invariance is broken by the same amount. This clearly indicates that the collinear approach should not be used, as the full treatment is easily implemented and does not introduce much more computational effort.
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Affiliation(s)
- Christoph Van Wüllen
- Technische Universität Berlin, Sekr. C3, Strasse des 17. Juni 115, D-10623 Berlin, Germany.
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37
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Hirata S, Ivanov S, Grabowski I, Bartlett RJ. Time-dependent density functional theory employing optimized effective potentials. J Chem Phys 2002. [DOI: 10.1063/1.1460869] [Citation(s) in RCA: 92] [Impact Index Per Article: 4.2] [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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A new quasi-relativistic approach for density functional theory based on the normalized elimination of the small component. Chem Phys Lett 2002. [DOI: 10.1016/s0009-2614(01)01357-4] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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39
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Yanai T, Iikura H, Nakajima T, Ishikawa Y, Hirao K. A new implementation of four-component relativistic density functional method for heavy-atom polyatomic systems. J Chem Phys 2001. [DOI: 10.1063/1.1412252] [Citation(s) in RCA: 63] [Impact Index Per Article: 2.7] [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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Mayer M, Krüger S, Rösch N. A two-component variant of the Douglas–Kroll relativistic linear combination of Gaussian-type orbitals density-functional method: Spin–orbit effects in atoms and diatomics. J Chem Phys 2001. [DOI: 10.1063/1.1390509] [Citation(s) in RCA: 75] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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42
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Varga S, Fricke B, Hirata M, Baştuǧ T, Pershina V, Fritzsche S. Total Energy Calculations of RfCl4 and Homologues in the Framework of Relativistic Density Functional Theory. J Phys Chem A 2000. [DOI: 10.1021/jp993980m] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- S. Varga
- Department of Material Sciences, JAERI, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan
| | - B. Fricke
- Department of Material Sciences, JAERI, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan
| | - M. Hirata
- Department of Material Sciences, JAERI, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan
| | - T. Baştuǧ
- Department of Material Sciences, JAERI, Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan
| | - V. Pershina
- Gesellschaft für Schwerionenforschung (GSI) D-64291 Darmstadt, Germany
| | - S. Fritzsche
- Fachbereich Physik, Universität Kassel, D-34109 Kassel, Germany
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Varga S, Fricke B, Nakamatsu H, Mukoyama T, Anton J, Geschke D, Heitmann A, Engel E, Baştuǧ T. Four-component relativistic density functional calculations of heavy diatomic molecules. J Chem Phys 2000. [DOI: 10.1063/1.480934] [Citation(s) in RCA: 76] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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44
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Gilardoni F, Weber J, Hauser A, Daul C. A comparison of ground- and excited-state properties of [Ru(bz)2]2+ and bis(?6-benzene)ruthenium(II)p-toluenesulfonate using the density functional theory. J Comput Chem 1999. [DOI: 10.1002/(sici)1096-987x(199910)20:13<1343::aid-jcc2>3.0.co;2-u] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Gilardoni F, Weber J, Hauser A, Daul C. A comparison of ground- and excited-state properties of gas phase and crystalline ruthenocene using density functional theory. J Chem Phys 1998. [DOI: 10.1063/1.476693] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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49
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Full potential linearized-augmented-plane-wave calculations for 5d transition metals using the relativistic generalized gradient approximation. ADVANCES IN QUANTUM CHEMISTRY 1998. [DOI: 10.1016/s0065-3276(08)60437-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/22/2023]
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50
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Mayer M, Häberlen OD, Rösch N. Relevance of relativistic exchange-correlation functionals and of finite nuclei in molecular density-functional calculations. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1996; 54:4775-4782. [PMID: 9914042 DOI: 10.1103/physreva.54.4775] [Citation(s) in RCA: 64] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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