1
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Majumder R, Sokolov AY. Consistent Second-Order Treatment of Spin-Orbit Coupling and Dynamic Correlation in Quasidegenerate N-Electron Valence Perturbation Theory. J Chem Theory Comput 2024; 20:4676-4688. [PMID: 38795071 DOI: 10.1021/acs.jctc.4c00458] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/27/2024]
Abstract
We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wave functions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the nonrelativistic complete active-space wave functions, and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.
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Affiliation(s)
- Rajat Majumder
- Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States
| | - Alexander Yu Sokolov
- Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States
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2
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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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3
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Nakashima H, Nakatsuji H. Inverse Hamiltonian method assisted by the complex scaling technique for solving the Dirac-Coulomb equation: Helium isoelectronic atoms. Chem Phys Lett 2020. [DOI: 10.1016/j.cplett.2020.137447] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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4
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Smith DGA, Burns LA, Simmonett AC, Parrish RM, Schieber MC, Galvelis R, Kraus P, Kruse H, Di Remigio R, Alenaizan A, James AM, Lehtola S, Misiewicz JP, Scheurer M, Shaw RA, Schriber JB, Xie Y, Glick ZL, Sirianni DA, O’Brien JS, Waldrop JM, Kumar A, Hohenstein EG, Pritchard BP, Brooks BR, Schaefer HF, Sokolov AY, Patkowski K, DePrince AE, Bozkaya U, King RA, Evangelista FA, Turney JM, Crawford TD, Sherrill CD. Psi4 1.4: Open-source software for high-throughput quantum chemistry. J Chem Phys 2020; 152:184108. [PMID: 32414239 PMCID: PMC7228781 DOI: 10.1063/5.0006002] [Citation(s) in RCA: 403] [Impact Index Per Article: 100.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2020] [Accepted: 04/12/2020] [Indexed: 12/13/2022] Open
Abstract
PSI4 is a free and open-source ab initio electronic structure program providing implementations of Hartree-Fock, density functional theory, many-body perturbation theory, configuration interaction, density cumulant theory, symmetry-adapted perturbation theory, and coupled-cluster theory. Most of the methods are quite efficient, thanks to density fitting and multi-core parallelism. The program is a hybrid of C++ and Python, and calculations may be run with very simple text files or using the Python API, facilitating post-processing and complex workflows; method developers also have access to most of PSI4's core functionalities via Python. Job specification may be passed using The Molecular Sciences Software Institute (MolSSI) QCSCHEMA data format, facilitating interoperability. A rewrite of our top-level computation driver, and concomitant adoption of the MolSSI QCARCHIVE INFRASTRUCTURE project, makes the latest version of PSI4 well suited to distributed computation of large numbers of independent tasks. The project has fostered the development of independent software components that may be reused in other quantum chemistry programs.
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Affiliation(s)
| | - Lori A. Burns
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Andrew C. Simmonett
- National Institutes of Health – National Heart,
Lung and Blood Institute, Laboratory of Computational Biology, Bethesda,
Maryland 20892, USA
| | - Robert M. Parrish
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Matthew C. Schieber
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | | | - Peter Kraus
- School of Molecular and Life Sciences, Curtin
University, Kent St., Bentley, Perth, Western Australia 6102,
Australia
| | - Holger Kruse
- Institute of Biophysics of the Czech Academy of
Sciences, Královopolská 135, 612 65 Brno, Czech
Republic
| | - Roberto Di Remigio
- Department of Chemistry, Centre for Theoretical
and Computational Chemistry, UiT, The Arctic University of Norway, N-9037
Tromsø, Norway
| | - Asem Alenaizan
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Andrew M. James
- Department of Chemistry, Virginia
Tech, Blacksburg, Virginia 24061, USA
| | - Susi Lehtola
- Department of Chemistry, University of
Helsinki, P.O. Box 55 (A. I. Virtasen aukio 1), FI-00014 Helsinki,
Finland
| | - Jonathon P. Misiewicz
- Center for Computational Quantum Chemistry,
University of Georgia, Athens, Georgia 30602, USA
| | - Maximilian Scheurer
- Interdisciplinary Center for Scientific
Computing, Heidelberg University, D-69120 Heidelberg,
Germany
| | - Robert A. Shaw
- ARC Centre of Excellence in Exciton Science,
School of Science, RMIT University, Melbourne, VIC 3000,
Australia
| | - Jeffrey B. Schriber
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Yi Xie
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Zachary L. Glick
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Dominic A. Sirianni
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Joseph Senan O’Brien
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
| | - Jonathan M. Waldrop
- Department of Chemistry and Biochemistry, Auburn
University, Auburn, Alabama 36849, USA
| | - Ashutosh Kumar
- Department of Chemistry, Virginia
Tech, Blacksburg, Virginia 24061, USA
| | - Edward G. Hohenstein
- SLAC National Accelerator Laboratory, Stanford
PULSE Institute, Menlo Park, California 94025,
USA
| | | | - Bernard R. Brooks
- National Institutes of Health – National Heart,
Lung and Blood Institute, Laboratory of Computational Biology, Bethesda,
Maryland 20892, USA
| | - Henry F. Schaefer
- Center for Computational Quantum Chemistry,
University of Georgia, Athens, Georgia 30602, USA
| | - Alexander Yu. Sokolov
- Department of Chemistry and Biochemistry, The
Ohio State University, Columbus, Ohio 43210, USA
| | - Konrad Patkowski
- Department of Chemistry and Biochemistry, Auburn
University, Auburn, Alabama 36849, USA
| | - A. Eugene DePrince
- Department of Chemistry and Biochemistry,
Florida State University, Tallahassee, Florida 32306-4390,
USA
| | - Uğur Bozkaya
- Department of Chemistry, Hacettepe
University, Ankara 06800, Turkey
| | - Rollin A. King
- Department of Chemistry, Bethel
University, St. Paul, Minnesota 55112, USA
| | | | - Justin M. Turney
- Center for Computational Quantum Chemistry,
University of Georgia, Athens, Georgia 30602, USA
| | | | - C. David Sherrill
- Center for Computational Molecular Science and
Technology, School of Chemistry and Biochemistry, School of Computational Science and
Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0400,
USA
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5
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Souissi H, Mejrissi L, Habli H, Alsahhaf M, Oujia B, Xavier Gadéa EF. Ab initio diabatic and adiabatic calculations for francium hydride FrH. NEW J CHEM 2020. [DOI: 10.1039/c9nj06391a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Abstract
Explicit ab initio diabatic and adiabatic calculations of potential energy curves (PECs) of the states 1,3Σ+, 1,3Π, and 1,3Δ of francium hydride FrH have been carried out with several approaches.
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Affiliation(s)
- Hanen Souissi
- Laboratoire de Physique Quantique
- Faculté des Sciences de Monastir
- Université of Monastir
- Monastir
- Tunisia
| | - Leila Mejrissi
- Laboratoire de Physique Quantique
- Faculté des Sciences de Monastir
- Université of Monastir
- Monastir
- Tunisia
| | - Hela Habli
- Laboratoire de Physique Quantique
- Faculté des Sciences de Monastir
- Université of Monastir
- Monastir
- Tunisia
| | - Maarib Alsahhaf
- Physics Department
- Faculty of Science
- Princess Nourah Bint Abdulrahman University
- Riyadh
- Kingdom of Saudi Arabia
| | - Brahim Oujia
- University of Jeddah
- Faculty of Science
- Physics Department
- Jeddah
- Kingdom of Saudi Arabia
| | - et Florent Xavier Gadéa
- Laboratoire de Chimie et Physique Quantique
- UMR5626 du CNRS
- Université de Toulouse
- UPS
- Toulouse Cedex 4
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6
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Kirsch T, Engel F, Gauss J. Analytic evaluation of first-order properties within the mean-field variant of spin-free exact two-component theory. J Chem Phys 2019; 150:204115. [DOI: 10.1063/1.5095698] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
Affiliation(s)
- Till Kirsch
- Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany
| | - Franziska Engel
- Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany
| | - Jürgen Gauss
- Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany
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7
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Koseki S, Matsunaga N, Asada T, Schmidt MW, Gordon MS. Spin–Orbit Coupling Constants in Atoms and Ions of Transition Elements: Comparison of Effective Core Potentials, Model Core Potentials, and All-Electron Methods. J Phys Chem A 2019; 123:2325-2339. [DOI: 10.1021/acs.jpca.8b09218] [Citation(s) in RCA: 29] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Shiro Koseki
- Department of Chemistry, Graduate School of Science, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan
- The Research Institute for Molecular Electronic Devices (RIMED), Osaka Prefecture University, 1-1 Gakuen-cho,
Naka-ku, Sakai 599-8531, Japan
| | - Nikita Matsunaga
- Department of Chemistry & Biochemistry, Long Island University, Brooklyn, New York 11201, United States
| | - Toshio Asada
- Department of Chemistry, Graduate School of Science, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan
- The Research Institute for Molecular Electronic Devices (RIMED), Osaka Prefecture University, 1-1 Gakuen-cho,
Naka-ku, Sakai 599-8531, Japan
| | - Michael W. Schmidt
- Department of Chemistry, Iowa State University and Ames Laboratory-USDOE, Ames, Iowa 50011, United States
| | - Mark S. Gordon
- Department of Chemistry, Iowa State University and Ames Laboratory-USDOE, Ames, Iowa 50011, United States
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8
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Franzke YJ, Middendorf N, Weigend F. Efficient implementation of one- and two-component analytical energy gradients in exact two-component theory. J Chem Phys 2018; 148:104110. [DOI: 10.1063/1.5022153] [Citation(s) in RCA: 41] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yannick J. Franzke
- Institute of Physical Chemistry, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany
| | - Nils Middendorf
- Institute of Physical Chemistry, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany
| | - Florian Weigend
- Institute of Physical Chemistry, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany
- Institute of Nanotechnology, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
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9
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Parrish RM, Burns LA, Smith DGA, Simmonett AC, DePrince AE, Hohenstein EG, Bozkaya U, Sokolov AY, Di Remigio R, Richard RM, Gonthier JF, James AM, McAlexander HR, Kumar A, Saitow M, Wang X, Pritchard BP, Verma P, Schaefer HF, Patkowski K, King RA, Valeev EF, Evangelista FA, Turney JM, Crawford TD, Sherrill CD. Psi4 1.1: An Open-Source Electronic Structure Program Emphasizing Automation, Advanced Libraries, and Interoperability. J Chem Theory Comput 2017; 13:3185-3197. [PMID: 28489372 PMCID: PMC7495355 DOI: 10.1021/acs.jctc.7b00174] [Citation(s) in RCA: 782] [Impact Index Per Article: 111.7] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Psi4 is an ab initio electronic structure program providing methods such as Hartree-Fock, density functional theory, configuration interaction, and coupled-cluster theory. The 1.1 release represents a major update meant to automate complex tasks, such as geometry optimization using complete-basis-set extrapolation or focal-point methods. Conversion of the top-level code to a Python module means that Psi4 can now be used in complex workflows alongside other Python tools. Several new features have been added with the aid of libraries providing easy access to techniques such as density fitting, Cholesky decomposition, and Laplace denominators. The build system has been completely rewritten to simplify interoperability with independent, reusable software components for quantum chemistry. Finally, a wide range of new theoretical methods and analyses have been added to the code base, including functional-group and open-shell symmetry adapted perturbation theory, density-fitted coupled cluster with frozen natural orbitals, orbital-optimized perturbation and coupled-cluster methods (e.g., OO-MP2 and OO-LCCD), density-fitted multiconfigurational self-consistent field, density cumulant functional theory, algebraic-diagrammatic construction excited states, improvements to the geometry optimizer, and the "X2C" approach to relativistic corrections, among many other improvements.
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Affiliation(s)
- Robert M Parrish
- Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology , Atlanta, Georgia 30332-0400, United States
| | - Lori A Burns
- Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology , Atlanta, Georgia 30332-0400, United States
| | - Daniel G A Smith
- Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology , Atlanta, Georgia 30332-0400, United States
| | - Andrew C Simmonett
- National Institutes of Health , National Heart, Lung and Blood Institute, Laboratory of Computational Biology, 5635 Fishers Lane, T-900 Suite, Rockville, Maryland 20852, United States
| | - A Eugene DePrince
- Department of Chemistry and Biochemistry, Florida State University , Tallahassee, Florida 32306-4390, United States
| | - Edward G Hohenstein
- Department of Chemistry and Biochemistry, The City College of New York , New York, New York 10031, United States
| | - Uğur Bozkaya
- Department of Chemistry, Hacettepe University , Ankara 06800, Turkey
| | - Alexander Yu Sokolov
- Division of Chemistry and Chemical Engineering, California Institute of Technology , Pasadena, California 91125, United States
| | - Roberto Di Remigio
- Department of Chemistry, Centre for Theoretical and Computational Chemistry, UiT, The Arctic University of Norway , N-9037 Tromsø, Norway
| | - Ryan M Richard
- Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology , Atlanta, Georgia 30332-0400, United States
| | - Jérôme F Gonthier
- Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology , Atlanta, Georgia 30332-0400, United States
| | - Andrew M James
- Department of Chemistry, Virginia Tech , Blacksburg, Virginia 24061, United States
| | - Harley R McAlexander
- Department of Chemistry, Virginia Tech , Blacksburg, Virginia 24061, United States
| | - Ashutosh Kumar
- Department of Chemistry, Virginia Tech , Blacksburg, Virginia 24061, United States
| | - Masaaki Saitow
- Department of Chemistry and Research Center for Smart Molecules, Rikkyo University , 3-34-1 Nishi-ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
| | - Xiao Wang
- Department of Chemistry, Virginia Tech , Blacksburg, Virginia 24061, United States
| | - Benjamin P Pritchard
- Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology , Atlanta, Georgia 30332-0400, United States
| | - Prakash Verma
- Department of Chemistry, Emory University , Atlanta, Georgia 30322, United States
| | - Henry F Schaefer
- Center for Computational Quantum Chemistry, University of Georgia , Athens, Georgia 30602, United States
| | - Konrad Patkowski
- Department of Chemistry and Biochemistry, Auburn University , Auburn, Alabama 36849, United States
| | - Rollin A King
- Department of Chemistry, Bethel University , St. Paul, Minnesota 55112, United States
| | - Edward F Valeev
- Department of Chemistry, Virginia Tech , Blacksburg, Virginia 24061, United States
| | | | - Justin M Turney
- Center for Computational Quantum Chemistry, University of Georgia , Athens, Georgia 30602, United States
| | - T Daniel Crawford
- Department of Chemistry, Virginia Tech , Blacksburg, Virginia 24061, United States
| | - C David Sherrill
- Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, School of Computational Science and Engineering, Georgia Institute of Technology , Atlanta, Georgia 30332-0400, United States
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10
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Bučinský L, Jayatilaka D, Grabowsky S. Importance of Relativistic Effects and Electron Correlation in Structure Factors and Electron Density of Diphenyl Mercury and Triphenyl Bismuth. J Phys Chem A 2016; 120:6650-69. [PMID: 27434184 DOI: 10.1021/acs.jpca.6b05769] [Citation(s) in RCA: 48] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
This study investigates the possibility of detecting relativistic effects and electron correlation in single-crystal X-ray diffraction experiments using the examples of diphenyl mercury (HgPh2) and triphenyl bismuth (BiPh3). In detail, the importance of electron correlation (ECORR), relativistic effects (REL) [distinguishing between total, scalar and spin-orbit (SO) coupling relativistic effects] and picture change error (PCE) on the theoretical electron density, its topology and its Laplacian using infinite order two component (IOTC) wave functions is discussed. This is to develop an understanding of the order of magnitude and shape of these different effects as they manifest in the electron density. Subsequently, the same effects are considered for the theoretical structure factors. It becomes clear that SO and PCE are negligible, but ECORR and scalar REL are important in low- and medium-order reflections on absolute and relative scales-not in the high-order region. As a further step, Hirshfeld atom refinement (HAR) and subsequent X-ray constrained wavefunction (XCW) fitting have been performed for the compound HgPh2 with various relativistic and nonrelativistic wave functions against the experimental structure factors. IOTC calculations of theoretical structure factors and relativistic HAR as well as relativistic XCW fitting are presented for the first time, accounting for both scalar and spin-orbit relativistic effects.
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Affiliation(s)
- Lukáš Bučinský
- Institute of Physical Chemistry and Chemical Physics FCHPT, Slovak University of Technology , Radlinskeho 9, Bratislava SK-812 37, Slovakia
| | - Dylan Jayatilaka
- School of Chemistry and Biochemistry, The University of Western Australia , 35 Stirling Highway, Perth WA 6009, Australia
| | - Simon Grabowsky
- Fachbereich 2 - Biologie/Chemie, Universität Bremen , Leobener Straβe NW2, 28359 Bremen, Germany
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11
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Insights into the value of statistical models and relativistic effects for the investigation of halogenated derivatives of fluorescent probes. Theor Chem Acc 2016. [DOI: 10.1007/s00214-016-1862-4] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
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12
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Cheng L, Gauss J. Perturbative treatment of spin-orbit coupling within spin-free exact two-component theory. J Chem Phys 2014; 141:164107. [DOI: 10.1063/1.4897254] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Lan Cheng
- Institute for Theoretical Chemistry, Department of Chemistry
and Biochemistry, The University of Texas at Austin, Austin,
Texas 78712, USA
| | - Jürgen Gauss
- Institut für Physikalische Chemie, Universität Mainz, D-55099 Mainz, Germany
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13
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Autschbach J. Relativistic calculations of magnetic resonance parameters: background and some recent developments. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2014; 372:20120489. [PMID: 24516182 DOI: 10.1098/rsta.2012.0489] [Citation(s) in RCA: 51] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
This article outlines some basic concepts of relativistic quantum chemistry and recent developments of relativistic methods for the calculation of the molecular properties that define the basic parameters of magnetic resonance spectroscopic techniques, i.e. nuclear magnetic resonance shielding, indirect nuclear spin-spin coupling and electric field gradients (nuclear quadrupole coupling), as well as with electron paramagnetic resonance g-factors and electron-nucleus hyperfine coupling. Density functional theory (DFT) has been very successful in molecular property calculations, despite a number of problems related to approximations in the functionals. In particular, for heavy-element systems, the large electron count and the need for a relativistic treatment often render the application of correlated wave function ab initio methods impracticable. Selected applications of DFT in relativistic calculation of magnetic resonance parameters are reviewed.
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Affiliation(s)
- Jochen Autschbach
- Department of Chemistry, State University of New York at Buffalo, , Buffalo, NY 14260-3000, USA
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14
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Aucar GA. Toward a QFT-based theory of atomic and molecular properties. Phys Chem Chem Phys 2014; 16:4420-38. [DOI: 10.1039/c3cp52685b] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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15
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Peng D, Middendorf N, Weigend F, Reiher M. An efficient implementation of two-component relativistic exact-decoupling methods for large molecules. J Chem Phys 2013; 138:184105. [PMID: 23676027 DOI: 10.1063/1.4803693] [Citation(s) in RCA: 141] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
Abstract
We present an efficient algorithm for one- and two-component relativistic exact-decoupling calculations. Spin-orbit coupling is thus taken into account for the evaluation of relativistically transformed (one-electron) Hamiltonian. As the relativistic decoupling transformation has to be evaluated with primitive functions, the construction of the relativistic one-electron Hamiltonian becomes the bottleneck of the whole calculation for large molecules. For the established exact-decoupling protocols, a minimal matrix operation count is established and discussed in detail. Furthermore, we apply our recently developed local DLU scheme [D. Peng and M. Reiher, J. Chem. Phys. 136, 244108 (2012)] to accelerate this step. With our new implementation two-component relativistic density functional calculations can be performed invoking the resolution-of-identity density-fitting approximation and (Abelian as well as non-Abelian) point group symmetry to accelerate both the exact-decoupling and the two-electron part. The capability of our implementation is illustrated at the example of silver clusters with up to 309 atoms, for which the cohesive energy is calculated and extrapolated to the bulk.
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Affiliation(s)
- Daoling Peng
- ETH Zurich, Laboratorium für Physikalische Chemie, Wolfgang-Pauli-Str. 10, CH-8093 Zurich, Switzerland
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16
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Kelley MS, Shiozaki T. Large-scale Dirac–Fock–Breit method using density fitting and 2-spinor basis functions. J Chem Phys 2013; 138:204113. [DOI: 10.1063/1.4807612] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/15/2023] Open
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17
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Autschbach J, Peng D, Reiher M. Two-Component Relativistic Calculations of Electric-Field Gradients Using Exact Decoupling Methods: Spin–orbit and Picture-Change Effects. J Chem Theory Comput 2012; 8:4239-48. [DOI: 10.1021/ct300623j] [Citation(s) in RCA: 53] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023]
Affiliation(s)
- Jochen Autschbach
- Department of Chemistry, University
at Buffalo, State University of New York, Buffalo, New York 14260-3000, United States
| | - Daoling Peng
- ETH Zürich, Laboratorium für Physikalische Chemie, Wolfgang-Pauli-Strasse
10, CH-8093 Zürich, Switzerland
| | - Markus Reiher
- ETH Zürich, Laboratorium für Physikalische Chemie, Wolfgang-Pauli-Strasse
10, CH-8093 Zürich, Switzerland
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18
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Ishikawa A, Nakashima H, Nakatsuji H. Accurate solutions of the Schrödinger and Dirac equations of , HD+, and HT+: With and without Born–Oppenheimer approximation and under magnetic field. Chem Phys 2012. [DOI: 10.1016/j.chemphys.2011.09.013] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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19
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20
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Theoretical modelling of the adsorption of thallium and element 113 atoms on gold using two-component density functional methods with effective core potentials. Chem Phys 2012. [DOI: 10.1016/j.chemphys.2011.04.029] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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21
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22
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23
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Cheng L, Gauss J. Analytic second derivatives for the spin-free exact two-component theory. J Chem Phys 2011; 135:244104. [DOI: 10.1063/1.3667202] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
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24
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Szalay PG, Müller T, Gidofalvi G, Lischka H, Shepard R. Multiconfiguration Self-Consistent Field and Multireference Configuration Interaction Methods and Applications. Chem Rev 2011; 112:108-81. [DOI: 10.1021/cr200137a] [Citation(s) in RCA: 470] [Impact Index Per Article: 36.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Péter G. Szalay
- Laboratory for Theoretical Chemistry, Institute of Chemistry, Eötvös Loránd University, P. O. Box 32, H-1518 Budapest, Hungary
| | - Thomas Müller
- Jülich Supercomputer Centre, Institute of Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany
| | - Gergely Gidofalvi
- Department of Chemistry and Biochemistry, Gonzaga University, 502 East Boone Avenue, Spokane, Washington 99258-0102, United States
| | - Hans Lischka
- Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061, United States
- Institute for Theoretical Chemistry, University of Vienna, Waehringerstrasse 17, A-1090 Vienna, Austria
| | - Ron Shepard
- Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, United States
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25
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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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26
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Cheng L, Gauss J. Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian. J Chem Phys 2011; 135:084114. [DOI: 10.1063/1.3624397] [Citation(s) in RCA: 124] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
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27
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Reiher M. Relativistic Douglas–Kroll–Hess theory. WILEY INTERDISCIPLINARY REVIEWS-COMPUTATIONAL MOLECULAR SCIENCE 2011. [DOI: 10.1002/wcms.67] [Citation(s) in RCA: 97] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/01/2023]
Affiliation(s)
- Markus Reiher
- Laboratorium für Physikalische Chemie, ETH Zurich, Zurich, Switzerland
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28
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Mizukami W, Nakajima T, Hirao K, Yanai T. A dual-level approach to four-component relativistic density-functional theory. Chem Phys Lett 2011. [DOI: 10.1016/j.cplett.2011.04.031] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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29
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Comparison of restricted, unrestricted, inverse, and dual kinetic balances for four-component relativistic calculations. Theor Chem Acc 2011. [DOI: 10.1007/s00214-010-0876-6] [Citation(s) in RCA: 38] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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30
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Affiliation(s)
- Wenjian Liu
- a Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, and Center for Computational Science and Engineering , Peking University , Beijing 100871, People's Republic of China
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31
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Four-Component Electronic Structure Methods. CHALLENGES AND ADVANCES IN COMPUTATIONAL CHEMISTRY AND PHYSICS 2010. [DOI: 10.1007/978-1-4020-9975-5_7] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
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32
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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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33
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Affiliation(s)
- MICHAEL FILATOV
- a Department of Theoretical Chemistry , Göteborg University , Reutersgatan 2, S-41320 , Göteborg , Sweden
| | - DIETER CREMER
- a Department of Theoretical Chemistry , Göteborg University , Reutersgatan 2, S-41320 , Göteborg , Sweden
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34
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Sikkema J, Visscher L, Saue T, Iliaš M. The molecular mean-field approach for correlated relativistic calculations. J Chem Phys 2009; 131:124116. [DOI: 10.1063/1.3239505] [Citation(s) in RCA: 160] [Impact Index Per Article: 10.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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36
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van Wüllen C, Klopper W, Mukherjee D. Electron correlation, molecular properties and relativity – A tribute to Werner Kutzelnigg. Chem Phys 2009. [DOI: 10.1016/j.chemphys.2009.01.011] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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37
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38
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Peng D, Hirao K. An arbitrary order Douglas–Kroll method with polynomial cost. J Chem Phys 2009; 130:044102. [DOI: 10.1063/1.3068310] [Citation(s) in RCA: 59] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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39
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de Macedo LGM, Angelotti WFD, Sambrano JR, de Souza AR. Fully relativistic prolapse-free Gaussian basis sets: the actinides and 81Tl-88Ra. J Chem Phys 2008; 129:106101. [PMID: 19044942 DOI: 10.1063/1.2976155] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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40
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Matveev AV, Rösch N. Atomic approximation to the projection on electronic states in the Douglas-Kroll-Hess approach to the relativistic Kohn-Sham method. J Chem Phys 2008; 128:244102. [DOI: 10.1063/1.2940352] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.3] [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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Matrix formulation of direct perturbation theory of relativistic effects in a kinetically balanced basis. Chem Phys 2008. [DOI: 10.1016/j.chemphys.2008.01.056] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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42
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Ishikawa A, Nakashima H, Nakatsuji H. Solving the Schrödinger and Dirac equations of hydrogen molecular ion accurately by the free iterative complement interaction method. J Chem Phys 2008; 128:124103. [DOI: 10.1063/1.2842068] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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43
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Peng D, Liu W, Xiao Y, Cheng L. Making four- and two-component relativistic density functional methods fully equivalent based on the idea of “from atoms to molecule”. J Chem Phys 2007; 127:104106. [PMID: 17867736 DOI: 10.1063/1.2772856] [Citation(s) in RCA: 174] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
It is shown that four- and two-component relativistic Kohn-Sham methods of density functional theory can be made fully equivalent in all the aspects of simplicity, accuracy, and efficiency. In particular, this has been achieved based solely on physical arguments rather than on mathematical tricks. The central idea can be visualized as "from atoms to molecule," reflecting that the atomic information is employed to "synthesize" the molecular no-pair relativistic Hamiltonian. That is, the molecular relativistic Hamiltonian can, without loss of accuracy, be projected onto the positive energy states of the isolated Dirac atoms with the projector approximated simply by the superposition of the atomic ones. The dimension of the four-component Hamiltonian matrix then becomes the same as that of a two-component one. Another essential ingredient is to formulate quasirelativistic theory on matrix form rather than on operator form. The resultant quasi-four-component, normalized elimination of the small component, and symmetrized elimination of the small component approaches are critically examined by taking the molecules of MH and M(2) (M=At, E117) as examples.
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Affiliation(s)
- Daoling Peng
- Beijing National Laboratory for Molecular Sciences, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, People's Republic of China
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44
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Sadlej AJ. Explaining the derivation and functioning of the exact infinite-order two-component Dirac theory in terms of the Hückel model. Struct Chem 2007. [DOI: 10.1007/s11224-007-9228-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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45
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Ke¸dziera D, Barysz M. Non-iterative approach to the infinite-order two-component (IOTC) relativistic theory and the non-symmetric algebraic Riccati equation. Chem Phys Lett 2007. [DOI: 10.1016/j.cplett.2007.08.006] [Citation(s) in RCA: 54] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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46
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Baranowska A, Siedlecka M, Sadlej AJ. Reduced-size polarized basis sets for calculations of molecular electric properties. IV. First-row transition metals. Theor Chem Acc 2007. [DOI: 10.1007/s00214-007-0379-2] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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47
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Barysz M, Leszczyński J. Relativistic two-component infinite order method for atomic core ionization potentials. J Chem Phys 2007; 126:154106. [PMID: 17461613 DOI: 10.1063/1.2711194] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this paper the authors have applied the infinite-order two-component method (IOTC) to compute the valence and inner shell ionization potentials for the Ne, Ar, Kr, and Xe elements. The obtained results show the very good performance of the recently defined relativistic IOTC method. They also confirm the importance of the relativistic effects in the determination of the inner shell ionization potentials.
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Affiliation(s)
- Maria Barysz
- Department of Quantum Chemistry, Nicolaus Copernicus University, Gagarina 7, PL-87 100 Toruń, Poland.
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48
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van Wüllen C, Langermann N. Gradients for two-component quasirelativistic methods. Application to dihalogenides of element 116. J Chem Phys 2007; 126:114106. [PMID: 17381195 DOI: 10.1063/1.2711197] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The authors report the implementation of geometry gradients for quasirelativistic two-component Hartree-Fock and density functional methods using either the zero-order regular approximation Hamiltonian or spin-dependent effective core potentials. The computational effort of the resulting program is comparable to that of corresponding nonrelativistic calculations, as it is dominated by the evaluation of derivative two-electron integrals, which is the same for both types of calculations. Besides the implementation of derivatives of matrix elements of the one-particle Hamiltonian with respect to nuclear displacements, the calculation of the derivative exchange-correlation energy for the open shell case involves complicated expressions because of the noncollinear approach chosen to define the spin density. A pilot application to dihalogenides of element 116 shows how spin-orbit coupling strongly affects the chemistry of the superheavy p-block elements. While these molecules are bent at a scalar-relativistic level, spin-orbit coupling is so strong that only the 7p3/2 atomic orbitals of element 116 are involved in bonding, which favors linear molecular geometries for dihalogenides with heavy terminal halogen atoms.
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Affiliation(s)
- Christoph van Wüllen
- Institut für Chemie Sekretariat C3, Technische Universität Berlin, Strasse des 17, Juni 135, D-10623 Berlin, Germany
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49
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Abstract
The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to the matrix representation of a quasirelativistic Hamiltonian that has the same electronic eigenstates as the original Dirac matrix (but no positronic eigenstates). This transformation involves a matrix X, for which an exact identity is derived and which can be constructed either in a noniterative way or by various iteration schemes, not requiring an expansion parameter. Both linearly convergent and quadratically convergent iteration schemes are discussed and compared numerically. The authors present three rather different schemes, for each of which even in unfavorable cases convergence is reached within three or four iterations, for all electronic eigenstates of the Dirac operator. The authors present the theory both in terms of a non-Hermitian and a Hermitian quasirelativistic Hamiltonian. Quasirelativistic approaches at the matrix level known from the literature are critically analyzed in the frame of the general theory.
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Affiliation(s)
- Wenjian Liu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, Peking University, Beijing 100871, People's Republic of China
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50
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Ilias M, Saue T. An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation. J Chem Phys 2007; 126:064102. [PMID: 17313208 DOI: 10.1063/1.2436882] [Citation(s) in RCA: 379] [Impact Index Per Article: 22.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Abstract
The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.
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Affiliation(s)
- Miroslav Ilias
- Laboratoire de Chimie Quantique, Institut de Chimie de Strasbourg, LC3-UMR7177 CNRS/Université Louis Pasteur, 4 Rue Blaise Pascal, F-67000 Strasbourg, France.
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