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Inoue N, Watanabe Y, Nakano H. Generalized Foldy-Wouthuysen transformation for relativistic two-component methods: Systematic analysis of two-component Hamiltonians. J Comput Chem 2024; 45:523-535. [PMID: 37997192 DOI: 10.1002/jcc.27251] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2023] [Revised: 10/24/2023] [Accepted: 10/26/2023] [Indexed: 11/25/2023]
Abstract
The generalized Foldy-Wouthuysen (GFW) transformation was proposed as a generic form that unifies four types of transformations in relativistic two-component methods: unnormalized GFW(UN), and normalized form 1, form 2, and form 3 (GFW(N1), GFW(N2), and GFW(N3)). The GFW transformation covers a wide range of transformations beyond the simple unitary transformation of the Dirac Hamiltonian, allowing for the systematic classification of all existing two-component methods. New two-component methods were also systematically derived based on the GFW transformation. These various two-component methods were applied to hydrogen-like and helium-like ions. Numerical errors in energy were evaluated and classified into four types: the one-electron Hamiltonian approximation, the two-electron operator approximation, the newly defined "picture difference error (PDE)," and the error in determining the transformation, and errors in multi-electron systems were discussed based on this classification.
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Affiliation(s)
- Nobuki Inoue
- Department of Chemistry, Graduate School of Science, Kyushu University, Fukuoka, Japan
| | - Yoshihiro Watanabe
- Department of Chemistry, Graduate School of Science, Kyushu University, Fukuoka, Japan
| | - Haruyuki Nakano
- Department of Chemistry, Graduate School of Science, Kyushu University, Fukuoka, Japan
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2
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Keller L, Blum V, Rinke P, Golze D. Relativistic correction scheme for core-level binding energies from GW. J Chem Phys 2020; 153:114110. [DOI: 10.1063/5.0018231] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Levi Keller
- Department of Applied Physics, Aalto University, Otakaari 1, FI-02150 Espoo, Finland
| | - Volker Blum
- Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708,
USA
| | - Patrick Rinke
- Department of Applied Physics, Aalto University, Otakaari 1, FI-02150 Espoo, Finland
| | - Dorothea Golze
- Department of Applied Physics, Aalto University, Otakaari 1, FI-02150 Espoo, Finland
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Stopkowicz S, Gauss J. A one-electron variant of direct perturbation theory for the treatment of scalar-relativistic effects. Mol Phys 2018. [DOI: 10.1080/00268976.2018.1536812] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
Affiliation(s)
- Stella Stopkowicz
- Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, Mainz, Germany
| | - Jürgen Gauss
- Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, Mainz, Germany
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4
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Cheng L, Wang F, Stanton JF, Gauss J. Perturbative treatment of spin-orbit-coupling within spin-free exact two-component theory using equation-of-motion coupled-cluster methods. J Chem Phys 2018; 148:044108. [DOI: 10.1063/1.5012041] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022] Open
Affiliation(s)
- Lan Cheng
- Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Fan Wang
- Institute of Atomic and Molecular Physics, Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Sichuan University, Chengdu, China
| | - John F. Stanton
- Department of Chemistry, University of Florida, Gainesville, Florida 32611, USA
| | - Jürgen Gauss
- Institut für Physikalische Chemie, Universität Mainz, D-55099 Mainz, Germany
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Li Z, Xiao Y, Liu W. On the spin separation of algebraic two-component relativistic Hamiltonians: Molecular properties. J Chem Phys 2014; 141:054111. [DOI: 10.1063/1.4891567] [Citation(s) in RCA: 52] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Affiliation(s)
- Zhendong Li
- 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
| | - Yunlong Xiao
- 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
| | - Wenjian Liu
- 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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Stopkowicz S, Cheng L, Harding ME, Puzzarini C, Gauss J. The bromine nuclear quadrupole moment revisited. Mol Phys 2013. [DOI: 10.1080/00268976.2013.796072] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Stella Stopkowicz
- a Institut für Physikalische Chemie , Universität Mainz , Mainz , D-55099 , Germany
- b Centre for Theoretical and Computational Chemistry, Department of Chemistry , University of Oslo , Oslo , N-0315 , Norway
| | - Lan Cheng
- a Institut für Physikalische Chemie , Universität Mainz , Mainz , D-55099 , Germany
- c Institute for Theoretical Chemistry, Department of Chemistry and Biochemistry , The University of Texas at Austin , Austin , TX , 78712 , USA
| | - Michael E. Harding
- a Institut für Physikalische Chemie , Universität Mainz , Mainz , D-55099 , Germany
- d Institut für Nanotechnologie, Karlsruher Institut für Technologie (KIT) , Karlsruhe , D-76021 , Germany
| | - Cristina Puzzarini
- e Dipartimento di Chimica ‘Giacomo Ciamician’ , Università di Bologna , Via Selmi 2, Bologna , I-40126 , Italy
| | - Jürgen Gauss
- a Institut für Physikalische Chemie , Universität Mainz , Mainz , D-55099 , Germany
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Cheng L, Stopkowicz S, Stanton JF, Gauss J. The route to high accuracy inab initiocalculations of Cu quadrupole-coupling constants. J Chem Phys 2012; 137:224302. [DOI: 10.1063/1.4767767] [Citation(s) in RCA: 16] [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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Puzzarini C, Cazzoli G, López JC, Alonso JL, Baldacci A, Baldan A, Stopkowicz S, Cheng L, Gauss J. Rotational spectra of rare isotopic species of fluoroiodomethane: determination of the equilibrium structure from rotational spectroscopy and quantum-chemical calculations. J Chem Phys 2012; 137:024310. [PMID: 22803539 DOI: 10.1063/1.4731284] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
Abstract
Supported by accurate quantum-chemical calculations, the rotational spectra of the mono- and bi-deuterated species of fluoroiodomethane, CHDFI and CD(2)FI, as well as of the (13)C-containing species, (13)CH(2)FI, were recorded for the first time. Three different spectrometers were employed, a Fourier-transform microwave spectrometer, a millimeter/submillimter-wave spectrometer, and a THz spectrometer, thus allowing to record a huge portion of the rotational spectrum, from 5 GHz up to 1.05 THz, and to accurately determine the ground-state rotational and centrifugal-distortion constants. Sub-Doppler measurements allowed to resolve the hyperfine structure of the rotational spectrum and to determine the complete iodine quadrupole-coupling tensor as well as the diagonal elements of the iodine spin-rotation tensor. The present investigation of rare isotopic species of CH(2)FI together with the results previously obtained for the main isotopologue [C. Puzzarini, G. Cazzoli, J. C. López, J. L. Alonso, A. Baldacci, A. Baldan, S. Stopkowicz, L. Cheng, and J. Gauss, J. Chem. Phys. 134, 174312 (2011); G. Cazzoli, A. Baldacci, A. Baldan, and C. Puzzarini, Mol. Phys. 109, 2245 (2011)] enabled us to derive a semi-experimental equilibrium structure for fluoroiodomethane by means of a least-squares fit procedure using the available experimental ground-state rotational constants together with computed vibrational corrections. Problems related to the missing isotopic substitution of fluorine and iodine were overcome thanks to the availability of an accurate theoretical equilibrium geometry (computed at the coupled-cluster singles and doubles level augmented by a perturbative treatment of triple excitations).
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Affiliation(s)
- Cristina Puzzarini
- Dipartimento di Chimica G. Ciamician, Università di Bologna, Via Selmi 2, I-40126 Bologna, Italy.
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9
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Li Z, Xiao Y, Liu W. On the spin separation of algebraic two-component relativistic Hamiltonians. J Chem Phys 2012; 137:154114. [DOI: 10.1063/1.4758987] [Citation(s) in RCA: 104] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
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10
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Li Z, Shao S, Liu W. Relativistic explicit correlation: Coalescence conditions and practical suggestions. J Chem Phys 2012; 136:144117. [DOI: 10.1063/1.3702631] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
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11
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Saue T. Relativistic Hamiltonians for chemistry: a primer. Chemphyschem 2011; 12:3077-94. [PMID: 22076930 DOI: 10.1002/cphc.201100682] [Citation(s) in RCA: 306] [Impact Index Per Article: 23.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2011] [Indexed: 11/06/2022]
Affiliation(s)
- Trond Saue
- Laboratoire de Chimie et Physique Quantique (UMR 5626), CNRS/Université de Toulouse 3 (Paul Sabatier), 118 route de Narbonne, 31062 Toulouse, France.
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12
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Stopkowicz S, Gauss J. Fourth-order relativistic corrections to electrical first-order properties using direct perturbation theory. J Chem Phys 2011; 134:204106. [DOI: 10.1063/1.3587633] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
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13
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Puzzarini C, Cazzoli G, López JC, Alonso JL, Baldacci A, Baldan A, Stopkowicz S, Cheng L, Gauss J. Spectroscopic investigation of fluoroiodomethane, CH2FI: Fourier-transform microwave and millimeter-/submillimeter-wave spectroscopy and quantum-chemical calculations. J Chem Phys 2011; 134:174312. [DOI: 10.1063/1.3583498] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023] Open
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14
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Stopkowicz S, Gauss J. Direct perturbation theory in terms of energy derivatives: Fourth-order relativistic corrections at the Hartree–Fock level. J Chem Phys 2011; 134:064114. [DOI: 10.1063/1.3522766] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Rezabal E, Gauss J, Matxain JM, Berger R, Diefenbach M, Holthausen MC. Quantum chemical assessment of the binding energy of CuO+. J Chem Phys 2011; 134:064304. [DOI: 10.1063/1.3537797] [Citation(s) in RCA: 35] [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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Bučinský L, Biskupič S, Jayatilaka D. Picture change error correction of radon atom electron density. J Chem Phys 2010; 133:174125. [DOI: 10.1063/1.3489351] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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17
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Puzzarini C, Stanton JF, Gauss J. Quantum-chemical calculation of spectroscopic parameters for rotational spectroscopy. INT REV PHYS CHEM 2010. [DOI: 10.1080/01442351003643401] [Citation(s) in RCA: 188] [Impact Index Per Article: 13.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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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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19
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Kutzelnigg W, Liu W. Relativistic theory of nuclear magnetic resonance parameters in a Gaussian basis representation. J Chem Phys 2009; 131:044129. [DOI: 10.1063/1.3185400] [Citation(s) in RCA: 32] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
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20
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Klein K, Gauss J. Perturbative calculation of spin-orbit splittings using the equation-of-motion ionization-potential coupled-cluster ansatz. J Chem Phys 2008; 129:194106. [DOI: 10.1063/1.3013199] [Citation(s) in RCA: 50] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023] Open
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21
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Stopkowicz S, Gauss J. Relativistic corrections to electrical first-order properties using direct perturbation theory. J Chem Phys 2008; 129:164119. [DOI: 10.1063/1.2998300] [Citation(s) in RCA: 40] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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22
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Michauk C, Gauss J. Perturbative treatment of scalar-relativistic effects in coupled-cluster calculations of equilibrium geometries and harmonic vibrational frequencies using analytic second-derivative techniques. J Chem Phys 2007; 127:044106. [PMID: 17672680 DOI: 10.1063/1.2751161] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
An analytic scheme for the computation of scalar-relativistic corrections to nuclear forces is presented. Relativistic corrections are included via a perturbative treatment involving the mass-velocity and the one-electron and two-electron Darwin terms. Such a scheme requires mixed second derivatives of the nonrelativistic energy with respect to the relativistic perturbation and the nuclear coordinates and can be implemented using available second-derivative techniques. Our implementation for Hartree-Fock self-consistent field, second-order Moller-Plesset perturbation theory, as well as the coupled-cluster level is used to investigate the relativistic effects on the geometrical parameters and harmonic vibrational frequencies for a set of molecules containing light elements (HX, X=F, Cl, Br; H2X, X=O, S; HXY, X=O, S and Y=F, Cl, Br). The focus of our calculations is the basis-set dependence of the corresponding relativistic effects, additivity of electron correlation and relativistic effects, and the importance of core correlation on relativistic effects.
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Affiliation(s)
- Christine Michauk
- Institut für Physikalische Chemie, Universität Mainz, D-55099 Mainz, Germany
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23
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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: 357] [Impact Index Per Article: 21.0] [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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Helgaker T, Hennum AC, Klopper W. A second-quantization framework for the unified treatment of relativistic and nonrelativistic molecular perturbations by response theory. J Chem Phys 2006; 125:24102. [PMID: 16848572 DOI: 10.1063/1.2198527] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A formalism is presented for the calculation of relativistic corrections to molecular electronic energies and properties. After a discussion of the Dirac and Breit equations and their first-order Foldy-Wouthuysen [Phys. Rev. 78, 29 (1950)] transformation, we construct a second-quantization electronic Hamiltonian, valid for all values of the fine-structure constant alpha. The resulting alpha-dependent Hamiltonian is then used to set up a perturbation theory in orders of alpha(2), using the general framework of time-independent response theory, in the same manner as for geometrical and magnetic perturbations. Explicit expressions are given to second order in alpha(2) for the Hartree-Fock model. However, since all relativistic considerations are contained in the alpha-dependent Hamiltonian operator rather than in the wave function, the same approach may be used for other wave-function models, following the general procedure of response theory. In particular, by constructing a variational Lagrangian using the alpha-dependent electronic Hamiltonian, relativistic corrections can be calculated for nonvariational methods as well.
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Affiliation(s)
- Trygve Helgaker
- Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.
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Pito[nbreve]ák M, Neogrády P, Kellö V, Urban M. Optimized virtual orbitals for relativistic calculations: an alternative approach to the basis set construction for correlation calculations. Mol Phys 2006. [DOI: 10.1080/00268970600662390] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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