1
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Stein F, Hutter J. Massively parallel implementation of gradients within the random phase approximation: Application to the polymorphs of benzene. J Chem Phys 2024; 160:024120. [PMID: 38214385 DOI: 10.1063/5.0180704] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2023] [Accepted: 12/15/2023] [Indexed: 01/13/2024] Open
Abstract
The Random-Phase approximation (RPA) provides an appealing framework for semi-local density functional theory. In its Resolution-of-the-Identity (RI) approach, it is a very accurate and more cost-effective method than most other wavefunction-based correlation methods. For widespread applications, efficient implementations of nuclear gradients for structure optimizations and data sampling of machine learning approaches are required. We report a well scaling implementation of RI-RPA nuclear gradients on massively parallel computers. The approach is applied to two polymorphs of the benzene crystal obtaining very good cohesive and relative energies. Different correction and extrapolation schemes are investigated for further improvement of the results and estimations of error bars.
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Affiliation(s)
- Frederick Stein
- Center for Advanced Systems Understanding (CASUS), Helmholtz-Zentrum Dresden, Rossendorf (HZDR), Untermarkt 20, 02826 Görlitz, Germany
| | - Jürg Hutter
- Department of Chemistry, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
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2
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Drontschenko V, Graf D, Laqua H, Ochsenfeld C. Efficient Method for the Computation of Frozen-Core Nuclear Gradients within the Random Phase Approximation. J Chem Theory Comput 2022; 18:7359-7372. [PMID: 36331398 DOI: 10.1021/acs.jctc.2c00774] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
A method for the evaluation of analytical frozen-core gradients within the random phase approximation is presented. We outline an efficient way to evaluate the response of the density of active electrons arising only when introducing the frozen-core approximation and constituting the main difficulty, together with the response of the standard Kohn-Sham density. The general framework allows to extend the outlined procedure to related electron correlation methods in the atomic orbital basis that require the evaluation of density responses, such as second-order Møller-Plesset perturbation theory or coupled cluster variants. By using Cholesky decomposed densities─which reintroduce the occupied index in the time-determining steps─we are able to achieve speedups of 20-30% (depending on the size of the basis set) by using the frozen-core approximation, which is of similar magnitude as for molecular orbital formulations. We further show that the errors introduced by the frozen-core approximation are practically insignificant for molecular geometries.
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Affiliation(s)
- Viktoria Drontschenko
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), 81377 Munich, Germany
| | - Daniel Graf
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), 81377 Munich, Germany
| | - Henryk Laqua
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), 81377 Munich, Germany
| | - Christian Ochsenfeld
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), 81377 Munich, Germany.,Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany
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3
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Franzke YJ, Holzer C, Mack F. NMR Coupling Constants Based on the Bethe-Salpeter Equation in the GW Approximation. J Chem Theory Comput 2022; 18:1030-1045. [PMID: 34981925 DOI: 10.1021/acs.jctc.1c00999] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
We present the first steps to extend the Green's function GW method and the Bethe-Salpeter equation (BSE) to molecular response properties such as nuclear magnetic resonance (NMR) indirect spin-spin coupling constants. We discuss both a nonrelativistic one-component and a quasi-relativistic two-component formalism. The latter describes scalar-relativistic and spin-orbit effects and allows us to study heavy-element systems with reasonable accuracy. Efficiency is maintained by the application of the resolution of the identity approximation throughout. The performance is demonstrated using conventional central processing units (CPUs) and modern graphics processing units (GPUs) for molecules involving several thousand basis functions. Our results show that a large amount of Hartree-Fock exchange is vital to provide a sufficient Kohn-Sham starting point to compute the GW quasi-particle energies. As the GW-BSE approach is generally less accurate for triplet excitations or related properties such as the Fermi-contact interaction, the admixture of the Kohn-Sham correlation kernel through the contracted BSE (cBSE) method improves the results for NMR coupling constants. This leads to remarkable results when combined with the eigenvalue-only self-consistent variant (evGW) and Becke's half and half functional (BH&HLYP) or the CAM-QTP family. The developed methodology is used to calculate the Karplus curve of tin molecules, illustrating its applicability to extended chemically relevant molecules. Here, the GW-cBSE method improves upon the chosen BH&HLYP Kohn-Sham starting points.
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Affiliation(s)
- Yannick J Franzke
- Fachbereich Chemie, Philipps-Universität Marburg, 35032 Marburg, Germany
| | - Christof Holzer
- Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
| | - Fabian Mack
- Institute of Physical Chemistry, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany
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4
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Fauser S, Trushin E, Neiss C, Görling A. Chemical accuracy with σ-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues. J Chem Phys 2021; 155:134111. [PMID: 34624971 DOI: 10.1063/5.0059641] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Recently, a new type of orbital-dependent functional for the Kohn-Sham (KS) correlation energy, σ-functionals, was introduced. Technically, σ-functionals are closely related to the well-known direct random phase approximation (dRPA). Within the dRPA, a function of the eigenvalues σ of the frequency-dependent KS response function is integrated over purely imaginary frequencies. In σ-functionals, this function is replaced by one that is optimized with respect to reference sets of atomization, reaction, transition state, and non-covalent interaction energies. The previously introduced σ-functional uses input orbitals and eigenvalues from KS calculations with the generalized gradient approximation (GGA) exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE). Here, σ-functionals using input orbitals and eigenvalues from the meta-GGA TPSS and the hybrid-functionals PBE0 and B3LYP are presented and tested. The number of reference sets taken into account in the optimization of the σ-functionals is larger than in the first PBE based σ-functional and includes sets with 3d-transition metal compounds. Therefore, also a reparameterized PBE based σ-functional is introduced. The σ-functionals based on PBE0 and B3LYP orbitals and eigenvalues reach chemical accuracy for main group chemistry. For the 10 966 reactions from the highly accurate W4-11RE reference set, the B3LYP based σ-functional exhibits a mean average deviation of 1.03 kcal/mol compared to 1.08 kcal/mol for the coupled cluster singles doubles perturbative triples method if the same valence quadruple zeta basis set is used. For 3d-transition metal chemistry, accuracies of about 2 kcal/mol are reached. The computational effort for the post-self-consistent evaluation of the σ-functional is lower than that of a preceding PBE0 or B3LYP calculation for typical systems.
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Affiliation(s)
- Steffen Fauser
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91 058 Erlangen, Germany
| | - Egor Trushin
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91 058 Erlangen, Germany
| | - Christian Neiss
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91 058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91 058 Erlangen, Germany
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5
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Pernal K, Hapka M. Range‐separated multiconfigurational density functional theory methods. WIRES COMPUTATIONAL MOLECULAR SCIENCE 2021. [DOI: 10.1002/wcms.1566] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
| | - Michał Hapka
- Lodz University of Technology Institute of Physics Lodz Poland
- Faculty of Chemistry University of Warsaw Warsaw Poland
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6
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Trushin E, Thierbach A, Görling A. Toward chemical accuracy at low computational cost: Density-functional theory with σ-functionals for the correlation energy. J Chem Phys 2021; 154:014104. [PMID: 33412877 DOI: 10.1063/5.0026849] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We introduce new functionals for the Kohn-Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct random phase approximation (dRPA), σ-functionals require as input exclusively eigenvalues σ of the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-constant and frequency integrations contained in the ACFD theorem. We optimize σ-functionals with the help of reference sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn-Sham calculations employing the exchange-correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chemical accuracy. For the 10 966 reactions of the W4-11RE reference set, the mean absolute deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non-covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self-consistent calculation of the total energy requires less computational time than a density-functional calculation with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort.
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Affiliation(s)
- Egor Trushin
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Adrian Thierbach
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
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7
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Graf D, Ochsenfeld C. A range-separated generalized Kohn-Sham method including a long-range nonlocal random phase approximation correlation potential. J Chem Phys 2020; 153:244118. [PMID: 33380112 DOI: 10.1063/5.0031310] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022] Open
Abstract
Based on our recently published range-separated random phase approximation (RPA) functional [Kreppel et al., "Range-separated density-functional theory in combination with the random phase approximation: An accuracy benchmark," J. Chem. Theory Comput. 16, 2985-2994 (2020)], we introduce self-consistent minimization with respect to the one-particle density matrix. In contrast to the range-separated RPA methods presented so far, the new method includes a long-range nonlocal RPA correlation potential in the orbital optimization process, making it a full-featured variational generalized Kohn-Sham (GKS) method. The new method not only improves upon all other tested RPA schemes including the standard post-GKS range-separated RPA for the investigated test cases covering general main group thermochemistry, kinetics, and noncovalent interactions but also significantly outperforms the popular G0W0 method in estimating the ionization potentials and fundamental gaps considered in this work using the eigenvalue spectra obtained from the GKS Hamiltonian.
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Affiliation(s)
- Daniel Graf
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), D-81377 Munich, Germany
| | - Christian Ochsenfeld
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), D-81377 Munich, Germany
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8
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Rishi V, Perera A, Bartlett RJ. A route to improving RPA excitation energies through its connection to equation-of-motion coupled cluster theory. J Chem Phys 2020; 153:234101. [DOI: 10.1063/5.0023862] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Affiliation(s)
- Varun Rishi
- Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
| | - Ajith Perera
- Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA
| | - Rodney J. Bartlett
- Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA
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9
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Erhard J, Fauser S, Kalaß S, Moerman E, Trushin E, Görling A. Lieb-Oxford bound and pair correlation functions for density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem. Faraday Discuss 2020; 224:79-97. [PMID: 32935700 DOI: 10.1039/d0fd00047g] [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
Compliance with the Lieb-Oxford bound for the indirect Coulomb energy and for the exchange-correlation energy is investigated for a number of density-functional methods based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem to treat correlation. Furthermore, the correlation contribution to the pair density resulting from these methods is compared with highly accurate reference values for the helium atom and for the hydrogen molecule at several bond distances. For molecules, the Lieb-Oxford bound is obeyed by all considered methods. For the homogeneous electron gas, it is violated by all methods for low electron densities. The simplest considered ACFD method, the direct random phase approximation (dRPA), violates the Lieb-Oxford bound much earlier than more advanced ACFD methods that, in addition to the simple Hartree kernel, take into account the exchange kernel and an approximate correlation kernel in the calculation of the correlation energy. While the dRPA yields quite poor correlation contributions to the pair density, those from more advanced ACFD methods are physically reasonable but still leave room for improvements, particularly in the case of the stretched hydrogen molecule.
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Affiliation(s)
- Jannis Erhard
- Lehrstuhl für Theoretische Chemie, Egerlandstraße 3, 91058 Erlangen, Germany.
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10
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Thierbach A, Görling A. Analytic energy gradients for the self-consistent direct random phase approximation. J Chem Phys 2020; 153:134113. [DOI: 10.1063/5.0021809] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Adrian Thierbach
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
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11
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Thierbach A, Görling A. Analytic energy gradients for the exact exchange Kohn–Sham method. J Chem Phys 2020; 152:114113. [DOI: 10.1063/1.5142711] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Affiliation(s)
- Adrian Thierbach
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
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12
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Thierbach A, Schmidtel D, Görling A. Robust and accurate hybrid random-phase-approximation methods. J Chem Phys 2019; 151:144117. [DOI: 10.1063/1.5120587] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Adrian Thierbach
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Daniel Schmidtel
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
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13
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Kalai C, Mussard B, Toulouse J. Range-separated double-hybrid density-functional theory with coupled-cluster and random-phase approximations. J Chem Phys 2019; 151:074102. [PMID: 31438697 DOI: 10.1063/1.5108536] [Citation(s) in RCA: 16] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We construct range-separated double-hybrid (RSDH) schemes which combine coupled-cluster or random-phase approximations (RPAs) with a density functional based on a two-parameter Coulomb-attenuating-method-like decomposition of the electron-electron interaction. We find that the addition of a fraction of short-range electron-electron interaction in the wave-function part of the calculation is globally beneficial for the RSDH scheme involving a variant of the RPA with exchange terms. Even though the latter scheme is globally as accurate as the corresponding scheme employing only second-order Møller-Plesset perturbation theory for atomization energies, reaction barrier heights, and weak intermolecular interactions of small molecules, it is more accurate for the more complicated case of the benzene dimer in the stacked configuration. The present RSDH scheme employing a RPA thus represents a new member in the family of double hybrids with minimal empiricism which could be useful for general chemical applications.
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Affiliation(s)
- Cairedine Kalai
- Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France
| | - Bastien Mussard
- Department of Chemistry and Biochemistry, University of Colorado Boulder, Boulder, Colorado 80302, USA
| | - Julien Toulouse
- Laboratoire de Chimie Théorique (LCT), Sorbonne Université and CNRS, F-75005 Paris, France
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14
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Shahbaz M, Szalewicz K. Evaluation of methods for obtaining dispersion energies used in density functional calculations of intermolecular interactions. Theor Chem Acc 2019. [DOI: 10.1007/s00214-019-2414-5] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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15
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Beuerle M, Ochsenfeld C. Low-scaling analytical gradients for the direct random phase approximation using an atomic orbital formalism. J Chem Phys 2018; 149:244111. [DOI: 10.1063/1.5052572] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Affiliation(s)
- Matthias Beuerle
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), Butenandtstr. 7, D-81377 München, Germany and Center for Integrated Protein Science (CIPSM) at the Department of Chemistry, University of Munich (LMU), Butenandtstr. 5–13, D-81377 München, Germany
| | - Christian Ochsenfeld
- Chair of Theoretical Chemistry, Department of Chemistry, University of Munich (LMU), Butenandtstr. 7, D-81377 München, Germany and Center for Integrated Protein Science (CIPSM) at the Department of Chemistry, University of Munich (LMU), Butenandtstr. 5–13, D-81377 München, Germany
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16
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Chen GP, Voora VK, Agee MM, Balasubramani SG, Furche F. Random-Phase Approximation Methods. Annu Rev Phys Chem 2017; 68:421-445. [DOI: 10.1146/annurev-physchem-040215-112308] [Citation(s) in RCA: 88] [Impact Index Per Article: 12.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Guo P. Chen
- Department of Chemistry, University of California, Irvine, California 92697-2025;,
| | - Vamsee K. Voora
- Department of Chemistry, University of California, Irvine, California 92697-2025;,
| | - Matthew M. Agee
- Department of Chemistry, University of California, Irvine, California 92697-2025;,
| | | | - Filipp Furche
- Department of Chemistry, University of California, Irvine, California 92697-2025;,
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17
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Grundei MMJ, Burow AM. Random Phase Approximation for Periodic Systems Employing Direct Coulomb Lattice Summation. J Chem Theory Comput 2017; 13:1159-1175. [DOI: 10.1021/acs.jctc.6b01146] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Martin M. J. Grundei
- Department of Chemistry, Ludwig-Maximilians-Universität (LMU) Munich, Butenandtstrasse 7, D-81377 Munich, Germany
| | - Asbjörn M. Burow
- Department of Chemistry, Ludwig-Maximilians-Universität (LMU) Munich, Butenandtstrasse 7, D-81377 Munich, Germany
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18
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Bleiziffer P, Krug M, Görling A. Self-consistent Kohn-Sham method based on the adiabatic-connection fluctuation-dissipation theorem and the exact-exchange kernel. J Chem Phys 2015; 142:244108. [DOI: 10.1063/1.4922517] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Patrick Bleiziffer
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Marcel Krug
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
| | - Andreas Görling
- Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany
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19
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Mussard B, Reinhardt P, Ángyán JG, Toulouse J. Spin-unrestricted random-phase approximation with range separation: Benchmark on atomization energies and reaction barrier heights. J Chem Phys 2015; 142:154123. [DOI: 10.1063/1.4918710] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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20
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Franck O, Mussard B, Luppi E, Toulouse J. Basis convergence of range-separated density-functional theory. J Chem Phys 2015; 142:074107. [DOI: 10.1063/1.4907920] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Odile Franck
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
- CNRS, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
- Sorbonne Universités, UPMC Univ Paris 06, Institut du Calcul et de la Simulation, F-75005 Paris, France
| | - Bastien Mussard
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
- CNRS, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
- Sorbonne Universités, UPMC Univ Paris 06, Institut du Calcul et de la Simulation, F-75005 Paris, France
| | - Eleonora Luppi
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
- CNRS, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
| | - Julien Toulouse
- Sorbonne Universités, UPMC Univ Paris 06, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
- CNRS, UMR 7616, Laboratoire de Chimie Théorique, F-75005 Paris, France
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