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Spadetto E, Philipsen PHT, Förster A, Visscher L. Toward Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy. J Chem Theory Comput 2023; 19:1499-1516. [PMID: 36787494 PMCID: PMC10018742 DOI: 10.1021/acs.jctc.2c01201] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/16/2023]
Abstract
Pair atomic density fitting (PADF) has been identified as a promising strategy to reduce the scaling with system size of quantum chemical methods for the calculation of the correlation energy like the direct random-phase approximation (RPA) or second-order Møller-Plesset perturbation theory (MP2). PADF can however introduce large errors in correlation energies as the two-electron interaction energy is not guaranteed to be bounded from below. This issue can be partially alleviated by using very large fit sets, but this comes at the price of reduced efficiency and having to deal with near-linear dependencies in the fit set. One posibility is to use global density fitting (DF), but in this work, we introduce an alternative methodology to overcome this problem that preserves the intrinsically favorable scaling of PADF. We first regularize the Fock matrix by projecting out parts of the basis set which gives rise to orbital products that are hard to describe by PADF. After having thus obtained a reliable self-consistent field solution, we then also apply this projector to the orbital coefficient matrix to improve the precision of PADF-MP2 and PADF-RPA. We systematically assess the accuracy of this new approach in a numerical atomic orbital framework using Slater type orbitals (STO) and correlation consistent Gaussian type basis sets up to quintuple-ζ quality for systems with more than 200 atoms. For the small and medium systems in the S66 database we show the maximum deviation of PADF-MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA reference results to be 0.07 and 0.14 kcal/mol, respectively. When the new projector method is used, the errors only slightly increase for large molecules and also when moderately sized fit sets are used the resulting errors are well under control. Finally, we demonstrate the computational efficiency of our algorithm by calculating the interaction energies of large, non-covalently bound complexes with more than 1000 atoms and 20000 atomic orbitals at the RPA@PBE/CC-pVTZ level of theory.
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Affiliation(s)
- Edoardo Spadetto
- Software for Chemistry and Materials NV, NL-1081HV Amsterdam, The Netherlands
| | | | - Arno Förster
- Software for Chemistry and Materials NV, NL-1081HV Amsterdam, The Netherlands.,Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands
| | - Lucas Visscher
- Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands
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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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Förster A. Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies. J Chem Theory Comput 2022; 18:5948-5965. [PMID: 36150190 PMCID: PMC9558381 DOI: 10.1021/acs.jctc.2c00366] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
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With increasing interelectronic distance, the screening
of the
electron–electron interaction by the presence of other electrons
becomes the dominant source of electron correlation. This effect is
described by the random phase approximation (RPA) which is therefore
a promising method for the calculation of weak interactions. The success
of the RPA relies on the cancellation of errors, which can be traced
back to the violation of the crossing symmetry of the 4-point vertex,
leading to strongly overestimated total correlation energies. By the
addition of second-order screened exchange (SOSEX) to the correlation
energy, this issue is substantially reduced. In the adiabatic connection
(AC) SOSEX formalism, one of the two electron–electron interaction
lines in the second-order exchange term is dynamically screened (SOSEX(W, vc)). A
related SOSEX expression in which both electron–electron interaction
lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, vc), the
evaluation of this correlation energy expression does not require
an expensive numerical frequency integration and is therefore advantageous
from a computational perspective. We compare the accuracy of the statically
screened variant to RPA and RPA+SOSEX(W, vc) for a wide range of chemical
reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, vc) for
charged excitations and noncovalent interactions where they lead to
major improvements over RPA.
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Affiliation(s)
- Arno Förster
- Theoretical Chemistry, Vrije Universiteit, De Boelelaan 1083, NL-1081 HV, Amsterdam, The Netherlands
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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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Wong ZC, Ungur L. Exploring vibronic coupling in the benzene radical cation and anion with different levels of the GW approximation. Phys Chem Chem Phys 2021; 23:19054-19070. [PMID: 34612443 DOI: 10.1039/d1cp02795f] [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
The linear vibronic coupling constants of the benzene radical cation and anion have been obtained with different levels of the GW approximation, including G0W0, eigenvalue self-consistent GW, and quasiparticle self-consistent GW, as well as DFT with the following exchange-correlation functionals: BLYP, B3LYP, CAM-B3LYP, tuned CAM-B3LYP, and an IP-tuned CAM-B3LYP functional. The vibronic coupling constants were calculated numerically using the gradients of the eigenvalues of the degenerate HOMOs and LUMOs of the neutral benzene molecule for DFT, while the numerical gradients of the quasiparticle energies were used in the case of GW. The results were evaluated against those of high level wave function methods in the literature, and the approximate self-consistent GW methods and G0W0 with long-range corrected functionals were found to yield the best results on the whole.
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Affiliation(s)
- Zi Cheng Wong
- Department of Chemistry, National University of Singapore, Block S8 Level 3, 3 Science Drive 3, 117543, Singapore.
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6
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Drontschenko V, Graf D, Laqua H, Ochsenfeld C. Lagrangian-Based Minimal-Overhead Batching Scheme for the Efficient Integral-Direct Evaluation of the RPA Correlation Energy. J Chem Theory Comput 2021; 17:5623-5634. [PMID: 34431662 DOI: 10.1021/acs.jctc.1c00494] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
A highly memory-efficient integral-direct random phase approximation (RPA) method based on our ω-CDGD-RI-RPA method [Graf, D. J. Chem. Theory Comput. 2018, 14, 2505] is presented that completely alleviates the memory bottleneck of storing the multidimensional three-center integral tensor, which severely limited the tractable system sizes. Based on a Lagrangian formulation, we introduce an optimized batching scheme over the auxiliary and basis-function indices, which allows to compute the optimal number of batches for a given amount of system memory, while minimizing the batching overhead. Thus, our optimized batching constitutes the best tradeoff between program runtime and memory demand. Within this batching scheme, the half-transformed three-center integral tensor BiμM is recomputed for each batch of auxiliary and basis functions. This allows the computation of systems that were out of reach before. The largest system within this work consists of a DNA fragment comprising 1052 atoms and 11 230 basis functions calculated on a single node, which emphasizes the new possibilities of our integral-direct RPA method.
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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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Chedid J, Jocelyn N, Eshuis H. Energies, structures, and harmonic frequencies of small water clusters from the direct random phase approximation. J Chem Phys 2021; 155:084303. [PMID: 34470345 DOI: 10.1063/5.0059343] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Abstract
The binding energies, structures, and vibrational frequencies of water clusters up to 20 molecules are computed at the direct random phase approximation (RPA) level of theory and compared to theoretical benchmarks. Binding energies of the WATER27 set, which includes neutral and positively and negatively charged clusters, are predicted to be too low in the complete basis set limit by an average of 7 kcal/mol (9%) and are worse than the results from the best density functional theory methods or from the Møller-Plesset theory. The RPA shows significant basis set size dependence for binding energies. The order of the relative energies of the water hexamer and dodecamer isomers is predicted correctly by the RPA. The mean absolute deviation for angles and distances for neutral clusters up to the water hexamer are 0.2° and 0.6 pm, respectively, using quintuple-ζ basis sets. The relative energetic order of the hexamer isomers is preserved upon optimization. Vibrational frequencies for these systems are underestimated by several tens of wavenumbers for large basis sets, and deviations increase with the basis set size. Overall, the direct RPA method yields accurate structural parameters but systematically underestimates binding energies and shows strong basis set size dependence.
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Affiliation(s)
- Julianna Chedid
- Department of Chemistry and Biochemistry, Montclair State University, Montclair, New Jersey 07043, USA
| | - Nedjie Jocelyn
- Department of Chemistry and Biochemistry, Montclair State University, Montclair, New Jersey 07043, USA
| | - Henk Eshuis
- Department of Chemistry and Biochemistry, Montclair State University, Montclair, New Jersey 07043, USA
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Ai W, Fang WH, Su NQ. The Role of Range-Separated Correlation in Long-Range Corrected Hybrid Functionals. J Phys Chem Lett 2021; 12:1207-1213. [PMID: 33482068 DOI: 10.1021/acs.jpclett.0c03621] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The extensive application of long-range corrected hybrid functionals highlights the importance of further improving their accuracy. Unlike common long-range corrected hybrid functionals mainly focusing on the exchange part, range-separated correlation and its role in long-range corrected hybrid functionals are the main concerns of this work. To this end, we present theory on the derivation of the range-separated correlation, whose reliability and validity are proved by the agreement with the full CI on the test of the short-range correlation energy. The tests on various properties indicate that the long-range part of the LYP functional cannot effectively capture the long-range correlation effect required in LC-BLYP, whose absence instead results in a better XC functional. This new functional significantly improves LC-BLYP on all the tests in this work, with an accuracy on par with or even greater than the widely recognized CAM-B3LYP method for some applications, while maintaining the important -1/r asymptotic behavior of the XC potential. The advances and insights gained in this work are useful for the application and development of long-range corrected hybrid functionals, while emphasizing the significance of developing effective and low-cost long-range correlation functionals.
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Affiliation(s)
- Wenna Ai
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
| | - Wei-Hai Fang
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
| | - Neil Qiang Su
- Department of Chemistry, Key Laboratory of Advanced Energy Materials Chemistry (Ministry of Education) and Renewable Energy Conversion and Storage Center (RECAST), Nankai University, Tianjin 300071, China
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9
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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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10
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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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