Fitting the Coulomb potential variationally in linear-combination-of-atomic-orbitals density-functional calculations

Mintmire, J. W.; Dunlap, Brett I.

doi:10.1103/physreva.25.88

Cited by 284 publications

(171 citation statements)

References 21 publications

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“…(5) we introduce an auxiliary basis set, the resolutionof-the-identity (RI) basis set, [67][68][69][70][71] by the unsymmetric resolutions of the identity…”

Section: A Correlation Energy In the Random Phase Approximationmentioning

confidence: 99%

Efficient self-consistent treatment of electron correlation within the random phase approximation

Bleiziffer¹,

Heßelmann²,

Görling³

2013

The Journal of Chemical Physics

101

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A self-consistent Kohn-Sham (KS) method is presented that treats correlation on the basis of the adiabatic-connection dissipation-fluctuation theorem employing the direct random phase approximation (dRPA), i.e., taking into account only the Coulomb kernel while neglecting the exchangecorrelation kernel in the calculation of the Kohn-Sham correlation energy and potential. The method, denoted self-consistent dRPA method, furthermore treats exactly the exchange energy and the local multiplicative KS exchange potential. It uses Gaussian basis sets, is reasonably efficient, exhibiting a scaling of the computational effort with the forth power of the system size, and thus is generally applicable to molecules. The resulting dRPA correlation potentials in contrast to common approximate correlation potentials are in good agreement with exact reference potentials. The negatives of the eigenvalues of the highest occupied molecular orbitals are found to be in good agreement with experimental ionization potentials. Total energies from self-consistent dRPA calculations, as expected, are even poorer than non-self-consistent dRPA total energies and dRPA reaction and non-covalent binding energies do not significantly benefit from self-consistency. On the other hand, energies obtained with a recently introduced adiabatic-connection dissipation-fluctuation approach (EXXRPA+, exact-exchange random phase approximation) that takes into account, besides the Coulomb kernel, also the exact frequency-dependent exchange kernel are significantly improved if evaluated with orbitals obtained from a self-consistent dRPA calculation instead of an exact exchange-only calculation. Total energies, reaction energies, and noncovalent binding energies obtained in this way are of the same quality as those of high-level quantum chemistry methods, like the coupled cluster singles doubles method which is computationally more demanding. © 2013 AIP Publishing LLC.

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“…(5) we introduce an auxiliary basis set, the resolutionof-the-identity (RI) basis set, [67][68][69][70][71] by the unsymmetric resolutions of the identity…”

Section: A Correlation Energy In the Random Phase Approximationmentioning

confidence: 99%

Efficient self-consistent treatment of electron correlation within the random phase approximation

Bleiziffer¹,

Heßelmann²,

Görling³

2013

The Journal of Chemical Physics

101

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“…3 so that the two-electron operator, g, is not specified. Whereas the broadly accepted choice is the two-electron repulsion operator (18,19), g(r 1 , r 2 ) ϭ ͉r 1 Ϫ r 2 ͉ Ϫ1 , other choices of metric are also possible. For example, the overlap operator, g(r 1 , r 2 ) ϭ ␦(r 1 Ϫ r 2 ), has been explored (8,12,16), and the use of an anti-Coulomb operator, g(r 1 , r 2 ) ϭ Ϫ͉r 1 Ϫ r 2 ͉, has also been advocated (20), because it is optimal for representing the potential caused by a charge distribution.…”

mentioning

confidence: 99%

“…Thus in realistic applications, different choices for the fitting metric will lead to different fitting coefficients and therefore different results. Present-day auxiliary basis calculations are generally performed with the Coulomb operator (9) for the fit, which was shown to be superior to the overlap metric both in theory and practice (12,16,19,21). Standardized auxiliary basis sets have been developed by the Ahlrichs group for Coulomb fitting (13,14) in self-consistent field calculations, and for second-order perturbation (MP2) calculations (15,17) of the correlation energy.…”

mentioning

confidence: 99%

Auxiliary basis expansions for large-scale electronic structure calculations

Jung

Sodt

Gill

et al. 2005

Proc. Natl. Acad. Sci. U.S.A.

200

207

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One way to reduce the computational cost of electronic structure calculations is to use auxiliary basis expansions to approximate four-center integrals in terms of two-and three-center integrals, usually by using the variationally optimum Coulomb metric to determine the expansion coefficients. However, the long-range decay behavior of the auxiliary basis expansion coefficients has not been characterized. We find that this decay can be surprisingly slow. Numerical experiments on linear alkanes and a toy model both show that the decay can be as slow as 1͞r in the distance between the auxiliary function and the fitted charge distribution. The Coulomb metric fitting equations also involve divergent matrix elements for extended systems treated with periodic boundary conditions. An attenuated Coulomb metric that is short-range can eliminate these oddities without substantially degrading calculated relative energies. The sparsity of the fit coefficients is assessed on simple hydrocarbon molecules and shows quite early onset of linear growth in the number of significant coefficients with system size using the attenuated Coulomb metric. Hence it is possible to design linear scaling auxiliary basis methods without additional approximations to treat large systems.linear scaling ͉ resolution of the identity ͉ density fitting E lectronic structure calculations are normally performed by using basis set expansions to allow approximations to the Schrödinger equation to be expressed as algebraic rather than differential equations. Molecular electronic structure calculations (1) of either the density functional theory or wave-function type typically use standardized atom-centered basis sets, {͉ ͘}, whose functions are fixed linear combinations of Gaussian functions. With Gaussian basis functions, two-electron matrix elements,can be efficiently evaluated (2), normally with g(r 1 , r 2 ) ϭ ͉r 1 Ϫ r 2 ͉ Ϫ1 for Coulomb interactions. There are formally O(N 4 ) of these integrals for an atomic orbital basis set of size N. However, for a given choice of basis set, the number of nonnegligible integrals grows as only O(N 2 ) with increases in the size of the molecule. This growth arises from the rapid (Gaussian) decay of the amplitude of the product charge distribution ͉ ͘ ϵ (r 1 ) (r 1 ) with separation of the basis function centers. In density functional theory calculations, even this reduced bottleneck can be overcome for construction of the Coulomb matrix, J ϭ ͚ ͗ ͉ ͘P , from the density matrix by use of linearscaling fast multipole (3-5) and tree code methods (6).However, for a molecule of fixed size, increasing the number of basis functions per atom, n, does inexorably lead to O(n 4 ) growth in the number of significant integrals. This growth follows directly from the fact that the number of nonnegligible product charge distributions, ͉ ͘, grows as O(n 2 ). As a result, the use of large (high-quality) basis expansions is computationally costly. This article revisits perhaps the most practical way around this ''basis set quality'' bo...

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“…23 Because a Cholesky decomposition of the matrix V is computed upon program start and kept in memory, by far the most CPU time goes into steps A and C, so these have also been parallelized using the recipe explained in the last section (see, e.g., Fig. 2) with a slight modification: for a given (i, j) shell pair, the required computing time may be as small as the latency of the global counter.…”

Section: Density-fitting (Ri) Calculationsmentioning

confidence: 99%

Shared‐memory parallelization of the TURBOMOLE programs AOFORCE, ESCF, and EGRAD: How to quickly parallelize legacy code

Wüllen

2010

J Comput Chem

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Abstract:The programs ESCF, EGRAD, and AOFORCE are parts of the TURBOMOLE program package and compute excited-state properties and ground-state geometric hessians, respectively, for Hartree-Fock and density functional methods. The range of applicability of these programs has been extended by allowing them to use all CPU cores on a given node in parallel. The parallelization strategy is not new and duplicates what is standard today in the calculation of ground-state energies and gradients. The focus is on how this can be achieved without needing extensive modifications of the existing serial code. The key ingredient is to fork off worker processes with separated address spaces as they are needed. Test calculations on a molecule with about 80 atoms and 1000 basis functions show good parallel speedup up to 32 CPU cores.

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Fitting the Coulomb potential variationally in linear-combination-of-atomic-orbitals density-functional calculations

Cited by 284 publications

References 21 publications

Efficient self-consistent treatment of electron correlation within the random phase approximation

Efficient self-consistent treatment of electron correlation within the random phase approximation

Auxiliary basis expansions for large-scale electronic structure calculations

Shared‐memory parallelization of the TURBOMOLE programs AOFORCE, ESCF, and EGRAD: How to quickly parallelize legacy code

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