Linear scaling computation of the Fock matrix

Challacombe, Matt; Schwegler, Eric

doi:10.1063/1.473575

Cited by 201 publications

(155 citation statements)

References 78 publications

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“…This fact should be connected with similar results obtained by [28], later refined by [10,29]. The concept of degrees of freedom was used successfully by Michielssen and Boag with the construction of the matrix decomposition algorithm (see [45,46]). …”

Section: Sparsifying the Transfer Matrixmentioning

confidence: 70%

The Fast Multipole Method: Numerical Implementation

Darve¹

2000

Journal of Computational Physics

304

296

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We study integral methods applied to the resolution of the Maxwell equations where the linear system is solved using an iterative method which requires only matrix-vector products. The fast multipole method (FMM) is one of the most efficient methods used to perform matrix-vector products and accelerate the resolution of the linear system. A problem involving N degrees of freedom may be solved in CN iter N log N floating operations, where C is a constant depending on the implementation of the method. In this article several techniques allowing one to reduce the constant C are analyzed. This reduction implies a lower total CPU time and a larger range of application of the FMM. In particular, new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms. Several numerical tests are also described. These confirm the efficiency and the theoretical complexity of the FMM.

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Section: Sparsifying the Transfer Matrixmentioning

confidence: 70%

The Fast Multipole Method: Numerical Implementation

Darve¹

2000

Journal of Computational Physics

304

296

View full text Add to dashboard Cite

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“…25,26 For appropriate insulating systems and compact basis sets, significant sparsity is present and may be exploited to formulate efficient conditionally linear-scaling integral-driven algorithms. [27][28][29] Such methods make hybrid DFT calculations possible on very large insulating systems, provided the basis set is compact.…”

Section: Introductionmentioning

confidence: 99%

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

Manzer

Horn

Mardirossian

et al. 2015

The Journal of Chemical Physics

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Construction of the exact exchange matrix, K, is typically the rate-determining step in hybrid density functional theory, and therefore, new approaches with increased efficiency are highly desirable. We present a framework with potential for greatly improved efficiency by computing a compressed exchange matrix that yields the exact exchange energy, gradient, and direct inversion of the iterative subspace (DIIS) error vector. The compressed exchange matrix is constructed with one index in the compact molecular orbital basis and the other index in the full atomic orbital basis. To illustrate the advantages, we present a practical algorithm that uses this framework in conjunction with the resolution of the identity (RI) approximation. We demonstrate that convergence using this method, referred to hereafter as occupied orbital RI-K (occ-RI-K), in combination with the DIIS algorithm is well-behaved, that the accuracy of computed energetics is excellent (identical to conventional RI-K), and that significant speedups can be obtained over existing integral-direct and RI-K methods. For a 4400 basis function C 68 H 22 hydrogen-terminated graphene fragment, our algorithm yields a 14× speedup over the conventional algorithm and a speedup of 3.3× over RI-K. C 2015 AIP Publishing LLC. [http://dx

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“…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)(4)(5) and tree code methods (6).…”

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

210

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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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Linear scaling computation of the Fock matrix

Cited by 201 publications

References 78 publications

The Fast Multipole Method: Numerical Implementation

The Fast Multipole Method: Numerical Implementation

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

Auxiliary basis expansions for large-scale electronic structure calculations

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