A fast algorithm for computing the Boys function

Beylkin, Gregory; Sharma, Sandeep

doi:10.1063/5.0062444

Cited by 4 publications

(2 citation statements)

References 24 publications

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“…To include such Gaussians one can split the solution of Poisson's equation between real space and reciprocal space, 80 with real-space calculations requiring the evaluation of mixed-Gaussians-plane wave integrals which we have recently developed. 100 Alternatively, one has to use an irregular grid, where wavelet basis with multiresolution analysis is an attractive approach. Several wavelet bases such as interpolating wavelets, Coiflets, and Gausslets allow one to use the diagonal approximation that is needed for our algorithm to work.…”

Section: Discussionmentioning

confidence: 99%

“…Fourth, if one wants to avoid the use of pseudo-potential, then sharp Gaussians are needed. To include such Gaussians one can split the solution of Poisson’s equation between real space and reciprocal space, with real-space calculations requiring the evaluation of mixed-Gaussians-plane wave integrals which we have recently developed . Alternatively, one has to use an irregular grid, where wavelet basis with multiresolution analysis is an attractive approach.…”

Section: Discussionmentioning

confidence: 99%

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Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Sharma

White

Beylkin

2022

J. Chem. Theory Comput.

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In this article, we present an algorithm to efficiently evaluate the exchange matrix in periodic systems when a Gaussian basis set with pseudopotentials is used. The usual algorithm for evaluating exchange matrix scales cubically with the system size because one has to perform O(N 2) fast Fourier transform (FFT). Here, we introduce an algorithm that retains the cubic scaling but reduces the prefactor significantly by eliminating the need to do FFTs during each exchange build. This is accomplished by representing the products of Gaussian basis function using a linear combination of an auxiliary basis the number of which scales linearly with the size of the system. We store the potential due to these auxiliary functions in memory, which allows us to obtain the exchange matrix without the need to do FFT, albeit at the cost of additional memory requirement. Although the basic idea of using auxiliary functions is not new, our algorithm is cheaper due to a combination of three ingredients: (a) we use a robust pseudospectral method that allows us to use a relatively small number of auxiliary basis to obtain high accuracy; (b) we use occ-RI exchange, which eliminates the need to construct the full exchange matrix; and (c) we use the (interpolative separable density fitting) ISDF algorithm to construct these auxiliary basis sets that are used in the robust pseudospectral method. The resulting algorithm is accurate, and we note that the error in the final energy decreases exponentially rapidly with the number of auxiliary functions.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Sharma

White

Beylkin

2022

J. Chem. Theory Comput.

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Gaussian product rule for two-electron wave functions

Kovačević

2022

The Journal of Chemical Physics

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The Gaussian product rule for two-electron wave functions is introduced. The two-electron Gaussian product rule enables a new way for solving two-electron integrals. The solution is demonstrated with an example of the two-centre two-electron integral in solid harmonic Gaussian basis. The solution is obtained by expanding inverse inter-electron separation and integrating in spherical coordinates. Resulting integral separates into four integrals, three of which are straightforward to solve. The remaining integral can be solved with Boys-like functions. It is demonstrated that the solution can deliver results with accuracy comparable with the McMurchie-Davidson scheme.

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Expansion of multicenter Coulomb integrals in terms of two-center integrals

Kovačević

2022

The Journal of Chemical Physics

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Three- and four-centre Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-centre integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-centre Coulomb integrals into a linear combination of two-centre Coulomb integrals and one-centre overlap integrals. With this approach, three- and four-centre Coulomb integrals can be reduced to the same form of two-centre integrals. Resulting two-centre Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-centre Coulomb integrals are solved hierarchically: simple two-centre Coulomb integrals are used for calculation of more complicated two-centre Coulomb integrals, which are used in the calculation of multicentre integrals.

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A fast algorithm for computing the Boys function

Cited by 4 publications

References 24 publications

Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Fast Exchange with Gaussian Basis Set Using Robust Pseudospectral Method

Gaussian product rule for two-electron wave functions

Expansion of multicenter Coulomb integrals in terms of two-center integrals

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