The three-dimensional convolution of reduced Bessel functions and other functions of physical interest

Filter, Eckhard; Steinborn, E. Otto

doi:10.1063/1.523517

Cited by 84 publications

(18 citation statements)

References 8 publications

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“…The Fourier transformation method seems to be considerably simpler than the approach proposed in Ref. 5 . It leads not only to a more compact proof of the remarkable Filter and Steinborn result for the particular case of the H convolution, but also to an extension to more complicated types of functions.…”

Section: Discussionmentioning

confidence: 98%

Fourier transforms of atomic orbitals. II. Convolution theorems

Niukkanen

1984

Int J of Quantum Chemistry

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Section: Discussionmentioning

confidence: 98%

Fourier transforms of atomic orbitals. II. Convolution theorems

Niukkanen

1984

Int J of Quantum Chemistry

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“…The overlap integral of B functions is denoted by [16,17] It is the special case p = 0 of the Fourier transform of a two-center product of B functions, which is defined as [19] In the case of equal scaling parameters, the following so-called convolution theorem holds (compare Eq. (4.1) of Filter and Steinborn [30]) where A112 = (II + 12 -lI2)/2 is a nonnegative integer in consequence of the selection rules [31] for the Gaunt coefficients [32] (~I m i l~~m 2 1 1 j m~)…”

Section: Definitions and Basic Formulasmentioning

confidence: 99%

Improved quadrature methods for the Fourier transform of a two‐center product of exponential‐type basis functions

Homeier

Steinborn

1992

Int J of Quantum Chemistry

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The numerical properties of a one-dimensional integral representation [H. P. Trivedi and E. 0. Steinborn, Phys. Rev. A 27, 670 (19S3)] for the Fourier transform of a two-center product of certain exponential-type orbitals (ETOS), the B functions [E. Filter and E. 0. Steinborn, Phys. Rev. A 18, 1 (1978)], are examined. These functions span the space of ETOS. Hence, molecular integrals for other ETOS, like the more common Slater-type orbitals, may be found as finite linear combinations of integrals with B functions. The main advantage of B functions is the simplicity of their Fourier transform that makes the derivation of relatively simple general formulas for molecular integrals with the Fourier transform method possible. The integrand of the integral representation mentioned above shows sharp peaks, making, in the cases of highly asymmetric charge distributions and/or large momentum vectors, usual quadrature methods rather slow. New quadrature schemes are presented that utilize Mobius-transformation-based quadrature rules. These rules are well suited for the numerical quadrature of functions possessing a sharp peak at or near one boundary of integration [H. H. H. Homeier and E. 0. Steinborn, J. Comput. Phys. 87, 61 (1990)l. Numerical results are presented that illustrate the fact that the new quadrature schemes are much more efficient than are automatic quadrature routines.

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“…Detailed discussions of the mathematical properties of reduced Bessel functions and of their anisotropic generalizations can be found in [48]. Furthermore, B functions have much more appealing properties applicable to multi-center integral problems, compared to other exponentially decaying functions [49][50][51][52][53]. The multi-center molecular integrals over B functions can be computed much more easily than the corresponding integrals of other exponentially decaying functions.…”

Section: Introductionmentioning

confidence: 99%

Bessel, sine and cosine functions and extrapolation methods for computing molecular multi-center integrals

Safouhi

2009

Numer Algor

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In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the SlevinskySafouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the threecenter nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations.

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The three-dimensional convolution of reduced Bessel functions and other functions of physical interest

Cited by 84 publications

References 8 publications

Fourier transforms of atomic orbitals. II. Convolution theorems

Fourier transforms of atomic orbitals. II. Convolution theorems

Improved quadrature methods for the Fourier transform of a two‐center product of exponential‐type basis functions

Bessel, sine and cosine functions and extrapolation methods for computing molecular multi-center integrals

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