A new algorithm for accurate and fast numerical evaluation of hybrid and three-centre two-electron Coulomb integrals over Slater-type functions

Berlu, Lilian; Safouhi, Hassan

doi:10.1088/0305-4470/36/44/007

Cited by 29 publications

(16 citation statements)

References 28 publications

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“…Moreover, it is easy to show that the regular solid harmonics are for all r ∈ R 3 solutions of the homogeneous three-dimensional Laplace equation 8) whereas the irregular solid harmonics are solutions for all r ∈ R 3 \ {0}.…”

Section: A Appendix: Terminology and Definitionsmentioning

confidence: 99%

The Spherical Tensor Gradient Operator

Weniger

2005

Collect. Czech. Chem. Commun.

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Section: A Appendix: Terminology and Definitionsmentioning

confidence: 99%

The Spherical Tensor Gradient Operator

Weniger

2005

Collect. Czech. Chem. Commun.

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“…It is noted that the S transformation have increased the efficacy ofD [62,63] and G [64] in that the zeros of the integrand are the zeros of the sine function and the characteristics of exponential decay of the integrand are significantly amplified, aiding the convergence. In this work, we present another approach based on the Slevinsky-Safouhi formulae for higher derivatives [65] applied to spherical Bessel functions, practical properties of sine and cosine functions and extrapolation methods.…”

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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“…In previous work [8][9][10][11][12][13][14][15][16][17][18][19], we showed the efficiency of combining quadrature rules with extrapolation methods for improving convergence of the so-called molecular multi-center integrals over exponential type functions for molecular electronic structure calculations. These integrals are extremely difficult to evaluate accurately and rapidly due to the strong oscillations of their integrands, which involve spherical Bessel functions.…”

Section: Introductionmentioning

confidence: 99%

Numerical treatment of a twisted tail using extrapolation methods

Slevinsky

Safouhi

2008

Numer Algor

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Highly oscillatory integral, called a twisted tail, is proposed as a challenge in The SIAM 100-digit challenge. A Study in High-Accuracy Numerical Computing, where Drik Laurie developed numerical algorithms based on the use of Aitken's 2 -method, complex integration and transformation to a Fourier integral. Another algorithm is developed by Walter Gautschi based on Longman's method; Newton's method for solving a nonlinear equation; Gaussian quadrature; and the epsilon algorithm of Wynn for accelerating the convergence of infinite series. In the present work, nonlinear transformations for improving the convergence of oscillatory integrals are applied to the integration of this wildly oscillating function. Specifically, theD transformation and its companion the W algorithm, and the G transformation are all used in the analysis of the integral. A Fortran program is developed employing each of the methods, and accuracies of up to 15 correct digits are reached in double precision.

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A new algorithm for accurate and fast numerical evaluation of hybrid and three-centre two-electron Coulomb integrals over Slater-type functions

Cited by 29 publications

References 28 publications

The Spherical Tensor Gradient Operator

The Spherical Tensor Gradient Operator

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

Numerical treatment of a twisted tail using extrapolation methods

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