Multicentre two-electron Coulomb and exchange integrals over Slater functions evaluated using a generalized algorithm based on nonlinear transformations

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.

“…(V) We used the combination S withD. See [61,78] for more details on this methods. The errors are listed in the tables with the superscript "e".…”

Section: Numerical Discussionmentioning

confidence: 99%

“…n c is the order of the transformation. d The S andD transformations were used [78]. applied the Slevinsky-Safouhi formula I for higher order derivatives [65] to the RHS of (9) with (μ, ν, m, n) = (1, 1, 0, 0) (see Theorem 1 in [65]).…”

Section: Development Of the Methodsmentioning

confidence: 99%

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

2009

Numer Algor

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“…The above integral transformation, which we caled the S transformation, was successfully applied to all molecular integrals leading to an unprecedented accuracy and efficiency [19,20]. However, this transformation requires the boundary terms to vanish at both limits of integration and was only applied to spherical Bessel integrals [21].…”

Section: Introductionmentioning

confidence: 99%

A Generalized Technique in Numerical Integration

2018

EPJ Web Conf.

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Abstract. Integration by parts is one of the most popular techniques in the analysis of integrals and is one of the simplest methods to generate asymptotic expansions of integral representations. The product of the technique is usually a divergent series formed from evaluating boundary terms; however, sometimes the remaining integral is also evaluated. Due to the successive differentiation and anti-differentiation required to form the series or the remaining integral, the technique is difficult to apply to problems more complicated than the simplest. In this contribution, we explore a generalized and formalized integration by parts to create equivalent representations to some challenging integrals. As a demonstrative archetype, we examine Bessel integrals, Fresnel integrals and Airy functions.

“…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%

“…These integrals are extremely difficult to evaluate accurately and rapidly due to the strong oscillations of their integrands, which involve spherical Bessel functions. With the help of nonlinear transformations and extrapolation techniques the improvement of the convergence of these molecular integrals is remarkable (see [16][17][18][19]). …”

Section: Introductionmentioning

confidence: 99%

Numerical treatment of a twisted tail using extrapolation methods

Slevinsky

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.