The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Abstract

Three-center nuclear attraction integrals over exponential-type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two-dimensional integral representations, involving spherical Bessel integral functions. The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three-center nuclear attraction integrals.

Fourier Transformation Method for Computing NMR Integrals over Exponential Type Functions

“…In the case of the three-center integrals, we need to compute semi-infinite integrals involving oscillatory functions. These oscillatory integrals can be computed to a high pre-determined accuracy using existing methods and algorithms based on extrapolation methods and numerical quadrature [Berlu & Safouhi (2003); Duret & Safouhi (2007);Safouhi (2001b;2004;2010a); Slevinsky & Safouhi (2009)]. …”

“…The obtained analytic expressions turned out to be similar to those obtained for the so-called three-center nuclear attraction integrals (zeroth order integrals). The latter were the subject of significant research [Berlu & Safouhi (2003); Duret & Safouhi (2007);Fernández et al (2001); Grotendorst & Steinborn (1988) ;Homeier & Steinborn (1993); Niehaus et al (2008); Rico et al (1998;; Safouhi (2001b;2004); Slevinsky & Safouhi (2009)]. In our research, we used techniques based on extrapolation methods combined with numerical quadratures to compute the analytic expressions of the NMR integrals.…”

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

“…(IV) We used the W algorithm [75,76] combined withD [75]. In [77], we developed an algorithm for the three-center nuclear attraction integrals, using W algorithm andD. The errors are listed in the tables with the superscript "d".…”

Numerical treatment of a twisted tail using extrapolation methods

“…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]). …”

“…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.…”