An extremely efficient and rapid algorithm for numerical evaluation of three-centre nuclear attraction integrals over Slater-type functions

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.…”

A Generalized Technique in Numerical Integration

“…Another example of the use of integration by parts in numerical integration arises in [19], applied to the oscillatory spherical Bessel integral function involved in molecular integrals. Through a reformalized integration by parts with respect to x dx, we transformed the initial spherical Bessel integral into an integral involving the simple sine function:…”

“…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].…”

Numerical treatment of a twisted tail using extrapolation methods

“…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.…”