An extremely efficient approach for accurate and rapid evaluation of three-centre two-electron Coulomb and hybrid integrals over \(B\) functions. (Q2718494)

This paper continues a series of previous studies by the author concerning the rapid and accurate evaluation of multicentre integrals which appear in molecular calculations. In this respect, a special class of exponentially declining functions, the so-called \(B\)-functions \(B^m_{n,l} (\alpha,r)\) [see \textit{E. Filter} and \textit{E. O. Steinborn}, Phys. Rev. A (3) 18, No. 1, 1--11 (1978)] have some suitable mathematical properties particularly advantageous for the molecular integrals. Likewise, an atomical orbital basis of Slater-type functions can be expressed as \(B\)-functions that enables to deal with analytic expressions for those integrals using the Fourier transform method. The case here treated of three-centre two-electron Coulomb and hybrid integrals over \(B\)-functions, involves analytic expressions whose numerical evaluation offers serious computational difficulties due to the presence of semiinfinite integrals including products of hypergeometric series and spherical Bessel functions. In a previous work and for solving this drawback, the author proved that these hypergeometric functions can be reduced to finite sums and that the integrands satisfy all the conditions required to apply a certain convergence accelerating \(H\overline D\)-procedure for its numerical evaluation [\textit{H. Safouhi}, J. Comput. Phys. 165, No. 2, 473--495 (2000; Zbl 0992.30018); see also \textit{H. Safouhi} and \textit{P. E. Hoggan} J. Phys. A, Math. Gen. 32, No. 34, 6203--6217 (1999; Zbl 0991.23693)]. In the paper under review, the author presents an efficient and rapid evaluation of the integrals in question using a new approach called \(S\overline D\) which is based on the \(H\overline D\) method giving rise to a sustantial simplification in the calculations while maintaining the same high accuracy.