Improved quadrature methods for the Fourier transform of a two‐center product of exponential‐type basis functions

Abstract

The numerical properties of a one-dimensional integral representation [H. P. Trivedi and E. 0. Steinborn, Phys. Rev. A 27, 670 (19S3)] for the Fourier transform of a two-center product of certain exponential-type orbitals (ETOS), the B functions [E. Filter and E. 0. Steinborn, Phys. Rev. A 18, 1 (1978)], are examined. These functions span the space of ETOS. Hence, molecular integrals for other ETOS, like the more common Slater-type orbitals, may be found as finite linear combinations of integrals with B functions. The main advantage of B functions is the simplicity of their Fourier transform that makes the derivation of relatively simple general formulas for molecular integrals with the Fourier transform method possible. The integrand of the integral representation mentioned above shows sharp peaks, making, in the cases of highly asymmetric charge distributions and/or large momentum vectors, usual quadrature methods rather slow. New quadrature schemes are presented that utilize Mobius-transformation-based quadrature rules. These rules are well suited for the numerical quadrature of functions possessing a sharp peak at or near one boundary of integration [H. H. H. Homeier and E. 0. Steinborn, J. Comput. Phys. 87, 61 (1990)l. Numerical results are presented that illustrate the fact that the new quadrature schemes are much more efficient than are automatic quadrature routines.

Convergence analysis of the addition theorem of Slater orbitals and its application to three-center nuclear attraction integrals

“…35͒, which in spite of their complexity in coordinate space have a very simple Fourier transform. [36][37][38][39][40][41][42][43][44][45][46][47] In the present work, use is made of the two-range addition theorem. Within the framework of such an approach, multicenter integrals are usually expressed as infinite series, the summation of which may become a challenging problem in numerical analysis.…”

New method of rapid and accurate evaluation for multicenter bielectronic integrals over B functions

“…(30), (32), (34), and (36) and the symbolic programing language Axiom. The zeros of the spherical Bessel function j λ (x) are computed also to 20 correct decimals using the symbolic programing language Axiom.…”

“…By applying the epsilon algorithm of Wynn [40] or Levin's utransform [41], we accelerate the convergence of the infinite series, Eqs. (30), (32), (34), and (36), but the calculation times are still prohibitive for a sufficient accuracy, in particular for two-electron integrals, as we showed in [35]. Therefore new numerical integration techniques are required.…”

The strange history of B functions or how theoretical chemists and mathematicians do (not) interact

“…The Fourier transform (6.1) of B functions is also a convenient starting point for the evaluation of more complicated multicenter integrals of B functions by a combination of the Fourier transform method with numerical quadratures. This was first done in articles by Trivedi and Steinborn [44], Grotendorst and Steinborn [45,46], Homeier and Steinborn [47,48,49], and Steinborn and Homeier [50], and in the PhD theses of Grotendorst [51] and Homeier [52].…”