M�bius-Type quadrature of electron repulsion integrals withB functions

Abstract

The numerical properties of a three-dimensional integral representation [J. Grotendorst and E. O. Steinborn, Phys. Rev. A 38, 3857 (1988)] of the electron repulsion integral with a special class of exponential-type orbitals (ETO's), the B functions [E. Filter and E. O. Steinborn, Phys. Rev. A 18, 1 (1978)], are examined. B functions span the space of ETO's. The commonly occurring ETO's can be expressed in terms of simple finite sums of B functions. Hence molecular integrals for other ETO', 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 which 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 causing, in the case of highly asymmetric charge distributions, slow convergence of the quadrature method used by Grotendorst and Steinborn. Quadrature schemes are presented which utilize quadrature rules based upon Mobius transformations. These rules are well suited for the numerical quadrature of functions which possess a sharp peak at or near a single boundary of integration [H. H. H. Homeier and E. O. Steinborn, J. Comput. Phys., 87, 61 (1990)]. Numerical results are presented which illustrate the fact that the new quadrature schemes are also applicable in case of highly asymmetric charge distributions.