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Improved quadrature methods for three‐center nuclear attraction integrals with exponential‐type basis functions
Author(s) -
Homeier Herbert H. H.,
Otto Steinborn E.
Publication year - 1991
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560390409
Subject(s) - quadrature (astronomy) , numerical integration , mathematics , basis function , tanh sinh quadrature , mathematical analysis , fourier transform , exponential type , gauss–kronrod quadrature formula , exponential function , boundary value problem , physics , nyström method , optics
The numerical properties of a two‐dimensional integral representation [J. Grotendorst and E.O. Steinborn, Phys. Rev. A 38 , 3857 (1988)] of the three‐center nuclear attraction 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 'S, 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 causing, in the case of highly asymmetric charge distributions, slow convergence of the quadrature method used by Grotendorst and Steinborn. New quadrature schemes are presented that use quadrature rules based on Möbius transformations. These rules are well suited for the numerical quadrature of functions that 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 that illustrate the fact that convergence of the new quadrature schemes is about a factor two faster in case of highly asymmetric charge distributions.