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Improved quadrature methods for the Fourier transform of a two‐center product of exponential‐type basis functions
Author(s) -
Homeier Herbert H. H.,
Steinborn E. Otto
Publication year - 1992
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.560410303
Subject(s) - quadrature (astronomy) , fourier transform , numerical integration , basis function , mathematics , mathematical analysis , gauss–kronrod quadrature formula , tanh sinh quadrature , boundary value problem , physics , nyström method , optics
The numerical properties of a one‐dimensional integral representation [H. P. Trivedi and E. O. Steinborn, Phys. Rev. A 27 , 670 (1983)] for the Fourier transform of a two‐center product of certain exponential‐type orbitals ( ETO s), the B functions [E. Filter and E. O. Steinborn, Phys. Rev. A 18 , 1 (1978)], are examined. These functions span the space of ETO s. 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, 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 Möbius‐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. O. Steinborn, J. Comput. Phys. 87 , 61 (1990)]. Numerical results are presented that illustrate the fact that the new quadrature schemes are much more efficient than are automatic quadrature routines.