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On the evaluation of overlap integrals with 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.560420416
Subject(s) - quadrature (astronomy) , numerical integration , mathematics , basis function , fourier transform , exponential type , mathematical analysis , volume integral , slater integrals , exponential function , tanh sinh quadrature , orbital overlap , type (biology) , gauss–kronrod quadrature formula , boundary value problem , atomic orbital , physics , nyström method , quantum mechanics , integral equation , electron , ecology , optics , biology
The numerical properties of a one‐dimensional integral representation [H.P. Trivedi and E. O. Steinborn, Phys. Rev. A 27 , 670 (1983)] for the overlap integral 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. B functions possess a very simple Fourier transform that results in relatively simple general formulas for molecular integrals derivable using the Fourier transform method. In addition, molecular integrals for other ETO s, like the more common Slater‐type orbitals, can be written as finite linear combinations of integrals with B functions because these functions span the space of ETO s. The integrand of the integral representation mentioned above shows peculiarities requiring special quadrature methods, especially in the case of highly asymmetric charge distributions. It is shown that good results can be obtained using Möbiustransformation‐based quadrature rules well suited for the numerical integration 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)]. The method based on Möbius‐type quadrature is compared to several other methods — based on other quadrature rules, finite sums, and infinite series representations together with convergence acceleration — to evaluate overlap integrals with B functions. The numerical results indicate that the new quadrature schemes are more efficient than are the other methods.