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Symmetrical “nonproduct” quadrature rules for a fast calculation of multicenter integrals
Author(s) -
Daul Claude,
Daul Stéphane
Publication year - 1997
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/(sici)1097-461x(1997)61:2<219::aid-qua4>3.0.co;2-z
Subject(s) - numerical integration , quadrature (astronomy) , gauss–kronrod quadrature formula , tanh sinh quadrature , mathematics , grid , mathematical analysis , physics , integral equation , nyström method , geometry , optics
A complete method of numerical integration, designed especially for density functional theory, is presented. We first refer to already known methods and then present a new development of the angular integration. A set of symmetrical quadrature rules which is equivalent to the popular Lebedev scheme has been developed for any arbitrary point group. In case of octahedral symmetry our method turns out to be exactly identical to Lebedev's. These formulas integrate exactly spherical harmonics of the highest possible order with, most probably, the least possible number of grid points. Nevertheless a rigorous mathematical proof of this statement has not yet been found. Examples of quadrature rules for noncubic point groups (not covered by Lebedev's grid), e.g., the icosahedral, pentagonal, or hexagonal ones are given. The application of this method to the resolution of the Poisson's equation is also presented. © 1997 John Wiley & Sons, Inc.