Molecular integrals by numerical quadrature. I. Radial integration |
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Authors: | Roland Lindh Per-Åke Malmqvist Laura Gagliardi |
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Institution: | (1) Department of Chemical Physics, Chemical Center, P.O. Box 124, 221 00 Lund, Sweden, SE;(2) Department of Theoretical Chemistry, Chemical Center, P.O. Box 124, 221 00 Lund, Sweden, SE;(3) Dipartimento di Chimica Fisica ed Inorganica, Università di Bologna, Viale Risorgimento 4, 40136 Bologna, Italy, IT |
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Abstract: | This article presents a numerical quadrature intended primarily for evaluating integrals in quantum chemistry programs based
on molecular orbital theory, in particular density functional methods. Typically, many integrals must be computed. They are
divided up into different classes, on the basis of the required accuracy and spatial extent. Ideally, each batch should be
integrated using the minimal set of integration points that at the same time guarantees the required precision. Currently
used quadrature schemes are far from optimal in this sense, and we are now developing new algorithms. They are designed to
be flexible, such that given the range of functions to be integrated, and the required precision, the integration is performed
as economically as possible with error bounds within specification. A standard approach is to partition space into a set of
regions, where each region is integrated using a spherically polar grid. This article presents a radial quadrature which allows
error control, uniform error distribution and uniform error reduction with increased number of radial grid points. A relative
error less than 10−14 for all s-type Gaussian integrands with an exponent range of 14 orders of magnitude is achieved with about 200 grid points.
Higher angular l quantum numbers, lower precision or narrower exponent ranges require fewer points. The quadrature also allows controlled
pruning of the angular grid in the vicinity of the nuclei.
Received: 30 August 2000 / Accepted: 21 December 2000 / Published online: 3 April 2001 |
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Keywords: | : Numerical quadrature Molecular orbital theory Density functional theory |
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