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Journal of Chemical Physics / Volume 128 / Issue 22 / ARTICLES / Theoretical Methods and Algorithms

J. Chem. Phys. 128, 224103 (2008); http://dx.doi.org/10.1063/1.2931563 (10 pages)

Numerical integration of exchange-correlation energies and potentials using transformed sparse grids

Juan I. Rodríguez1, David C. Thompson1, Paul W. Ayers1, and Andreas M. Köster2

1Department of Chemistry, McMaster University, Hamilton, Ontario, L8S 4M1, Canada
2Departamento de Química, CINVESTAV, Avenida Instituto Politécnico Nacional 2508, A.P. 14-740 México Distrito Federal 07000, Mexico

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(Received 26 March 2008; accepted 28 April 2008; published online 12 June 2008)

A new numerical integration procedure for exchange-correlation energies and potentials is proposed and “proof of principle” results are presented. The numerical integration grids are built from sparse-tensor product grids (constructed according to Smolyak’s prescription [Dokl. Akad. Nauk. 4, 240 (1963)] ) on the unit cube. The grid on the unit cube is then transformed to a grid over real space with respect to a weight function, which we choose to be the promolecular density. This produces a “whole molecule” grid, in contrast to conventional integration methods in density-functional theory, which use atom-in-molecule grids. The integration scheme was implemented in a modified version of the DEMON2K density-functional theory program, where it is used to evaluate integrals of the exchange-correlation energy density and the exchange-correlation potential. Ground-state energies and molecular geometries are accurately computed. The biggest advantages of the grid are its flexibility (it is easy to change the number and distribution of grid points) and its whole molecule nature. The latter feature is potentially helpful for basis-set-free computational algorithms.

© 2008 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. DESCRIPTION OF METHOD
    1. Overview
    2. Grids on the unit cube
      1. Theory of multidimensional integration
      2. One-dimensional integration
      3. Tensor product integration formulas
      4. Sparse-tensor products
    3. Transformation of coordinates
      1. The conditional distribution transformation method
      2. The promolecular density
    4. Interpretation
  3. RESULTS AND DISCUSSION
  4. CONCLUSIONS

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KEYWORDS and PACS

Keywords

density functional theory, electron correlations, ground states, integration, molecular configurations, numerical analysis, potential energy surfaces, tensors

PACS

  • 31.15.eg

    Exchange-correlation functionals (in current density functional theory)

  • 31.50.Bc

    Potential energy surfaces for ground electronic states

  • 33.15.Bh

    General molecular conformation and symmetry; stereochemistry

  • 02.60.Jh

    Numerical differentiation and integration

  • 02.10.Ud

    Linear algebra

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.2931563

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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