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Numerical Methods for Solving the Hartree-Fock Equations of Diatomic Molecules II

Part of: Partial differential equations, boundary value problems Numerical linear algebra

Published online by Cambridge University Press:  16 March 2016

John C. Morrison ,
Kyle Steffen ,
Blake Pantoja ,
Asha Nagaiya ,
Jacek Kobus  and
Thomas Ericsson
John C. Morrison
Affiliation:
Department of Physics and Astronomy, University of Louisville, Louisville, KY, 40292, USA
Kyle Steffen
Affiliation:
Department of Physics and Astronomy, University of Louisville, Louisville, KY, 40292, USA
Blake Pantoja
Affiliation:
Department of Physics and Astronomy, University of Louisville, Louisville, KY, 40292, USA
Asha Nagaiya
Affiliation:
Department of Physics and Astronomy, University of Louisville, Louisville, KY, 40292, USA
Jacek Kobus*
Affiliation:
Institute of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland
Thomas Ericsson
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology , SE-412 96 Göteborg, Sweden
*
*Corresponding author. Email address:jacek.kobus@fizyka.umk.pl (J. Kobus)
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Abstract

In order to solve the partial differential equations that arise in the Hartree- Fock theory for diatomicmolecules and inmolecular theories that include electron correlation, one needs efficient methods for solving partial differential equations. In this article, we present numerical results for a two-variablemodel problem of the kind that arises when one solves the Hartree-Fock equations for a diatomic molecule. We compare results obtained using the spline collocation and domain decomposition methods with third-order Hermite splines to results obtained using the more-established finite difference approximation and the successive over-relaxation method. The theory of domain decomposition presented earlier is extended to treat regions that are divided into an arbitrary number of subregions by families of lines parallel to the two coordinate axes. While the domain decomposition method and the finite difference approach both yield results at the micro-Hartree level, the finite difference approach with a 9- point difference formula produces the same level of accuracy with fewer points. The domain decompositionmethod has the strength that it can be applied to problemswith a large number of grid points. The time required to solve a partial differential equation for a fine grid with a large number of points goes down as the number of partitions increases. The reason for this is that the length of time necessary for solving a set of linear equations in each subregion is very much dependent upon the number of equations. Even though a finer partition of the region has more subregions, the time for solving the set of linear equations in each subregion is very much smaller. This feature of the theory may well prove to be a decisive factor for solving the two-electron pair equation, which – for a diatomic molecule – involves solving partial differential equations with five independent variables. The domain decomposition theory also makes it possible to study complex molecules by dividing them into smaller fragments that are calculated independently. Since the domain decomposition approachmakes it possible to decompose the variable space into separate regions in which the equations are solved independently, this approach is well-suited to parallel computing.

Keywords

Pair equationHartree-Fock equationdomain decompositionHermite splineshighorder finite differencessuccessive over-relaxation

MSC classification

Secondary: 65N35: Spectral, collocation and related methods 65N22: Solution of discretized equations 65F10: Iterative methods for linear systems
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 632 - 647
Copyright
Copyright © Global-Science Press 2016 

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