| Abstract: | In order to solve the partial differential equations that arise in the Hartree-Focktheory for diatomic molecules and in molecular theories that include electron correlation,one needs efficient methods for solving partial differential equations. In thisarticle, we present numerical results for a two-variable model problem of the kind thatarises when one solves the Hartree-Fock equations for a diatomic molecule. We compareresults obtained using the spline collocation and domain decomposition methodswith third-order Hermite splines to results obtained using the more-established finitedifference approximation and the successive over-relaxation method. The theory ofdomain decomposition presented earlier is extended to treat regions that are dividedinto an arbitrary number of subregions by families of lines parallel to the two coordinateaxes. While the domain decomposition method and the finite difference approachboth 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. Thedomain decomposition method has the strength that it can be applied to problems witha large number of grid points. The time required to solve a partial differential equationfor a fine grid with a large number of points goes down as the number of partitionsincreases. The reason for this is that the length of time necessary for solving a set oflinear 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 forsolving the set of linear equations in each subregion is very much smaller. This featureof the theory may well prove to be a decisive factor for solving the two-electron pairequation, which – for a diatomic molecule – involves solving partial differential equationswith five independent variables. The domain decomposition theory also makesit possible to study complex molecules by dividing them into smaller fragments thatare calculated independently. Since the domain decomposition approach makes it possibleto decompose the variable space into separate regions in which the equations aresolved independently, this approach is well-suited to parallel computing. |
