A multi-domain weighted residual method for the one-electron Schrödinger equation: Application to H+2

The Schrödinger equation is solved for a single electron moving in the coulombic field of some arbitrary configuration of nuclei. Space is partitioned by centering a sphere on each of the individual nuclei without any overlap or touching of the spheres, i.e., muffin-tin spheres. All regions are treated by a weighted residual technique, which is a more general approach than the variational method. Outside the spheres, both the wavefunction and its product with the potential energy function are expanded as a linear combination of solutions taken from the modified Helmholtz equation (M.H.E.). A basis set is prepared by solving the M.H.E. repeatedly for a select set of eigenvalues and boundary conditions, using a boundary integral technique. Inside any sphere, the wavefunction is written as a linear combination of terms, each a product of a radial function and a spherical harmonic. The radial factor is written as product of an exponential and a power series. For either region, an alternate basis set is chosen to supply the weight functions required by the weighted residual approach. Weight functions are chosen according to their ability to provide increased efficiency and accuracy. Only simple integrals over the sphere surfaces are involved in calculating matrix coefficients. In order to demonstrate the method, the H+2 molecule is considered as a test case, with the potential energy function treated in full.