Abstract
With the aim of describing bound and continuum states for diatomic
molecules, we develop and implement a spectral method that makes use of
Generalized Sturmian Functions (GSF) in prolate spheroidal coordinates.
In order to master all computational issues, we apply here the method to
one-electron molecular ions and compare it with benchmark data for both
ground and excited states. We actually propose two different
computational schemes to solve the two coupled differential equations.
The first one is an iterative 1d procedure in which one solves
alternately the angular and the radial equations, the latter yielding
the state energy. The second, named direct $2d$ method, consists in
representing the Hamiltonian matrix in a two–dimensional GSF basis set,
and its further diagonalization. Both spectral schemes are timewise
computationally efficient since the basis elements are such that no
derivatives have to be calculated numerically. Moreover, very accurate
results are obtained with minimal basis sets. This is related on one
side to the use of the natural coordinate system and, on the other, to
the intrinsic good property of all GSF basis elements that are
constructed as to obey appropriate physical boundary conditions. The
present implementation for bound states paves the way for the study of
continuum states involved in ionization of one or two-electron diatomic
targets.