Boundary conditions for the wave equation

A single electron in the field of two fixed nuclei, constituting the idealized hydrogen molecular ion, provides the simplest case for the application of wave mechanics to molecular, as distinct from atomic, problems. The most extensive theoretical discussion of the corresponding wave equation has been given by A. H. Wilson in these 'Proceedings.’ He was led to conclude that this equation possesses no eigen-solutions satisfying the usual boundary conditions for an atomic problem. Subsequent investigators have succeeded, however, in obtaining by numerical methods eigen-values in good agreement with observed values of the energy. But, with the exception of Teller, they appear not to have taken account of Wilson’s result. It is therefore worth while to investigate the existence of their solutions and to clear up, if possible, any doubt as to the applicability of the familiar boundary conditions to this type of problem. The usual existence theorems for eigen-values apply only to boundary conditions at ordinary points of the differential equation. The difficulty in cases like Wilson’s equation is that the conditions are given at singular points.