The ionised hydrogen molecule

Abstract

The model which has been proposed for the ion of the hydrogen molecule H2+, consists of one electron and two protons. Since the mass of the electron is negligible compared with that of the protons, we may, to a first approximation, consider the protons as at rest. The system is then a particular case of the problem of three bodies, and can be solved completely classically. This has been done by Pauli,* and more recently by Niessen. † The value obtained by Pauli for the energy of the normal state is not in agreement with the experimental result inferred from the ionisation potential and heat of dissociation of the molecule. Niessen obtains the experimental result by the introduction of half integer quantum numbers. The classical problem is separable in elliptic co-ordinates, and so if we apply Schrodinger’s method to the system we shall obtain a wave equation which is separable in the same co-ordinates. The resulting differential equations can be solved exactly. This is the only three-body problem which admits of an exact solution, and it is of interest to obtain an analytical result, and not merely one obtained by a perturbation method, which it may be difficult to justify.