Exact Formula for the Dipole Moment of a One-Electron Diatomic Molecule

It is shown that the dipole moment function, μ(R, Z_a, Z_b), for an arbitrary bound electronic state of a one-electron diatomic molecule, with inter-nuclear distance R and atomic numbers Z_a, Z_b may be expressed exactly in terms of the separation eigenconstant C and the electronic energy eigenvalue W of the Schrodinger equation by means of the Hellmann-Feynman theorem and a new recursion relation. The formula is used to investigate the behaviour of μ in the vicinity of the united atom and when the nuclei are far apart. The generalization required to extend the relation to other expectation values is derived in an appendix.