Asymptotically exact calculation of the exchange energies of one-active-electron diatomic ions with the surface integral method

We present a general procedure, based on the Holstein–Herring method, for calculating exactly the leading term in the exponentially small exchange energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulae we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Σ and Π states of H⁺₂. We then apply our general formulae to calculate the exchange energy splittings between the lowest states of the diatomic alkali cations K⁺₂, Rb⁺₂ and Cs⁺₂, which are isovalent to H⁺₂. Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.