The formulation of Bardsley et al (Phys. Rev. A 11 1911) for evaluating the asymptotic electron exchange interaction between an alkali ion and its parent atom is extended by the use of a more accurate representation of the asymptotic wavefunction of the isolated atom and also by the inclusion of a further term in the series that accounts for the presence of the ion. The Bardsley analysis is compared with a purely numerical approach in which the Holstein–Herring integrand is constructed from a polarized wavefunction built from numerical solutions of the differential equations of Rayleigh–Schrödinger perturbation theory and the required quadrature is performed numerically. Values of the asymptotic exchange interactions are given for the molecular ions Li+2, Na+2, K+2, Rb+2 and Cs+2. As an example, exchange interactions for Cs+2 evaluated by an ab initio method are compared to those predicted by the Bardsley formula. Potentials for the lowest states of Cs+2 are constructed from the ab initio values matched to the asymptotic exchange and long-range polarization interactions, and these potentials are used to predict the 12Σ+g and 12Σ+u scattering lengths and low-energy charge transfer cross sections.