We have recently introduced a model of the dispersion interaction based on the position-dependent dipole moment of the exchange hole [J. Chem. Phys. 122, 154104 (2005)]. The original derivation, involving simple dipole-induced-dipole electrostatics, was somewhat heuristic, however, and lacking in rigor. Here we present a much more satisfying derivation founded on second-order perturbation theory in the closure approximation and a semiclassical evaluation of the relevant interaction integrals. Expressions for C6, C8, and C10 dispersion coefficients are obtained in a remarkably straightforward manner. Their values agree very well with ab initio reference data on dispersion coefficients between the atoms H, He, Ne, Ar, Kr, and Xe. We also highlight the importance of the exchange-hole contribution to the dispersion coefficients, especially to C6.