A generalization of the Kramers-Heisenberg dispersion formula

A generalization of Dirac's transformation of the off-resonance single-photon scattering amplitude that leads to the Kramers-Heisenberg dispersion formula is described. The amplitude is first calculated to order $e^2$ in perturbation theory with the minimal-coupling interaction Hamiltonian as perturbing energy, and is then transformed so that the effects of multipole transition moments of all orders and types (electric, magnetic, and diamagnetic) readily become manifest. No appeal is made to any multipolar expansions, however, as exact integral expressions are used for the summed series. It is shown that the same dispersion formula is obtained by using the multipolar Hamiltonian partitioned in the usual way, and this despite the fact that the unperturbed Hamiltonians are different in the two calculations.