Exact low-energy expansion of the Rayleigh scattering cross-section by atomic hydrogen

We present an exact low-energy expansion redward of Lyα of the Rayleigh scattering cross-section by atomic hydrogen, which is given by the Kramers–Heisenberg formula. The expansion is expressed as a power series of  (ω/ω_l)  , where ω_l and ω are the angular frequencies corresponding to the Lyman limit and the incident radiation, respectively. The expansion coefficients are represented as infinite sums over all the intermediate states and they can also be expressed as the regular particular solutions of inhomogeneous differential equations, which is known as the Dalgarno and Lewis method. In this paper, using a software capable of symbolic calculations, we obtain the exact values of these coefficients. We provide brief discussions on the accuracies of approximate expressions for Rayleigh scattering by a hydrogen atom found in the literature.