Exact nonrelativistic expressions for the tensor for scattering of light by atoms

Exact nonrelativistic analytical expressions are derived for dipole two-photon transitions between arbitrary multiplets of the hydrogen atom and positive hydrogenlike ions. The result is expressed in terms of a single Gauss hypergeometric function and polynomials whose degrees increase linearly with the number of nodes of the bound states of the quantum system. The cross sections of elastic scattering of light by K -and L -shells of the hydrogen atom are given as an example. It is demonstrated that by expanding the discrete-spectrum wave functions in ultraspherical polynomials it is also possible to obtain analytical expressions of the cross sections of two-photon transitions between states described by the Simons model potential. The basis consisting of Chebyshev polynomials is shown to be the best expansion basis, and the coefficients of such an expansion are given for a broad range of parameters of the problem. Calculation of the polarizability of the 5 S -state of the rubidium atom is chosen as an example. Finally, the results are compared with the experimental data and the theoretical results of other researchers.