Efficient calculation of Rayleigh and Raman scattering

We present two methods for computing the Rayleigh and Raman scattering cross sections for photon scattering on atomic hydrogen or hydrogenlike systems. Both methods are applicable for incident photon energies above the ionization threshold. The first method implements the well-known Gaussian quadrature approach to deal with principal value integration and relies on evaluation of the exact eigenfunctions of hydrogen. The second, more computationally efficient approach uses a finite-${L}^{2}$ basis expansion of the target and applies complex exterior scaling methods to accurately account for the contribution of the intermediate continuum states. This method is much more general in that it does not rely on analytic solutions to the Hamiltonian, or evaluation of any special functions, and is expected to be applicable to more complex systems where exact wave functions are cumbersome to evaluate. Both methods are in complete agreement with previous work based on analytical representations of the Green's function or the dipole matrix elements. Rayleigh, Raman, and photoionization cross sections for scattering on the first few excited states of atomic hydrogen are presented and compared with previous results where available.