Exact semi-classical light–matter interaction operator applied to two-photon processes with strong relativistic effects

X-ray processes involve interactions with high-energy photons. For these short wavelengths, the perturbing field cannot be treated as constant, and there is a need to go beyond the electric-dipole approximation. The exact semi-classical light-matter interaction operator offers several advantages compared to the multipole expansion such as improved stability and ease of implementation. Here, the exact operator is used to model x-ray scattering in metal K pre-edges. This is a relativistic two-photon process where absorption is dominated by electric-dipole forbidden transitions. With the restricted active space state-interaction approach, spectra can be calculated even for the multiconfigurational wavefunctions including second-order perturbation. However, as the operator itself depends on the transition energy, the cost for evaluating integrals for hundreds of thousands unique transitions becomes a bottleneck. Here, this is solved by calculating the integrals in a molecular-orbital basis that only runs over the active space, combined with a grouping scheme where the operator is the same for close-lying transitions. This speeds up the calculations of single-photon processes and is critical for the modeling of two-photon scattering processes. The new scheme is used to model Kα resonant inelastic x-ray scattering of iron-porphyrin complexes with relevance to studies of heme enzymes, for which the total computational time is reduced by several orders of magnitude with an effect on transition intensities of 0.1% or less.