Simulation of Resonance Raman X-Ray Scattering Signals from Nonadiabatic Quantum Dynamics. Approximated Protocols in the Limit of Short Lifetimes of Core-Excited States

Time-resolved resonant inelastic X-ray scattering (RIXS) provides a sensitive probe of photoinduced nonadiabatic dynamics. Its direct simulation, however, is computationally demanding, as it formally requires nested loops of nuclear wave-packet propagation on the coupled valence states and on the  core-excited states. Here, we derive a hierarchy of computationally efficient approximations to the time-dependent RIXS signal by exploiting the separation of timescales between the ultrashort core-hole lifetime and the slower nuclear and electronic dynamics. Starting from the nonstationary generalization of the Kramers-Heisenberg expression, we approximate the Raman transition operator through short-time expansions of the intermediate-state propagators, replacing explicit propagation on the core-excited manifold by instantaneous expectation values of valence-to-core energy gaps. We further derive first- and second-order approximations for the scattering correlation function in terms of the evolving valence-state population and low-order moments of the valence-state energy gap. In the energy-gain region, the resulting RIXS signal reduces to closed-form lineshape functions whose positions follow the time-dependent valence-state energy gaps and whose intensities track the population of the upper valence state. The proposed formulation therefore reduces time-resolved RIXS simulations to an inexpensive post-processing of nonadiabatic quantum dynamics while retaining a direct connection between the signal and the underlying electronic and nuclear dynamics.