Nonlinear light scattering in molecules triggered by an impulsive X-ray Raman process

The time-and-frequency resolved nonlinear light scattering (NLS) signals from a time evolving charge distribution of valence electrons prepared by impulsive X-ray pulses are calculated using a superoperator Green's function formalism. The signal consists of a coherent $\sim N^2$-scaling difference frequency generation and an incoherent fluorescence $\sim N$-scaling component where $N$ is the number of active molecules. The former is given by the classical Larmor formula based on the time-dependent charge density. The latter requires additional information about the electronic structure and may be recast in terms of transition amplitudes representing quantum matter pathways.