Wave packet theory for non-resonant x-ray emission and non-resonant Auger electron emission in molecules

We present a time-dependent theory for non-resonant x-ray emission spectrum (XES) and normal Auger spectrum (NAS) calculation, based on a fully quantum description of nuclear dynamics using the vibrational wave packet concept. We compare two formulations of the time-dependent theory, either employing a two-time propagation scheme or using spectral integration over the electron energy continuum. We find that the latter formulation is more efficient for numerical simulations, providing a reasonable accuracy when the integration step is shorter than the lifetime broadening of the core-ionized state. We demonstrate our approach using the example of non-resonant x-ray emission from a water molecule, considering the lowest core-ionized K-1 and first core-ionized shake-up K-1V-1V1 intermediate states. These channels exemplify the developed theory on bound-bound, bound-continuum, continuum-bound, and continuum-continuum transitions. Our results suggest that the time-dependent approach is efficient for simulating XES involving dissociative states, whereas the time-independent approach, based on Franck-Condon factors, is more efficient for bound-bound transitions expressed as discrete frequency dependence in the energy domain. The methods and discussion have general applicability, including both NAS and more complex systems, such as liquid water.