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.