  We systematically investigate the underlying relations among different levels
of approximation for simulating electronic de-excitations, with a focus on
modeling X-ray emission spectroscopy (XES). Using Fermi's golden rule and
explicit modeling of the initial, core-excited state and the final,
valence-hole state, we show that XES can be accurately modeled by using orbital
optimization for the various final states within a Slater-determinant
framework. However, in this paper, we introduce a much cheaper approach reliant
only on a single self-consistent field for all the final states, and show that
it is typically sufficient. Further approximations reveal that these
fundamentally many-body transitions can be reasonably approximated by
projections of ground state orbitals, but that the ground state alone is
insufficient. Furthermore, except in cases where the core-ionization induces
negligible changes in polarization, linear-response approaches within the
adiabatic approximation will have difficulty in accurately modeling
de-excitation to the core level. Therefore, change in the net dipole moment of
the valence electrons can serve as a metric for the validity of the
linear-response approximation.
