Through approximate solutions to the equation of motion for
density matrices coupled with the first-principles Hamiltonian,
we derive a rate equation describing time evolutions of
core-valence photoexcitation and collisional relaxation in metals
irradiated by VUV/x-ray free-electron lasers.
We discuss how to treat many-body effects such as energy-level
renormalization, absorption-edge shift and electron screening
caused by interactions among electrons and core holes.
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