  X-ray spectroscopy is an important tool for the investigation of matter. X
rays primarily interact with inner-shell electrons creating core (inner-shell)
holes that will decay on the time scale of attoseconds to few femtoseconds
through electron relaxations involving the emission of a photon or an electron.
The advent of femtosecond x-ray pulses expands x-ray spectroscopy to the time
domain and will eventually allow the control of core-hole population on
timescales comparable to core-vacancy lifetimes. For both cases, a theoretical
approach that accounts for the x-ray interaction while the electron relaxations
occur is required. Here we describe a time-dependent framework, based on
solving the time-dependent Schr\"odinger equation, that is suitable for
describing the induced electron and nuclear dynamics.
