  Resonant inelastic X-ray scattering (RIXS) is a promising technique for
obtaining electron-phonon coupling constants. However, the ability to extract
these coupling constants throughout the Brillouin zone for crystalline
materials remains limited. To address this need, we developed a Green's
function formalism to capture electron-phonon contributions to core-level
spectroscopies without explicitly solving the full vibronic problem. Our
approach is based on the cumulant expansion of the Green's function combined
with many-body theory calculated vibrational coupling constants. The
methodology is applied to X-ray photoemission spectroscopy, X-ray absorption
spectroscopy (XAS), and RIXS. In the case of the XAS and RIXS, we use a
2-particle exciton Green's function, which accounts implicitly for
particle-hole interference effects that have previously proved difficult. To
demonstrate the methodology and gain a deeper understanding of the experimental
technique, we apply our formalism to small molecules, for which unambiguous
experimental data exist. This comparison reveals that the vibronic coupling
constant probed by RIXS is in fact related to exciton-phonon coupling rather
than electron-phonon coupling, challenging the conventional interpretation of
the experiment.
