  To fully capitalize on the potential and versatility of resonant inelastic
x-ray scattering (RIXS), it is essential to develop the capability to interpret
different RIXS contributions through calculations, including the dependence on
momentum transfer, from first-principles for correlated materials. Toward that
objective, we present new methodology for calculating the full RIXS response of
a correlated metal in an unbiased fashion. Through comparison of measurements
and calculations that tune the incident photon energy over a wide portion of
the Fe L$_3$ absorption resonance of the example material BaFe$_2$As$_2$, we
show that the RIXS response in BaFe$_2$As$_2$ is dominated by the direct
channel contribution, including the Raman-like response below threshold, which
we explain as a consequence of the finite core-hole lifetime broadening.
Calculations are initially performed within the first-principles Bethe-Salpeter
framework, which we then significantly improve by convolution with an effective
spectral function for the intermediate-state excitation. We construct this
spectral function, also from first-principles, by employing the cumulant
expansion of the Green's function and performing a real-time time dependent
density functional theory calculation of the response of the electronic system
to the perturbation of the intermediate-state excitation. Importantly, this
allows us to evaluate the indirect RIXS response from first-principles,
accounting for the full periodicity of the crystal structure and with
dependence on the momentum transfer.
