LAUSR

This paper presents a simple formalism for calculating X-ray absorption (XAS) and resonant inelastic x-ray scattering (RIXS) that has as input orbital-resolved density of states from a single-particle or many-body \textit{ab initio} calculation and is designed to capture itinerant-like features. We use this formalism to calculate both the XAS and RIXS with input from DFT and DFT+DMFT for the recently studied reduced valence nickelates $R_4$Ni$_3$O$_8$ and $R$NiO$_2$ ($R$ a rare earth), and these results are then contrasted with those for the cuprate CaCuO$_2$ and the unreduced nickelate $R_4$Ni$_3$O$_{10}$. In contrast to the unreduced $R_4$Ni$_3$O$_{10}$, the reduced valence nickelates as well as the cuprate show strong orbital polarization due to the dominance of $x^2-y^2$ orbitals for the unoccupied $3d$ states. We also reproduce two key aspects of a recent RIXS experiment for $R_4$Ni$_3$O$_8$: (i) a charge transfer feature between $3d$ and oxygen $2p$ states whose energy we find to decrease as one goes from $R$NiO$_2$ to $R_4$Ni$_3$O$_8$ to the cuprate, and (ii) an energy-dependent polarization reversal of the fluorescence line that arises from hybridization of the unoccupied $z^2$ states with $R$ 5d states. We end with some implications of our results for the nature of the $3d$ electrons in reduced valence nickelates.