Core excitations and ionizations of uranyl in Cs$_{2}$UO$_{2}$Cl$_{4}$ from relativistic embedded damped response time-dependent density functional theory and equation of motion coupled cluster calculations

X-ray spectroscopies, by their high selectivity and sensitivity to the chemical environment around the atoms probed, provide significant insight into the electronic structure of molecules and materials. Interpreting experimental results requires reliable theoretical models, accounting for environment, relativistic, electron correlation, and orbital relaxation effects in a balanced manner. = In this work, following up on prior work for valence processes [A. S. P. Gomes et al., Phys.\ Chem.\ Chem.\ Phys. 15, 15153 (2013)], we present a protocol for the simulation of core excited spectra (with damped response time-dependent density functional theory, 4c-DR-TD-DFT) and ionization energies (with the core-valence separated equation of motion coupled cluster theory, 4c-CVS-EOM-IP) based on the Dirac-Coulomb Hamiltonian, while environment effects are accounted for through the frozen density embedding (FDE) method. We showcase this approach for the uranium M$_4$-, L$_3$-edge and oxygen K-edge of uranyl tetrachloride UO$_2$Cl$_4^{2-}$ in a host Cs$_{2}$UO$_{2}$Cl$_{4}$ crystal. We have found that the 4c-DR-TD-DFT simulations yield excitation spectra that very closely match the experiment for the uranium M$_4$ and oxygen K-edges, with good agreement for the broad experimental spectra for the L$_3$-edge. By investigating the sensitivity of spectral shapes to the lifetimes, and by decomposing each peak into its components, we have been able to correlate our results with angle-resolved spectra. We have observed that for the uranium M$_4$ edge, a simplified model for uranyl tetrachloride-in which an embedding potential replaces the chloride equatorial ligands-closely matches the spectral profile obtained for the uranyl tetrachloride system.