Core excitations of uranyl in Cs$_{2}$UO$_{2}$Cl$_{4}$ from relativistic embedded damped-response time-dependent density functional theory 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, we present a protocol for the simulation of core excited spectra with damped response time-dependent density functional theory based on the Dirac-Coulomb Hamiltonian (4c-DR-TD-DFT), in which 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$_4^{2-}$) unit as found 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 decomposing the complex polarizability in terms of its components we have been able to correlate our results with angle-resolved spectra. We have observed that for all edges, but in particular the uranium M$_4$-edge, an embedded model in which the chloride ligands are replaced by an embedding potential, reproduces rather well the spectral profile obtained for UO$_2$Cl$_4^{2-}$. Our results underscore the importance of the equatorial ligands to simulating core spectra at both uranium and oxygen edges.