Toward Accurate RIXS Spectra at Heavy Element Edges: A Relativistic Four-Component and Exact Two-Component TDDFT Approach.

We present a relativistic time-dependent density functional theory (TDDFT) approach for the simulation of resonant inelastic X-ray scattering (RIXS) spectra, based on both a full four-component (4c) Dirac-Coulomb Hamiltonian and a modern atomic mean-field exact two-component (amfX2C) Hamiltonian model. The approach builds on the pseudo-wavefunction formalism and a core-valence separation scheme, enabling the efficient evaluation of couplings between two manifolds of excited states relative to a common ground state, as required for solving the Kramers-Heisenberg equation for RIXS. The relativistic formulation provides a variational description of scalar and spin-orbit relativistic effects, which are essential for accurately describing inner-shell excitations involved in RIXS processes. Its transformation to the 2c regime via the amfX2C Hamiltonian significantly reduces the computational cost while offering 4c-quality results by accounting for two-electron and exchange-correlation picture-change effects arising from the X2C transformation. In addition to two-dimensional RIXS maps, the methodology enables the direct evaluation of high-energy-resolution fluorescence detection (HERFD) and resonant X-ray emission spectra (RXES). Applications to 2p3d and 3d4f RIXS maps of selected ruthenium and uranium complexes demonstrate that the amfX2C approach reproduces reference 4c results and experimental spectra with high accuracy, capturing all key spectral features and providing reliable peak assignments.