Local electronic structure of rutile RuO$_2$

Recently, rutile RuO$_2$ has raised interest for its itinerant antiferromagnetism, crystal Hall effect, and strain-induced superconductivity. Understanding and manipulating these properties demands resolving the electronic structure and the relative roles of the rutile crystal field and $4d$ spin-orbit coupling (SOC). Here, we use O-K and Ru $M_3$ x-ray absorption (XAS) and Ru $M_3$ resonant inelastic x-ray scattering (RIXS) to disentangle the contributions of crystal field, SOC, and electronic correlations in RuO$_2$. The locally orthorhombic site symmetry of the Ru ions introduces significant crystal field contributions beyond the approximate octahedral coordination yielding a crystal field energy scale of $\Delta(t_{2g})\approx 1$ eV breaking the degeneracy of the $t_{2g}$ orbitals. This splitting exceeds the Ru SOC ($\approx160$ meV) suggesting a more subtle role of SOC, primarily through the modification of itinerant (rather than local) $4d$ electronic states, ultimately highlighting the importance of the local symmetry in RuO$_2$. Remarkably, our analysis can be extended to other members of the rutile family, thus advancing the comprehension of the interplay among crystal field symmetry, electron correlations, and SOC in transition metal compounds with the rutile structure.