Unraveling the electronic structure and the oxygen $K$-edge x-ray absorption near-edge structure spectrum of DyFeO$_3$
Rare-earth orthoferrites such as DyFeO$_3$ exhibit a rich interplay between localized rare-earth and transition-metal moments, giving rise to complex magnetic phases and magnetoelectric phenomena. Understanding their electronic and spectroscopic properties from first principles requires an accurate description of the localized Fe-$3d$ and Dy-$4f$ states and their hybridization with O-$2p$ states. Here, we combine first-principles calculations and x-ray absorption near-edge structure (XANES) measurements to investigate the electronic structure and the O $K$-edge spectrum of DyFeO$_3$. We employ density-functional theory (DFT) with Hubbard $U$ corrections (DFT+$U$) determined from first principles using density-functional perturbation theory, the HSE06 hybrid functional, and orbital-resolved DFT+$U$ with Hubbard parameters calibrated to reproduce the electronic structure computed using HSE06. We find that standard DFT+$U$, despite improving the band gap, substantially underestimates the crystal-field splitting of the unoccupied Fe-$3d$ states and consequently fails to accurately reproduce the separation of the two lowest-energy features in the O $K$-edge spectrum. HSE06 provides a more balanced description of the relevant electronic states, including the Fe-$3d$ crystal-field splitting. Mapping the corresponding electronic structure onto orbital-resolved DFT+$U$ yields a spectrum that remarkably reproduces the two lowest-energy experimental features and captures the main characteristics of the spectrum at higher energies. These results establish the low-energy O $K$-edge features as a sensitive probe of the Fe-$3d$ crystal-field splitting, while showing that the localized Dy-$4f$ states leave no distinct spectral fingerprints despite their importance for the magnetic properties.