$L_2$,3 x-ray-absorption edges of $d^0$ compounds: ${\mathrm{K}}^{+}$, ${\mathrm{Ca}}^{2+}$, ${\mathrm{Sc}}^{3+}$, and ${\mathrm{Ti}}^{4+}$ in $O_h$ (octahedral) symmetry

The $L_2$,3 x-ray-absorption edges of 3$d^0$ compounds are calculated with use of an atomic description of the 2$p^6$3$d^0$ to 2$p^5$3$d^1$ excitation, with the inclusion of the crystal field. For reasons of clarity, we confine ourselves to $d^0$ compounds in octahedral symmetry, but the same approach is applicable to all other $d^N$ compounds in any point-group symmetry. The experimental spectra of ${\mathrm{FeTiO}}_3$, ${\mathrm{Sc}}_2$ ${\mathrm{O}}_3$, ${\mathrm{ScF}}_3$, ${\mathrm{CaF}}_2$, and the potassium halides are well reproduced by the present calculations, including the previously misinterpreted small leading peaks. The splitting between the two main peaks in both the $L_3$ and $L_2$ edge are related, though not equal, to the crystal-field splitting. Comparison to experiment showed that the broadening of the main multiplet lines is different. This can be related to Coster-Kronig Auger processes for the $L_2$ edge and to a solid-state broadening which is a combination of vibrational (phononic) and dispersional broadenings. With the full treatment of the atomic multiplets, the atomic effects can be separated from solid-state effects, which offers a better description of the latter. This includes vibrational broadenings, the covalent screening of the intra-atomic Coulomb and exchange interactions, via the position of small leading peaks, and surface effects. The same general framework can be used to discuss crystal-field effects in both lower symmetries, with the possibility of polarization-dependent spectra (e.g., ${\mathrm{TiO}}_2$), and partly filled d bands.