The augmented-plane-wave (APW) band structure of Re${\mathrm{O}}_3$ is analyzed in terms of the Slater-Koster linear-combination-of-atomic-orbitals (LCAO) interpolation scheme with nonorthogonal orbitals. This approach has several advantages over an earlier treatment involving orthonormal basis functions. First, it provides insight into the physical origin of the crystal-field splittings in Re${\mathrm{O}}_3$ and other transition-metal compounds. Second, it leads to more physically meaningful LCAO parameters. Finally, it provides a direct relationship between the crystal-field levels of an isolated transition-metal ion or molecular complex and the band structure of the periodic crystal. In the case of Re${\mathrm{O}}_3$, it is shown that the crystal-field effects are due to overlap and covalency between the rhenium $5d$ orbitals and the $2s$, $2p\sigma$, and $2p\pi$ orbitals of the neighboring oxygen ligands. The splitting between the $e_g$ and $t_{2g}$ bands is due to the $2s$ contribution ${\Delta }_s$. The difference between the $2s$ and $2p\sigma$ contributions ${\Delta }_{\sigma }-{\Delta }_s$ is responsible for the $e_g$ bandwidth. This same difference is proportional to the effective transfer integral $b$ in Anderson's theory of superexchange. The $t_{2g}$ bandwidth is due to ${\Delta }_{\pi }$, the $2p\pi$ overlap-covalency parameter. This LCAO method is applied to KNi${\mathrm{F}}_3$, using LCAO integrals determined from the Sugano-Shulman molecular-orbital calculation for the ${\left(\mathrm{N}\mathrm{i}{\mathrm{F}}_6\right)}^{4-}$ complex. The resulting KNi${\mathrm{F}}_3$ band structure is qualitatively similar to the APW results for Re${\mathrm{O}}_3$, except the bandwidths are narrower by about a factor of 4. In the limit where the Coulomb energy $U$ is large compared with the $e_g$ and $t_{2g}$ bandwidths so that the electrons localize, it is shown that the crystal-field splitting between the localized $e_g$ and $t_{2g}$ Wannier functions is identical with that obtained by Sugano and Shulman via the molecular-orbital method.

• Received 22 April 1970

DOI:https://doi.org/10.1103/PhysRevB.2.3918