dBands in Cubic Lattices. II

The results of a previous, perturbation theoretic treatment of $d$ bands in the body-centered cubic lattice are extended in several respects: The methods of the previous calculation are applied to determine energy levels at the points $\Gamma$ and $X$ in the Brillouin zone of the face-centered cubic lattice. As before, the crystal potential is that of a lattice of point charges, screened by a uniform distribution of electrons. The perturbation expansion of the wave function of a $d$ electron is developed for the body-centered cubic lattice. Calculations are reported for two states near the top and bottom of the $d$ band, including terms of first order in the potential. These functions have the characteristic property that the wave function of a state near the top of the $d$ band is more compact than that belonging to a state near the bottom. The energies of four states for the body-centered lattice are computed as a function of the binding parameter $\mathrm{Za}$ by a more accurate method than that employed in the previous work, making possible an estimation of the accuracy of perturbation theory and the dependence of bandwidth on binding parameter. The role of crystal field effects in the tight-binding limit is discussed, and the circumstances are determined under which the $d$ band may split into sub-bands based on functions of different cubic symmetries. Estimation of the value of the binding parameter for which such separation occurs strongly suggests that this split does not occur for the actual transition metals. Finally, the effects of spin-orbit coupling on the band structure are studied in the tight-binding approximation. A formulation of k·p perturbation theory for $d$ bands is given.