dBands in the Body-Centered Cubic Lattice

A model of a crystal, consisting of positive point charges neutralized by a uniform distribution of negative charge, is employed to study the form of the $d$ bands in a body-centered cubic lattice as a function of the lattice spacing. The wave functions are expressed as linear combinations of plane waves and the potential treated as a perturbation. It is shown that the perturbation series for the energy is a power series in $\mathrm{Za}$, where $Z$ is the atomic number and $a$ is the lattice parameter. The leading term in the series is of the order ${\left(\frac{1}{a}\right)}^2$, and the coefficients of successive terms in the series decrease rapidly. The first three terms are evaluated for the states of predominantly $d$ symmetry at the center of the Brillouin zone, and the corner $H$.