The assumption of a muffin-tin zero equal to the $d$-state energy, taken from Anderson's muffin-tin-orbital theory, is used in transition-metal pseudopotential theory to derive a simple form for the hybridization matrix element $〈\overrightarrow{\mathrm{k}}\left|\Delta \right|d〉$ in terms of a parameter $r_d$, given for all 27 transition elements. Using this form, matrix elements between $d$ states on neighboring atoms are found to be given by $V_{\mathrm{ddm}}=\frac{{\eta }_{\mathrm{ddm}}{\hslash }^2{r}_d^3}{\left(m{d}^5\right)}$ with ${\eta }_{\mathrm{dd}\sigma }=-\frac{45}{\pi }$, ${\eta }_{\mathrm{dd}\pi }=\frac{30}{\pi }$, and ${\eta }_{\mathrm{dd}\delta }=-\frac{15}{\left(2\mathrm{}\pi \right)}$. Thus for any structure and nearest-neighbor distance $d$, we need in addition only $〈\overrightarrow{\mathrm{k}}\mathrm{}\left|W\right|\mathrm{}\overrightarrow{\mathrm{k}}〉$ (fitted to $d$-band energies calculated earlier) to provide all parameters needed for an elementary band calculation for any transition metal. Results are displayed for all. By replacing the free-electron band in this description by an $\mathrm{sp}$ band and equating appropriate band energies, we identify expressions for matrix elements coupling $d$ states to $s$ and $p$ states. The universal form $V_{\mathrm{ldm}}=\frac{{\eta }_{\mathrm{ldm}}{\hslash }^2{r}_d^{\frac{3}{2}}}{\left(m{d}^{\frac{7}{2}}\right)}$ is obtained but the coefficients obtained are sensitive to the matching procedure. Empirical values ${\eta }_{\mathrm{sd}\sigma }=-3.11\mathrm{}$, ${\eta }_{\mathrm{pd}\sigma }=-2.82\mathrm{}$, and ${\eta }_{\mathrm{pd}\pi }=+1.35\mathrm{}$ accord well with the bands given by Mattheiss for five cubic perovskites.

• Received 2 November 1979

DOI:https://doi.org/10.1103/PhysRevB.21.3214