Efficacious Form for Model Pseudopotentials

A simple way has been discovered to put model pseudopotentials, $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})={\ensuremath{\Sigma}}_{\mathrm{lm}}|{Y}_{\mathrm{lm}}〉{V}_{l}(r)\ifmmode\times\else\texttimes\fi{}〈{Y}_{\mathrm{lm}}|$, into a form which reduces the number of integrals of $V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ required for an energyband calculation from $\frac{\mathrm{mn}(n+1)}{2}$ to $\mathrm{mn}$ for each $l$ in the sum (where $n$ is the number of plane waves used in the expansion and $m$ the number of points in the Brillouin zone at which the calculation is performed). The new form may be chosen to improve the accuracy of the pseudopotential when used in other chemical environments.