A modification of the point-ion model is proposed which provides an approximate correction for ion-size effects. The difference between the optimum pseudopotential for the smoothest pseudo-wave-function and the point-ion potential is treated as an ion-size correction to the Hamiltonian, appropriate to a smooth variational trial function. By neglecting the variation of the trial function over the ion cores in the evaluation of matrix elements, one obtains a simple, approximate form for the pseudopotential: $V_p=V_{\mathrm{PI}}+{\Sigma }_{\gamma }[A_{\gamma }+(\overline{V}-U_{\gamma }\left)B_{\gamma }\right]\delta (\mathrm{r}-{\mathrm{r}}_{\gamma })$, where $V_{\mathrm{PI}}$ is the point-ion potential, $U_{\gamma }$ is the potential at ion $\gamma$ due to the other ions, and $\overline{V}$ is the average potential. The coefficients $A_{\gamma }$ and $B_{\gamma }$ are properties of the ions alone, and have been computed for a large set of ions. The approximate pseudopotential is applied to the calculation of ionization potentials of alkali atoms, where it works well, and of $F$-band energies in alkali halides and alkaline-earth fluorides, where it is found that all of the coefficients $A_{\gamma }$ must be reduced by a factor of 0.53 in order to obtain agreement with experiment. With the adjusted pseudopotential coefficients, the theory accounts well not only for the Ivey law in alkali halides, but also for deviations from the Ivey law. In addition, the seemingly anomalous $F$-band energy in Ba${\mathrm{F}}_2$ is accounted for. The empirically adjusted constants may be useful in other color-center problems as well.

• Received 29 July 1968

DOI:https://doi.org/10.1103/PhysRev.176.1014