Wave Functions forF′Centers in Alkali Halides

The point-ion-lattice model of Gourary and Adrian is used to investigate $F^{\prime }$ centers in alkali halides. Ground-state energies and wave functions are determined by series solution of the Hartree-Fock equation for two electrons in a square-well potential of depth $-\frac{{\alpha }_M{e}^2}{b}$ and range $b$, where ${\alpha }_M$ is the Madelung constant and $b$ the interionic distance. The difference between the point-ion and square-well potentials is subsequently treated as a perturbation, and an additional correction for polarization is derived from the semi-continuum model. An analogous calculation of ground-state energies for $F$ centers is performed in order to determine $F^{\prime }$-band energies. The theoretical $F^{\prime }$-band energies which can be compared with experiment are: NaCl, 2.40 eV; KCl, 2.42 eV; KBr, 1.91 eV. These are in fair agreement with Pick's experimental values: NaCl, 2.43 eV; KCl, 1.65 eV; KBr, 1.38 eV. The wave functions are more diffuse than Pekar's but are still concentrated primarily in the vacancy. In order to verify the method, $F$-band energies were also calculated, in good agreement with experiment.