Total energies in the tight-binding theory

The density-functional theory of interaction between closed-shell atoms is simplified sufficiently to be carried out analytically. The resulting form is $V_0\mathrm{}\left(d\right)=-{\eta }_0{\epsilon }_p\mu d\exp \left(\frac{-5\mathrm{}\mu d}{3}\right)$, where the valence $p$-state energy is related to $\mu$ by ${\epsilon }_p=\frac{-{\hslash }^2{\mu }^2}{2m}$. In universal-parameter tight-binding theory of ionic crystals a band-structure energy is added, but no Madelung energy. ${\eta }_0$ is adjusted from its theoretical value of 71 to give the correct equilibrium spacing for the potassium halides, giving ${\eta }_0=44,86,103\mathrm{}$, and 146 for $2p$, $3p$, $4p$, and $5p$ anions, respectively. Then using average values of $\mu$ based upon Herman-Skillman term values the equilibrium spacing, cohesive energy, and elastic constants are predicted for the monovalent, divalent, and trivalent compounds in the rocksalt structure. Such a theory based upon tight-binding energies rather than Madelung energies provides an alternative to the Born theory. It is of comparable accuracy but has much wider applicability. The theory also predicts an approximate equality between the band gap and the cohesive energy per ion pair for the alkali halides and between twice the band gap and the cohesive energy for their divalent counterparts, in reasonable accord with experiment. By identifying the overlap interaction in covalent solids with nonorthogonality terms in tight-binding theory it is shown that direct application of the closed-shell theory to covalent solids underestimates the kinetic-energy term by a factor of $2^{2/3}$. Making this correction to the earlier calculations for Si and Ge by Harrison and Sokel brings them into reasonable accord with experiment. This overlap interaction is then approximated by the same analytic form as for closed-shell systems but based upon hybrid, rather than $p$-state, parameters. The coefficient ${\eta }_0$ is adjusted, from its theoretical value of 47, to 46 for C, 57 for Si, 61 for Ge, and 73 for Sn in order to fit the observed atomic spacing. The resulting overlap interaction plus the bonding energy then gives direct predictions of the cohesion and bulk modulus for these elements, reproducing the observed trends. The theory is extended to polar semiconductors by again identifying the overlap interaction with the nonorthogonality terms of tight-binding theory. This suggests a nonorthogonality $S=\left(\frac{1}{2}\right){\alpha }_c^h$, where ${\alpha }_c^h$ is the hybrid covalency of the compound, and a replacement of ${\eta }_0$ by ${\eta }_0{\alpha }_c^h$ for the overlap interaction of the polar semiconductors. The latter leads to reasonable predictions, without further parameters, of the equilibrium spacing, cohesion, and bulk modulus for these systems. The dependence of $S$ on covalency is not supported by detailed calculation. The identification of the overlap interaction with the nonorthogonality terms also suggests an approximate relation between the average of the