Estimation of bond-bending force constants in tetrahedral semiconductors and their variation with pressure obtained from spectroscopic data

The bond-bending force constant $k_{\theta }$ of the Keating-Martin (KM) model for the elastic constants of tetrahedral semiconductors is related to the optical spectrum according to $k_{\theta }{r}^2=4(1-f_i)\Delta E_m$, where $r$ is the bond length, $f_i$ is the Phillips fraction of ionic character, and $\Delta E_m$ is the reduction in single-particle energy per valence electron resulting from the over-all band gap. As the energy to bend a directed orbital is unrelated to the energy of competing phases, there is no $d$-core dehybridization or bottom of the band correction. The above formula is accurate to the limit of the KM model when $f_i\le 0.5$. For more ionic compounds, it seems that $k_{\theta }{r}^2=2\mathrm{}\Delta U_I$, where $\Delta U_I$ is the change in internal energy per atom pair upon transformation to the NaCl structure. The observed variation of $k_{\theta }$ with pressure is indeed equal to the value found by differentiation of the above formula. An approximate relation for Keating's third-order elastic parameter is derived: $\epsilon =-\frac{3k_{\theta }}{2r}$.