The Vibrations of Pentatonic Tetrahedral Molecules

Andrews has suggested that the restoring forces in polyatomic molecules can best be chosen as harmonic restoring forces along the directions of the chemical bonds and prependicular to them. In order to test the suggestion, we have calculated the vibrational frequencies of tetrahedral molecules with this choice of forces. The agreement between calculated and observed values is unsatisfactory. It seems that there may be repulsive forces between the corner atoms of the type between ions of crystals or the inert gas atoms. The introduction of terms in the potential energy proportional to $\frac{1}{{r_j}^n}$, where $r_j$ is the distance between two corner atoms, makes it possible to secure very good agreement between calculated and observed frequencies, in the case of C${\mathrm{Cl}}_4$, Si${\mathrm{Cl}}_4$, Sn${\mathrm{Cl}}_4$, C${\mathrm{Br}}_4$, and Sn${\mathrm{Br}}_4$ but not such close agreement in that of Ti${\mathrm{Cl}}_4$. The calculated frequencies are not very sensitive to the value of $n$ which may be anywhere from 5 to 9. The repulsive forces necessary are of the same order of magnitude as those calculated from crystal properties and the viscosities of the inert gases. In the case of the S$\mathrm{O}_4^{}{}_{}{}^{=}$ and Cl$\mathrm{O}_4^{}{}_{}{}^{-}$, the inverse high power repulsive force is not sufficient, but the addition of terms, $\frac{e^2}{r_j}$, $e$ being the electronic charge, as well as an inverse higher power term does give very good agreement between calculated and observed frequencies of these molecules.