Knowledge Bank

The observed zero-point inertial defect $\Delta 0=I0c-I0a-I0n$ is often used as a test of the planarity of a $molecule1-3$, but it is difficult to make this test quantitative when there is limited information. In this work the general behavior of the inertial defect is studied for a variety of molecules. The anharmonic contribution to $\Delta 0$ vanishes, and $\Delta 0$ consists of harmonic and Coriolis contributions (plus small contributions of order $\kappa 1Ic)$: \begin{eqnarray*}\Delta^{hat}{0}&=&\frac{3K}{4I_aI_bI_c}\sum_k\frac{[(I_ba^{aa}k-I_aa^{bb}k)^{2}+I^{2}c(a^{ab}k)^{2}]}{\omega_k},\Delta^{Cor}{0}\ &=&K\sum{k<1}\frac{(\zeta^c{hi})^{2}-(\zeta^a{ki})^{2}-(\zeta^b{ki})^{2}](\omega_k-\omega_i)^{2}}{\omega_k\omega_i(\omega_k+\omega_i)}\end{eqnarray*} where $K=h2/2hc$ $\Delta 0bar$ is always positive. It depends on the same vibrations as the centrifugal constants, and can be estimated fairly accurately from them. The Coriolis part $\Delta 0Cor$ is small and positive (or zero in the case of $X3$) for triatomic molecules, but tends to become increasingly negative for larger molecules. Near-degenerate pairs of vibration make only a small contribution to $\Delta 0Cor$, which is dominated by well-separated pairs of vibrations with large Coriolis coupling constants.