Knowledge Bank

The vibrational-rotational energy levels of the various isotopic forms of a diatomic molecule, usually to a very good approximation, may be written in terms of a single function $E(\eta ,\xi )$, where \begin{eqnarray*} \eta = \frac{v+\frac{1}{2}}{\sqrt{a}} = \hbox{mass reduced vibrational quantum number} \ \xi=\frac{J(J+1)}{\mu} = \hbox{mass-reduced rotational quantum number}. \end{eqnarray*} Most generally, $\xi$ occurs in the effective potential in the radial Schr\”{o}dinger equation as a natural parameter (e.g. in a perturbation expansion) while $\eta$ arises from the semiclassical quantization condition (in atomic units: $\eta =me=e=1)$ $$ \eta = \frac{v+\frac{1}{2}}{\sqrt{\mu}}=\sqrt 2 \int \limits ^{^{r}E,\ \xi+}_{^{r}E,\ \xi-} dr\left[E- V(r) - \frac{\xi}{2r^{2}}\right]^{\frac{1}{2}}+0(\mu^{-1}).$$ More explicitly, the Dunham expansion and the behavior of energy levels near dissociation can be straightforwardly written in mass-reduced form; and G($\eta$), Birge-Sponer $(\Delta G(\eta ))$, and $\mu B(\eta )$ plots seem to indicate smooth behavior for intermediate energy levels. The corresponding isotopically-combined results (e.g. spectroscopic constants, potential energy curves) are more precisely determined than are the corresponding results for the various isotopic forms individually.