The Energy Levels of a Rotating Vibrator

TLDR

Correction terms modify the standard vibrational and rotational coefficients (ωe, Be) in the energy expression. The authors employ the WKB method to calculate energy levels, derive formulas for the correction terms and isotope shifts, and present a technique to extract potential functions from band spectra using Morse potentials. The study identifies new correction terms of order \(B_e^2/\omega_e^2\) that slightly alter ωe and Be, detectable only in light molecules such as H₂ and NaH.

Abstract

The energy levels of a rotating vibrator are calculated in considerable detail by means of the Wentzel-Brillouin-Kramers method. The new terms determined are ${\ensuremath{\omega}}_{e}z$ and a set of correction terms which appear in the earlier members of the equation. These correction terms enter in such a way that ${\ensuremath{\omega}}_{e}$ is not exactly the coefficient of ($v+\frac{1}{2}$); ${B}_{e}$ is not exactly the coefficient of $K(K+1)$, etc. However the differences are small and are detectable only in the case of light molecules. The correction terms are of the magnitude of $\frac{{{B}_{e}}^{2}}{{{\ensuremath{\omega}}_{e}}^{2}}$. Formulas for the effect of the correction terms on isotope shifts are given, and for the calculation of the correction terms themselves. Also a method is given for obtaining actual potential functions from band spectrum data, based on Morse's potential function. Finally the numerical magnitude of the correction terms for several states of ${\mathrm{H}}_{2}$ and for NaH is discussed.