On the Asymmetrical Top in Quantum Mechanics

Abstract

An elaboration and a more complete analysis of Witmer's work on the asymmetrical top treated as a perturbation of the symmetrical one show that one can deduce the rigorous solution of the problem from those of the algebraic equations of degree $2j+1$ or less. Without actually solving such equations, we find the terms divisible into even and odd groups just as in the case of the sigma-type doubling in diatomic molecules treated by Kronig, Van Vleck and others. For the case where the asymmetry is slight, an explicit expression for the separation of such similar doublets is obtained. The selection rules, which are rigorous for any degree of asymmetry, consist of the following: (a) Kronig's rule; (b) $\Delta j=0, \pm 1$; $\Delta m=0, \pm 1$; and (c) rule for the quantum number sigma, $\Delta \sigma =\mathrm{even}\mathrm{for}\mathrm{electric}\mathrm{moment}$ in $z$ direction and $\Delta \sigma =\mathrm{odd}\mathrm{for}\mathrm{moment}$ in $x-y$ plane. The effect of the electronic motions on the rotation of a polyatomic molecule as a whole is also briefly discussed.