The Vibration-Rotation Energy Levels of Polyatomic Molecules I. Mathematical Theory of Semirigid Asymmetrical Top Molecules

Abstract

The exact classical kinetic energy for a system of point masses is obtained. From this the correct form for the quantum-mechanical Hamiltonian operator is derived. If the assumption of small vibrations is applied to this operator, the familiar approximation of a rigid top plus normal coordinate vibrator is obtained. In order to secure better approximations, in which larger amplitudes of vibration are admitted, a perturbation method is introduced which permits the change of moment of inertia with vibration, the coupling of rotation and vibration, and the centrifugal stretching effects to be taken into account. If the stretching terms alone are neglected, it is possible to reduce the secular equation for the rotational energy levels to the Wang form, except that ``effective moments of inertia'' must be used whose magnitude depends on the vibrational quantum state. The relation of these quantities to the equilibrium moments of inertia or to the instantaneous moments of inertia averaged over the vibrational motion is not simple, although the numerical deviation from them may not be great. In addition, for molecules with less than orthorhombic symmetry there is the further possibility that the orientation of the principal axes of inertia will vary with the vibrational quantum number. It is also pointed out that the Wang equation should not fit the data when a nearby vibrational state perturbs the state under examination or when the centrifugal effects are large. A method is indicated whereby the latter terms may in principle be calculated.