Vibration-Rotation Energies of the Planar XY3 Molecular Model

Abstract

The vibration-rotation energies of the planar XY3 molecular model are investigated to the second order of approximation, including the cubic and quartic anharmonic terms of the potential energy, the Coriolis interactions between the degenerate oscillations and the total angular momentum, the dependence of the moments of inertia upon the vibrational state, and the centrifugal expansion terms. The contact transformation developed by Shaffer, Nielsen, and Thomas is applied to the Wilson-Howard Hamiltonian, and it is found that the energies are simply the diagonal elements of the transformed Hamiltonian. The energies are expressed in term value form, E/hc=Gvib+Frot—ζh/4π2cIzz0(l2—l4)K, Gvib, Frot, and ζ being given in terms of the constants appearing in the Hamiltonian. The dependence of the moments of inertia, Ixx and Izz, upon the vibrational states is determined, and it is found that the quantity Δ=Izz—2Ixx is independent of the anharmonic constants, depending only on the dimensions of the molecule, the normal frequencies, and the vibrational state.