The vibration-rotation energies of tetrahedral XY4 molecules : Part I. Theory of spherical top molecules

Abstract

The theory of vibration-rotational perturbations in tetrahedral XY 4 molecules has been reexamined in the light of the modern theory of angular momentum coupling. It is shown that, even to third order of approximation, the splitting of a vibration rotation level into its tetrahedral sublevels is governed only by perturbation terms of one basic symmetry in all states in which vibrational quanta of ν 1, ν 3, and ν 4 are excited and to a certain approximation in many of the infrared active states in which quanta of both ν 2 and ν 3 or ν 4 are excited. The perturbation term is identified as the tetrahedrally symmetric linear combination of fourth rank spherical tensor operators. In dominant approximation the rotational fine structure splitting patterns are characterized solely by the rotational angular momentum of the state. Only the overall extent of the patterns depends on the vibrational and total angular momentum quantum numbers and the vibrational character of the state. In next approximation the basic splitting patterns are all deformed to a certain extent by matrix elements off-diagonal in the rotational angular momentum quantum number. These cannot be neglected if theory is to account for the modern high resolution spectra. The terms of the vibration-rotation Hamiltonian to third order of approximation are classified according to their symmetry. Explicit expressions are given for the pure vibrational energies of the simpler bands. Explicit numerical values are also given for the matrix elements of the rotational sublevels of types A 1, A 2, E, F 1, and F 2 from which the rotational energies of the vibrational ground state and the infrared active fundamentals can be computed. These matrix elements also give the numbers for the basic splitting patterns of the dominant approximation for any state involving combinations of ν 1, ν 3, and ν 4.