Interactions between Ordinary Vibrations and Hindered Internal Rotation. I. Rotational Energies

Abstract

The effect of vibration‐hindered rotation interactions on the rotational energy levels of molecules containing a symmetric internal rotor has been re‐examined theoretically. The complete Hamiltonian has been derived and includes terms which are due to Coriolis coupling and to the explicit dependence of the kinetic energy on the angle of internal rotation. Over‐all and internal rotation are separated from vibration, in zeroth order, by means of the Eckart and Sayvetz conditions. Difficulties in previous treatments are traced back to the application of these conditions. For each vibrational state, an approximate Hamiltonian for over‐all and internal rotation is obtained by a Van Vleck perturbation treatment in which torsional denominators are assumed to be negligible. A correction for these denominators is determined by taking the first term in a power series expansion. A preliminary transformation, similar to that described by Hecht and Dennison, is used to remove zeroth‐order coupling between the angular momenta of over‐all and internal rotation. Simple product wavefunctions, then, form a suitable basis for evaluating first‐order energy corrections due to higher‐order coupling of internal and over‐all rotation. Symmetric rotors, slightly asymmetric rotors, and asymmetric rotors with relatively high barriers to internal rotation are each treated separately. Empirical frequency expressions for rotational transitions of all three types of molecule are given. The symmetric rotor formula is identical in form to Kivelson's but the empirical constants must be interpreted differently. A symmetric rotor analysis was done on CH₃SiF₃ and CD₃SiF₃. K→K, J—1→J transitions of slightly asymmetric rotors obey a particularly simple frequency relation. Two groups of asymmetric rotor transitions are tractable: (1) The 0_0,0→1_0,1, ½[1_1,0→2_1,1+1_1,1→2_1,2], and 2_2,1→3_2,2 transitions, where the designation of levels is for a prolate rotor, and (2) transitions between the two members of an asymmetry doublet.