A kinetic theory of the temperature dependence of methyl tunnelling spectra

The spectral density function associated with methyl group reorientation at low temperatures is relatively complicated as a result of quantum tunnelling motion, and due to the existence of three proton spin symmetry species. To account for different aspects of the observed temperature dependence of this spectral density function, two models have been proposed (by J. Haupt, 1971 and by P.S. Allen, 1974) both of which assume that the spectral density is modified by thermally excited transitions between different methyl group torsional levels. The models are incompatible in that the torsional transitions of different spin symmetry species are uncorrelated in one case (Haupt) and correlated in the other (Allen). Both models fail in that each explains only one of the two main features of the spectrum. In the present discussion it is shown that the torsional states are mixed by thermal fluctuations in a way which cannot be described adequately by torsional transitions. A description of the mixing is given which yields an explanation similar to that of Allen for the temperature dependence of the tunnel frequency, but since the evolution of mixed torsional states belonging to different spin symmetry species may be essentially uncorrelated, a broadening similar to that invoked by Haupt is also obtained.