Rotational tunnelling of N coupled methyl groups

Abstract

The influence of orientational coupling between rotational tunnelling molecular groups on the tunnelling spectrum is studied in the limit of strong coupling. In previous work contradicting results were obtained depending on the calculation methods: exact numerical computations for two or three rotors find an increasing relative splitting of the tunnelling lines (i.e. measured on the scale of the tunnelling energy itself) with increasing coupling strength, whereas Hartree approximations aways yield a vanishing splitting in this limit. In order to clarify this point, a model of N equivalently coupled XH3 groups is investigated. The relative splitting is calculated for two types of tunnelling elements prevailing in the strong coupling limit. One of the matrix elements is connected with the motion of only one rotor and is shown to lead to unsplit tunnelling lines. The other matrix element involves the collective motion of all rotors and it is demonstrated that it causes the relative splittings of the tunnelling lines. Within a WKB-type argument the importance of these two matrix elements for the tunnelling spectrum are compared; for N<or=3 a dominance of the coupling influence, and for N>or=5 a dominance of the single-particle influence is found. The case N=4 turns out to be marginal, insofar as neither the contributions from the classical paths nor the contributions from the quadratic fluctuations around the classical paths distinguish between both matrix elements. Therefore even in the strong coupling limit a finite relative splitting of the tunnelling line should remain observable for N=4.