The symmetry groups of non-rigid molecules as semi-direct products

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Abstract
The possibility of formulating the symmetry groups of non-rigid molecules (as defined by Longuet-Higgins) as products of subgroups is investigated. For the most general case of non-rigidity, no such formulation is possible. When the non-rigidity results from internal rotation about one or more axes, the molecular symmetry group may be written as a semi-direct product of an invariant torsional subgroup, which is a direct product of cyclic groups, and a ‘frame subgroup’, which is isomorphous with the point group of the ‘frame’ formed by detaching all the nuclei undergoing internal rotation. A correlation table may be drawn up linking the representations of the torsional and frame subgroups with those of the molecular symmetry group. This table may be used to classify the various factors of the molecular wave-function, without the need to perform the tedious construction of the molecular symmetry group. The method is extended to molecules containing rotating groups which in turn contain more rotating groups (‘nested’ internal rotation). In one of the appendices, a similar treatment is applied to N.M.R. symmetry groups.

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