  In many situations the energy levels for a quantum system whose Hamiltonian
is invariant under a specific symmetry group are split when the Hamiltonian is
replaced by a new one with lower symmetry. In non-rigid molecules quantum
tunnelling processes allow the molecule to change between different geometrical
configurations related by permutations of identical nuclei (or with inversion
as well), resulting in the splitting of the energy levels for the rigid
molecule case where tunnelling is absent. However, for non-rigid molecules
there is apparently a paradoxical situation where although the original rigid
molecule energy levels are associated with a symmetry group isomorphic to the
point group for the geometrical configuration, the split non-rigid molecule
energy levels are associated with a symmetry group consisting of all
permutations and inversions related to the quantum tunnelling between
configurations, and for which the point group is a sub-group. The resolution of
this paradox, where energy level splitting is evidently accompanied by an
enlargement of the symmetry group, is the subject of this article.
