Uniform quantization conditions in the presence of symmetry: The rotational spectrum ofSF6

Uniform semiclassical quantization conditions are obtained for a one-dimensional Hamiltonian possessing octahedral symmetry. The Hamiltonian describes the rotational dynamics of ${\mathrm{SF}}_6$, a system in which tunneling plays an important role. Quantization conditions are obtained for each symmetry class. These are shown to agree with previously obtained primitive quantization conditions in the small tunneling limit and to reproduce a characteristic periodicity in the symmetries labels. Quantum and semiclassical eigenvalues are computed numerically. Near the classical separatrix the uniform quantization provides orders of magnitude improvement in accuracy over primitive quantization. Our calculation is based on periodic orbit theory, modified here to include classically forbidden reflections and transmissions, and completes a study undertaken in a previous paper. The methods used may be generalized to other Hamiltonians and symmetry groups.