A correspondence principle interpretation of the dynamics of classical non-separable Hamiltonian systems has led Percival (1973) to predict that for such systems there are two regions of the quantal energy spectrum with contrasting properties. Polyatomic molecules are described by non-separable Hamiltonians and with advancing experimental methods the relevance of Percival’s predictions to molecular spectra is apparent. In this thesis eigenvalues for the Hénon-Heiles non-separable Hamiltonian are obtained using the full symmetry of the potential and it is shown that two regions of the quantal energy spectrum exist which behave differently under a slowly changing perturbation. Such behaviour is required by Percival for the existence of regular and irregular spectra.
This thesis also tackles the problem of determination of regular energy levels of polyatomic molecules. Semiclassical methods based on Einstein-Brillouin-Keller-Maslov (E3KM) quantization and a classical variational principle are described. They are applied to model potentials of up to two coordinates and promise to be effective for the determination from potential surfaces of large numbers of vibrational energy levels of suitable polyatomic molecules at energies intermediate between equilibrium and dissociation
