3.1 Qualitative analysis of classial limits of quantum structures began with the late 1970's work on rotational energy surfaces and has since been extended to vibrational polyads and electronic states. These techniques will be developed further in this project, using recent results on singular reduction, and extended to mixed quantum-classical rotational energy surfaces and models which combine rotational energy surfaces with vibrational polyads. The links between re-arrangements of these structures and quantum monodromy will also be explored. Quantum monodromy has been shown to be present in physical systems, such as water molecules and hydrogen atoms in electric and magnetic fields, and current indications are that it will become as widely accepted as a universal phenomenon as the quantum Hall effect.
3.2 The realisation that classical REs are important organizing centres for molecular spectra has been growing over the past few years culminating in the computation of global RE bifurcation diagrams for a number of small molecules and quantum predictions based on these. Periodic orbits have similarly been used as organizing centres for highly excited spectra with angular momentum J = 0 and global bifurcation diagrams computed. However there has been virtually no analogous work on RPOs for J > 0, one of the aims of this project.
3.3 Semiclassical methods for obtaining quantum spectra and wavefunctions from classical inputs have attracted renewed interest in recent years. They are especially useful for unveiling `hidden' structures in quantum spectra due to stable periodic orbits, approximate tori and short unstable periodic orbits. Quantitative results can be obtained by direct semiclassical quantisation or scaled Fourier transformation of quantum spectra. However relatively little work has been done on classical systems with more than 2 degrees of freedom. Semiclassical studies of the integrable and near-integrable dynamics (including Arnold diffusion) of atomic and molecular systems such as helium, hydrogen in electric and/or magnetic fields and water will be undertaken in this part of the project.