At low energies, close to the bottom of the potential energy surface, the vibrational motion of a molecule is well described by the normal modes formalism. However, when the energy increases, non-linear couplings (like the Fermi or the Darling-Dennison resonances) set in, which eventually lead to bifurcations. Bifurcations are real discontinuities of the classical phase space, where fundamental families of periodic orbits are created, destroyed, or change their stability properties. Since the classical periodic orbits form the backbones of the quantum mechanical wave functions, classical bifurcations obviously make the quantum spectrum more complicate... and more interesting (see the figure above).

Since experimental set-ups and computer allow the investigation of molecules at increasingly higher energies, the description of bifurcations and the corresponding changes in the quantum spectrum has become an important issue of modern molecular physics : all the molecules we have studied over extended energy ranges (e.g. all the way up to dissociation for HOCl and HOBr and to isomerization for HCP and DCP) were shown to display several bifurcations, essentially of the saddle-node type. As a consequence, new modes are born at intermediate energies, which do not exist close to the bottom of the surfaces and play a special role as precursors of the isomerization and dissociation reactions.

Although I sometimes perform classical calculations on ab initio potential energy surfaces, like for the HOCl and ozone molecules, my favorite tool for studying bifurcations at high vibrational energies is the semi-classical analysis of simple integrable models, obtained for example from canonical perturbation theory, which are in very good agreement with the ab initio Hamiltonian up to close to the thresholds. Most of the time, these calculations are compared to exact quantum calculation performed in R. Schinke's group in Göttingen and to classical calculations performed in S. Farantos' group in Iraklion.