It is well known that experimental information on spectral transitions of simple atoms was one of the keystone facts leading to formulation of quantum mechanics. Nowadays, the initial understanding of the energy level systems of molecules is based on several initial quantum models. Electronic structure is largely related to hydrogen atom structure with characteristic excitations being signicantly larger than typical vibrational excitations, normally described by a system of harmonic oscillators. More fine structure due to rotational excitations is described in the simplest approximation by a rigid rotator model.
The purpose of my lecture is to go beyond these simple models and to describe the qualitative approach which enables one to relate generic patterns of energy levels typically observed for rovibrating molecules, as well as a generic evolution of these patterns with variation of the integral of motion values (energy, angular momenta, etc) and external parameters (field strengths, etc). The main information used in construction of such models is the symmetry and topology.
I start with simple examples of cluster structure observed for rotational multiplets of asymmetrical, symmetrical and spherical type molecules and of cluster structure modifications under rotational excitation. Topology of corresponding classical phase space, symmetry group action and equivariant Morse theory are the main ingredients of the theory in this case. Typical qualitative modifications of the cluster structure (quantum bifurcations) are described in the spirit of equivariant catastrophe theory.
Symmetry breaking and spontaneously symmetry breaking phenomena are illustrated with such molecular examples.
Vibrational problems are generically related with reduced phase spaces being (weighted) projective spaces. This is the consequence of the vibrational degeneracy due to symmetry and resonances typically appearing for a wide class of molecular systems. The concept of non-linear normal modes and transitions from normal to local modes are illustrated on molecular examples with different symmetry and different resonance conditions.
The generating functions counting the number of states are introduced and analyzed for vibrational problems. The relation between combinatorics and topology are illustrated on these examples.
Classical phase space, S^2×S^2 being the product of two two-dimensional spheres, is introduced for electronic problems using as example the hydrogen atom and Rydberg states of molecules. Qualitative modifications of the internal structure of Rydberg multiplets associated with Hamiltonian Hopf bifurcations are discussed and related to Hamiltonian monodromy.