Until recently, physicists were unable to compute the exact spectra of most systems, because this requires the diagonalization of (usually) large Hamiltonian matrices. Fortunately, theoretical tools like Van Vleck's Canonical Perturbation Theory (CPT), which allow a precise determination of quantum spectra without direct diagonalization, were derived very early and played a fundamental role in the development of quantum mechanics.

Although modern computers now make it possible to calculate the whole spectrum of small molecules, CPT is still an interesting tool in the field of molecular physics. Indeed, CPT basically consists of a series of canonical (or unitary) transformations, which are aimed at rewriting the Hamiltonian of the system in terms of as complete as possible a set of good quantum numbers (or corresponding classical constants of the motion). These Hamiltonians are in turn  very convenient starting points for the investigation of the non-linear dynamics of these systems, especially when they are subjected to semi-classical quantization.

We adapted CPT to the study of 

1. highly excited vibrational states of molecules with a single equilibrium position (semi-rigid molecules)

2. highly excited vibrational states of molecules with several equilibrium positions (floppy molecules)

3. highly excited vibrational states of molecules with several coupled electronic surfaces (non-adiabatic dynamics)

This provided us a deep understanding of the non-linear dynamics of small molecules, like for example bifurcations, monodromy, and non-adiabatic dynamics. We also showed that CPT leads to the definition of meaningful local modes for intra-molecular vibrational energy redistribution (IVR) studies.