The concept of monodromy was first introduced in 1980 by J.J. Duistermaat in the context of classical mechanics, as the basic topological obstruction to the existence of global action-angle variables in integrable Hamiltonian systems. Nonetheless, the concept of monodromy is perhaps more intuitively grabbed in the world of quantum mechanics. Quantum states form a lattice of points in the space of quantum numbers. If this lattice is regular, then quantum numbers can be defined globally, that is, over all the space. In contrast, if this lattice has defects (point defects, line defects, etc...), then quantum numbers can only be defined locally : the system has monodromy.
Since 1980, monodromy had been shown to take place in few atomic and molecular systems, like the perturbed hydrogen atom, the Stark effect in rotating dipolar molecules, rotating quasi-linear triatomic molecules, systems with coupled angular momenta, H2+, ... D. Sadovskii involved me in the search for additional molecular systems with monodromy.
To this end, we first applied Canonical Perturbation Theory to several ab initio surfaces, in order to check the existence of monodromy in the dynamics of floppy molecules (i.e. molecules with several equilibrium positions), like HCN-CNH and LiNC-LiCN. We found that the existence of monodromy in these systems depends on the geometry of the isomerization pathway, which connects the different equilibrium positions : if the pathway is convex, as is the case for LiNC-LiCN, then the system has monodromy (see the figure above : monodromy manifests itself by the fact that a losange propagated around the lattice defect (the green line) has a different shape when coming back to the starting point). Monodromy is instead cancelled whenever the isomerization pathway has a waist, as is the case for HCN-CNH.
Then, we again applied Canonical Perturbation Theory to an ab initio surface for CO2 and showed that this molecule displays monodromy in the 3-dimensional space of the symmetric CO stretch, the bend, and the vibrational angular momentum, which are coupled by a strong 2:1:1 Fermi resonance. This is probably the first example in molecular physics of monodromy taking place in a 3-dimensional place. We finally described and put in evidence the appropriate conditions leading to °plane-switching in CO2, a rather curious non-linear effect closely related to monodromy.