Umbrella inversion processes. Disentanglement of the vibration-rotation problem and eigenfunctions

The umbrella inversion mode is a large amplitude motion characteristic of four atom molecules of type AB3 (such as ammonia and the hydronium ion) and is a mechanism for chirality inversion of pyramidal molecules. A reduced dimension approach to the modelling of the umbrella inversion can be obtained by enforcing C3v symmetry, that is the minimum symmetry group of the equilibrium configuration of the AB3 systems. The conservation of the C3v symmetry along the inversion path, is obtained working with a particular hyper spherical representation.