Semi-classical quantization of the shuttling eigenstates in a [2]rotaxane

The essential mechanical behavior in a switchable rotaxane can be reasonably reduced to the relative motion of the two components along a single "shuttling" dimension. Since the simplest rotaxanes are composed of a few hundred atoms, the reduced mass for this motion is very large from the standpoint of quantum mechanics and therefore the density of quantum eigenstates in this shuttling dimension is very large. To meet the challenge of computing the energies for such a large number of eigenstates, we apply a semiclassical technique and show that variations in the state density with energy have implications for classical-quantum correspondence in these systems.