On the geometric quantization of the ro-vibrational motion of homonuclear diatomic molecules

In the framework of geometric quantization we extend the harmonic oscillator rules to rotation and vibration of molecules. First, the geometric quantization of the rigid rotor is introduced. The novelty is that we found an explicit relation giving the impulse along the third axes, which takes discrete values and found that it depends not only on the principal quantum number, but also on the orbital and magnetic quantum numbers. Geometric quantization is further extended to the class of vibrational (damped) systems. The theory is sought as mathematical requirement, which can be traced in the analysis of integrable systems. It is shown that properties of the critical points of the momentum map can be the key to the integrability. A $C^{⁎}$ isomorphism of the momentum map between the Lie algebra of physical observables and that of the harmonic oscillator induces the geometric quantization on former. We show how the method works, taking into account the Lennard-Jones' potential, which characterizes the interaction between molecules. A second example shows a physical system having the energy levels depending on the magnitude of the frequencies.