A Study of Energy Band Rearrangement in Isolated Molecules by Means of the Dirac Oscillator Approximation

Energy band rearrangement along a control parameter in isolated molecules is studied through axially symmetric Hamiltonians describing the coupling of two angular momenta $\mathbf{S}$ and $\mathbf{L}$ of fixed amplitude. We focus our attention on the case $S=1$ which, albeit nongeneric, describes the global rearrangement of a system of energy bands between two well-defined limits corresponding to uncoupled and coupled momenta. The redistribution of energy levels between bands is closely related to the degeneracy of the eigenvalues of the corresponding semiquantum Hamiltonian at isolated points of the three-dimensional Cartesian product of the two-dimensional phase space and the one-dimensional control parameter space. The present paper shows that the band rearrangement for the full quantum system can be quantitatively, rather than qualitatively, reproduced with Dirac oscillator approximations. We also interpret the energy band rearrangement by comparing the evolution of the joint spectra of commuting observables (i. e., energy and axial angular momentum) with that of the image of the energy-momentum map of the completely classical limit of the Dirac oscillator approximations.