Knowledge Bank

The first diagonalization step in a rho-axis-method treatment of methyl-top internal rotation problems involves finding eigenvalues and eigenvectors of 3 torsional Hamiltonian which depends on the rotational projection quantum number K as a parameter. Traditionally die the torsional quantum number $\nu t=0.1.2\dots$ is assigned to eigenfunctions of given K in order of increasing energy. In this talk we propose an alternative labeling scheme using the torisional quantum number $\nu T$ which is based on properties of the K-dependent torsional overlap integrals . In particular, the quantum number $\nu T$ is assigned in such a way that torsional wavefunctions $|\nu T,K>$ vary as slowly as possible when K changes by unity. Roughly speaking, $\nu T=\nu t$, for torsional levels below the barrier, whereas $\nu T$ is more closely related lo the free-rotor quantum number for levels above the barrier. Because of the latter fact, we believe $\nu T$ will in general be a physically more meaningful torsional quantum number for levels above the barrier. The usefulness of overlap integrals for qualitative prediction of torsion- rotation hand intensities and for rationalizing the magnitudes of perturbations involving some excitation of the small-amplitude vibrations in an internal rotor problem is also discussed.