Some Properties and Uses of Torsional Overlap Integrals

The first diagonalization step in a rho-axis-method treatment of methyl-top internal rotation problems involves finding eigenvalues and eigenvectors of a torsional Hamiltonian, which depends on the rotational projection quantum number K as a parameter. Traditionally the torsional quantum number vt = 0, 1, 2···is assigned to eigenfunctions of given K in order of increasing energy. In this paper we propose an alternative labeling scheme, using the torsional quantum number vT, which is based on properties of the K-dependent torsional overlap integrals . In particular, the quantum number vT is assigned in such a way that torsional wavefunctions |vT, K> vary as slowly as possible when K changes by unity. Roughly speaking, vT = vt for torsional levels below the barrier, whereas vT is more closely related to the free-rotor quantum number for levels above the barrier. Because of the latter fact, we believe vT 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 band intensities and for rationalizing the magnitudes of perturbations involving some excitation of the small-amplitude vibrations in an internal rotor problem is also discussed. Copyright 1998 Academic Press. Copyright 1998Academic Press