Some Properties and Uses of Torsional Overlap Integrals

ISSN： 0022-2852

Source： Journal of Molecular Spectroscopy, Vol.187, Iss.1, 1998-01, pp. : 49-60

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Abstract

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 numberKas a parameter. Traditionally the torsional quantum numberv_t= 0, 1, 2···is assigned to eigenfunctions of givenKin order of increasing energy. In this paper we propose an alternative labeling scheme, using the torsional quantum numberv_T, which is based on properties of theK-dependent torsional overlap integrals 〈v_T,K|v^′_T,K′〉. In particular, the quantum numberv_Tis assigned in such a way that torsional wavefunctions |v_T,K〉 vary as slowly as possible whenKchanges by unity. Roughly speaking,v_T=v_tfor torsional levels below the barrier, whereasv_Tis more closely related to the free-rotor quantum number for levels above the barrier. Because of the latter fact, we believev_Twill in general be a physically more meaningful torsional quantum number for levels above the barrier. The usefulness of 〈v_T,K|v^′_T,K′〉 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.