Knowledge Bank

Linear combination of vibrational wave functions centered at the periodic potential minima is used to approximate the torsional wavefunction normally obtained by Floquet's Theorem in Mathieu function form. The Huckel type $\Sigma kExp(2\pi 1\Lambda K/n)\varphi k$ and Mubius type $\Sigma kExp[\pi 1(2A+1)k/n]\varphi k$ combinations are shown to have the correct pseudo-angular momentum upon rotation of one or both parts of the coaxial (torsional) rotor. Therefore, they may be correlated to wave functions with free internal rotation. The combination of Huckel and Mobius energy levels then forms the cluster of properly alternating non-degenerate levels for the high-barrier case. For example, in $X2Y6$, the cluster is AEEA and for $X2Y10$ the cluster is AEEEEA. The even vibrational wavefunctions in general forms a Huckel energy system and Mobius when the system has 3 (such as $X2Y6$), 5,7 fold symmetry. This treatment is compared with spin double group and spectroscopic double group of Hougen and Bunker. It is suggested that the ratio of splitting between different vibrational levels might be roughly estimated by the ratio of Frank-Condon overlap integrals for displaced oscillators.