Quantum mechanics of pseudorotation in ring systems: An application of the MuMATH symbolic algebra system

A variety of quantum mechanical models have guided the interpretation of the far infrared (IR) spectrum of easily deformed ring systems, but an explicit guide to such modelling would ease further analysis. The coordinates introduced by Cremer and Pople provide a starting point for description of puckering, separate from other internal motions of a ring system. For a ring of N atoms there are N ‐3 puckering modes, composed of [ N ‐ (2)/2] pseudorotation “amplitudes” and [( N ‐ 3)/2] “angles.” (The brackets [] mean “truncate to the integer.”) Separation of the Schroedinger equation is possible for the “free puckerer,” the “puckerer in a box,” and for puckering opposed by a separable harmonic potential; in this latter case the energy is determined and the state is labeled by a set of pseudorotation quantum numbers M k and radial quantum numbers n k :\documentclass{article}\pagestyle{empty}\begin{document}$$E_k = \sum\limits_k {hv_k (2n_k + M_k + 1)}$$\end{document}Here v k is the harmonic frequency for the k ‐th mode, and h is Planck's constant. Since most ring systems are nonharmonic, and require a distinct “quartic‐puckering” potential for each puckering mode, a perturbation treatment of the quartic terms is required. We provide formulas and a symbolic algebra computer program to generate expressions for integrals needed for the perturbation or linear variation modeling.