COORDINATES, HAMILTONIAN AND SYMMETRY OPERATIONS FOR THE SMALL-AMPLITUDE VIBRATIONAL PROBLEM IN INTERNAL-ROTOR MOLECULES LIKE $CH_{3}CHO$
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Date
1996
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Ohio State University
Abstract
The small-amplitude vibrational problem in molecules with a periodic large-amplitude vibration exhibits some features not found in the traditional vibration-rotation $formalism^{1}$. Motivated by the long-term goal of trying to understand the mechanism(s) by which internal rotation of methyl tops strongly enhances intramolecular vibrational energy redistribution, we have constructed a vibration-torsion-rotation Hamiltonian for molecules containing one $C_{3v}$ rotor, which has many analogies with that for quassi-linear molecules. In this formalism, internal rotation motion is taken to be slow compared to all other vibrational motions, so that (i) the small-amplitude vibrational problem is solved for each value of the torsional angle (in the spirit of the Born-Oppenheimer approximation), and (ii) the angular momentum operator associated with internal rotation motion is grouped with the three components of the total angular momentum operator rather than with the vibrational linear momentum operators. One surprising group-theoretical result has emerged. Even though the molecular symmetry group for basis set functions in the principal axis system (or rho axis system) is isomorphic with $C_{3v}$, small-amplitude vibrational eigenvactors resulting from diagonalization of the GF matrix do not necessarily belong to irreducible representations of $C_{3v}$ but may in fact change sign when the methyl top is rotated by 2$\pi$. As a result, a double group of $C_{3v}$ must be used to classify these eigenvectors. This sigh change in vibrational wavefunctions under one torsional revolution, which is reminiscent of the sigh change in vibrational wavefunctions under one torsional revolution, which is reminiscent of the sigh change in vibrational wavefunctions under one torsional revoluiton, which is reminscent of the sighn change in electronic wavefunctions under one vibrational ""revolution"" when a Jahm-Teller effect is $present^{2}$ seems to be related to the fact that the $CH_{3}CHO$ molecule passes through two chemically different $C_{3}$ configurations (staggered and eclipsed with respect to the aldehyde H) during the internal rotation motion. The usefulness of the present theoretical formalism has not yet been determined, and a number of questions concerning its application still remain. Further work is thus planned, in which this formalism will be tested by using it to study various torsionally mediated perturbations in mathanol and acetaldehyde.
Description
$^{1}$E. B. Wilson, JR., J. C. Decius and P. C. Cross, ""Molecular Vibrations,"" McGraw-Hill, New York, 1955. $^{2}$H. C. Longuet-Higgins, Advances in Spectroscopy 2, 429-472 (1961).
Author Institution: Molecular Physics Division, National Institute of Standards and Technology
Author Institution: Molecular Physics Division, National Institute of Standards and Technology