A Ubiquitous Kinetic Coupling Between Torsion And In- And Out-of-plane Xh₃ Wagging Vibrations For An -xh₃ Group Attached To A Planar Frame

As a large amplitude motion, the internal rotation of a methyl group, methyl torsion, is traditionally treated as being separate from the remaining small amplitude vibrations.[1] However, recent studies have shown that this is not the case in toluene,[2] several substituted toluenes[3,4] and \textit{N}-methylpyrrole,[5] where methyl torsion interacts with vibrations involving out-of-plane wagging of the methyl group. Likewise, interaction between silyl torsion and the out-of-plane silyl wagging mode has been observed in phenylsilane.[6] The origin of this interaction is discussed from a kinetic energy perspective in a curvilinear framework. The model considered consists of three vibrational motions when an XH$_{3}$ group is attached to a planar frame: the in-plane and out-of-plane wagging motions of the XH$_{3}$ group and XH$_{3}$ group internal rotation. Tilting of the XH$_{3}$ group relative to the frame is included. When the “equilibrium position” of the XH$_{3}$ group at each torsional angle involves it being tilted relative to the planar frame, the XH$_{3}$ wag vibrations kinetically couple with the torsional motion, with the coupling terms being linearly dependent on the magnitude of the respective XH$_{3}$ tilt angles. The coupling only vanishes when the tilt angle is zero. XH$_{3}$ tilt also leads to the -XH$_{3}$ group precessing relative to the frame during the torsional motion. Quantum chemistry calculations on a range of methylated molecules show the ubiquitous nature of in-plane and out-of-plane methyl group tilt, and hence the pervasive nature of the interaction between methyl torsion and in-plane and out-of-plane wag vibrations of the methyl group. 1. D. G. Lister, J. N. Macdonald and N. L. Owen, Internal rotation and inversion: an introduction to large amplitude motions in molecules. (Academic Press, London, 1978). 2. J. R. Gascooke, E. A. Virgo and W. D. Lawrance, J. Chem. Phys. 142, 024315 (2015). 3. J. R. Gascooke, L. D. Stewart, P. G. Sibley and W. D. Lawrance, J. Chem. Phys. 149, 074301 (2018). 4. L. D. Stewart, J. R. Gascooke and W. D. Lawrance, J. Chem. Phys. 150, 174303 (2019). 5. J. R. Gascooke and W. D. Lawrance, unpublished results. 6. M. Griggs, J. R. Gascooke and W. D. Lawrance, unpublished results