✍ Scribed by W.V.F. Brooks; S.J. Cyvin
No coin nor oath required. For personal study only.
The theory of harmonic vibrations of the benzene molecular model (symmetry De~ ) is considered. For this model the forms of the G and Z matrices are given, based on a set of symmetry co-ordinates. Both in-plane and out-of-plane vibrations are studied. A particular study of the ten types of interatomic distances is made, and the corresponding mean-square amplitudes of vibration are given as linear combinations of the in-plane Z matrix elements. THEORETICAL studies of the vibrations of benzene-type molecules have been reported by several authors [1-8]. In the present article some theoretical results on mean-square amplitude quantities for this molecular model are reported. Symmetry co-ordi-ates The thirty modes of vibration (twenty-one in-plane and nine out-of-plane) are distributed among the symmetry species of the appropriate group (Deh) according to Fin.plane : 2Alg -~-A~s ~-2B1~ ~-2B~ + 4E~ + 3E1~ I~out.of.plane = 2Bgg ~-Elg ~-A~ -~-2E~ In most of the cited works [4-8], together with the quite recently published study of normal co-ordinates [9], in principal the same types of symmetry co-ordinates are used. All the occurring differences may be attributed to (i) different signs, (ii) multiplying the angle bending co-ordinates by constant factors, or (iii) different ways of removing the redundancy of EI~ species. In the present work, following CALI~A~O and CRAWFORD [8] in their choice of in-plane co-ordinates, the angle deviations have been multiplied by appropriate equilibrium distance values, and the redundancy of E,~ has been removed by producing the zero co-ordinates. The
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