Vibrational parameters in crystals: Rigid and non-rigid molecules

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In 1956, Cruickshank developed the theory for and made the first applications of the analysis of molecular motion in crystals on the basis of observed anisotropic atomic vibration (displacement) parameters. In this review of some recent work in this area, I consider first the calculation of the mean square displacement amplitude (MSDA) of an atom in any direction and then consider the use of differences in MSDA for pairs of atoms along their interatomic line. When the data are good, these differences will be smaller than about 1 × 10 −3 A = for bonded pairs (the Hirshfeld criterion), unless one of the atoms is hydrogen or deuterium, or a transition metal atom in a complex in which certain static or dynamic distortions are present. The MSDA difference will also be small for all pairs of atoms in a rigid portion of the molecule. Thus scrutiny of patterns of MSDA differences can reveal information about patterns of internal motion. The rigid-body model of molecular motion, developed originally by Cruickshank, is reviewed briefly, and modifications of it to allow for internal motion (e.g., torsion) of groups of atoms that are themselves reasonably rigid are discussed. Estimation of the amplitudes of internal motion can, in favorable cases lead to plausible estimates of frequencies, force constants and barriers for such motion. Examples of such an analysis, and its limitations, are considered for torsional motions of groups such as methyl, trifluoromethyl and t-butyl.