Cruickshank's analysis of the rigid-body motion of molecules in crystals in terms of two symmetric tensors, one for libration and one for translation, has been widely adopted in the interpretation of the results of anisotropic refinements of the structures of molecular crystals. In recent years it has been recognized by several people that this treatment is inadequate when there is no pre-ordained center of libration (e.g. a center of symmetry), but in each instance the two-tensor description of molecular motion has been retained, an effective center of libration has been assumed to exist, and its location has been sought by one means or another.Actually, an additional tensor (which we call S) is needed to account for correlations of libration and translation. For a molecule at a sufficiently unsymmetrical site, S has eight independent components, one of its diagonal elements being arbitrary, and the contribution to the anisotropic displacement tensors by the rigid-body part of the enormously various actual motions can be described in terms of six uncorrelated simple motions: the three familar principal mean-square translations plus three screw (helical) motions about and along three mutually perpendicular, non-intersecting axes.The problem of fitting the observed atomic displacement tensors in terms of rigid-body translation and screw motion involves in the general case a least-squares fit of twenty independent parameters. If the molecule is at a symmetrical site, some or all of the components of S are subject to special restrictions; for example, if the site-symmetry is i, S vanishes completely and the treatment is identical with Cruickshank's. In any event, the fit is always independent of the origin assumed in the description of the motion and is found by a straightforward linear least-squares process. Corrections to intramolecular interatomic distances foreshortened by rigid-body motion are shown to depend only on the libration tensor, which is independent of the assumed origin.Examples of the application of this analysis are given.

Modern X-ray and neutron diffraction techniques can give precise parameters that describe dynamic or static displacements of atoms in crystals. However, confusing and inconsistent terms and symbols for these quantities occur in the crystallographic literature. This report discusses various forms of these quantities, derived from probability density functions and based on Bragg diffraction data, both when the Gaussian approximation is appropriate and when it is not. The focus is especially on individual atomic anisotropic displacement parameters (ADPs), which may represent atomic motion and possible static displacive disorder. The first of the four sections gives background information, including definitions. The second concerns the kinds of parameter describing atomic displacements that have most often been used in crystal structure analysis and hence are most commonly found in the literature on the subject. It includes a discussion of graphical representations of the Gaussian mean-square displacement matrix. The third section considers the expressions used when the GaussJan approximation is not adequate. The final section gives recommendations for symbols and nomenclature.

Simonetta, M.; Gavezzotti, A. "The Application of the Extended Hiickel Method to Surface Chemistry and Crystallography: Chemisorption of C2H2 on Pt(l 11)".

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[2.2]Paracyclophane (I), C16H16, crystallizes in the tetragonal system with a= 7.781 (1), c = 9.290 (2)/~, space group P42/mnm, and two molecules per unit cell. 1,1,2,2,9,9,10,10-Octafluoro-[2.2]paracyclophane (II), C16HsFs, crystallizes in the monoclinic system with a=7"994 (2), b=7.986 (2), c= 10"855 (3)/~, fl= 97.84 (1) °, space group P21/n, and two molecules per unit cell. The known structure of (I) was refined with new three-dimensional diffractometer data by full-matrix least-squares methods to R = 0.029 (360 reflections). The structure of (II) was solved by direct methods with three-dimensional diffractometer data and was refined by full-matrix least-squares to R=0.037 (956 reflections). In the initial refinement of each molecule, the substituent atoms on the bridging carbon atoms showed high anisotropy of apparent thermal motion normal to the bridge-substituent bond, this pattern being remarkably similar in the two molecules. Final refinement was done with a model in which each molecule is statistically disordered, to simulate a dynamic disorder that occurs by a twist of the aromatic rings in opposite directions about their common normal; this model is in accord with spectroscopic and thermodynamic measurements on the hydrocarbon. The amplitude of this motion is found to be about 3 ° for each molecule. The carbon skeletons of (I) and (II) are very similar. The bridge C-C distance is about 1.59 ,~. The average C-F distance is 1"35 /~ and the average F-C-F angle is 106 °. The planes of the four unsubstituted C atoms of each ring are 3.09 A apart. The para carbon atoms are bent about 12 ° out of the plane of the other four atoms of the aromatic ring.