Effects of displacements on molecular geometry

Thermal motion and disorder are two of the main factors limiting our ability to obtain accurate distances and angles by X-ray diffraction methods. As quoted earlier in this Chapter, if only the atoms kept still it would make the task much easier. An understanding of the effects of thermal motion and disorder is necessary so that they may be corrected for when determining bond distances. In order to obtain the most accurate set of bond distances, the X-ray diffraction experiment should be performed at as low a temperature as possible, thus reducing all atomic motion. 

Another effect of thermal motion is the riding motion of one atom on another, like that of a man (B) bouncing as he moves along riding a horse (A). In this modeF there are two bonded atoms, A and R, one of which, B, is much lighter than the other, A. Atom B then rides on atom A in such a way that the translational motion of B contains all the motion of A plus an additional motion that is not correlated with that of A. It is assumed that the center of motion is located on atom A, and that the second bonded atom B is riding on that point. 

FIGURE 13.15. Libration causing apparent (not real) bond shortening, (a) Because the atom is vibrating but the bond length stays the same, the atom (b) vibrates in an arc (librates) (c). The electron density is interpreted as an ellipsoid (d), but its major axis is displaced as shown in that the bond appears shortened. 

The anisotropic rigid-body displacements of a molecule can be described in terms of translation (T, vibration along a straight-line path), libration (L, vibration along an arc), and screw (S, a combination of vibration and translation that may be regarded as vibration along a helical path). The mean-square amplitude of translational vibration is usually referred to as a system of Cartesian coordinates and unit vectors. S is the mean correlation between libration about an axis and translation parallel to this axis. Each of these three components can be expressed as 

The set of anisotropic displacement parameters, obtained from the least-squares refinement of the crystal structure (as described by Chapter 10) can be analyzed to obtain T, L and S. It has been assumed that there is no correlation between the motion of different atoms. Values of Uij are analyzed (again by an additional least-squares analysis) in such a way that good agreement is obtained between the refined values and those predicted when constants have been obtained for the T, L, and S tensors. The total number of anisotropic displacement parameters (6 per atom) is the input, and a total of 12 parameters for a centrosymmetric structure, or 20 parameters for a noncentrosymmetric structure, is the output of this least-squares analysis. The results consist of the molecular translational (T), librational (L), and screw (S) tensors. This treatment leads to estimates of corrections that should be made to bond distances. On the other hand, this type of analysis cannot be used for intermolec-ular distances because the correlation between the motion of different molecules is not known. 

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