Ellipsoid of thermal motion

Unit cell of /3-cristobalite. Left snapshot the numbers indicate the height of the atoms in the direction of view as multiples of. Right with ellipsoids of thermal motion at 300 °C... 

Fig. 6. A view of the structure of RbPaFe (6/). The atoms are represented by their ellipsoids of thermal motion.

Figure 1 Structure of BsH9(PMe3)2, showing ellipsoids of thermal motion (Reproduced by permission from J. Amer. Chem. Soc., 1974, 96, 3013)...

It should be noted that more complicated forms of the temperature factor term can be employed when the crystal structure analysis is particularly precise and the resolution high. These expressions take into account the possible anisotropy of the thermal motion or statistical disorder. In the most sophisticated cases, six parameters are used to define the three-dimensional ellipsoids of thermal motion, which serve to describe anisotropic temperature factors. These should not be a source of concern to the reader at this time. 

Figure 1. Stereoscopic drawings of the molecular configuration of Mo2 r C H )2 CO)4t 2-H) g2-P CH )2) showing (a) thermal ellipsoids of nuclear motion for all atoms scaled to enclosed 50% probability (b) the atom labeling. The entire molecule possesses a pseudotwofold axis passing through the bridging hydrogen and phosphorus atoms.

Figure 8. The molecular structure of the [Cr2(CO)io(M2-tf)] anion for the bis(triphenylphosphine)-iminium salt showing (a) a view normal to the Cr-Cr axis (b) a view looking down the Cr-Cr axis. The Cr-Cr intemuclear separation is 3.349(13) A. The thermal ellipsoids of nuclear motion for all atoms are scaled to enclosed 50% probability.

Figure 7. The two analogous Ni H fragments (a and b) in [P/i4P]2+-[Ni 2(CO)2 H2]2 (3), and the Ni H fragment (c) in [Ph4As]3+-[Nii2(CO)2itf]3-.Me2CO (4), showing hydrogen atoms in the octahedral interstices. The mean Ni-H distances and 20% isotropic thermal ellipsoids of nuclear motion were obtained from the neutron diffraction experiments.

ORTEP is an acronym for "Oak Ridge Thermal Ellipsoid Program," a computer program frequently used in structural analysis. The acronym is often used as a short label to indicate a drawing in which ellipsoids indicate the extent of thermal motions of the atoms. 

The next level of approximation accounts for the anisotropy of thermal motions in a harmonic approximation and describes atoms as ellipsoids in one of the three following forms, which are, in fact, equivalent to one another ... 

It would be very hard to be certain that such an unusual structure was correct without the X-ray crystallographic data shown in Fig. 3.2. This shows the compound is a 5-coordinate monomer with a distorted Y geometry (see Eq. 4.30). The atom positions arc uncertain as a result of thermal motion and experimental error, so probability ellipsoids are used to represent the atoms. There is a 50% probability that the atom is located within its ellipsoid. The atoms furthest from the central heavy atom tend to show larger ellipsoids, probably as a result of greater thermal librational motion. 

Figure 4. A perspective drawing of C.jH NjSi.B, with nonhydrogen atoms represented by thermal vibration ellipsoids drawn to encompass 50% of their electron density hydrogen atoms are represented by arbitrarily small spheres which are in no way representative of their true thermal motion.

Molecular motion in solids has been the object of many studies in the field of physical chemistry of polymers , but dynamic processes in molecular crystals of organic and inorganic compounds are less well investigated. In fact, the average chemist is not aware of the fact that processes like internal rotation or ring inversion proceed in solids quite often with barriers which are not very different from those found for these types of internal motion in the liquid state. Thus, for the equatorial axial ring inversion of fluorocyclohexane values of 42.4 and 43.9 kJ mol have been measured in the liquid and the solid, respectively. The familiar thermal ellipsoids of individual atoms obtained from X-ray studies are qualitative indicators of molecular motion in the crystal, but a more quantitative study of such processes is only possible after appropriate solid state NMR techniques are applied. 

Figure 3. Analcime. Apparent thermal-motion probability ellipsoids of the T and 0 atoms in the A structure and the displacements from the symmetrized position obtained by DLS. Ellipsoids are based on thermal parameters reported by Knowles, Rinaldi, and Smith (7) and are scaled to enclose 50% probability. The diagrams were generated with the aid of computer program ortep by C.K. Johnson...

Fig. 3.2. Thermal ellipsoids (at 99% probability) for 1,2,4-triazole by neutron diffraction at 15 K illustrating the relative thermal motion of hydrogen and nonhydrogen atoms. That of the hydrogen bonded H(l) is only slightly less than that of H(3) and H(5), and the corrections of the X-H bond lengths are +0.005 A for N-H versus +0.006 for the C-H bonds at 15 K [199]...

Fio. 1. A stereodiagram showing the structure and conformation of reserpine (6), one of the first crystal-structure determinations without the presence of a heavy atom. The ellipsoids represent the thermal motions of the atoms, drawn at the 50% probability level. 

Fig. 3. A stereodiagram of an acetoxylactam acid from 4-amino-4-deoxymethylene anhy-drolycoctonam (17). The thermal ellipsoids, at the 50% probability level, show the large increase in thermal motion of the atoms on the periphery of the molecule as compared with the atoms in the more rigid, fused-ring system. The O atoms are crosshatched.

The atomic environment within the crystal is usually far from isotropic, and the next simplest model of atomic motion (after the isotropic model just described) is one in which the atomic motion is represented by the axes of an ellipsoid this means that the displacements have to be described by six parameters (three to define the lengths of three mutually perpendicular axes describing the displacements in these directions, and three to define the orientation of these ellipsoidal axes relative to the crystal axes), rather than just one parameter, as in the isotropic case. Atomic displacement parameters, and their relationship to thermal vibrations and spatial disorder in crystals are covered in more detail in Chapter 13. 

The anisotropic displacement parameters can be visualized as ellipsoids Figure 2.54) that delineate the volume where atoms are located most of the time, typically at the 50 % probability level. The magnitude of the anisotropy and the orientations of the ellipsoids may be used to validate the model of the crystal structure and the quality of refinement by comparing thermal motions of atoms with their bonding states. Because of this, when new structural data are published, the ellipsoid plot is usually required when the results are based on single crystal diffraction data. 

The shapes of the thermal ellipsoids of the C-atoms, which contain information on the dynamic process in the solid state [18-20], are quite different between the parallel and perpendicular Cp rings. The shape difference indicates that an activation energy for the jumping motion of parallel Cp rings is higher than that for perpendicular Cp rings. 

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