Thermal motion probability ellipsoids

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...

The ORTEP2 program draws thermal motion probability ellipsoids at atomic sites and produces stereoscopic pairs of molecules. Overlap is eliminated automatically. ORTEP2 is rewritten from the version of ORTEP-II (1976). The ORTEP3 program is the same version of ORTEP-III (1996). New capabilities are (i) ability to use the input data of DS SYSTEM (ii) addition of HP-GL files (HP7580 file) -f screen output items (iii) expansion of the number of atoms (1500) and bonds (2000). 

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]...

More complex mathematical treatment is necessary when the thermal motion is very large, as for hydrogen atoms in a room-temperature neutron structure analysis, or when it is curvilinear as in a hindered-rotor. The Uy second-rank tensor does not adequately describe the nuclear or electron-scattering density when the motion is far from harmonic or when it deviates from the familiar ellipsoidal probability form. Tb deal with such examples, more complex mathematic expressions using Gram Charlier or Edgeworth expansions are available [210]. 

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 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. 

Fig. 18. DHDK (5) 0.4 CHCI3 stereodiagram of. the crystal structure. The thermal motion ellipsoids represent 40 % probability distributions (taken from Ref.

Fig. 14.13 Anisotropic harmonic lithium vibration shown as the green thermal ellipsoids with 95% probability refined by Rietveld analysis for room-temperature neutron diffraction data measured for LiFeP04. Expected curved one-dimensional continuous chains of lithium motion were drawn by the dashed lines to show how the motions of Li atoms evolve from vibrations to diffusion...

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 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.

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