Thermal vibrations bond length determination

The presence of well-defined peaks and valleys in I-V curves indicates that LEED is indeed not a purely two-dimensional surface diffraction technique. There is a finite penetration and diffraction takes place in the first 3 to 5 atomic layers. The depth of penetration affects peak widths markedly the shallower the penetration, the broader is the diffraction peak. By simulating such I-V curves numerically with the help of a suitable theory, it is often possible to determine the relative positions of surface atoms (including therefore bond lengths and bond angles) " it may also be possible to indicate roughly the thermal vibration state of surface atoms l However, a chemical identification of the surface atoms is not possible with LEED. 

With small molecules, it is usually possible to obtain anisotropic temperature factors during refinement, giving a picture of the preferred directions of vibration for each atom. But a description of anisotropic vibration requires six parameters per atom, vastly increasing the computational task. In many cases, the total number of parameters sought, including three atomic coordinates, one occupancy, and six thermal parameters per atom, approaches or exceeds the number of measured reflections. As mentioned earlier, for refinement to succeed, observations (measured reflections and constraints such as bond lengths) must outnumber the desired parameters, so that least-squares solutions are adequately overdetermined. For this reason, anisotropic temperature factors for proteins have not usually been obtained. The increased resolution possible with synchrotron sources and cryocrystallography will make their determination more common. With this development, it will become possible to obtain better estimates of uncertainties in atom positions than those provided by the Luzzati method. 

Finally, the mean values of the bond lengths, that would be determined by crystallography, were calculated assuming coupling to the (CuOg) vibration only with components Qe 2Az2 — Ajc — Ay2 and Qe V3(Ax2 — Ay2). This was achieved by first evaluating the expectation values of qe and qe in each of the thermally populated vibronic wavefunctions, as outlined by Riley and coworkers [17] ... 

It should be noted that the values given in Table 4 reflect both the different qualities of the crystal structure analyses and of the crystals. The bond lengths are not corrected for anisotropic thermal vibrations. From the differences foimd in two independent structure analyses of DCH polymer it can be assumed that in some cases the standard deviations given may be underestimated. In all cases the quality of the analyses does not allow the determination of the electron density distribution along the polymer chain which has been possible for the two model compounds and for the resonance structure (I)... 

Bond lengths, amplitndes of vibration, and asymmetries in the distribntion of nonbonded atom pairs were determined by Bartell et al. at temperatnres ranging from 300 to 1200 K the effect of thermal excitation was analyzed to extract iirformation abont anharmonicity. The nozzle temperatnre was 15 °C. 

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