Libration tensor

The simulation of pure crystals at room temperature shows little, except a validation of the force field if the stmcture is not distorted in the mn, and perhaps a picture of molecular average displacements that can be related to librational tensors. Phase changes are obviously more interesting. Generally speaking, the simulation of melting is easy because, as temperature increases and density decreases. 

Certain special features to be imposed on a model may be expressed by more complicated constraint equations. We note as an example the assumption of a rigid molecule with prescribed dimensions whose position and orientation are to be refined. The position may be described by the coordinates of the centre of mass and the orientation by three Euler angles with respect to a unitary coordinate system. The atomic coordinates and thus the structure factor. Equation [1], are expressed as functions of these six parameters. The latter may then be adjusted to optimize the deviance. A similar procedure can be used to constrain the atomic displacement parameters of a molecule to rigid-body movements described by a translation tensor, a libration tensor and a transla-tion/libration-correlation tensor (TLS model). This model neglects intramolecular vibrations. 

When a body undergoes vibrations, the displacements vary with time, so time averages must be taken to derive the mean-square displacements, as we did to obtain the lattice-dynamical expression of Eq. (2.58). If the librational and translational motions are independent, the cross products between the two terms in Eq. (2.69) average to zero, and the elements of the mean-square displacement tensor of atom n, U"j, are given by... 

The excellent resolution of the 0-tensor components at W band has been used to measure the relaxation properties of QA in the Zn-substituted bRC of R. sphaeroides.m The experiment showed, in contrast to the respective ubiquinone radical in organic solution, an anisotropic relaxation behavior in the pulse high field ESE experiments. From the analysis of the T2 experiments a motional anisotropy of Q% in the protein pocket was deduced with a preferred libration about the C-O symmetry axis. Recently, similar experiments were also performed on Qb- in ZnbRCs. Compared to QA different echo decay time constants were found. A model was proposed in which the relaxation is related to reorientational fluctuations around the quinones specific H-bonds to the protein.142... 

This is a general scheme, which can allow for additional considerations and further approximations. First, the average with respect to picosecond dynamic processes is carried out, in practice, together with the average with respect to solvent coordinates to allow the QM evaluation of magnetic tensors corrected for solvent effects and for fast vibrational and solvent librational motions. The effective treatment of these aspects represents the heart of this contribution. 

Then, as case study, we consider the glycine and glycyl radicals (Fig. 6.2) in solution. As mentioned above, the calculation of magnetic tensors needs to take into account the several factors such as the geometries, environmental effects, and dynamical effects (vibrational averaging from intramolecular vibrations and/or solvent librations). We use an integrated computational approach where the molecular... 

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. 

For further confirmation of the mode-softening and a possible identification of the molecular nature of the over-damped mode, we used the rigid-body motion analysis of the thermal- parameters of the room temperature x-ray diffraction study. A thermal-motion analysis (TMA) program was used to calculate the components of the librational (L) and the translational (T) tensors with a least-square fit of the published thermal parameters ( ) of all nonhydrogen atoms of the molecule. The librational frequencies were calculated by the method of Cruickshank (7), using the appropriate eigenvalues of the L-tensor and the corresponding moments of inertia. 

However, for all of these interactions it is important to have the dynamics effectively modeled. Librational dynamics of significant amplitude can average the tensor elements. Such motions are present even in polycrystalline samples [18]. Shown in Fig. 6.4.5 are powder pattern spectra at 276 K, below the gel to liquid crystalline phase transition that quenches global dynamics, but retains local dynamics [19] and at 143 K, well below the temperature that quenches most librational motions [20]. 

The motional axis must be close to the 0-22 element, since this element of the tensor is not averaged very much. In fact, the data can be very clearly fit to a motional axis consistent with the X2 torsional axis and an amplitude of 19° of librational diffusion in a xi rotameric potential energy well. 

The hyperfine tensor (Table III) is now far closer to that expected, but the anisotropy is still less than that for gas-phase radicals or that calculated using simple theory (22). This may be because of a residual movement or libration and it is hoped that studies at 4.2°K. will shed further light on this. 

If the molecule, or part of the molecule is vibrating iu a nonquantum mechanical way, then the magnitude of the anisotropic hyperfine tensor is reduced. If the libration is about one of the principal directions, then the effective value of the coupling in that direction will be the coupling appropriate to the mean angle of deviation from the principle direction. 

The transformation T" connects the symmetry coordinates with the direction cosine librational displacement coordinates 2.i(lk) as before. We are now in a position to calculate the polarizability derivative tensor components if the symmetry coordinates S or the normal coordinates Q are known in terms of the librational displacement coordinates. 

The situation described above applies to complexes packed in a crystal lattice. We have recently found that the situation is drastically different for the Cu -(2,2 6, 6"-terpyridine)2 complex in glassy matrices of frozen ethanol or frozen ethanol-dichloromethane mixtures and the same complex isolated in a matrix of crystalline ethanol. In these cases, only small amplitude librations along the Jahn-Teller active mode are observable below the melting point of ethanol. Above the melting point, averaging of the g and hyperfine tensor by the Jahn-Teller effect is complete [45]. 

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