Thermal motion analysis

The three-dimensional Gram-Charlier expansion, first applied to thermal motion analysis by Johnson and Levy (1974), is a statistical expansion in terms of the zero and higher derivatives of a normal distribution (Kendal and Stuart 1958). If Dj is the operator d/du], the expansion is defined by... 

In neutron diffraction structure analyses, the correction for thermal motion is much more complex for hydrogen atoms than for nonbydrogen atoms. It has long been recognized that vibrational thermal motion causes an apparent reduction in interatomic distances relative to those for the atoms at rest, and the methods for thermal motion analysis and geometry correction are well developed [190—192]. 

Although these effects are small, they affect not only the covalent X-H bond lengths, but also the H - A hydrogen-bond lengths. For very high-precision low-temperature neutron diffraction analyses [186, 187], they have to be taken into account. An adequate way to do this, without the complexity of a complete anhar-monic thermal motion analysis, is to use the experimentally determined anisotropic temperature factors. As shown by comparisons between experimental and theoretical X-H values from low temperature neutron analyses and theoretical ab-initio molecular orbital calculations, one can assume that the motion of the two... 

Johnson CK (1970) An introduction to thermal motion analysis. In Ahmed FR (ed) Crystallographic computing. Munksgaard, Copenhagen, pp 207 -254... 

Ellison RD, Johnson CK, Levy HA (1971) Glycolic acid direct neutron diffraction determination of crystal structure and thermal motion analysis. Acta Cryst B 27 333-344... 

Some criteria have been suggested in order to decide when a thermal motion analysis can be expected to yield meaningful results. Two of them, the rigid-bond test and the rigid-molecule test, investigate whether or not bonds or portions of molecules are considered to move in a physically reasonable way. They will be described in turn. 

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. 

Maverick, E. F. Trueblood, K. N. THMA11 Program for Thermal Motion Analysis. UCLA, 1988. 

G. Filippini and C. M. Gramaccioli, Acta Crystallogr., Sect. B, 42,605 (1986). Thermal Motion Analysis in Tetraphenylmethane A Lattice-Dynamical Approach. 

This program is for a thermal motion analysis of the anisotropic displacement parameters (ADPs), is carried out in terms of the LST rigid-body approximation, and considers the correlations in the internal and overall motions. THMA is rewritten from the 1992 version of THMA14. ... 

K. N. Trueblood, THMA14 The Computer Program for Thermal Motion Analysis including Internal Torsion, University of California, Los Angeles, CA, 1992. 

Liidemann et al., 1997] Liidemann, S. K., Carugo, O., and Wade, R. C. Substrate access to cytochrome P450cam A comparison of a thermal motion pathway analysis with moleculM dynamics simulation data. J. Mol. Model. 3 (1997) 369-374... 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

Vibrational spectroscopy has played a very important role in the development of potential functions for molecular mechanics studies of proteins. Force constants which appear in the energy expressions are heavily parameterized from infrared and Raman studies of small model compounds. One approach to the interpretation of vibrational spectra for biopolymers has been a harmonic analysis whereby spectra are fit by geometry and/or force constant changes. There are a number of reasons for developing other approaches. The consistent force field (CFF) type potentials used in computer simulations are meant to model the motions of the atoms over a large ranee of conformations and, implicitly temperatures, without reparameterization. It is also desirable to develop a formalism for interpreting vibrational spectra which takes into account the variation in the conformations of the chromophore and surroundings which occur due to thermal motions. 

Although all molecules are in constant thermal motion, when all of their atoms are at their equilibrium positions, a specific geometrical structure can usually be assigned to a given molecule. In this sense these molecules are said to be rigid. The first step in the analysis of the structure of a molecule is the determination of the group of operations that characterizes its symmetry. Each symmetry operation (aside from the trivial one, E) is associated with an element of symmetry. Thus for example, certain molecules are said to be planar. Well known examples are water, boron trifluoride and benzene, whose structures can be drawn on paper in the forms shown in Fig. 1. 

Hg(CN)2 in the solid state has a structure (I42d neutron diffraction), completely different from that of Cd(CN)2 Almost-linear molecules (r(Hg—C) 201.9, r(C—N) 116.0pm (corrected for thermal motion) a(C—Hg—C) 175.0°) are arranged such that four secondary bonds N" Hg (274.2 pm) yield the often-occurring 2 + 4 coordination around Hg.103 Analysis of the 199Hg MAS NMR spectrum of Hg(CN)2 has yielded the chemical shift and shielding tensor parameters.104... 

Thermally stimulated creep (TSCr) method, 21 742-743 Thermally stimulated current spectrometry (TSC), 21 743 Thermal mass meters, 20 681 Thermal mechanical analysis (TMA), of polyester fibers, 20 21 Thermal motion, in silicon-based semiconductors, 22 237-238 Thermal noise, silicon-based semiconductors and, 22 237 Thermal oxidation, 10 77-78, 79 in VOC control, 20 683-685 Thermal oxidation rates, silicon, 22 490 Thermal oxidizers... 

Small molecule crystallographers are familiar with these concepts, since it is routine to measure data at low temperature to improve precision by reduction of thermal motion, and structures are often done at multiple temperatures to assess the origins of disorder in atomic positions. Albertsson et al. (1979) have reported the analysis of the crystal structure of Z)(-l-)-tartaric acid at 295, 160, 105, and 35 K. Figure 22 shows the individual isotropic. S-factors for the atoms in the structure at each of these temperatures the smooth variation of B with T is apparent. Below 105 K, B is essentially identical for all atoms and is also temperature independent the value of B = 0.7 agrees well with the expected zero-point vibradonal value. However, even for this simple structure, not all of the atoms show B vs T behavior at high temperature which extrapolates to 0 A at 0 K. 

The (/-orbital population analysis was performed with both the harmonic and a more complete anharmonic thermal motion treatment, as discussed in section 10.7.3. The harmonic map is shown here. 

The glass transition temperature can be measured in a variety of ways (DSC, dynamic mechanical analysis, thermal mechanical analysis), not all of which yield the same value [3,8,9,24,29], This results from the kinetic, rather than thermodynamic, nature of the transition [40,41], Tg depends on the heating rate of the experiment and the thermal history of the specimen [3,8,9], Also, any molecular parameter affecting chain mobility effects the T% [3,8], Table 16.2 provides a summary of molecular parameters that influence the T. From the point of view of DSC measurements, an increase in heat capacity occurs at Tg due to the onset of these additional molecular motions, which shows up as an endothermic response with a shift in the baseline [9,24]. 

An x-ray analysis will measure the diffraction pattern (positions and intensities) and the phases of the waves that formed each spot in the pattern. These parameters combined result in a three-dimensional image of the electron clouds of the molecule, known as an electron density map. A molecular model of the sequence of amino acids, which must be previously identified, is fitted to the electron density map and a series of refinements are performed. A complication arises if disorder or thermal motion exist in areas of the protein crystal this makes it difficult or impossible to discern the three-dimensional structure (Perczel et al. 2003). 

In the preceding Section (5.3) the CD bands in the near ultraviolet region are assigned to those of tyrosyl and phenylalanyl residues, whose thermal motions are restricted sterically keeping their geometry constant around the main chain in the proteins. The CD spectrum of aromatic moieties provides a good measure for the analysis of motional freedom around a given amino acid residue. Better examples have been reported on several kinds of amino acids and their polymers. 

Heat and temperature were poorly understood prior to Carnot s analysis of heat engines in 1824. The Carnot cycle became the conceptual foundation for the definition of temperature. This led to the somewhat later work of Lord Kelvin, who proposed the Kelvin scale based upon a consideration of the second law of thermodynamics. This leads to a temperature at which all the thermal motion of the atoms stops, By using this as the zero point or absolute zero and another reference point to determine the size of the degrees, a scale can be defined. The Comit e Consultative of the International Committee of Weights and Measures selected 273.16 K as the value lor the triple point for water. This set the ice-point at 273.15 K. 

According to the theoretical analysis of this model24,25), even with the single bond energy which only little exceeds that of the thermal motion, in practice an oligomer exists in equilibrium only in two states — free and completely bound, i.e., the interaction scheme being... 

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