Interatomic mean-square

Note Gold neighbor, coordination number N), interatomic distance [i (A)], and mean square relative displacement... 

Often, one needs to compare different 3D structures or conformations of a molecule. That is done internally by the 3D stmcture generation program to weed out too similar conformations of fragments. Another aspect is the need of the computational chemist to compare different generated or experimental structures. A well-established measure is the so-called root mean square (RMS) value of all atom-atom distances between two 3D structures. The RMS value needed here is a minimum value achieved by superimposing the two 3D structures optimally. Before calculating the RMS, the sum of interatomic distances is minimized by optimizing the superimposition in 3D. 

Finally, if the temperature increases, becomes larger until the crystal melts. The Lindemann criterion predicts that melting sets in when becomes about 0.25 a2, where a is the interatomic distance of the metal. Because the mean squared displacements of surface atoms is higher we expect that the surface melts at lower temperatures than the bulk does [2]. Indeed, evidence has been presented that the (110) surface of lead starts to melt at 560 K whereas the bulk melting temperature is about 600 K [13]. 

Nj and Oj in equation (2.4) represent the number of atoms in the jth shell and root-mean-square deviation of the interatomic distances over Rj which results both from static and dynamic (thermal) disordering effects respectively. The scattering amplitude, Fj(k) is given by... 

Hindered rotation is studied for the disaccharides composed of basic p-glucopyranose units. The van der Waals Interactions are calculated for the Lennard-Jones, Buckingham, and Kitaygorodsky interatomic potential functions. Values of the ratio of unperturbed to free-rotation root-mean-square end-to-end distance are calculated for chains composed of the unsolvated disaccharide repeating units. 

Figure 3. Ratio of the mean square root displacements derivatives along directions of weak and strong coupling, calculated in the model of a highly anisotropic layered crystal. Its anisotropy of interatomic interaction and elastic properties correspond to those of NbSe2. The pronounced maximum on this curve corresponds to a minimum on the temperature dependence of the thermal expansion along the layers.

The average of these converged structures is taken as the protein structure, whose precision can be assessed by the deviations of the individual structures from the average. The quality of the final structure can be described in terms of this root mean square deviation, for both the peptide backbone and side chains, and to some extent by the extent to which it conforms to limitations of dihedral bond angles and interatomic contacts anticipated from thousands of previously known structures (the Ramachandran plot ). By all criteria, NMR structures of proteins that are determined in this way are comparable to structures determined by x-ray crystallography. In addition, NMR methods can be applied to evaluate the... 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

Equation (2.2.12) may be directly obtained from minimizing the elastic free energy under the constraint that the mean-square radius of gyration has a fixed value [see Eqs. (2.1.63) and (2.1.39), C q) -+ ot q)C q) [10]. The physical meaning of this result is that under chain compression the free energy due to the interatomic contacts is basically a function of only, no matter what are the individual values of the a (q). As a consequence, all the mean-square distances (r (k)) may be expressed under a general form [53]. Defining... 

Abbreviations used in table MC - Monte Carlo aa - amino acid vdW - van der Waals potential Ig - immunoglobulin or antibody CDR - complementarity-determining regions in antibodies RMS -root-mean-square deviation r-dependent dielectric - distance-dependent dielectric constant e - dielectric constant MD - molecular dynamics simulation self-loops - prediction of loops performed by removing loops from template structure and predicting their conformation with template sequence bbdep - backbone-dependent rotamer library SCMF - self-consistent mean field PDB - Protein Data Bank Jones-Thirup distances - interatomic distances of 3 Ca atoms on either side of loop to be modeled. 

Here fl,- is the force constant for atom i and is the thermally averaged mean-square displacement for atom i in the protein the latter quantity is proportional to the crystallographically determined Debye-Waller factor if static disorder is neglected (see Chapt. VI). To simplify the treatment, average mean-square displacements can be used to represent the different types of atoms. The factor 5(r,) is an empirical scaling function that accounts for the interatomic screening of particles which are away from the RZ-RR boundary, 108 it varies from 0.5 at the reaction zone boundary to zero at the reaction region (see Fig. 8). 

Because of the spherical electronic shells of their constituent atoms, rare gas solids (R.G.S.) are the best candidates for the comparison of experimental results with theoretical predictions using various interaction potentials. XAS gives information about the interatomic distances and their mean square deviation which can be compared to data obtained by calculation or other experimental techniques. This section presents results on solid krypton under high pressure up to 20 GPa. 

Using the Debye model, compute the root-mean-square displacement of nickel at 300 K and at its melting point. What is the fractional displacement of the metal atoms relative to the interatomic distance at the melting temperature ... 

Nj and Rj are the most important structural data that can be determined in an EXAFS analysis. Another parameter that characterizes the local structine aroimd the absorbing atom is the mean square displacement aj that siunmarizes the deviations of individual interatomic distances from the mean distance Rj of this neighboring shell. These deviations can be caused by vibrations or by structural disorder. The simple correction term exp [ 2k c ] is valid only in the case that the distribution of interatomic distances can be described by a Gaussian function, i.e., when a vibration or a pair distribution function is pmely harmonic. For the correct description of non-Gaussian pair distribution functions or of anhar-monic vibrations, different special models have been developed which lead to more complicated formulae [15-18]. This term, exp [-2k cj], is similar to the Debye-Waller factor correction used in X-ray diffraction however, the term as used here relates to deviations from a mean interatomic distance, whereas the Debye-Waller factor of X-ray diffraction describes deviations from a mean atomic position. 

At the present time, of all EXAFS-like methods of analysis of local atomic structure, the SEES method is the least used. The reason is that the theory of the SEES process is not sufficiently developed. However the standard EXAES procedure of the Fourier transformation has been applied also to SEES spectra. The Fourier transforms of MW SEES spectra of a number of pure 3d metals have been compared with the corresponding Fourier transforms of EELFS and EX-AFS spectra. Besides the EXAFS-like nature of SEES oscillations shown by this comparison, parameters of the local atomic structure of studied surfaces (the interatomic distances and the mean squared atomic deviations from the equilibrium positions [12, 13, 15-17, 21, 23, 24]) have been obtained from an analysis of Fourier transforms of SEES spectra. The results obtained have, at best, a semi-quantitative character, since the Fourier transforms of SEES spectra differ qualitatively from both the bulk crystallographic atomic pair correlation functions and the relevant Fourier transforms of EXAFS and EELFS spectra. 

The dependence of the interference terms on the sample temperature is determined by the Debye-Waller factors, which in turn depending on the mean squared change of the interatomic distance, In the range of room temperatures we... 

As an example of the capabilities of EXAFS spectroscopy, the mean Cd-S distances as a function of the diameters of a series of CdS nanocrystals are depicted in Figure 3.15 [163]. These data are gained from a thorough temperature-depen-dent study of the size dependence of various structural and dynamic properties of CdS nanoparticles ranging in size from 1.3 to 12.0 nm. The properties studied include the static and the dynamic mean-square relative displacement, the asymmetry of the interatomic Cd-S pair potential, with conclusions drawn as to the crystal structure of the nanoparticles, the Debye temperatures, and the Cd-S bond lengths. As seen from Figure 3.15, the thiol-stabilized particles (samples 1-7) show an expansion of the mean Cd-S distance, whereas the phosphate-stabilized particles (samples 8-10) are slightly contracted with respect to the CdS bulk values. 

Thus, the alloying of the intermetaUic compound NisAl results in the decrease of mean-square ampHtudes of the atom vibration. This is a direct experimental evidence of increase in strength of interatomic bonding. 

The bulk modulus decreases and mean-square vibration amplitude increases when going from GaP to InSb. The band gap Eg decreases at the same time. We see that the strength of the interatomic bonding reduces with increasing the sum Ha + Hs, where Ha and wb are the principal quantum numbers of the outer shells for the a " and elements, respectively. The sum jia + ub equals for C, Si, and Ge to 4, 6, and 8, respectively. It changes from 7 for the semiconductor GaP to 9 for InSb. 

Table 153 Interatomic distances and mean-square amplitudes of atomic vibrations in the Ih ice. Experimental data of X-ray investigation of Kuhs and Lehman (after [72]).

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