Displacement of atoms

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

The observation of these linear relations is of interest since the energy changes of the initial states are obtained from a process involving displacements of atoms which are effectively equal to two-electron changes. On the other hand, electron transfer involves the movement of a single electron without atom transfer. There is no a priori reason therefore for the correlation. Observation of the correlations also... 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

The coordination polyhedron results when the centers of mutually adjacent coordinated atoms are connected with one another. For every coordination number typical coordination polyhedra exist (Fig. 2.2). In some cases, several coordination polyhedra for a given coordination number differ only slightly, even though this may not be obvious at first glance by minor displacements of atoms one polyhedron may be converted into another. For example, a trigonal bipyramid can be converted into a tetragonal pyramid by displacements of four of the coordinated atoms (Fig. 8.2, p. 71). 

Using DFT calculations to predict a phonon density of states is conceptually similar to the process of finding localized normal modes. In these calculations, small displacements of atoms around their equilibrium positions are used to define finite-difference approximations to the Hessian matrix for the system of interest, just as in Eq. (5.3). The mathematics involved in transforming this information into the phonon density of states is well defined, but somewhat more complicated than the results we presented in Section 5.2. Unfortunately, this process is not yet available as a routine option in the most widely available DFT packages (although these calculations are widely... 

A crystal with n atoms per unit cell has 3nN degrees of freedom, N being the number of unit cells in the crystal. Thus, subtracting the translations and rotations of the crystal as a whole, there are 3nJV — 6 ( 3nN) normal modes. Since the displacements of atoms in different cells are correlated, the normal modes are waves, or phonons, extending over the crystal, with force constants d>, obtained from a sum over the interactions between atoms in all unit cells, and wavevector q. 

The tensor mean-square displacements of atom j, is the time average , where u is the 3 x 1 column matrix of the displacements of atom j along the Cartesian axes, and T indicates the transpose. Since the normal modes are independent of each other, cross terms between modes disappear in the averaging. The result is... 

This suggests a simple rule for the calculation of the characters of the matrices in Ftot Each displacement that is taken into itself contributes -fl to the character of the operation each displacement that is taken into its negative contributes—1 and all displacements of atoms that change position contribute 0. 

Stable adsorption complexes are characterized by local minima on the potential energy hypersurface. The reaction pathway between two stable minima is determined by computation of a transition state structure, a saddle point on the potential energy hypersurface, characterized by a single imaginary vibrational mode. The Cartesian displacements of atoms that participate in this vibration characterize movements of these atoms along the reaction coordinate between sorption complexes. 

As has been well established, piezoelectricity in a non-polar crystal is brought about by the internal strain in the crystal. The internal strain means the displacement of atoms which is not affine to the deformation of crystal lattice. In the case of a polymer film which is not electrically conductive and where the charges are possibly embedded, a description of piezoelectricity can be reached by considering not only the internal strain in the lattice but also the displacement of these charges which is not affine to the average deformation of the whole system. 

These results suggest that tight fit is needed to orient molecules within reaction cavities having passive walls. Too tight a fit will leave no free volume within a reaction cavity that would be needed to accommodate displacement of atoms during the course of a reaction. This limits the number of transformations that can be achieved within a reaction cavity wherein the reactants are held tightly. 

There are three significant possible effects when radiation interacts with matter (5,6). First, the radiation can interact with the nucleus and induce radioactivity as in the case of neutrons. Second, displacement of atoms can occur. This has happened in a number of uranium- and thorium-containing minerals over geological periods. The outstanding example is zircon, which can contain over 10% Th and 2% U. The internal bombardment from these materials and their decay products over geological periods produces low or metamict zircon, where the disorder gives an amorphous state having a low density. 

Shuffles are small displacements of atoms (usually smaller than an atomic spacing) in a local region, such as the displacements that occur in the core of a gliding dislocation. 

Under moderate energies, the primary event of incident particle interaction with crystals of any nature is very simple it is an elastic pair collision resulting in displacement of atoms (ions) into an interstitial position [1, 7, 10] provided they received energy exceeding a threshold value [11] which is typically 10-20 eV. 

In this way the uniformity of the surface, although violated instantaneously, is retained on the average, provided the time element is large compared with the time of relaxation of the thermal motion on the surface. We should immediately note that this time of relaxation is by no means necessarily of the order of 10-13 sec (the period of atomic oscillations in the lattice) since not only is simple displacement of atoms about their equilibrium positions possible, but also much more complicated and slower processes—for example, exchange of foreign dissolved atoms between the surface and the bulk of the crystal. From our point of view such a surface can be called uniform if each atom of the surface has the same probability of being replaced... 

Here qy = M fxy, Mt is the mass of atom i, and xy is the /th component of the displacement of atom i. The procedure must be repeated for each of the IRs (labeled here by Tj N(T /) is a normalization factor. The projection needs to be carried out for a maximum of three times for each IR, but in practice this is often performed only once, if we are able to write down by inspection the other components Q(Xl) of degenerate representations. It is, in fact, common practice, instead of using eq. (5), to find the transformed basis... 

However, eq. (24) does not convey the information that the displacement of atom 2 in the unit cell is n/2 out of phase with that of atom 1. 

Theoretical calculations (see Section II, B) predict that the overcrowding in phenanthrene, caused by the close approach of the hydrogen atoms at positions 4 and 5 (see Fig. 2), will mainly be relieved by displacement of atoms out of the mean molecular plane and by bond-angle deformation. 

From (8.23) and (8.24) one can see two special cases when the potential becomes separable. In the first case c12 = 0, we have two independent anharmonic modes, each having two equilibrium positions. In the second case, the angular part of the potential (8.23) Vr is zero, and the motion breaks up into radial vibration in the double well V0(q) and a free rotation, i.e. propagation of the waves of transverse displacements along the ring. The latter case is called free pseudorotation. Since the displacements of atomic groups in the wave are purely transverse, they do not contribute to the total angular momentum. 

The displacements of atoms near a dislocation from their normal lattice positions are the same as the displacements that would be caused by some external stress. Therefore, we can think of the dislocation as causing a stress field around it. Around an edge dislocation, there is a state of hydrostatic stress, crH = ax + <7y + ct-, at a location x, y,... 

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