Nearest neighbor distance contributions

Neon, and the elements directly below it in the periodic table or the Solid State Table, form the simplest closed-shell systems. The electronic structure of the inert-gas solid, which is face-ccntercd cubic, is essentially that of the isolated atoms, and the interactions between atoms are well described by an overlap interaction that includes a correlation energy contribution (frequently described as a Van der Waals interaction). The total interaction, which can be conveniently fitted by a two-parameter Lennard-Jones potential, describes the behavior of both the gas and the solid. Electronic excitations to higher atomic states become excitons in the solid, and the atomic ionization energy becomes the band gap. Surprisingly, as noted by Pantelides, the gap varies with equilibrium nearest-neighbor distance, d, as d... 

For most distortions, both overlap interactions and Madelung terms contribute to the elastic constants. However, for the distortion associated with C44 in the rocksalt structure, only the Madelung term enters. (There are no changes in nearest-neighbor distance to first order in the strain.) Thus the experimental values can be compared with the Madelung term, and by virtue of the Cauchy relation, c, 2 should take the same value. That contribution has been calculated by Kellerman (1940) and is Values of this expression arc given in Table... 

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

As metal atoms interact with nearest neighbors at relatively short distance, orbital overlap results in electron density being shared. As mentioned earlier, that electron density is delocalized in orbitals that are essentially molecular orbitals encompassing all of the atoms. The number of atoms that contribute an orbital to the molecular orbital scheme approaches the number of atoms present. As two atoms... 

The second term in (6-9) expresses that nearest and next nearest neighbors dominate scattering contributions to the EXAFS signal, while contributions from distant shells are weak. The dependence of the amplitude on 1/r2 reflects that the outgoing electron is a spherical wave, the intensity of which decreases with the distance squared. The term exp(-2r/X) represents the exponential attenuation of the electron when it travels through the solid, as in the electron spectroscopies of Chapter 3. The factor 2 is there because the electron has to make a round trip between the emitting and the scattering atom in order to cause interference. 

Figure 9.5 EXAFS of Rh/AKO, catalysts after reduction at 200 °C (left) and 400 °C (right) top the magnitude of the Fourier transform of the measured EXAFS signal, bottom the back transformed EXAFS corresponding to distances from Rh atoms of between 0.8 and 3.2 nm. The lower Fourier transform contains a dominant contribution from Rh nearest neighbors at 0.27 nm and a minor contribution from oxygen neighbors in the metal-support interface. After correction for the Rh-O phase shift, the oxygen ions are at a distance of 0.27 nm (from Koningsberger et at. 119]).

Closest-packed Structures.—If the stability of a metal crystal were determined by the number of bonds formed at a minimum interatomic distance, with no contribution of longer bonds, the structures with closest packing would be the most stable for the metallic elements. These structures, which have been described in the preceding section, involve contact between each atom and the 12 nearest neighbor. (The next interatomic distances are 41 percent larger and presumably have little significance.)... 

The quantities B (Rq) and are treated as adjustable parameters and Rq is a reference distance that can be chosen arbitrarily. In principle, the number of ligand shells considered for the calculation of intrinsic parameters is not limited, however, it is usually assumed that only the nearest neighbors of the rare-earth ion contribute significantly to the crystal-field potential. Thus, especially long-range interactions like electrostatic interactions are not accounted for explicitly. Because these interactions are most important for k = 2 parameters, in many cases only the k = 4, 6 intrinsic parameters have been considered. 

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