Neighbor contributions

These are typical of ionic liquids and are familiar in simulations and theories of molten salts. The indications of structure in the first peak show that the local packing is complex. There are 5 to 6 nearest neighbors contributing to this peak. More details can be seen in Figure 4.3-3, which shows a contour surface of the three-dimensional probability distribution of chloride ions seen from above the plane of the molecular ion. The shaded regions are places at which there is a high probability of finding the chloride ions relative to any imidazolium ion. 

In the limit that nearest-neighbor contributions dominate, such a one-center nb automatically leads to an approximate 21/2-fold increase in overlap and two-fold increase in interaction energy, compared with a two-center 7tb donor. A corresponding enhancement results when the pi-acceptor is reduced from two-center (7tb ) to one-center (nb ) form, i.e., a valence p-type vacancy. Unlike the intrinsically bidirectional character of conjugation between two pi bonds (7ta->-7tb, 7tb 7ta ), the interactions of a pi bond with a nonbonding center are intrinsically mono-directional and lead to uncompensated transfer of pi charge from one moiety to the other. 

A close estimate of the dynamic correlation energy was obtained by a simple formula in terms of pair populations and correlation contributions within and between localized molecular orbitals. The orbital and orbital-pair correlation strengths rapidly decrease with the distance between the orbitals in a pair. For instance, the total valence correlation energy of diamond per carbon atom, estimated as 164 mh, is the result of 84 mh from intra-orbital contributions, 74.5 mh from inter-orbital closest neighbors contributions, and 6.1 mh from interorbital vicinal contributions. The rapid decay of the orbital correlation contributions with the distance between the localized orbitals explains the near-... 

There are an even number of lattice points equispaced on a ring, which are represented by the integers 1,2,..., A. The restriction U — v = 1 limits the summation to nearest neighbor contributions, and since A is periodic the identity A + 1 = 1 holds. Along a bond the pairs interact through an electrostatic force the magnitude and sign determined by the coefficient oce- The... 

Interior Cells. To formulate the theory we must consider cells at boundary separately from interior cells that are surrounded by other cells. We take such an interior cell to be crudely described by a sphere of radius R. The concentrations C in this cell change due to reaction rates °R and due to flux across its surface. This latter contribution is simply 3T appropriately summed over all neighbor contributions. Thus we get a rate equation for th< concentration of each species,... 

The nearest- and next-nearest-neighbor contributions to the energy, and e2, can be written as (see, Ref. 45 for more details)... 

For a pairwise potential, Vy, the energy can be decomposed into contributions due to the nn nearest neighbor interactions, the strain due to deviations from the optimum pair separation in such pairs, and the non-nearest neighbor contribution, as represented, respectively, by the three terms in the expression [208]... 

In an attempt to parametrize the crystal-field interaction, Bradbury and Newman (1967) introduced the superposition model. In this model, it is assumed that the A4 and are dominated by ligand (nearest-neighbor) contributions, which may be written as follows... 

By inspection of a layout of the shield structure, a danger point is. selected and a line along the path of minimum attenuation determined. This line is generally inadirection from the outer to the inner mouth of the duct. The intensity contribution of the surface element at the source/of the minimum, path is calculated, as well as the contribution of all neighboring surface elements which can be "seen through the effectively thinner j>ortion of the shield. These neighboring contributions usually fall off very rapidly as elements further from the minimum line are considered. 

Another case where the theory does not change occurs when the conditional solvation Helmholtz energy of all the units cannot be factorized as in (8.6.16), but can be factorized into nearest-neighbor contributions i.e.. 

In Chapter 3, it was pointed out that the contribution of a certain class of retardation mechanisms to the equilibrium compliance can be obtained by integrating over the appropriate portion of the retardation spectrum if a maximum appears there which can be separated from neighboring contributions the integration is performed in accordance with equation 33 of Chapter 3, and it is illustrated in Fig. 3-7. Alternatively, an integration can be performed over the corresponding maximum in J" by equation 53 of Chapter 3 with finite limits, as illustrated in Fig. 3-8. 

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