Virtual charge density

Generalized Reaction Fields from Surface Charge Densities Rather than centering attention on the charges or distributed multipoles, at various positions inside the solute cavity, that induce the reaction field, one can instead focus on the cavity surface. In particular, the effect of the reaction field may be modeled by an appropriately distributed set of induced polarization charges (virtual charges) on the surface S of the dielectric, as already mentioned. The virtual charge density, a(r), for each location r on S is... 

Complex Ion Formation. Phosphates form water-soluble complex ions with metallic cations, a phenomenon commonly called sequestration. In contrast to many complexing agents, polyphosphates are nonspecific and form soluble, charged complexes with virtually all metallic cations. Alkali metals are weakly complexed, but alkaline-earth and transition metals form more strongly associated complexes (eg, eq. 16). Quaternary ammonium ions are complexed Htde if at all because of their low charge density. The amount of metal ion that can be sequestered by polyphosphates generally increases... 

One of the unusual features of tris-dithiolenes is their preference for trigonal prismatic rather than octahedral coordination geometries.7 Examples of this structure are the dithiolenes (8) of Cr, Mo and W. Distortions towards octahedral structures are known to occur especially in anionic tris-dithiolenes, but the preference for trigonal prismatic arrangements is clearly established for the neutral species. The main reason is seen in favorable interligand interactions between the sulfur atoms.7 In Mo(edt)3 the intra- and inter-ligand S—S distances of 3.10 and 3.11 A are virtually identical.14 In the tris-dithiolene anions, the increased charge density on the S atoms leads to repulsive interactions for which the octahedral coordination is preferred. 

Second, there is a line of charge-ordering in the T -p plane, where the charge-charge correlation function begins to oscillate. This line, as established from GDH theory, passes close to the critical point and may generate a virtual tricritical state. A charge-density wave scenario also arises from r-dependent cavity interactions. 

Even with these useful results from statistical mechanics, it is difficult to specify straightforward criteria delineating when the Poisson-Boltzmann or linear Poisson-Boltzmann equations can be expected to yield quantitatively accurate results for particle-wall interactions. As we have seen, such criteria vary greatly with different types of boundary conditions, what type of electrolyte is present, the electrolyte concentration and the surface-to-surface gap and double layer dimensions. However, most of the evidence supports the notion that the nonlinear Poisson-Boltzmann equation is accurate for surface potentials less than 100 mV and salt concentrations less than 0.1 M, as stated in the Introduction. Of course, such a statement might not hold when, for example, the surface-to-surface separation is only a few ion diameters. We have also seen that the linear Poisson-Boltzmann equation can yield results virtually identical with the nonlinear equation, particularly for constant potential boundary conditions and with surface potentials less than about 50 mV. Even for constant surface charge density conditions the linear equation can be useful, particularly when Ka < 1 or Kh > 1, or when the particle and wall surfaces have comparable charge densities with opposite signs. 

Figure 15. Radial charge density plot for the resonant p-type virtual orbital for dilation angles 9 = 0.0 and 6 = 90pt (0.42 radians) in e-Be scattering. The role of optimal theta in the accumulation of electron density near the nucleus is clearly seen. In the inset, the maximum is seen to occur at rmaz — 2.5 a.u., very close to that for the rmax of the outer valence 2s orbital, seen in fig. 14- Though a cursory look at the nodal pattern identifies this as a 4P orbital, the dominant contribution to the charge density distribution is mainly of 2p-iype.

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