Dielectric theory nonlocal

The nonlocal dielectric theory has as a special case the standard local theory. Its fuller formulation permits the introduction in a natural way of statistical concepts, such as the correlation length which enters as a basic parameter in the susceptibility kernel For brevity we do not cite many other features making this approach quite useful for the whole field of material systems, not only for solutions. 

As noted above, nonlocal dielectric theory provides the starting point for continuum approaches to SD. The derivation given below follows that presented by Song et al. [47], The solvation energy change due to solute electronic transition occurring at t = 0 is given by... 

The reorganization free energy is usually split in two parts. The local mode contribution is obtained in standard routines which require local potentials (say harmonic potentials) and vibrational frequencies in the reactants and products states. The collective modes associated with the proteins and the solvent, however, pose complications. One complication arises because classical electrostatics needs modification when the spatial extension of the electric field and charge distributions are comparable with the local structure extensions of the environment. Other complications are associated with the presence of interfaces such as metal/solution, protein/solution, and metal/film/solution interfaces. These issues are only partly resolved, say by nonlocal dielectric theory and dielectric theory of anisotropic media. 

On the assumption that = 2, the theoretical values of the ion solvation energy were shown to agree well with the experimental values for univalent cations and anions in various solvents (e.g., 1,1- and 1,2-dichloroethane, tetrahydrofuran, 1,2-dimethoxyethane, ammonia, acetone, acetonitrile, nitromethane, 1-propanol, ethanol, methanol, and water). Abraham et al. [16,17] proposed an extended model in which the local solvent layer was further divided into two layers of different dielectric constants. The nonlocal electrostatic theory [9,11,12] was also presented, in which the permittivity of a medium was assumed to change continuously with the electric field around an ion. Combined with the above-mentioned Uhlig formula, it was successfully employed to elucidate the ion transfer energy at the nitrobenzene-water and 1,2-dichloroethane-water interfaces. 

The dielectric theory may be expressed in a nonlocal form based on the definition of the susceptibility and permittivity in a form that makes these physical quantities the kernel of appropriate integral equations. 

The technique of complex-valued dielectric functions was originally applied to solvation problems by Ovchinnikov and Ovchinnikova [2] in the context of the electron transfer theory. They reformulated in terms of s(k, to) the familiar golden rule rate expression for electron transfer [3], This idea, thoroughly elaborated and extended by Dogonadze, Kuznetsov and their associates [4-7], constitutes a background for subsequent nonlocal solvation theories. 

When the change in the solute-solvent interactions results mainly from changes in the solute charge distribution, one can employ the theory of electric polarization to formulate the dynamic response of the system. This formulation involves the nonlocal dielectric susceptibility m(r, r, i) of the solution. While this first step might lead to either the molecular or the continuum theory of solvation, in the continuum approach (r, r, t) is related approximately to the pure solvent susceptibility (r, r, t) in the portions of... 

As was discussed in the previous section, continuum theories of solvation dynamics often require as input the nonlocal dielectric susceptibility of the solvent, (r,to), or equivalently, its Fourier transform [54]... 

Two points should be mentioned here. First, the effect of solutes on the solvent dielectric response can be important in solvents with nonlocal dielectric properties. In principle, this problem can be handled by measuring the spectrum of the whole system, the solvent plus the solutes. Theoretically, the spatial dependence of the dielectric response function, s(r, co), which includes the molecular nature of the solvent, is often treated by using the dynamical mean spherical approximation [28, 36a, 147a, 193-195]. A more advanced approach is based on a molecular hydrodynamic theory [104,191, 196, 197]. These theoretical developments have provided much physical insight into solvation dynamics. However, reasonable agreement between the experimentally measured Stokes shift and emission line shape can be... 

The dielectric response of the interface can be described in a unified manner in terms of the nonlocal electrostatic theory [88, 89]. Indeed, it vras shown to be possible to express the electric properties of the interface through the dielectric function of the metal/solvent system, not applying a particular form of this function, for any structure of the interface. Such an approach allows revealing general properties of the double layer and expressing the parameters involved via the nonlocal dielectric function. We briefly... 

This example shows the degree of complication inherent in the nonlocal extension of the continuum theory even for the simplest Born-like case. In accord with Equation (1.141), the dimensionless parameter A/a measures the importance of nonlocality effects the local Born limit is recovered when A/a - 0. The opposite strongly nonlocal limit a/A - 0 corresponds to the unscreened solvation Usoly = -Q2( 1 - 1 /e /la. For the general form of the dielectric function a(k) a numerical solution for one-dimensional Equation (1.144) is straightforward [19]. However, there exists a principal difficulty hindering such solution when a(k) has poles on the real A-axis (see Sections 1.6.7 and 1.6.8). This creates oscillating kernels a f — r ) in the real space. 

The Lifshitz theory uses only the so-called "local" dielectric and magnetic responses. That is to say, the electric field at a place polarizes that place and that place only. What if the field is from a wave sinusoidally oscillating in space Then the material polarization must oscillate in space to follow the field. What if that oscillation in space is of such a short wavelength that the structure of the material cannot accommodate the spatial variation of the wave We are confronted with what is referred to as a "nonlocal" response a polarization at a particular place is constrained by polarizations and electric fields at other places. 

The theory of van der Waals (vdW) surface interactions is presented here in terms of correlation-self energies of the constituent parts involved in the interaction due to their mutual polarization in the electrostatic limit. In this description the van der Waals interactions are exhibited using the dynamic, nonlocal and inhomogeneous screening functions of the constituent parts. In regard to the van der Waals interaction of a single molecule and a substrate, this problem is substantially the same as that of the van der Waals interaction of an atom and a substrate, in which the atomic aspects of the problem are subsumed in a multipole expansion based on spatial localization of the atom/molecule. As we (and others) have treated this in detail in the past we will not discuss it further in this paper. Here, our attention will be focussed on the van der Waals interaction of an adsorbate layer with a substrate, with the dielectric properties of the adsorbate layer modeled as a two-dimensional plasma sheet, and those of the substrate modeled by a semi-infinite bulk plasma. This formulation can be easily adapted to an... 

The theory of nonlinear optical processes in crystals is based on the phenomenological Maxwell equations, supplemented by nonlinear material equations. The latter connect the electric induction vector D(r,t) with the electric field vector E(r, t). In general, the relations are both nonlocal and nonlinear. The property of nonlocality leads to the so-called spatial dispersion of the dielectric tensor. The presence of nonlinearity leads to the interaction between normal electromagnetic waves in crystals, i.e. makes conditions for the appearance of nonlinear optical effects. 

Various difficulties with the form of e(w) or the procedure used by NINHAM and PARSEGIAN may arise when conductors or very small particles are examined. In the first case, nonlocal dispersion has not been adequately treated in van der Waals theory in general, but even if the nonlocal effects could be ignored, interband transitions may need to be accommodated. LANDAU and LIFSHITZ [5.50] propose the form e(o)) = 4iria)/a, where a is the conductivity, for the very low-frequency dielectric permeability of conductors. Small particles and clusters also must be treated with caution if they are metallic due to surface-scattering and size quantization effects [5.59]. 

For the time being, there is no generally accepted theory of the repulsive hydration force. It has been attributed to various effects solvent polarization and H-bonding [323], image charges [324], nonlocal electrostatic effects [325], and the existence of a layer of lower dielectric constant, e, in a vicinity of the interface [326,327]. It seems, however, that the main contribution to the hydration repulsion between two charged interfaces originates from the finite size of the hydrated counterions [328], an effect which is not taken into account in the DLVO theory (the latter deals with point ions). 

The functional form for Wpp reveals the complicated dependence of excluded volume parameter on temperature and expansion of the exponential up to linear terms in Vcc (cf Eq. (6.57)) signifies the validity of the theory for weakly charged polyelectrolytes so that Vcc is small. The rightmost term in Eq. (6.56) is the electrostatic interaction energy (Vcc), which is written after describing the response of the inhomogeneous systems to an applied electric field by a nonlocal response function (also known as the inverse dielectric function [71-73]), (r,r )) defined by... 

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