Dielectric response nonlocal

Although quite obvious, it is important to remark that a perfect conductor-like model completely neglects the chemical nature of the metal (i.e., all the metals behave in the same way). This is different to what happens for solvents, where even the simplest models include at least one solvent-specific parameter, the static dielectric constant. However, the chemical nature of the metal is relevant for another aspect of the static dielectric response that is neglected by the conductor model the nonlocal effects. They will be discussed in the following Section, The Response Properties of a Molecule Close to a Metal Specimen Surface Enhanced Phenomena and Related Continuum Models. 

The use of a bulk-like dielectric constant, such as those in Equations (2.334)-(2.336), neglects the specific contribution given by the surface to the dielectric response of the metal specimen. For metal particles, such a contribution is often introduced in the model by considering the surface as an additional source of scattering for the metal conduction electrons, which consequently affects the relaxation time r [69], Experiments indicate that the precise chemical nature of the surface also plays a role [70], The presence of a surface affects the nonlocal part of the metal response as well, giving rise to surface-assisted excitations of electron-hole pairs. The consequences of these excitations appear to be important for short molecule-metal distances [71], It is worth remarking that, when the size of the metal particle becomes very small (2-3 nm), the electron behaviour is affected by the confinement, and the metal response deviates from that of the bulk (quantum size effects) [70],... 

X-ray diffraction is an instructive example of such a nonlocal response. The material is polarizable in proportion to the local density of electrons. It is not polarizable at all points along the sinusoidal wave. The structure factor of x-ray diffraction describes the nonlocal response to a wave that is only weakly absorbed but that is strongly bent by the way its spatial variation couples with that of the sample to which it is exposed. Reradiation from the acceleration of the electrons creates waves that reveal the electron distribution. In no way can the scattering of the original wave be described or formulated in the continuum limit of featureless dielectric response. Because x-ray frequencies are often so high that the material absorbs little energy, it is possible to interpret x-ray scattering to infer molecular structure. 

It was recently shown via molecular dynamics simulations14 that, in the close vicinity of a surface, water molecules exhibit an anomalous dielectric response, in which the local polarization is not proportional to the local electric field. The recent findings are also in agreement with earlier molecular dynamics simulations, which showed that the polarization of water oscillates in the vicinity of a dipolar surface,11,14 leading therefore to a nonmonotonic hydration force.15 Previous models for oscillatory hydration forces, based either on volume-excluded effects,18,19 or on a nonlocal dielectric constant,f4 predicted many oscillations with a periodicity of 2 A, which is inconsistent with these molecular dynamics simulations,11,18,14 in which the polarization exhibits only a few oscillations in the vicinity of the surface, with a larger periodicity. 

It is clear from the foregoing considerations that the surface plasmon is shifted by interaction with the oscillatory modes of the adsorbed layer, and new coupled modes are introduced. In fact, the adsorbed layer substantially changes all the dielectric response properties of the substrate in accordance with Eq.(22). In consequence of this, its optical properties are modified, in particular in surface plasmon resonance experiments (as well as in all other probes). Analysis of such modifications reflect on the nature of the oscillatoiy modes of the adsorbate, which can identify it for sensing purposes. It should be noted that the determination of the screening function K (Eq.(22), for example) not only provides the shifted coupled mode spectram in terms of its frequency poles, but it also provides the relative oscillator strengths of the various modes in terms of the residues at the poles. The analytic technique employed here for the adsorbate layer (in interaction with the substrate) can be extended to multiple layers, wire- and dot-like structures, lattices of such, as well as to the case of a few localized molecular oscillators. It can also take account of spatial nonlocality, phonons, etc., and the frequencies of the shifted surface (and other) plasmon resonances can be tuned by the application of a magnetic field. 

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... 

The account of the nonlocal character of the dielectric response modifies the Fresnel equations. In the framework of the jellium model for a metal, the reflection coefficients in s- and p-polarizations can be represented as °... 

One uses a simple CG model of the linear responses (n= 1) of a molecule in a uniform electric field E in order to illustrate the physical meaning of the screened electric field and of the bare and screened polarizabilities. The screened nonlocal CG polarizability is analogous to the exact screened Kohn-Sham response function x (Equation 24.74). Similarly, the bare CG polarizability can be deduced from the nonlocal polarizability kernel xi (Equation 24.4). In DFT, xi and Xs are related to each other through another potential response function (PRF) (Equation 24.36). The latter is represented by a dielectric matrix in the CG model. 

In much of the above analysis, the relative magnitude of the surface and bulk contribution to the nonlinear response has not been addressed in any detail. As noted in Section 3.1, in addition to the surface dipole terms of Eq. (3.9), there are also nonlocal electric-quadrupole-type nonlinearities arising from the bulk medium. The effective polarization is made of a combination of surface nonlinear polarization, PNS (2co) (Eq. (3.9)), and bulk nonlinear polarization (Eq. (3.8)) which contains bulk terms y and . The bulk term y is isotropic with respect to crystal rotation. Since it appears in linear combination with surface terms (e.g. Eq. (3.5)), its separate determination is not possible under most circumstances [83, 129, 130, 131]. It mimics a surface contribution but its magnitude depends only upon the dielectric properties of the bulk phases. For a nonlinear medium with a high index of refraction, this contribution is expected to be small since the ratio of the surface contribution to that from y is always larger than se2(2co)/y. The magnitude of the contribution from depends upon the orientation of the crystal and can be measured separately under conditions where the anisotropic contribution of vanishes. 

In going from static to dynamic descriptions we have to introduce an explicit dependence on time in the Hamiltonian. Both terms of the Hamiltonian (1.2) may exhibit time dependence. We limit our attention here to the interaction term. Formally, time dependence may be introduced by replacing the set of response operators collected into Q(r, r ) with Q(r, r, t) and maintaining the decomposition of this operator we presented in Section 1.1.2. For simplicity we reduce Q(r, r, t) to the dielectric component under the form P(r, t). With this simplification we discard both dielectric nonlocality and nonelec-trostatic terms, which actually play a role in dynamical processes, especially dispersion and nonlocality. 

In the language of reciprocal space, nonlocal metal response refers to the dependence of the metal dielectric constant on the wavevector k of the various plane waves into which any probing electric fields can be decomposed. Such an effect is often mentioned in reports on SERS, but it is usually neglected. One of the oldest papers addressing the importance of nonlocal effects on the polarizability of an adsorbed molecule is the article by Antoniewicz, who studied the static polarizability of a polarizable point dipole close to a linearized Thomas-Fermi metal [63], The static dielectric constant eTF(k) of such a model metal can be written as ... 

The molecule is often represented as a polarizable point dipole. A few attempts have been performed with finite size models, such as dielectric spheres [64], To the best of our knowledge, the first model that joined a quantum mechanical description of the molecule with a continuum description of the metal was that by Hilton and Oxtoby [72], They considered an hydrogen atom in front of a perfect conductor plate, and they calculated the static polarizability aeff to demonstrate that the effect of the image potential on aeff could not justify SERS enhancement. In recent years, PCM has been extended to systems composed of a molecule, a metal specimen and possibly a solvent or a matrix embedding the metal-molecule system in a molecularly shaped cavity [62,73-78], In particular, the molecule was treated at the Hartree-Fock, DFT or ZINDO level, while for the metal different models have been explored for SERS and luminescence calculations, metal aggregates composed of several spherical particles, characterized by the experimental frequency-dependent dielectric constant. For luminescence, the effects of the surface roughness and the nonlocal response of the metal (at the Lindhard level) for planar metal surfaces have been also explored. The calculation of static and dynamic electrostatic interactions between the molecule, the complex shaped metal body and the solvent or matrix was done by using a BEM coupled, in some versions of the model, with an IEF approach. 

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... 

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. 

Theoretical formulations of reorganization in the course of electron-transfer processes have undergone a number of advances in recent years. The relative importance of various solvent contributions (including translational as well as orientational response, and inductive and dispersion as well as elecrostatic interactions) can depend strongly on the polarity (i.e., dipolar, higher multipolar, or nonpolar) as well as other molecular features of the solvent [21, 47-49]. Molecular-level perspectives on solvent response are of great utility in helping to parameterize effective cavity models (e.g., in conjunction with conventional [50] or spatially nonlocal [47] dielectric models). Additivity relationships traditionally assumed to pertain to sol-... 

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... 

Problem 3.5. Prove that, if an interface between two media can be described by a transient layer with effective isotropic dielectric function and effective thickness, Eqs (3.75) and (3.76) derived for nonlocal optical response are reduced to Eqs (3.44) and (3.45), respectively. 

---