Nonlocal response

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

W. H. Weber and G. W. Ford, Enhanced Raman scattering by adsorbates including the nonlocal response of the metal and excitation of nonradiative modes, Phys. Rev. Lett. 44, 1774-1777 (1980). 

The contributions from the bulk, or nonlocal response, of the medium are proportional to the first spatial derivative of the field ... 

The nonlocal response from Cu was examined for Cu(100) in experiments similar to those described for Ag(100) above [131]. The s-polarized SH intensities from either p- or s-polarized excitation at 1064 nm showed very low signal levels. Any... 

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. 

Chemical Reactivity as a Nonlocal Response to a Nonlocal Perturbation... 

Thus far we have considered only the quantities f(r) and s(r) as indices of chemical reactivity. These quantities, however, are local responses to the global perturbations dJf and dfi, respectively. Chemical reactions proreed by nonlocal responses of one reactant to nonlocal perturbations generated in chemical attack by another reactant. Thus, a fifth issue emerges, how to define nonlocal reactivity indices. Berkowitz and Parr (BP) [15] addressed this issue by introducing the softness kernel,... 

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. 

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. 

We have discussed the symmetries of linear optical activity and sum-frequency generation. The former is an odd-order process that requires a nonlocal response tensor in order to be specific to chiral molecules in solution, whereas the latter is an even-order response where the dominant electric-dipolar susceptibility is a probe of chirality. These observations can be extended to pseudoscalars at third-and fourth-order. 

Advanced readers will notice that in case of nonlocal response where, for example, P(r) = f d r X (r — r E(r, a similar harmonic field Hamiltonian is obtained in k-space. 

In the early 1980s, phenomenological electrodynamics of smooth interfaces was developed with a capability to parameterize the main types of electro-modulated optical signals. The signals were expressed through irreducible integrals of the nonlocal response function, which characterizes the interface in the optical frequency range. This had opened opportunities for model microscopic theories that could calculate or... 

A further step in the analysis of reactivity patterns is performed by considering the so-called nonlocal descriptors. These indicators constitute measures of molecular local responses upon local perturbations. These indices are dependent of two ormore spatial positions r, r, r ",. Nonlocal descriptors are essentially associated to polarization effects between the interacting chemical species. Among the nonlocal responses, pair site descriptors (e.g., only depending on two coordinate positions) have shown great relevance for chemical reactivity in the context of spin free conceptual DFT. A key expression relating the static pair site linear response of density, to local and global DFT indicators, was first presented by Berkowitz and Parr in 1988, namely, ... 

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... 

For flat metallic surfaces nonlocal response theory has shed some light on the resonance mechanism (Liebsch 1989 Jensen et al. 1997). However, it has been shown that due to a large contribution of bulk... 

Another interesting and potentially useful properly of nematic liquid crystal is that it is capable of nonlocal photorefractive response, as in C60 or carbon nanotube-doped NLC. The nonlocal response allows one to simulate neural net operation of smart pixels where cormections to nearest neighbors are made. In photorefractive materials, the space-charge field distribution is spatially shifted from the incident optical intensity function as the fields originate from a gradient function... 

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