Coupled Local/Nonlocal

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

External stress, locally applied, can have nonlocal static effects in ferroelastics (see Fig. 4 of Ref. [7]). Dynamical evolution of strains under local external stress can show striking time-dependent patterns such as elastic photocopying of the applied deformations, in an expanding texture (see Fig.5 of Ref. [8]). Since charges and spins can couple linearly to strain, they are like internal (unit-cell) local stresses, and one might expect extended strain response in all (compatibility-linked) strain-tensor components. Quadratic coupling is like a local transition temperature. The model we consider is a (scalar) free energy density term... 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

Fig. 5. Coupling functions H in Eq. (24) for ring working electrodes with circumference L for two different values of / . Solid line large aspect ratio jl resulting in nonlocal spatial coupling. Dashed line very small aspect ratio / resulting in local (diffusive) spatial coupling. (Courtesy of J. Christoph.)...

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 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. 

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