Nonlocal relation

We have shown that a frequency-dependent susceptibility implies temporal dispersion the polarization at time t depends on the electric field at all times previous to t. It is also possible under some circumstances to have spatial dispersion the polarization at point x depends on the values of the electric field at points in some neighborhood of x. This nonlocal relation between P and E... 

This is evidently a nonlocal relation between J and V, the kernel Gp being given explicitly by [51]... 

March, N. H. Santamaria, R. Nonlocal relation between kinetic and exchange energy densities in Hartree-Fock theory. Int. J. Quantum Chem. 1991, 39 (4), 585-592. 

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

In LN, the bonded interactions are treated by the approximate linearization, and the local nonbonded interactions, as well as the nonlocal interactions, are treated by constant extrapolation over longer intervals Atm and At, respectively). We define the integers fci,fc2 > 1 by their relation to the different timesteps as Atm — At and At = 2 Atm- This extrapolation as used in LN contrasts the modern impulse MTS methods which only add the contribution of the slow forces at the time of their evaluation. The impulse treatment makes the methods symplectic, but limits the outermost timestep due to resonance (see figures comparing LN to impulse-MTS behavior as the outer timestep is increased in [88]). In fact, the early versions of MTS methods for MD relied on extrapolation and were abandoned because of a notable energy drift. This drift is avoided by the phenomenological, stochastic terms in LN. 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

Work on dimension reduction methods for both input and input-output modeling and for interpretation has produced considerable practical interest, development, and application, so that this family of nonlocal methods is becoming a mainstream set of technologies. This section focuses on dimension reduction as a family of interpretation methods by relating to the descriptions in the input and input-output sections and then showing how these methods are extended to interpretation. 

The calculations were subsequently extended to moderate surface charges and electrolyte concentrations.8 The compact-layer capacitance, in this approach, clearly depends on the nature of the solvent, the nature of the metal electrode, and the interaction between solvent and metal. The work8,109 describing the electrodesolvent system with the use of nonlocal dielectric functions e(x, x ) is reviewed and discussed by Vorotyntsev, Kornyshev, and coworkers.6,77 With several assumptions for e(x,x ), related to the Thomas-Fermi model, an explicit expression6 for the compact-layer capacitance could be derived ... 

An inference of fundamental importance follows from Eqs. (2.3.9) and (2.3.11) When long axes of nonpolar molecules deviate from the surface-normal direction slightly enough, their azimuthal orientational behavior is accounted for by much the same Hamiltonian as that for a two-dimensional dipole system. Indeed, at sin<9 1 the main nonlocal contribution to Eq. (2.3.9) is provided by a term quadratic in which contains the interaction tensor V 2 (r) of much the same structure as dipole-dipole interaction tensor 2B3 > 0, B4 < 0, only differing in values 2B3 and B4. For dipole-dipole interactions, 2B3 = D = flic (p is the dipole moment) and B4 = -3D, whereas, e.g., purely quadrupole-quadrupole interactions are characterized by 2B3 = 3U, B4 = - SU (see Table 2.2). Evidently, it is for this reason that the dipole model applied to the system CO/NaCl(100), with rather small values 0(6 25°), provided an adequate picture for the ground-state orientational structure.81 A contradiction arose only in the estimation of the temperature Tc of the observed orientational phase transition For the experimental value Tc = 25 K to be reproduced, the dipole moment should have been set n = 1.3D, which is ten times as large as the corresponding value n in a gas phase. Section 2.4 will be devoted to a detailed consideration of orientational states and excitation spectra of a model system on a square lattice described by relations (2.3.9)-(2.3.11). 

The (nonlocal) polarizabilities are important DFT reactivity descriptors. But, how are polarizabilities related to chemistry As stated above, an essential ingredient of the free energy surface is the potential energy surface and, in particular, its gradients. In a classical description of the nuclei, they determine the many possible atomic trajectories. Thanks to Feynman, one knows a very elegant and exact formulation of the force between the atoms namely [22,23]... 

For larger displacement UA-, the variation of i(r, r ) relative to UK can be computed by using the nonlinear polarizability kernels defined below [26] (see Section 24.4). Forces and nonlocal polarizabilities are thus intimately related. 

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. 

The physical meaning of Equation 24.28 is clear the local held felt by an atom is not the bare applied held but a held screened by M 1, which is the inverse of a molecular nonlocal held response. The bare nonlocal polarizability a(L,J) describing the response of the atom (dipole induced at) L to the bare held E applied at the atom J is also related toM 1 by... 

In general, (Eioc) 5 E, since the local electric field is averaged over the atomic sites and not over the spaces between these sites. In metals, where valence electrons are free (nonlocalized electrons), the assumption (Eioc) = E is reasonable, but for bound valence electrons (dielectrics and semiconductors) this relation needs to be known. However, for our purpose of a qualitative description of optical properties, we will still retain this assumption. 

The key physics of our model (see Eqs. (9) and (10)) is contained in the nonlocal diffusion kernels which occur after integrating over the atomic processes which produce step fluctuations. We have calculated these kernels for a variety of physically interesting cases (see Appendix C) and have related the parameters in those kernels to atomic energy barriers (see Appendix B). The model used here is close in spirit to the work of Pimpinelli et al. [13], who developed a scaling analysis based on diffusion ideas. The theory of Einstein and co-workers and Bales and Zangwill is based on an equihbrated gas of atoms on each terrace. The concentration of this gas of atoms obeys Laplace s equation just as our probability P does. To make complete contact between the two methods however, we would need to treat the effect of a gas of atoms on the diffusion probabilities we have studied. Actually there are two effects that could be included. (1) The effect of step roughness on P(J) - we checked this numerically and foimd it to be quite small and (2) The effect of atom interactions on the terrace - This leads to the tracer diffusion problem. It is known that in the presence of interactions, Laplace s equation still holds for the calculation of P(t), but there is a concentration... 

In the lineage of the methodology developed above, the ADA and the WDA are nonlocal extensions of the EDA formulation. In this sense, the TF model discussed in Section II. 1 is the EDA counterpart for the OF-KEDF. however, the vW model discussed in Section III. 1 is somewhat different, because the ansatz in Eq. (39) departs from the EDA ansatz in Eq. (23). For later convenience, we name the strategy in Section III. 1 the Semilocal-Density Approximation (SEDA). In the following, through a detailed analysis of the exchange energy density functional (XEDF) and the OF-KEDF, we shall see the classical WDA is actually closely related to the SEDA. 

Thus, a susceptibility that depends on frequency and wave vector implies that the relation between P(x, t) and E(x, t) is nonlocal in time and space. Such spatially dispersive media lie outside our considerations. However, spatial dispersion can be important when the wavelength is comparable to some characteristic length in the medium (e.g., mean free path), and it is well at least to be aware of its existence it can have an effect on absorption and scattering by small particles (Yildiz, 1963 Foley and Pattanayak, 1974 Ruppin, 1975, 1981). 

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