Nonlocal charge-density

The total electronic potential energy of a molecule depends on the averaged electronic charge density and the nonlocal charge-density susceptibility. The molecule is assumed to be in equilibrium with a radiation bath at temperature T, so that the probability distribution over electronic states is determined by the partition function at T. The electronic potential energy is given exactly by... 

Nuclear size corrections of order (Za) may be obtained in a quite straightforward way in the framework of the quantum mechanical third order perturbation theory. In this approach one considers the difference between the electric field generated by the nonlocal charge density described by the nuclear form factor and the field of the pointlike charge as a perturbation operator [16, 17]. 

The hole correction of the electrostatic energy is a nonlocal mechanism just like the excluded volume effect in the GvdW theory of simple fluids. We shall now consider the charge density around a hard sphere ion in an electrolyte solution still represented in the symmetrical primitive model. In order to account for this fact in the simplest way we shall assume that the charge density p,(r) around an ion of type i maintains its simple exponential form as obtained in the usual Debye-Hiickel theory, i.e.,... 

The charge-density susceptibility is a linear response function it is nonlocal because a perturbing potential applied at any point r affects the charge density throughoutthe molecule. Quantum mechanically,x(r, r co) is specified by (2)... 

Even if one assumes that the water near a surface has the same structure as it does in bulk, the oscillations of the short-range interactions between surfaces could be explained by a nonlocal dielectric constant for water.24 This model assumes that the dielectric displacement field (D = epE 4- P) at a position r not only depends on the local electric field [D(r) = r(r)E(r), but also depends on the electric field in the whole space D(r) = jr(r,r )E(r )dr. In this model, the oscillations of the interactions are due to charge overscreening25 and are analogous to the charge density waves in plasmas.24... 

Approximate methods such as nonlocal polarizability density models or label-free exchange perturbation theory are useful near the minimum of the interaction potential and go beyond the classical DID model [77]. Several papers deal with effective polarizabilities in liquids [88, 95-98, 140, 152, 391] ionic melts [14, 15, 58] and solids [64, 66, 67, 137-139, 661-663]. The case of charge transfer in atomic collisions has also been considered [71]. 

Figure 10. Slice of the eleetron density of MgO obtained from an optimization of the periodic stmctnre using a nonlocal DFT approach with plane-wave pseudopotentials. The development of charge density and assoeiated eritical points between Mg and O atoms indicates the existence of eovalent character in this material.

The parameter ff is determined at each point by requiring the sum rule (Eq. 39) be satisfied. The exchange-correlation hole needs no longer remain centered on the electron and depends nonlocally on the charge density which may be highly Inhomogeneous. 

The form of F(p, Vp) varies and often contains empirical parameters. F(p, Vp) is frequently termed a gradient or non-local correction, since the potential is computed not only as a funcion of the location but also as a function of the Laplacian of the charge density, Vp(r). Of course, even these nonlocal functionals are perfectly local in a mathematical sense. The development of nonlocal exchange functionals is dominated by Becke, who has published a number of increasingly refined mathematical expressions for F(p, Vp) since 1983 (B). Nonlocal correlation functionals have been proposed by Perdew (P), Lee, Yang, and Parr (LYP), and Perdew and Wang (PW). The most commonly used nonlocal functional combinations are BP, BLYP and BPW. Earlier correction schemes like the self-interaction correction by Stoll, PreuB, and Pavli-dou (SPP) have been found to be inferior to the gradient-corrected functionals in most cases and seldom appear in the literature. 

An alternative approach that avoids many of the problems associated with electric polarization in dielectrics is TD-DFT. In the time-dependent case, the time change of the polarization induces a current, which may be considered an ultra-nonlocal functional of the charge density, and has been successfully used as an alternative additional variable for the description of dielectric properties of both solids and molecular systems. 

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... 

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