Exchange-correlation effects

I drew attention in Chapter 12 to the fact that the Xa orbitals did not satisfy the nice properties of standard HF-LCAO ones the Koopmans theorem is not valid, and so on. The same is true of all density functional KS-LCAO calculations. In practice, it usually turns out that the KS-LCAO orbitals are very similar to ordinary HF-LCAO ones, which must mirror the fact that exchange-correlation effects are only a minor part of the total electronic energy. So the orbitals are often analysed as if they were ordinary HF orbitals (Figure 13.4). 

A system of fundamental theoretical importance in many-body theory is the uniform-density electron gas. After decades of effort, exchange-correlation effects in this special though certainly not trivial system are by now well understood. In particular, sophisticated Monte Carlo simulations have provided very useful information (5) and have been conveniently parametrized by several authors (6). If the exchange-correlation hole function at a given reference point r in an atomic or molecular system is approximated by the hole function of a uniform electron gas with spin-densities given by the local values of p (r) and Pp(C obtain an... 

Thus the self-energy , containing the exchange-correlation effects of the interaction. 

To summarize the application of DFT methods to M/C and ceramic interfaces, a significant noise level is apparent in predicted energetic features, originating from different basis set expansions, different degrees of ionic relaxation allowed for, and different approximations for electron exchange-correlation effects. However, qualitative trends from different studies are most often in agreement with each other and experiments. We... 

The spectra exhibits deviations from the single-particle picture of the potential for the ejected electron, which can be divided into two categories i) broadening of the spectra due to lifetime effects, and ii) energy shifts due to exchange correlation effects]... 

The last term includes the classical Coulomb repulsion and quantum-mechanical exchange-correlation effects. The separation of the classical and quantum-mechanical parts can be made explicit by writing the pair density as... 

Non-local exchange-correlation effects have been considered in the study of Na i [108]. The measured spectrum shows a surface plasmon at 2.64 eV [62]. The SAPS geometry is formed by a surface shell with eighteen atoms enclosing three others in the interior. The combination of geometrical (SAPS) corrections and nonlocal (WDA) effects leads to a peak position at 2.70 eV. 

Density functional calculations in which the exchange-correlation effects are treated beyond the LDA give the hope that Eq. (21) will be fulfilled better. This has been achieved to a large extent by the so called non-local weighted-density-approximation (WDA) [24, 29]. The results are also given in Table 2. IchomoI is very close to the experimental ionization potential and also rather close to the WDA ionization potential calculated from Eq. (20). The original paper [30] should be consulted for details. 

There are numerous types of short-range interactions between adsorbed ions such as exchange-correlation effects, an overlap of their electronic cores, mutual polari2ation, interaction via the electrons of the metal, interaction between their solvation shells (which may partially be retained in the adsorbed state), changing of the local structure of the compact layer, and so on. All these forces decay rapidly with ion-ion distances exceeding the molecular scale, and the electrostatic contribution becomes dominant. A crucially important feature of this long-range contribution is a surprisin y slow decay as a function of R,... 

However, the model in which the (valence) electrons are completely free and are neither feeling the attraction nor the repulsion is certain not properly describing the nature of the chemical bond. In fact, this limitation was also the main objection brought to Thomas-Fermi model and to the atomic or molecular approximation of the homogeneous electronic gas or helium model in solids. Nevertheless, the lesson is well served because Thomas-Fermi description may be regarded as the inferior extreme in quantum known structures while further exchange-correlation effects may be added in a perturbation manner. 

The physical meaning of the parabolic energy (1.98) since, through its correlation with the so called pi-energy (1.101) widely includes exchange-correlation effects, especially with its chemical hardness dependence though the explicit resonance relationship (1.100) ... 

Consider the pair-correlation function P(r) in Fig.5. For a rough estimate of the exchange-correlation effects in we can replace the bare Coulomb interaction v in the Hartree potential h in eq. (2.2) by... 

This short-range property of the ccsnbined exchange-correlation effects is a basic ingredient of the LDA of Kohn and Sham from the stationary properties of the groiond-statp functional E = E[p], ... 

It was pointed out j n Ref. 64 that the scp.qened electrostatic potential (i.e. V + V, not including any V ) rather than t e self-consistent potential should be used for determination of e -and this holds for other dielectric quantities, as well. In fact, the electromagnetic theory refers to classical test charges, the exchange-correlation effects enter only indirectly, through determination of the actual charge distribution. The screened potential = V ° + V plotted in Fie. 5.1.7 for comparison is... 

So far the standard method for d rain tion of e has consisted of Inverting a large matrix e Aq+G,q+G ) with elements computed through the Adler-Wiser formula " all the RPA-results in Tab. 8.1 were computed in this way. This approach requires, for every individual matrix element, the evaluation of extensive sums over all valence and conduction bands, and is limited to the RPA approximation. (The exchange-correlation effects can then be incorporated as corrections.) The method is rather cumbersome and meets convergence problems concerning both number of conduction bands to be included and dimension of the e matrix to be inverted. 

The DFT, briefiy summarized in Section 2, is a reformulation of the many-body problem of interacting-electrons into a simple problem of noninteracting electrons (the Kohn-Sham system, see subsection 2.2) and all the exchange-correlation effects are combined together in the exchange-correlation (XC) functional, which is however unknown and must be approximated. The DFT was initially employed by the solid-state physics community to study bulk systems, where the electron density shows small spatial variations. For such homogeneous systems the Local Density... 

Application of Time-Dependent Current-Density-Functional Theory to Nonlocal Exchange-Correlation Effects in Polymers. 

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