Energy exchange and correlation

Correlations include all the electron-electron interaction processes which are neglected in the Hartree-Fock mean field approximation. Their contribution to the interfacial energy comes from the changes in these processes which take place when the interface is formed. 

At a metal-vacuum interface the most important contribution comes from the electrons located outside the metal, which are less screened than in the bulk. Their interaction with the correlation hole left behind them yields a positive term to the surface energy - typically 1.46 J/m for a magnesium-vacuum interface, in the jellium approximation. 

At a metal-insulator interface, the correlation term arises from the mutual polarization of the two media, i.e. from the dispersion forces. Naidich (1981) has assumed that, besides a small chemical term, the adhesion energy involves mainly van der Waals pair interactions between atoms located in the close vicinity of the interface. By analogy with London s formula, he has written W db in the following way  

A possible reason for this discrepancy is the neglect of the band polarizability of both media, which involves virtual excitations across the insulator gap and across the metal Fermi level. Barrera and Duke (1976) have accounted for these excitations in a model similar to the one used by Inglesfield and Wikborg (1975) and Wikborg and Inglesfield (1977) for metal-metal van der Waals interactions. The estimated van der Waals 

This expression involves three integrations on the coupling strength g, on the component q of the wave vector parallel to the surface and on the frequencies co. The integral over o yields three terms associated with (i) the bulk plasmon poles cobn, for which the e (u ) vanish (n=l, 2) (ii) the bulk transverse excitations tom, for which e (o)) vanish and (iii) the two interfacial frequencies coi , for which + ezico) vanishes  

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The total density is the sum of die a and /3 contributions, p = Pa + Pp, and for a closed-shell singlet these are identical (p, = pp). Functionals for the exchange and correlation energies may be formulated in terms of separate spin-densities however, they are often given instead as functions of the spin polarization C, (normalized difference between p and pp), and the radius of the effective volume containing one electron, rs-... 

E [n] includes all many-body contributions to the total energy, in particular the exchange and correlation energies. 

Gritsenko, O. V., Schipper, P. R. T., Baerends, E. J., 1997, Exchange and Correlation Energy in Density Functional Theory. Comparison of Accurate DFT Quantities With Traditional Hartree-Fock Based Ones and Generalized Gradient Approximations for the Molecules Li2, N2, F2 , J. Chem. Phys., 107, 5007. 

The term Exc[p] is called the exchange-correlation energy functional and represents the main problem in the DFT approach. The exact form of the functional is unknown, and one must resort to approximations. The local density approximation (LDA), the first to be introduced, assumed that the exchange and correlation energy of an electron at a point r depends on the density at that point, instead of the density at all points in space. The LDA was not well accepted by the chemistry community, mainly because of the difficulty in correctly describing the chemical bond. Other approaches to Exc[p] were then proposed and enable satisfactory prediction of a variety of observables [9]. 

Since the Coulomb, exchange, and correlation energies are all consequences of the interelectronic 1 /r12 operator in the Hamiltonian, one can define the exchange energy functional Ex [p] in the same manner as... 

It is important to notice that the Coulomb-like term (the third term in the equation) is written in terms of the four-current, and the exchange and correlation energy is given by... 

Fig. 2-10. Profile of electron density and electronic potential energy across a metal/vacuum interface calculated by using the jellium model of metals MS = jellium surface of metals Xf = Fermi wave length p. - average positive charge density P- s negative charge density V = electron exchange and correlation energy V, - kinetic energy of electrons. [From Lange-Kohn, 1970.]...

Thus we have argued that the only hybrid form which is correct for the slowly-varying densities is that of Eq. (36), with a value of a between 0 and 0.2. Note that the approximation a = 1, which reproduces exact exchange, ignores the cancellation of nonlocalities between the exchange and correlation energies, and the fourth column of Table 3 shows how poor the latter approximation is. 

Expressions for the Exchange and Correlation Energy of Mixed-State Systems... 

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