Exchange-correlation energy averages

We do not distinguish here this density functional definition of exchange energy from that of Hartree-Fock (HF). This simplification is well-justified, if the HF electron density and the exact electron density differ only slightly [40]. Similarly, the coupling-constant averaged exchange-correlation hole is the usual... 

Note that the energy is minimized with respect to all choices of the orbital basis and subject to the (1, conditions on p, = F, ,- this ensures that there exists an ensemble of Slater determinants with the desired electron density. Because an ensemble average of Slater determinants does not describe electron correlation, these variational energy expressions include a correlation functional, Ec p, which corrects the energy for the effects of electron correlation. Reasonable approximations for Ec[p] exist, though they tend to work only in conjunction with approximate exchange-energy functionals, Ex p. ... 

The first-order parameters and k" are determined by a variational condition on the second-order quasi-energy, the result of which is an equation identical to Eq. (70). Differentiating the time-averaged exchange-correlation quasi-energy with respect to the field strengths e, according to Eq. (157), we obtain... 

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.]...

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