Exchange-correlation energy derivative

There is no explicit correlation-kinetic contribution to the KS exchange energy E s [p]. The contribution, however, is manifested viathe KS orbitals, so that E s[p] which is related to its functional derivative vx (r) by the virial theorem29 is expressed entirely in terms of the field KS(r) as... 

The Thomas-Fermi kinetic energy density Ckp(r)5/3 derives directly from the first term on the RHS of Eq. (17), the Dirac exchange energy density —cxp(r)4/3 coming from the second term. Many-body perturbation theory on this state, in which electrons are fully delocalized, yields a precise result [36,37] for the correlation energy Ec in the high-density limit as A In rs + B, where for present purposes the correlation energy is defined as the difference between the true... 

Here, i CT(r) and (r) represent the ordinary nonrelativistic electron annihilation and creation operators. An LDA-type approximation has recently been derived for the exchange-correlation free energy Fxc[n, xl leading to explicit expressions for the effective potentials Veg(r) and Aeff (r, r ) (Kurth et al. 1999). 

For high density system, the enhancement factor becomes unity, and exchange effects dominate over the correlation effects. When the density becomes lower, the enhancement factor kicks in and includes correlation effects into the exchange energies. The enhancement factor is not unique, but can be derived differently in different approximations. The most reliable ones are parameterizations of molecular Monte-Carlo data. Some well known, and regularly used, parameterizations have been made by Hedin and Lundqvist [29], von Barth and Hedin [22], Gun-narsson and Lundqvist [30], Ceperly and Adler [31], Vosko, Wilk, and Nusair [32], and Perdew and Zunger [27]. 

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