The exchange and correlation energies

The exchange energy Ex is defined in density functional theory as 

The correlation energy Ec on the other hand is defined as the remaining unknown piece of energy 

The unknown part of our energy functional is only this Exc[n] = Ex[n - -Ec[n and we shall now take a look at some ways to get approximations for this functional. 

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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. 

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]. 

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... 

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... 

Just as for the simple, spin compensated situation where pa(r) = pp(r) = j p(r), there are related expressions for the exchange and correlation energies per particle of the uniform electron gas characterized by p (r) pp(r), the so-called spin polarized case. The degree of spin polarization is often measured through the spin-polarization parameter... 

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