Exchange-correlation energy introduced

In principle, the KS equations would lead to the exact electron density, provided the exact analytic formula of the exchange-correlation energy functional E was known. However, in practice, approximate expressions of Exc must be used, and the search of adequate functionals for this term is probably the greatest challenge of DFT8. The simplest model has been proposed by Kohn and Sham if the system is such that its electron density varies slowly, the local density approximation (LDA) may be introduced ... 

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]. 

Integrals involving the exchange-correlation potential r,c or the exchange-correlation energy density cannot be evaluated analytically so that further sets of auxiliary functions are introduced. (In practice and 6,<, behave similarly so that a common set is used to fit both functions.) The exchange-correlation basis (XCB) also consists of Hermite Gaussians... 

A sometimes overlooked fact is that the Kohn-Sham equation is exact at this stage. It is much easier to solve than the coupled Schrodinger equations that would have to be solved for the original system, since it decouples into single particle equations. The only problem is that we have introduced the exchange-correlation energy, which is an unknown quantity, and which must be approximated. Fortunately, it will turn out to be relatively easy to find reasonably good local approximations for it. 

What is the relation between Haver and the exchange-correlation energy xc introduced earlier We find that immediately, comparing the total energy given in Eqs. (11.17) and (11.19), and now in Eq. (11.58). It is seen that the exchange-correlation energy is as follows ... 

In the theory presented thus far, DFT can be considered as an exact approach. Unfortunately, the exchange correlation energy is not known. It is at this point where approximations must be introduced in order to solve the electronic structure problem. 

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