Local density approximation exchange-correlation energy

Several authors " have attempted to use density functional type approaches for only the correlation energy. If the Hartree-Fock expression for exchange is kept, this of course ensures that the self-Coulomb integrals are properly cancelled by self-exchange one goes back (for better or worse) to the HF level as the point of reference. The computational demands are of the same order as those of the HF calculation itself Results of early attempts of this nature have been summarized by Stoll et al If the local density approximation to correlation. 

Here, exc(p(r)) is the exchange-correlation energy per particle of a uniform electron gas of density p( ). This energy per particle is weighted with the probability p(r) that there is in fact an electron at this position in space. Writing Exc in this way defines the local density approximation, LDA for short. The quantity exc(p(r)) can be further split into exchange and correlation contributions,... 

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]. 

Table 9.1 presents excitation energies for a few atoms and ions. Calculations were performed with the generalized KLI approximation [69,74], For comparison, experimental data and the results obtained with the local-spin-density (LSD) exchange-correlation potential [75] are shown. The KLI method contains only the exchange. 

The local density approximation (LDA)24 is often used to calculate Exc[n and Vxc(r). The LDA uses as input the exchange-correlation energy of an electron gas of constant density. In a homogeneous system the exchange energy per particle is known exactly and it has the expression... 

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