Localized exchange-correlation hole

While the LSD exchange-correlation hole is accurate for small interelec-tronic separations (Sect. 2.3), it is less satisfactory at large separations, as discussed in Sect. 2.5. For example, consider the hole for an electron which has wandered out into the classically-forbidden tail region around an atom (or molecule). The exact hole remains localized around the nucleus, and in Sect. 2.5 we give explicit results for its limiting form as the electron moves far away [19]. The LSD hole, however, becomes more and more diffuse as the density at the electron s position gets smaller, and so is quite incorrect. The weighted density approximation (WDA) and the self-interaction correction (SIC) both yield more accurate (but not exact) descriptions of this phenomenon. 

As mentioned above, LSD yields a reasonable description of the exchange-correlation hole, because it satisfies several exact conditions. However, since the correlation hole satisfies a zero sum rule, the scale of the hole must be set by its value at some value of . The local approximation is most accurate at points near the electron. In fact, while not exact at m = 0, LSD is highly accurate there. Thus the on-top hole provides the missing link between the uniform electron gas and real atoms and molecules [18]. 

Fig. 7. Local on-top exchange-correlation hole at full coupling strength divided by density as a function of r in the He atom. The nuclear cusp produces greater LSD error for r-0...

In this review we will give an overview of the properties (asymptotics, shell-structure, bond midpoint peaks) of exact Kohn-Sham potentials in atomic and molecular systems. Reproduction of these properties is a much more severe test for approximate density functionals than the reproduction of global quantities such as energies. Moreover, as the local properties of the exchange-correlation potential such as the atomic shell structure and the molecular bond midpoint peaks are closely related to the behavior of the exchange-correlation hole in these shell and bond midpoint regions, one might be able to construct... 

Weighted density approximation160-162 provides a strategy to construct non-local approximation to the exchange-correlation-energy functional using the relation between Exc[p] and the exchange-correlation hole nxc(r, r ) ... 

A system of fundamental theoretical importance in many-body theory is the uniform-density electron gas. After decades of effort, exchange-correlation effects in this special though certainly not trivial system are by now well understood. In particular, sophisticated Monte Carlo simulations have provided very useful information (5) and have been conveniently parametrized by several authors (6). If the exchange-correlation hole function at a given reference point r in an atomic or molecular system is approximated by the hole function of a uniform electron gas with spin-densities given by the local values of p (r) and Pp(C obtain an... 

ELF, Electron Localization Function (p. 701) exchange-correlation energy (p. 683) exchange-correlation hole (p. 696) exchange hole (p. 697) exchange-correlation potential (p. 689)... 

Based on PPLB s work, Perdew [6] generalized the sum rule for the exchange-correlation hole to the fractional number of electrons, and Perdew and Levy [7] reached the conclusion that the local density approximation (LDA) and the generalized gradient approximation (GGA) energies were too low for systems with fractional numbers of electrons by analyzing this generalized sum rule. 

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