Electron density exchange-correlation hole

This hole arises from the exchange and dynamical Coulomb interactions between electrons which lead to a local depletion of electron density around a give electron. The exchange-correlation hole is related to a pair correlation function... 

In this chapter we make first contact with the electron density. We will discuss some of its properties and then extend our discussion to the closely related concept of the pair density. We will recognize that the latter contains all information needed to describe the exchange and correlation effects in atoms and molecules. An appealing avenue to visualize and understand these effects is provided by the concept of the exchange-correlation hole which emerges naturally from the pair density. This important concept, which will be of great use in later parts of this book, will finally be used to discuss from a different point of view why the restricted Hartree-Fock approach so badly fails to correctly describe the dissociation of the hydrogen molecule. 

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. 

The exchange-correlation hole density at r around an electron at r for coupling strength 1, n ,x > ,> ), is then defined by the relation... 

We do not distinguish here this density functional definition of exchange energy from that of Hartree-Fock (HF). This simplification is well-justified, if the HF electron density and the exact electron density differ only slightly [40]. Similarly, the coupling-constant averaged exchange-correlation hole is the usual... 

Fig. 8. Full coupling-strength radial exchange-correlation hole density around an electron at X = 3, 5, 00, plotted along the direction of the departing electron...

We found that our method results in electron densities nx(r) which deviate from the reference density by less than 1%. This is shown in figure 1(a) where the density is plotted as a function of A along a line parallel to the direction in which the external potential varies (we call this the ( direction). While the density is fixed, all other physical quantities vary smoothly and monotically with A. As an example we consider the spherically-averaged exchange-correlation hole... 

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... 

If we separate out the Hartree energy from this, which can easily be done since it can be written as an explicit density functional, we can define the exchange-correlation hole, nxc(x,x ), around an electron as... 

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