Hole, exchange-correlation

We begin with some exact conditions on Exc based on its real space decomposition. We define the second order density matrix in terms of the wavefunction 

We can further decompose the hole into exchange and correlation contributions  

By exchange, we mean the density functional definition of exchange, in which the wavefunction is a Slater determinant whose density is the exact density of the interacting system, and which minimizes the energy of the non-interacting system in the Kohn-Sham external potential, iv,a=o- Another useful concept is the pair distribution function, defined as[14] 

The exchange-correlation hole may be written in terms of the pair distribution as 

The exchange-correlation hole is of considerable interest in density functional theory, as the exact exchange-correlation energy may be expressed in terms of this hole. By use of the Hellmann-Feynman theorem, one may write the exchange-correlation energy as the electrostatic interaction between the density and the hole, averaged over coupling constant[13], i.e., 

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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. 

The concept of the exchange-correlation hole is widely used in density functional theory and its most relevant properties are the subject of the following section. 

The exchange-correlation hole can formally be split into the Fermi hole, hx =°2 (r, r2)... 

Figure 2-2. Fermi, Coulomb and the resulting total exchange-correlation holes for H2 at three different intemuclear distances the position of the probe electron is marked with an arrow (adapted from Baerends and Gritsenko, J. Phys. Chem. A, 101, 5390 (1997), with permission by the American Chemical Society).

Exchange-Correlation Energy and Exchange-Correlation Hole.87... 

EXCHANGE-CORRELATION ENERGY AND 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. 

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... 

In the subsections below, we show how this idea has been refined by further study of the exchange-correlation hole since that work. 

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]. 

To see this in more detail, consider the ratio of the system-averaged on-top exchange-correlation hole to the system-average of the density itself, , as defined in Eq. (20). This ratio satisfies the inequalities... 

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...

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