The exchange-correlation hole

To understand why the LDA is such a surprisingly good approximation, it is helpful to rephrase the problem in terms of the exchange-correlation hole. To do 

The reason we defined this pair density, is that we can rewrite the electron-electron term W as 

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 

This hole is a measure of how other electrons are repelled by the electron around which it is centered, and integrates to -1 [8]  

This means that we may interpret the exchange-correlation energy as the Coulomb interaction between the charge density and its surrounding exchange-correlation hole. 

---

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)... 

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

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]. 

This long-range correlation effect shows up in both the first-order density matrix and the exchange-correlation hole for finite systems [19]. We concentrate here on the exchange-correlation hole. The general asymptotic form of the pair density is then... 

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... 

From the above we can expect that, if the functional derivatives of the GGA functionals give a good representation of the properties of the exchange-correlation hole, the near-degeneracy correlation properties responsible for the bond midpoint peak will show up in the exchange potentials rather than the correlation potentials. 

In this nonvariational approach for the first term represents the potential of the exchange-correlation hole which has long range — 1/r asymptotics. We recognize the previously introduced splitup into the screening and screening response part of Eq. (69). As discussed in the section on the atomic shell structure the correct properties of the atomic sheU structure in v arise from a steplike behavior of the functional derivative of the pair-correlation function. However the WDA pair-correlation function does not exhibit this step structure in atoms and decays too smoothly [94]. A related deficiency is that the intershell contributions to E c are overestimated. Both deficiencies arise from the fact that it is very difficult to represent the atomic shell structure in terms of the smooth function p. Substantial improvement can be obtained however from a WDA scheme dependent on atomic shell densities [92,93]. In this way the overestimated intershell contributions are much reduced. Although this orbital-depen-... 

Nonempirical GGA functionals satisfy the uniform density limit. In addition, they satisfy several known, exact properties of the exchange-correlation hole. Two widely used nonempirical functionals that satisfy these properties are the Perdew-Wang 91 (PW91) functional and the Perdew-Burke-Ernzerhof (PBE) functional. Because GGA functionals include more physical ingredients than the LDA functional, it is often assumed that nonempirical GGA functionals should be more accurate than the LDA. This is quite often true, but there are exceptions. One example is in the calculation of the surface energy of transition metals and oxides. 

Thus, we have come a long way from the exactly soluble problems of quantum mechanics, the free-electron gas and the hydrogen atom. The concept of the exchange-correlation hole linked with the LDA has allowed... 

The subscript zero on Xo refers to the fact that we have performed our perturbation theory as though the electrons were independent particles. In practice, as we have seen in 2.5, the motion of each electron is correlated through the exchange-correlation hole. This leads to an enhancement of the response function which can be written... 

---