Electrons 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. 

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).

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... 

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. 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 will discuss in detail in the next section this potential, which incorporates the main features of the atomic shell structure. The screening potential is just the potential of the coupling constant integrated exchange-correlation hole. Due to the fact that this hole integrates to one electron, the screening potential has... 

The conclusions reached here apply equally well to general multi-electron homonuclear molecules. In this case the value of in the dissociation limit becomes equal to the sum of the contributions of the atomic fragments. The A-integrated exchange-correlation hole for a reference electron on one of the atomic fragments will then be equal to the A-integrated hole of the atom itself. The properties of the exchange-correlation in heteronuclear molecules are discussed in the next section. 

Fig. 2.10 Schematic illustration of the mutual exclusion zone or exchange-correlation hole about a given electron within a free-electron gas. The hole has a radius, r8, corresponding to exactly one electron being excluded, thereby revealing one positive charge of underlying jellium background. The electron plus its positive hole move together through the gas of other electrons as though they are a neutral entity or quasi-particle.

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... 

---