Exchange-correlation hole charge

Table XVI summarizes the band energies at selected points for Si and Ge calculated using the WDA both with metallic screening and with semiconductor screening in the exchange-correlation functional. In this approach, the exchange-correlation energy of an interacting electron system is expressed in terms of the exchange-correlation hole charge n (in Ry units)...

Fig. 32. Contour plot of the exchange-correlation hole charge for an electron at the anti-bonding site in the (110) plane of Si (electron location denoted by +) for the LDA (a) and for the WDA(LL) (b). The contour interval is 1.25 electrons per unit cell. (from Ref. 101)... 

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.

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

This defines the important exchange-correlation hole density pxc (2 1) as the differenee between the conditional probability density and the ordinary electron density. The following total charge integral rules hold for parallel aa and a/3 antiparallel electron spins ... 

The LDA exchange-correlation hole is, therefore, spherically symmetric and centered on the electron by construction. The LDA is successful for ground-state properties despite these limitations because depends only on the spherical average of the hole charge and because the sum rule (Eq. 39) is satisfied leading to some systematic cancellation of errors. 7... 

The parameter ff is determined at each point by requiring the sum rule (Eq. 39) be satisfied. The exchange-correlation hole needs no longer remain centered on the electron and depends nonlocally on the charge density which may be highly Inhomogeneous. 

Equation 4.49 defines the exchange or Fermi hole. It is as if an electron of a given spin digs a hole around itself in space in order to exclude another electron of the same spin from coming near it (Pauli exclusion principle). The integrated hole charge is unity, i.e., there is exactly one electron inside the hole. Likewise, the correlation energy functional can be defined as... 

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