Fermi and Coulomb Holes

The exchange-correlation hole can formally be split into the Fermi hole, hx =r - (q, r2) and the Coulomb hole h 1 02 (q, r2), 

First of all we note that the Fermi hole - which is due to the antisymmetry of the wave function - dominates by far the Coulomb hole. Second, another, very important properly of the Fermi hole is that it, just like the total hole, integrates to -1 

This is easy to understand because it means that the conditional probability for electrons of spin o integrates to Na -1 instead of Na because there is one electron of the same spin o 

What can we say about the shape of the Fermi hole First, it can be shown that hx is negative everywhere, 

Second, if we recall the definition, equation (2-16), and modify it for the exchange-only case 

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With the KS theory Fermi and Coulomb holes defined by Eq. (27), the derivative vxc(r) can also be expressed in terms of its separate Pauli, Coulomb and correlation-kinetic contributions as... 

The symmetry problem is trivially solved in the local-scaling version of density functional theory because we can include the symmetry conditions in our choice of orbit-generating or initial wavefunction W [33]. Since it is from this initial wave-function that we obtain and xc,gy namely, the non-local quantities appearing in the energy functional of Eq. (50), it follows that the symmetry properties of the parent wavefunction are transferred to the variational functional. Notice, therefore, that symmetry is not as important for the density as it is for the Fermi and Coulomb holes, which are related, to and jtyc,gy respectively. 

Kohn-Sham Theory Fermi and Coulomb Holes, and Exchange and Correlation Energies... 

Fig. 1. Force fields P(r) and < (r) due to the Kohn-Sham theory Fermi and Coulomb holes, and the field (r) due to the quantum-mechanical Fermi-Conlomb hole charge distribution for the He atom. The function (— 1/r ) is also plotted...

Fe—S dimers, 38 443-445 map, four-iron clusters, 38 458 -functional theory, 38 423-467 a and b densities, 38 440 broken symmetry method, 38 425 conservation equation, 38 437 correlation for opposite spins and Coulomb hole, 38 439-440 electron densities, 38 436 exchange energy and Fermi hole, 38 438-439... 

M. A. Buijse and E. J. Baerends, in Electronic Density Functional Theory of Molecules, Clusters and Solids, D. E. Ellis, Ed., Kluwer, Dordrecht, 1995, pp. 1-46. Fermi Holes and Coulomb Holes. 

The functions / < and / describe the correlation between electrons of the same spin and between electrons of opposite spin respectively, and determine the so-called "Fermi-hole and "Coulomb hole . 

Besides that any two electrons avoid each other because of the same charge (Coulombic hole). Both holes (Fermi and Coulomb) have to be reflected in a good wave function. We will come back to this problem in Chapter 10. 

The XC-hole can be split into contributions from the exchange- or X-hole, which arises from the Fermion nature of an electron obeying the Pauli principle, and the correlation- or C-hole due to Coulomb repulsion within the pair of electrons. (The X-hole and C-hole are often referred to as Fermi hole and Coulomb hole, respectively). 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

As noted above, the curl of the expression on the right-hand side of Equation 7.47 vanishes. However, it does not mean that the Coulombic and non-Coulombic components—the former is the electric field produced by the Fermi-Coulomb hole and the latter arises from the kinetic energy tensor—of this field also have vanishing curl. Thus the potential Wxc of Equation 7.38 may sometimes be path dependent [21]. 

Fig. 8. Fermi hole, Coulomb hole and total hole in at various bond distances. In all plots the reference electron is placed at 0.3 bohr at the left of the right H atom...

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