The Fermi Hole

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. 

This vanishing of the probability density for rx — r2 and Ci == C2 means that it is unlikely for two electrons having parallel spins to be in the same place (rx = r2). The phenomenon is called the Fermi hole and we note that it is a direct consequence of the Pauli principle for electrons with the same spin. 

Fig. 1. The Fermi hole for electrons with parallel spins.

It is clear that, for electrons with parallel spins, the auxiliary condition (Eq. II.2) gives rise to a correlation effect which very closely resembles the correlation effect coming from the Coulomb repulsion in the Hamiltonian for = 2 the Fermi hole replaces to a certain degree the Coulomb hole. This means that, if... 

The exchange-correlation hole can formally be split into the Fermi hole, hx =°2 (r, 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 property of the Fermi hole is that it, just like the total hole, integrates to -1... 

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

Let us introduce another early example by Slater, 1951, where the electron density is exploited as the central quantity. This approach was originally constructed not with density functional theory in mind, but as an approximation to the non-local and complicated exchange contribution of the Hartree-Fock scheme. We have seen in the previous chapter that the exchange contribution stemming from the antisymmetry of the wave function can be expressed as the interaction between the charge density of spin o and the Fermi hole of the same spin... 

Hence, if we can construct a simple but reasonable approximation to the Fermi hole, the solution of equation (3-3) can be made considerably easier. Slater s idea was to assume that... 

The phenomenon of electron pairing is a consequence of the Pauli exclusion principle. The physical consequences of this principle are made manifest through the spatial properties of the density of the Fermi hole. The Fermi hole has a simple physical interpretation - it provides a description of how the density of an electron of given spin, called the reference electron, is spread out from any given point, into the space of another same-spin electron, thereby excluding the presence of an identical amount of same-spin density. If the Fermi hole is maximally localized in some region of space all other same-spin electrons are excluded from this region and the electron is localized. For a closed-shell molecule the same result is obtained for electrons of p spin and the result is a localized a,p pair [46]. 

Under the conditions of maximum localization of the Fermi hole, one finds that the conditional pair density reduces to the electron density p. Under these conditions the Laplacian distribution of the conditional pair density reduces to the Laplacian of the electron density [48]. Thus the CCs of L(r) denote the number and preferred positions of the electron pairs for a fixed position of a reference pair, and the resulting patterns of localization recover the bonded and nonbonded pairs of the Lewis model. The topology of L(r) provides a mapping of the essential pairing information from six- to three-dimensional space and the mapping of the topology of L(r) on to the Lewis and VSEPR models is grounded in the physics of the pair density. 

The second integral above is a standard integral in the HF theory and gives (N— 2) In the first integral, we can remove the restriction (j / i) in the summation because that term cancels. Second, the integration over the spin variables forces the spin of the /th and /th orbitals to be the same in the second term inside the curly brackets. Taking all these facts into account, the Fermi hole comes out to be... 

The key to understanding the difference between the Slater potential and the exact exchange potential lies in the explicit dependence of the Fermi hole pjr, r ) on... 

Spin density is found in the molecular plane because of spin polarization, which is an effect arising from exchange correlation. The Fermi hole that surrounds the unpaired electron allows other electrons of the same spin to localize above and below the molecular plane slightly more than can electrons of opposite spin. Thus, if the unpaired electron is a, we would expect there to be a slight excess of density in the molecular plane as a result, the hyperfine splitting should be negative (see Section 9.1.3), and this is indeed the situation observed experimentally. An ROHF wave function, because it requires the spatial distribution of both spins in the doubly occupied orbitals to be identical, cannot represent this physically realistic situation. 

Coulomb hole will remove half an electron from the nucleus where the reference electron is positioned and build a charge of half an electron at the other nucleus. Unlike the Fermi hole, which for H2 was found to be completely independent of where the reference electron is located, it should be clear from the foregoing discussion that the Coulomb hole has to switch abruptly if the reference electron moves from the left to the right nucleus. 

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