Fermi hole orbitals

The term between square brackets can be diagonalized snch that a set of natnral domain-averaged Fermi hole orbitals can be obtained ... 

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. 

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... 

Equation (4.49) indicates that for this wave function the classical Coulomb repulsion between the electron clouds in orbitals a and b is reduced by Kab, where the definition of this integral may be inferred from comparing the third equality to the fourth. This fascinating consequence of the Pauli principle reflects the reduced probability of finding two electrons of the same spin close to one another - a so-called Fermi hole is said to surround each electron. 

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. 

A consequence of the Pauli exclusion principle is that electrons with parallel spins tend to avoid one another it is said that about an electron there is a Fermi hole which other electrons tend to avoid. In consequence, four electrons with parallel spins occupying the four sp% orbitals of an atom tend to assume relative positions > corresponding to the corners of a tetrahedron about the nucleus.16 Hence the carbon atom in the state s22s2pz S may be described as tetrahedral. The effect of correlation is to increase the tetrahedral character by the assumption by the orbitals of some d, f, character, which concentrates them about the tetrahedral directions. 

The Fermi hole for the reference electron at a bonded maxima in the VSCC of the carbon atom has the appearance of the density of a directed sp hybrid orbital of valence bond theory or of the density of a localized bonding orbital of molecular orbital theory. Luken (1982, 1984) has also discussed and illustrated the properties of the Fermi hole and noted the similarity in appearance of the density of a Fermi hole to that for a corresponding localized molecular orbital. We emphasize here again that localized orbitals like the Fermi holes shown above for valence electrons are, in general, not sufficiently localized to separate regions of space to correspond to physically localized or distinct electron pairs. The fact that the Fermi hole resembles localized orbitals in systems where physical localization of pairs is not found further illustrates this point. 

It should be borne in mind that the resemblance of a Fermi hole density to that of a localized valence orbital is obtained only when the reference electron is placed in the neighbourhood of a local maximum in the VSCC. The Fermi hole and hence the density of the reference electron are much more delocalized for general positions throughout the valence region (see Fig. E7.4(f)). Localized molecular orbitals thus overemphasize electron localiz-ability and do not provide true representations of the extent to which electrons are spatially localized. 

Interestingly, and probably due to a very exciting connection between the Fermi-hole and the localized orbitals [28], various localization methods result in rather similar localized orbitals, except for the description of double bonds by a o- and 7r-orbital-pair or two equivalent r (banana) bonds. Boys localization gives r orbitals, while the Edmiston-Ruedenberg and the popula-... 

This is called a Fermi hole and is the first example we encounter of a particle being dressed (i.e. having its properties modified) by many-body forces. Strictly speaking, the Fermi hole differs for each electron, but the interaction can be made local by averaging it over different orbitals, and this is referred to as the Hartree-Slater approximation. ... 

We discussed (i) in Section VII above. The relative distribution in pQ itself depends only on molecular geometry and on the Fermi holes between electrons of like spin, i.e. on the exclusion principle. The H.F. sea potential Fj makes the H.F. orbitals diffuse as compared to hydrogen-like ones for example, but its effect on collisions is indirect and faint (see below for the dependence of the on the overall medium ). 

The pseudopotential/pseudo-orbital pair are linked and what is achieved by the formulation of the valence orbital problem is a replacement of the effect of the Pauli principle. The Pauli principle causes electrons (of like spin) to avoid each other independently of their mutual repulsion it generates the so-call Fermi hole around a particular electron. Now as the valence electron penetrates the core space it must have a distribution which reflects this Fermi hole it must avoid the phantom core electrons or they must avoid it. So the pseudopotential/pseudo-orbital pair must reflect this fact and this is why they are linked. If we choose to make the pseudo-orbital smooth then the local form of the pseudopotential becomes oscillatory and vice versa, so that the imposition of pseudo-orbital smoothness may have some ramifications for the choice of a model potential to simulate the effect of the pseudopotential. 

The Fermi hole in turn is defined in terms of the idempotent Dirac density matrix ys(r, r ) of Eq. (36) where the orbitals < j(x) are solutions of the Har-... 

The work Wnfr) is retained in the equation to ensure there is no self-interaction). In contrast to the Kohn-Sham equation, this differential equation can in practice be solved because the dependence of the Fermi hole p, (r, r ), and thus of the work W (r), on the orbitals is known. Furthermore, since the solution of this equation leads to the exact asymptotic structure of vj (r), and the fact that Coulomb correlation effects are generally small for finite systems, the highest occupied eigenvalue should approximate well the exact (nonrelativistic) removal energy. This conclusion too is borne out by results given in Sect. 5.2.2. 

We see that the Fermi hole for electrons of the same spin is an approximate substitute for the Coulomb hole and that the correlation problem for electrons of the same spin is, at least partially, eliminated. The situation is far more dramatic for electrons of different spin — and two electrons paired in the same molecular orbital enter this category — for which there is no Fermi hole. Two such electrons have a definite probability of being in the same volume element and the correlation problem here is particularly acute. 

Soncini and Lazzeritti calculated the one- and two-bonds nuclear spin-spin coupling densities and the Fermi hole densities for hydrogen fluoride, water, ammonia, and methane molecules. The pair density function p2(xi, X2) determines the probability of two electrons being found simultaneously at points Xi = fiT]i and X2 = tit] , where i and fi are coordinates in three-dimensional space, and rii and TI2 are the spin variables of the two electrons. For a system described by a one determinant wavefunction of occupied spin-orbitals < >, (x), that is, a wavefunction in the HF approximation, the pair density function becomes... 

And what would happen if we made the decision for electron 3 more difficult Let us put electron 1 (ci = j) in the center of the molecule and electron 2 (<72 = — 2) as before, at nucleus b. According to what we think about the whole machinery, electron 3 (with 0-3 = j) should run away from electron 1 because both electrons have the same spin coordinates, and this is what they hate most. But where should it run Will electron 3 select nucleus a or nucleus bl The nuclei do not look equivalent. There is an electron sitting at b. while the a center is empty. Maybe electron 3 will jump to a then Well, the function analyzed is the Haitree-Fock type, electron 3 ignores the Coulomb bole (it does not see electron 2 sitting on b) and therefore, it will not prefer the empty nucleus a to sit at. It looks like electron 3 will treat both nuclei on the same basis. In the case of two atomic orbitals, electron 3 has a choice either bonding orbital (pi or antibonding orbital 2 (either of these situations corresponds to equal electron densities on a and on b). Out of the two molecular orbitals, (p2 looks much more attractive to electron 3, because it has a node exactly, where electron 1 with its nasty spin is. This means that there is a chance for electron 3 to take care of the Fermi hole of electron 1 we predict that electron 3 will choose only (pj. Let us check this step by step ... 

EOM-CC method (p. 638) exchange hole (p. 597) explicit correlation (p. 584) exponentially correlated function (p. 594) Fermi hole (p. 597) frozen orbitals (p. 624)... 

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