Fermi hole density

It is demonstrated through a study of the Fermi hole density that the local decrease in the potential resulting from the approach of each ligand to the central atom does result in a partial condensation of the pair density to yield... 

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. 

Fig. E7.5. Contour maps of the Fermi hole density for pyramidal (a, b) and planar (c, d) ammonia, In maps (a) and (b) the reference electron is positioned at the non-bonded and bonded maxima, respectively, in the VSCC of the nitrogen atom. Note that the Fermi density is more contracted towards the core in NH3 than it is in CH4, as are the maxima in its VSCC. Maps (c) and (d) are corresponding plots for planar ammonia. The density of the non-bonded Fermi hole, map (c), is more delocalized than that for the pyramidal geometry, map (a). In the planar geometry, contours of the non-bonded Fermi hole density encompass the N-H internuclear axis. Clearly, maps (c) and (d) overlap one another to a greater extent than do maps (a) and (b)—the electron pairs are more localized in pyramidal than in planar iunmonia.

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... 

Fermi hole density functions have been computed by Soncini and Lazzeretti for the first-row hydrides. The relative maps have been compared with those of one- and two-bonds nuclear spin-spin coupling density. The interesting observation concerns geminal H-X-H coupling, which is determined mainly by the spin density near the heavy X nucleus. If it were not involved, no geminal coupling could arise. 

Malta, Hernandez-Trujillo < > aromaticity index based on the delocali zation of the Fermi hole density ... 

In a study in 1982, Luken and Culberson analyzed the change of the Fermi hole shape with respect to the position of reference electron to gain information about the spatial localizatirai of electrons [36], The Fermi hole density is derived from the same-spin pair density and describes the probability density to find an electron at given position, when another same-spin electron is localized at the reference position with all the other electros located somewhere in the space. Like in Sect. 2.2, it shows how the electronic motion of electrons creating a same-spin pair is correlated. For a closed-shell Hartree-Fock wave function, the so-called Fermi hole mobility function F(r) ... 

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. 

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... 

Figure 6-1. Fermi holes of different depths for the on-top density. 

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 XC energy represents the correction to the Coulomb energy for the self-energy of an electron in a many-electron system. The latter is due to both the direct self-energy of the electron as well as the redistribution of electronic density around each electron because of the Pauli exclusion principle and the Coulomb interaction. As an example, we now discuss the case of Fermi hole and the exchange energy in Hartree-Fock (HF) theory [16]. For brevity, we restrict ourselves to closed-shell cases. 

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... 

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. 

Figure 1 2 10. The reduced Lifshitz parameter"z" - (ET - EF)/(EA- ET), where (EA- Er) is the full energy band dispersion in the c-axis direction, as a function of the number of holes in the G subband in A1 doped MgB2. The quantum uncertainty in the z value is indicated by the error bars that are given by D (

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