Fermi hole potential

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... 

The motion of each electron in the Hartree-Fock approximation is solved for in the presence of the average potential of all the remaining electrons in the system. Because of this, the Hartree-Fock approximation, as discussed earlier, does not provide an adequate description of the repulsion between pairs of electrons. If the electrons have parallel spin, they are effectively kept apart in the Hartree-Fock method by the antisymmetric nature of the wavefunction, producing what is commonly known as the Fermi hole. Electrons of opposite spin, on the other hand, should also avoid each other, but this is not adequately allowed for in the Hartree-Fock method. The avoidance in this latter case is called the Coulomb hole. 

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 is attractive because two electrons described by similar wave functions tend to avoid each other through the Fermi hole, which occurs when spins are parallel, this hole being less repulsive when electrons are closer to each other. Anderson has designated this contribution as potential exchange. In the next subsection we shall give physical comments concerning the labels kinetic and potential exchanges. 

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 precursor to Kohn-Sham density-functional theory is Slater theory [12], In the latter theory, the nonlocal exchange operator of Hartree-Fock theory [25] is replaced by the Slater local exchange potential Vf(r) defined in terms of the Fermi hole p,(r, r ) as... 

For the nonuniform electron gas at a metal surface, the Slater potential has an erroneous asymptotic behavior both in the classically forbidden region as well as in the metal bulk. In the vacuum region, the Slater potential has the analytical [10] asymptotic structure [35,51] V r) = — Xs(p)/x, with the coefficient otsiP) defined by Eq. (103). In the metal bulk this potential approaches [35] a value of ( — 1) in units of (3kp/27r) instead of the correct Kohn-Sham value of ( — 2/3). Further, in contrast to finite systems, the Slater potential V (r) and the work W,(r) are not equivalent [31, 35, 51] asymptotically in the classically forbidden region. This is because, for asymptotic positions of the electron in the vacuum, the Fermi hole continues to spread within the crystal and thus remains a dynamic charge distribution [34]. 

In the work-interpretation derivation, the force field due to the pair-correlation density is first determined, and the potential then obtained as the work done to move an electron in this field. The force field and potential due to g (r,r p(r) are the Hartree field t nfr) and potential Wn(r) = Vnfr), respectively, since the spherically symmetric Fermi hole p r,F p(r) does not contribute to the field at the electron position. The field arises only due to the density p(r ) which is a charge distribution that is not spherically symmetric... 

The LDA is not as primitive as it looks. The electron density distribution for the homogeneous gas model satisfies the Pauli exelusion prineiple and, therefore, this approximation gives the Fermi holes that fulfill the boundary eonditions with Eqs. (11.63), (11.76) and (11.79). The LDA is often used beeause it is rather inexpensive, while still giving a reasonable geometry of molecules and vibrational fiequendes. The quantities that the LDA fails to reproduce are the binding energies, ionization potentials, and the intermolecular dispersion interaction. 

Riseborough (1986) re-examined the treatment of Liu and Ho and included both the hybridization term and the primary f-d Coulomb interaction. He found that if the f level is near the bottom of the conduction band, then the inclusion of the hybridization term makes it easier for the hole potential to split off a bound state from the band, as is necessary to explain the double-peak structure. Hybridization plays a less significant role for other locations of the f level. He emphasized that in this model the separation between the two peaks will be slightly larger than the width of the filled part of the conduction band and that the well-screened peak need not appear at the Fermi level. The model is thus compatible with the observation of similar peaks in Pr, Nd, and some of their compounds. 

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