The Coulomb Hole

From equations (2-17) and (2-21) it is obvious that the Coulomb hole must be normalized to zero, i. e. the integral over all space contains no charge  

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. 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the Xa method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

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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. 

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 problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

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... 

To illustrate the convergence of the FCI principal expansion with respect to short-range electron correlation, we have in Fig. 1.1 plotted the ground-state He wavefunction with both electrons fixed at a distance of 0.5 ao from the nucleus, as a function of the angle 0i2 between the position vectors ri and r2 of the two electrons. The thick grey lines correspond to the exact nonrelativistic wavefunction, whereas the FCI wavefunctions are plotted using black lines. Clearly, the description of the Coulomb cusp and more generally the Coulomb hole is poor in the orbital approximation. In particular, no matter how many terms we include in the FCI wavefunction, we will not be able to describe the nondifferentiability of the wavefunction at the point of coalescence. 

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),... 

As discussed in the text, the asymptotic structure of vxc(r) in the vacuum can be thought of as arising from that part of the Coulomb hole localized to the... 

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. 

This definition ensures that together with the Kohn-Sham theory Fermi plP(r, F), and the quantum-mechanical Fermi-Coulomb p (r, F) holes, the Coulomb hole pP(r, F) too corresponds to the system density p(r). The correlation energy EP[p] is then... 

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