Exchange-correlation energy exclusion principle

Equation 4.49 defines the exchange or Fermi hole. It is as if an electron of a given spin digs a hole around itself in space in order to exclude another electron of the same spin from coming near it (Pauli exclusion principle). The integrated hole charge is unity, i.e., there is exactly one electron inside the hole. Likewise, the correlation energy functional can be defined as... 

We see such a relation between J and E in the second row (KS ). The other rows report already various approximations computed by HF, BLYP, and BP, each of which gives its own Kohn-Sham spinorbitals and its own approximation of the density distrihution po. This density distribution was used for the calculation of the components of the total energy within each approximate method. Of course, the Hartree-Fbck method (third row) gave 0 for the correlation energy, because there is no correlation in it except that which follows fi-om the Pauli exclusion principle taken into account in the exchange energy (cf. Chapter 10). 

The restrictions introduced come from the Pauli exclusion principle (cf. Slater determinant), and hence have been related to the exchange energy. So far, no restriction has appeared that would stem from the Coulombic interactions of electrons. This made people think of differentiating the holes into two contributions exchange hole and correlation hole he (called the Coulombic... 

This way, the Fermi hole is the quantum measure of the Pauli Exclusion principle, at the level of exchange energy (because it takes into account the reciprocal orientation of spin) and correlation (because it generalizes the Coulombian classical interaction). 

It is worth commenting further on Exc- Like planets, electrons perturb one another when they approach closely and, as a consequence, their motions cannot possibly be independent. Moreover, unlike planets, electrons with the same spin avoid one another particularly keenly and, because of the Pauli Exclusion Principle, are never found at the same point in space. The non-independence of motion arising from the Pauli Principle is known as Fermi correlation. The consequent stabilization, along with a correction for the self-repulsions of the electrons, is known as the exchange energy Ex, and is the major part of Exc- A smaller component, Eq, arises primarily from correlation between the motions of electrons with diff erent spins. 

So far, within the Born-Oppenheimer approximation, the theory is exact. However, we do not know the precise form of E [n] and V (r) = into which we have placed all the complicated many-body physics. As the name suggests, P [w] arises from a combination of two quantum mechanical effects electron exchange and correlation. Briefly, electron exchange arises because a many-body wave function must by antisymmetric under exchange of any two electrons since electrons are fermions. This antisymmetry of the wave function, which is simply a general expression of the Pauli exclusion principle, reduces the Coulomb energy of the electronic system by increasing the spatial separation between electrons of like... 

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