Hole charge, Fermi-Coulomb, distribution

The quantum-mechanical exchange-correlation energy EXCD>] and the Pauli-Coulomb component Wxc(r) of the potential can be expressed in terms of a field c(r). This field is derived via Coulomb s law from the quantum-mechanical Fermi-Coulomb hole charge distribution pxc(r, r/) at r7 for an electron at r as... 

The pair-correlation density is a property that arises due to the Pauli and Coulomb correlations between electrons. Thus it can also be interpreted as the density p(r ) at r plus the reduction in this density at r due to the electron correlations. The reduction in density about an electron which occurs as a result of the Pauli exclusion principle and Coulomb repulsion is the quantum-mechanical Fermi-Coulomb hole charge distribution p (r, r ). Thus we may write the pair-correlation density as... 

The work Wn(r) is path-independent and V x = O Furthermore, the scalar potential WaCr) is recognized to be the density-functional theory Hartree potential Vnfr) of Eq. (34). Thus the functional derivative of the Coulomb self-energy functional En[p] has the physical interpretation of being the work done in the field of the electronic density. The component Wee(r) is then the sum of the Hartree potential and the work done to move an electron in the field of the quantum-mechanical Fermi-Coulomb hole charge distribution ... 

This work W,ie(r) is path-independent for the symmetrical density systems noted previously since the V x = 0 for these cases. It is important to note, however, that the corresponding Fermi-Coulomb hole charge distribution Pxc(r r ) which gives rise to the field need not possess the same symmetry for arbitrary electron position. For example, in either closed shell atoms or open-shell atoms in the central-field approximation for which the density is spherically symmetric, the Fermi-Coulomb hole is not, the only exception being when the electron is at the nucleus. 

Fig. 1. Force fields P(r) and < (r) due to the Kohn-Sham theory Fermi and Coulomb holes, and the field (r) due to the quantum-mechanical Fermi-Conlomb hole charge distribution for the He atom. The function (— 1/r ) is also plotted...

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