Density functional theory Fermi holes

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... 

M. A. Buijse and E. J. Baerends, in Electronic Density Functional Theory of Molecules, Clusters and Solids, D. E. Ellis, Ed., Kluwer, Dordrecht, 1995, pp. 1-46. Fermi Holes and Coulomb Holes. 

The symmetry problem is trivially solved in the local-scaling version of density functional theory because we can include the symmetry conditions in our choice of orbit-generating or initial wavefunction W [33]. Since it is from this initial wave-function that we obtain and xc,gy namely, the non-local quantities appearing in the energy functional of Eq. (50), it follows that the symmetry properties of the parent wavefunction are transferred to the variational functional. Notice, therefore, that symmetry is not as important for the density as it is for the Fermi and Coulomb holes, which are related, to and jtyc,gy respectively. 

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... 

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 ... 

Since the correlation between opposite spins has both intra- and interorbital 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 correlation 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, there is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, and the corresponding phenomenon for electrons of same spin is the Fermi hole. This hole picture is discussed in more detail in connection with density functional theory in Chapter 6. 

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... 

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