Slaters Approximation of Hartree-Fock Exchange

If this is combined with the classical expression for the nuclear-electron attractive potential and the electron-electron repulsive potential we have the famous Thomas-Fermi expression for the energy of an atom, 

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 

if we can construct a simple but reasonable approximation to the Fermi hole, the solution of equation (3-3) can be made considerably easier. Slater s idea was to assume that 

The radius rs is sometimes called the Wigner-Seitz radius and can be interpreted to a first approximation as the average distance between two electrons in the particular system. Regions of high density are characterized by small values of rs and vice versa. From standard electrostatics it is known that the potential of a uniformly charged sphere with radius rs is proportional to l/rs, or, equivalently, to p( j)1/3. Hence, we arrive at the following approximate expression for Ex (Cx is a numerical constant), 

What does this mean We have replaced the non-local and therefore fairly complicated exchange term of Hartree-Fock theory as given in equation (3-3) by a simple approximate expression which depends only on the local values of the electron density. Thus, this expression represents a density functional for the exchange energy. As noted above, this formula was originally explicitly derived as an approximation to the HF scheme, without any reference to density functional theory. To improve the quality of this approximation an adjustable, semiempirical parameter a was introduced into the pre-factor Cx which leads to the Xa or Hartree-Fock-Slater (HFS) method which enjoyed a significant amount of popularity among physicists, but never had much impact in chemistry, 

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