The Hartree-Fock approximation in jellium

In the homogeneous electron gas -or jellium- the electrons are assumed to be embedded in a uniform compensating positive background. It is well known that in this case the plane waves  

These states fill up the Fermi sphere, the radius kp of which is obtained by imposing that the number of states in the Fermi sphere equals the number of electrons, giving  

The exchange potential which enters in the Hartree-Fock equation (Eq. 10), can be evaluated analytically in this simple case  

Averaging the exchange potential (Eq. 20) over all the occupied states k kp (replacing the sum by an integral (2n/(27r) ) / because 

Dash-dotted line Kohn-Sham approximation. 

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