Exchange potential contribution

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

It is however possible to obtain a physically meaningful representation of 0(r) for cations, in the context of density functional theory. The basic expression here is the fundamental stationary principle of DFT, which relates the electronic chemical potential ju, with the electrostatic potential and the functional derivatives of the kinetic and exchange-correlation contributions [20] ... 

The KS exchange potential coefficient aKS x(0) is essentially the image-potential value of 1/4, ranging from 0.195 to 0.274 over the metallic range of densities. Its value is precisely 0.250 for 0 = yjl, which corresponds to a Wigner-Seitz radius of rg — 4.1. The jellium model is stable for approximately this value of rs. With the assumption that the asymptotic structure of vxc (r) is the image potential, we see that the correlation contribution to this structure is an... 

Hartree potential, (d) the Fock non-local exchange potential, (e-g) second-order contributions to the self-energy Xj(E) (e) direct (optical potential), (f) exchange and (g) Fermi sea correlation... 

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