Gradient term in the Kohn-Sham exchange-correlation potential

We show that the gradient expansion of the Kohn-Sham potential does not exist for the case of pure exchange but that when correlation is taken into account by screening the exchange, the expansion does exist and has a different dependence on the charge density than that assumed by Herman et al. and by Sham. We obtain the gradient term to all powers in $e^2$. Our gradient term vanishes in regions where the charge density vanishes, unlike the gradient term calculated by Herman et al. and Sham which becomes infinite.