  In this paper, we study a few theoretical issues in the discretized Kohn-Sham
(KS) density functional theory (DFT). The equivalence between either a local or
global minimizer of the KS total energy minimization problem and the solution
to the KS equation is established under certain assumptions. The nonzero charge
densities of a strong local minimizer are shown to be bounded below by a
positive constant uniformly. We analyze the self-consistent field (SCF)
iteration by formulating the KS equation as a fixed point map with respect to
the potential. The Jacobian of these fixed point maps is derived explicitly.
Both global and local convergence of the simple mixing scheme can be
established if the gap between the occupied states and unoccupied states is
sufficiently large. This assumption can be relaxed if the charge density is
computed using the Fermi-Dirac distribution and it is not required if there is
no exchange correlation functional in the total energy functional. Although our
assumption on the gap is very stringent and is almost never satisfied in
reality, our analysis is still valuable for a better understanding of the KS
minimization problem, the KS equation and the SCF iteration.
