On the Analysis of the Discretized Kohn-Sham Density Functional Theory

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.