  It is well known that the self-consistent field (SCF) iteration for solving
the Kohn-Sham (KS) equation often fails to converge, yet there is no clear
explanation. In this paper, we investigate the SCF iteration from the
perspective of minimizing the corresponding KS total energy functional. By
analyzing the second-order Taylor expansion of the KS total energy functional
and estimating the relationship between the Hamiltonian and the part of the
Hessian which is not used in the SCF iteration, we are able to prove global
convergence from an arbitrary initial point and local linear convergence from
an initial point sufficiently close to the solution of the KS equation under
assumptions that the gap between the occupied states and unoccupied states is
sufficiently large and the second-order derivatives of the exchange correlation
functional are uniformly bounded from above. Although these conditions are very
stringent and are almost never satisfied in reality, our analysis is
interesting in the sense that it provides a qualitative prediction of the
behavior of the SCF iteration.
