  In this paper, we propose and analyze a gradient flow based Kohn-Sham density
functional theory. First, we prove that the critical point of the gradient flow
based model can be a local minimizer of the Kohn-Sham total energy. Then we
apply a midpoint scheme to carry out the temporal discretization. It is shown
that the critical point of the Kohn-Sham energy can be well-approximated by the
scheme. In particular, based on the midpoint scheme, we design an orthogonality
preserving iteration scheme to minimize the Kohn-Sham energy and show that the
orthogonality preserving iteration scheme produces approximations that are
orthogonal and convergent to a local minimizer under reasonable assumptions.
Finally, we report numerical experiments that support our theory.
