27,845 research outputs found
On the Convergence of the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory
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
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.Comment: 29 page
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