On the Convergence of the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory

TL;DR

This paper provides a theoretical analysis of the convergence behavior of the self-consistent field iteration in Kohn-Sham density functional theory, revealing conditions under which convergence can be guaranteed.

Contribution

It offers the first rigorous proof of global and local convergence of SCF iteration based on energy functional analysis and Hessian estimates.

Findings

Global convergence from arbitrary initial points.

Local linear convergence near solutions.

Conditions involving energy gap and bounded derivatives.

Abstract

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.