A Trust Region Direct Constrained Minimization Algorithm for the Kohn–Sham Equation

Abstract

The self-consistent field (SCF) iteration, widely used for computing the ground state energy and the corresponding single particle wave functions associated with a many-electron atomistic system, is viewed in this paper as an optimization procedure that minimizes the Kohn–Sham (KS) total energy indirectly by minimizing a sequence of quadratic surrogate functions. We point out the similarity and difference between the total energy and the surrogate, and show how the SCF iteration can fail when the minimizer of the surrogate produces an increase in the KS total energy. A trust region technique is introduced as a way to restrict the update of the wave functions within a small neighborhood of an approximate solution at which the gradient of the total energy agrees with that of the surrogate. The use of trust regions in SCF is not new. However, it has been observed that directly applying a trust region-based SCF (TRSCF) to the KS total energy often leads to slow convergence. We propose to use TRSCF within a direct constrained minimization (DCM) algorithm we developed in [J. Comput. Phys., 217 (2006), pp. 709–721]. The key ingredients of the DCM algorithm involve projecting the total energy function into a sequence of subspaces of small dimensions and seeking the minimizer of the total energy function within each subspace. The minimizer of a subspace energy function, which is computed by the TRSCF, not only provides a search direction along which the KS total energy function decreases, but also gives an optimal “step length" that yields a sufficient decrease in total energy. A numerical example is provided to demonstrate that the combination of TRSCF and DCM is more efficient than SCF.