Direct Minimization for Ensemble Electronic Structure Calculations

Abstract

We consider a direct optimization approach for ensemble density functional theory electronic structure calculations. The update operator for the electronic orbitals takes the structure of the Stiefel manifold into account and we present an optimization scheme for the occupation numbers that ensures that the constraints remain satisfied. We also compare sequential and simultaneous quasi-Newton and nonlinear conjugate gradient optimization procedures, and demonstrate that simultaneous optimization of the electronic orbitals and occupation numbers improve performance compared to the sequential approach.

An introduction to Lie group integrators – basics, new developments and applications

“…where φ : M → R is a smooth discrete energy function to be minimised on M. The discrete energy φ considered here is the so called Kohn-Sham energy, [1]. For a related application of Lie group techniques in quantum control, see [23].…”

Mathematical Analysis and Numerical Approximations of Density Functional Theory Models for Metallic Systems

“…But there may be some fluctuation for the strategy (S1) when the energy almost converges. From the convergence curves for 1 2 ∇ Ψ F and ∇ η F sF , we see that the descent speed of the gradient obtained by the strategy (S1) slows down suddenly when the energy almost converges and then is much smaller than the strategy (S2). Finally, by comparing PCG-S1 and PCG-S1-r1, we find that the restarting approach does improve the convergence of the iteration for the strategy (S1).…”

“…To compare the three step size strategies more clearly, we take Au 92 as an example and show the convergence curves for F − F min , 1 2 ∇ Ψ F and ∇ η F sF in Figure 3, where F min is a high-accuracy approximation of the exact total energy. We also illustrate the benefit of the restarting approach for the strategy (S1).…”

“…Recently, Ulbrich et al [37] studied a proximal gradient method for the ensemble Kohn-Sham DFT with the Fermi-Dirac smearing. We may refer to [1,34] for more works on the direct minimization algorithms for the ensemble Kohn-Sham DFT model. To our knowledge, there is little mathematical analysis on the ensemble Kohn-Sham DFT and its approximations.…”