40 research outputs found
Accelerating self-consistent field iterations in Kohn-Sham density functional theory using a low rank approximation of the dielectric matrix
We present an efficient preconditioning technique for accelerating the fixed
point iteration in real-space Kohn-Sham density functional theory (DFT)
calculations. The preconditioner uses a low rank approximation of the
dielectric matrix (LRDM) based on G\^ateaux derivatives of the residual of
fixed point iteration along appropriately chosen direction functions. We
develop a computationally efficient method to evaluate these G\^ateaux
derivatives in conjunction with the Chebyshev filtered subspace iteration
procedure, an approach widely used in large-scale Kohn-Sham DFT calculations.
Further, we propose a variant of LRDM preconditioner based on adaptive
accumulation of low-rank approximations from previous SCF iterations, and also
extend the LRDM preconditioner to spin-polarized Kohn-Sham DFT calculations. We
demonstrate the robustness and efficiency of the LRDM preconditioner against
other widely used preconditioners on a range of benchmark systems with sizes
ranging from 100-1100 atoms ( 500--20,000 electrons). The
benchmark systems include various combinations of
metal-insulating-semiconducting heterogeneous material systems, nanoparticles
with localized orbitals near the Fermi energy, nanofilm with metal dopants,
and magnetic systems. In all benchmark systems, the LRDM preconditioner
converges robustly within 20--30 iterations. In contrast, other widely used
preconditioners show slow convergence in many cases, as well as divergence of
the fixed point iteration in some cases. Finally, we demonstrate the
computational efficiency afforded by the LRDM method, with up to 3.4
reduction in computational cost for the total ground-state calculation compared
to other preconditioners.Comment: Accepted in Physical Review
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
Periodic Pulay method for robust and efficient convergence acceleration of self-consistent field iterations
Pulay's Direct Inversion in the Iterative Subspace (DIIS) method is one of
the most widely used mixing schemes for accelerating the self-consistent
solution of electronic structure problems. In this work, we propose a simple
generalization of DIIS in which Pulay extrapolation is performed at periodic
intervals rather than on every self-consistent field iteration, and linear
mixing is performed on all other iterations. We demonstrate through numerical
tests on a wide variety of materials systems in the framework of density
functional theory that the proposed generalization of Pulay's method
significantly improves its robustness and efficiency.Comment: Version 2 (with minor edits from version 1