A subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization

TL;DR

This paper introduces a novel subspace projection method using spectral finite-element discretization for large-scale Kohn-Sham density functional theory calculations, achieving near-linear scaling and high accuracy for complex materials.

Contribution

The method combines spectral finite-elements, Chebyshev filtering, and localization to enable efficient, accurate large-scale DFT calculations with subquadratic scaling.

Findings

Achieves chemical accuracy in benchmark systems

Demonstrates near-linear to subquadratic scaling with system size

Significant computational savings over traditional methods

Abstract

We present a subspace projection technique to conduct large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization. The proposed method treats both metallic and insulating materials in a single framework, and is applicable to both pseudopotential as well as all-electron calculations. The key ideas involved in the method include: (i) employing a higher-order spectral finite-element basis that is amenable to mesh adaption; (ii) using a Chebyshev filter to construct a subspace which is an approximation to the occupied eigenspace in a given self-consistent field iteration; (iii) using a localization procedure to construct a non-orthogonal localized basis spanning the Chebyshev filtered subspace; (iv) using a Fermi-operator expansion in terms of the subspace-projected Hamiltonian represented in the non-orthogonal localized basis to compute relevant quantities like the density matrix, electron density and band energy. We demonstrate the accuracy and efficiency of the approach on benchmark systems involving pseudopotential calculations on metallic aluminum nano-clusters up to 3430 atoms and on insulating alkane chains up to 7052 atoms, as well as all-electron calculations on silicon nano-clusters up to 3920 electrons. The benchmark studies revealed that accuracies commensurate with chemical accuracy can be obtained, and a subquadratic-scaling with system size was observed for the range of materials systems studied. In particular, for the alkane chains---close to linear-scaling is observed, whereas, for aluminum nano-clusters---the scaling is observed to be $\mathcal{O} (N^{1.46})$. For all-electron calculations on silicon nano-clusters, the scaling with the number of electrons is computed to be $\mathcal{O} (N^{1.75})$. Furthermore, significant computational savings have been realized with the proposed approach with respect to reference calculations.