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

摘要

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