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

We present a subspace projection technique to conduct large-scale Kohn-Sham density functional theory calculations using higher-order 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 development of this 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 nonorthogonal localized basis spanning the Chebyshev filtered subspace; and (iv) using a Fermi-operator expansion in terms of the subspace-projected Hamiltonian represented in the nonorthogonal localized basis to compute relevant quantities like the density matrix, electron density, and band energy. We demonstrate the accuracy and efficiency of the proposed approach on benchmark systems involving pseudopotential calculations on aluminum nanoclusters up to 3430 atoms and on alkane chains up to 7052 atoms, as well as all-electron calculations on silicon nanoclusters up to 3920 electrons. The benchmark studies revealed that accuracies commensurate with chemical accuracy can be obtained with the proposed method, and a subquadratic-scaling with system size was observed for the range of materials systems studied. In particular, for the alkane chains-representing an insulating material-close to linear scaling is observed, whereas, for aluminum nanoclusters-representing a metallic material-the scaling is observed to be O(N-1.46). For all-electron calculations on silicon nanoclusters, the scaling with the number of electrons is computed to be O(N-1.75). In all the benchmark systems, significant computational savings have been realized with the proposed approach, with approximately tenfold speedups observed for the largest systems with respect to reference calculations.