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

Abstract

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 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 proposed approach on benchmark systems involving pseudopotential calculations on aluminum nano-clusters 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 nano-clusters-representing a metallic material-the scaling is observed to be O(N 1.46 ). For all-electron calculations on silicon nano-clusters, 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 ∼ 10−fold speedups observed for the largest systems with respect to reference calculations.

Real-space formulation of orbital-free density functional theory using finite-element discretization: The case for Al, Mg, and Al-Mg intermetallics

“…Moreover, the adaptive nature of the finite-element discretization also enables the consideration of all-electron orbital-free DFT calculations that are widely used in studies of warm dense matter 16,17,19 . Further, recent numerical studies have shown that by using a higher-order finite-element discretization significant computational savings can be realized for both orbital-free DFT 31 and Kohn-Sham DFT calculations 6,47 , effectively overcoming the degree of freedom disadvantage of the finite-element basis in comparison to the plane-wave basis.…”

“…We note that in a non-periodic setting, representing a finite atomic system, all the integrals in equations (5)-(6) are over R 3 and the summations in equations (6)- (7) include all the atoms. In the case of an infinite periodic crystal, all the integrals over x in equations (5)- (6) are over the unit cell whereas the integrals over x ′ are over R 3 . Similarly, in equations (6)- (7), the summation over I is on the atoms in the unit cell, and the summation over J extends over all lattice sites.…”

“…However, the Kohn-Sham approach requires the computation of single-electron wavefunctions to compute the kinetic energy of non-interacting electrons, whose computational complexity typically scales as O(N 3 ) for an N -electron system, thus, limiting standard calculations to materials systems containing few hundreds of atoms. While there has been progress in developing close to linear-scaling algorithms for the Kohn-Sham approach 4,5 , these are still limited to a few thousands of atoms, especially for metallic systems 6 . The orbital-free approach to DFT 7 , on the other hand, models the kinetic energy of non-interacting electrons as an explicit functional of the electron density, thus circumventing the computationally intensive step of computing the singleelectron wavefunctions.…”

Large-scale all-electron density functional theory calculations using an enriched finite-element basis

“…The proposed method offers a computationally efficient, systematically improvable, and scalable basis for large scale all-electron DFT calculations, applicable to both light and heavy atoms. The use of the enrichment in developing linear-scaling DFT algorithms for all-electron calculations based on finite element basis 57,98 or Tuckertensor basis 99 holds good promise, and is currently being investigated. Furthermore, the use of enrichment ideas in conjunction with reduced-order scaling DFT algorithms can also be effectively utilized in the evaluation of the exact exchange operator, and forms a future direction of interest.…”

“…Finite element basis 42,43 , on the other hand, being a local piecewise polynomial basis, retains the variational property of the plane-waves, and, in addition, has other desirable features such as locality of the basis that affords good parallel scalability, being easily amenable to adaptive spatial resolution, and the ease of handling arbitrary boundary conditions. While most studies employing the finite element basis in DFT calculations [44][45][46][47][48][49][50][51][52][53] have shown its usefulness in pseudopotential calculations, some of the works 44,[53][54][55][56][57] have also demonstrated its promise for all-electron calculations. In particular, the work of Motamarri et.…”