Tucker-tensor algorithm for large-scale Kohn-Sham density functional theory calculations

Abstract

In this work, we propose a systematic way of computing a low-rank globally adapted localized Tucker-tensor basis for solving the Kohn-Sham density functional theory (DFT) problem. In every iteration of the self-consistent field procedure of the Kohn-Sham DFT problem, we construct an additive separable approximation of the Kohn-Sham Hamiltonian. The Tucker-tensor basis is chosen such as to span the tensor product of the one-dimensional eigenspaces corresponding to each of the spatially separable Hamiltonians, and the localized Tucker-tensor basis is constructed from localized representations of these one-dimensional eigenspaces. This Tucker-tensor basis forms a complete basis, and is naturally adapted to the Kohn-Sham Hamiltonian. Further, the locality of this basis in real-space allows us to exploit reduced-order scaling algorithms for the solution of the discrete Kohn-Sham eigenvalue problem. In particular, we use Chebyshev filtering to compute the eigenspace of the Kohn-Sham Hamiltonian, and evaluate nonorthogonal localized wave functions spanning the Chebyshev filtered space, all represented in the Tucker-tensor basis. We thereby compute the electron-density and other quantities of interest, using a Fermi-operator expansion of the Hamiltonian projected onto the subspace spanned by the nonorthogonal localized wave functions. Numerical results on benchmark examples involving pseudopotential calculations suggest an exponential convergence of the ground-state energy with the Tucker rank. Interestingly, the rank of the Tucker-tensor basis required to obtain chemical accuracy is found to be only weakly dependent on the system size, which results in close to linear-scaling complexity for Kohn-Sham DFT calculations for both insulating and metallic systems. A comparative study has revealed significant computational efficiencies afforded by the proposed Tucker-tensor approach in comparison to a plane-wave basis.