Accelerating real-space methods by discontinuous projection*

By virtue of multiple advances in the past two decades, real-space electronic structure methods have surpassed planewave methods in large-scale calculations of isolated and extended systems alike. Combining advances in both finite-difference and finite-element methods over the decades, we discuss a new approach to accelerate real-space methods further still, while retaining the simplicity, systematic convergence, and parallelizability inherent in the methodology. The key idea is to compress the large, sparse real-space Hamiltonian by projection in a strictly local, systematically improvable, discontinuous basis spanning the occupied subspace. We show how this basis can be constructed and employed to reduce the dimension of the real-space Hamiltonian by up to three orders of magnitude. Molecular dynamics step times of a few minutes for systems containing thousands of atoms demonstrate the scalability of the methodology in a discontinuous Galerkin formulation [1]. Results for 1D, 2D, and 3D systems demonstrate the additional advantages afforded by the new projection formulation [2].
[1] J. Chem. Theory Comput. 14, 2930–2946 (2018).
[2] J. Chem. Phys. 149, 094104 (2018).