Divide-and-conquer density functional theory on hierarchical real-space grids: Parallel implementation and applications

Abstract

A linear-scaling algorithm based on a divide-and-conquer (DC) scheme has been designed to perform large-scale molecular-dynamics (MD) simulations, in which interatomic forces are computed quantum mechanically in the framework of the density functional theory (DFT). Electronic wave functions are represented on a real-space grid, which is augmented with a coarse multigrid to accelerate the convergence of iterative solutions and with adaptive fine grids around atoms to accurately calculate ionic pseudopotentials. Spatial decomposition is employed to implement the hierarchical-grid DC-DFT algorithm on massively parallel computers. The largest benchmark tests include $11.8\ifmmode\times\else\texttimes\fi{}{10}^{6}$-atom ($1.04\ifmmode\times\else\texttimes\fi{}{10}^{12}$ electronic degrees of freedom) calculation on 131 072 IBM BlueGene/L processors. The DC-DFT algorithm has well-defined parameters to control the data locality, with which the solutions converge rapidly. Also, the total energy is well conserved during the MD simulation. We perform first-principles MD simulations based on the DC-DFT algorithm, in which large system sizes bring in excellent agreement with x-ray scattering measurements for the pair-distribution function of liquid Rb and allow the description of low-frequency vibrational modes of graphene. The band gap of a CdSe nanorod calculated by the DC-DFT algorithm agrees well with the available conventional DFT results. With the DC-DFT algorithm, the band gap is calculated for larger system sizes until the result reaches the asymptotic value.