Fig. 1. Schematic of the divide-and-conquer algorithm in 2D. The physical space Ω is a union of overlapping domains, Ω=_αΩ_α. Each domain is further decomposed into the non-overlapping core, Ω_0α, the primary buffer layer, Γ_1α (see the shaded area), and the secondary buffer layer, Γ_2α (see the hatched area). The depth of the primary and total (= primary + secondary) buffer layers are d₁ and d, respectively.

Fig. 2. Schematic of hierarchical real-space grids in 2D. Coarse multigrids (gray) are used to accelerate iterative solutions of the DFT problem on the original real-space grid (corresponding to the grid refinement level, l=3). The bottom panel shows that fine grids are adaptively generated near the atoms (spheres) to accurately operate the ionic pseudopotentials on the wave functions.

Fig. 3. Schematic of the parallel divide-and-conquer algorithm in 2D. The physical system is divided into subsystems, P₀,…,P₃, of equal volume, and each subsystem is assigned to a processor in a parallel computer. Each subsystem, in turn, consists of multiple non-overlapping core domains, Ω₀₀,…,Ω₀₈. To perform electronic-structure calculations on overlapping domains, Ω₀,…,Ω₈, on processor P₀, the contributions to the total electron density, , within the primary buffer-layer depth d₁ (see the hatched area), need to be cached from the nearest-neighbor processors. In addition, the ionic positions within depth d+r_c (enclosed by dashed lines) need to be cached from the nearest-neighbor processors to compute ionic pseudopotentials. Here, d and r_c are the total buffer-layer depth and the range of nonlocal pseudopotentials, respectively.

Fig. 4. Potential energy per atom as a function of the domain size for an amorphous CdSe system (512 atoms in a cubic cell of side length, 45.664 a.u.). The buffer size is fixed as 2.854 a.u. The atomic units are used for both energy and length. Numerals in the figure indicate the number of self-consistent iterations required for the convergence of the electron density within , where is the electron density at ith iteration, ρ₀ is the average electron density, 0.0215 a.u., and the brackets denote the average over the grid in the entire system.

Fig. 5. Potential energy as a function of the buffer length, d, for an amorphous CdSe system (512 atoms in a cubic cell of side length, 45.664 a.u.). The domain size is fixed as 11.416 a.u. The atomic units are used for both energy and length. Numerals in the figure indicate the number of self-consistent iterations required for the convergence of the electron density within , where is the electron density at ith iteration, ρ₀ is the average electron density, 0.0215 a.u., and the brackets denote the average over the grid for the entire system.

Fig. 6. The potential energy as a function of the number of self-consistent iterations for an amorphous CdSe system (32,768 atoms in a cubic cell of side length, 182.656 a.u.), where E_min denotes the converged energy (or the potential energy obtained at the 50th iteration). The domain and buffer sizes are 11.416 and 5.708 a.u., respectively. The atomic units are used for both energy and length. The calculation was performed on 512 IBM POWER4 processors.

Fig. 7. (Left) Convergence of the electron density in the self-consistent iterations for an amorphous CdSe system (32,768 atoms in a cubic cell of side length, 182.656 a.u.). ρ_i(r) is the electron density at ith iteration, ρ₀ is the average electron density, 0.0215 a.u., and the brackets denote the average over the grid in the entire system. The domain and buffer sizes are 11.416 and 5.708 a.u., respectively. (Right) Convergence of the electron density in the self-consistent iterations for an amorphous CdSe system (512 atoms in a cubic cell of side length, 45.664 a.u.). The domain and buffer sizes are the same as in the left panel.

Fig. 8. Total (solid curve) and potential (dashed curve) energies in the atomic unit as a function of time in a molecular-dynamics simulation of liquid Rb at 1400 K (432 atoms in a cubic cell of side length, 65.696 a.u.). The domain and buffer sizes are 16.424 and 8.212 a.u., respectively.

Fig. 9. Wall-clock (circles) and communication (squares) times per self-consistent iteration of the parallel EDC-DFT algorithm, with scaled workloads—512 P atom CdSe system on P processors (P=1,…,128) on an Intel Xeon-based Linux cluster.