SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Extended systems

Abstract As the first component of SPARC (Simulation Package for Ab-initio Real-space Calculations), we present an accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory (DFT) for isolated clusters. Specifically, utilizing a local reformulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local component of the force, we develop a framework using the finite-difference representation that enables the efficient evaluation of energies and atomic forces to within the desired accuracies in DFT. Through selected examples consisting of a variety of elements, we demonstrate that SPARC obtains exponential convergence in energy and forces with domain size; systematic convergence in the energy and forces with mesh-size to reference plane-wave result at comparably high rates; forces that are consistent with the energy, both free from any noticeable ‘egg-box’ effect; and accurate ground-state properties including equilibrium geometries and vibrational spectra. In addition, for systems consisting up to thousands of electrons, SPARC displays weak and strong parallel scaling behavior that is similar to well-established and optimized plane-wave implementations, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for practical DFT simulations of isolated clusters. Program summary Program title: SPARC Catalogue identifier: AFBL_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AFBL_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU GPL v3 No. of lines in distributed program, including test data, etc.: 47525 No. of bytes in distributed program, including test data, etc.: 826436 Distribution format: tar.gz Programming language: C/C++. Computer: Any system with C/C++ compiler. Operating system: Linux. RAM: Problem dependent. Ranges from 80 GB to 800 GB for a system with 2500 electrons. Classification: 7.3. External routines: PETSc 3.5.3 ( http://www.mcs.anl.gov/petsc ), MKL 11.2 ( https://software.intel.com/en-us/intel-mkl ), and MVAPICH2 2.1 ( http://mvapich.cse.ohio-state.edu/ ). Nature of problem: Calculation of the electronic and structural ground-states for isolated clusters in the framework of Kohn–Sham Density Functional Theory (DFT). Solution method: High-order finite-difference discretization. Local reformulation of the electrostatics in terms of the electrostatic potential and pseudocharge densities. Calculation of the electronic ground-state using the Chebyshev polynomial filtered Self-Consistent Field (SCF) iteration in conjunction with Anderson extrapolation/mixing. Evaluation of boundary conditions for the electrostatic potential through a truncated multipole expansion. Reformulation of the non-local component of the force. Geometry optimization using the Polak–Ribiere variant of non-linear conjugate gradients with secant line search. Restrictions: System size less than ∼4000 electrons. Local Density Approximation (LDA). Troullier–Martins pseudopotentials without relativistic or non-linear core corrections. Running time: Problem dependent. Timing results for selected examples provided in the paper.