SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters

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. Since the system sizes studied in ab-initio calculations are relatively modest, the extra computation in the adopted procedure is negligible.Algorithm 1: Pseudocharge density generation and self energy calculationis interpolated on to the finite-difference grid in the overlap region Ω r b J ∩ Ω p = ∅ (and an additional 4n 0 points in each direction) using cubic-splines, 13 from which b (i,j,k) J is calculated using Eqn. 24. Subsequently, f h,p J,loc -contribution of the p th processor to the local component of the force-is calculated using Eqn. 41. Finally, the contributions from all processors are summed to simultaneously obtain f h J,loc for all the atoms. Algorithm 3: Calculation of the local component of the atomic force. Input: R, φ (i,j,k) , V J , and r b J for J ∈ P p r b J do Determine starting and ending indices i s , i e , j s , j e , k s , k e for Ω r b JAgain, the integral in Eqn. 17 has been approximated using the integration rule in Eqn. 22. The calculation of f h J,nloc in SPARC is summarized in Algorithm 4. We use P p r c J to denote the set of all atoms whose Ω r c Jcube with side of length 2r c J centered on the J th atom-has overlap with the processor domain Ω p . The value of r c J corresponds to the maximum cutoff radius amongst the non-local components of the pseudopotential for the J th atom. We have chosen a cube rather than a sphere due to its simplicity and efficiency within the Euclidean finite-difference discretization. While describing Algorithm 4, we use the subscripts s and e to denote the starting and ending indices of Ω r c J ∩ Ω p = ∅, respectively. In th...

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