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We present the Clenshaw–Curtis Spectral Quadrature (SQ) method for real-space O ( N ) Density Functional Theory (DFT) calculations. In this approach, all quantities of interest are expressed as bilinear forms or sums over bilinear forms, which are then approximated by spatially localized Clenshaw–Curtis quadrature rules. This technique is identically applicable to both insulating and metallic systems, and in conjunction with local reformulation of the electrostatics, enables the O ( N ) evaluation of the electronic density, energy, and atomic forces. The SQ approach also permits infinite-cell calculations without recourse to Brillouin zone integration or large supercells. We employ a finite difference representation in order to exploit the locality of electronic interactions in real space, enable systematic convergence, and facilitate large-scale parallel implementation. In particular, we derive expressions for the electronic density, total energy, and atomic forces that can be evaluated in O ( N ) operations. We demonstrate the systematic convergence of energies and forces with respect to quadrature order as well as truncation radius to the exact diagonalization result. In addition, we show convergence with respect to mesh size to established O ( N 3 ) planewave results. Finally, we establish the efficiency of the proposed approach for high temperature calculations and discuss its particular suitability for large-scale parallel computation. [ABSTRACT FROM AUTHOR]

We present SQDFT: a large-scale parallel implementation of the Spectral Quadrature (SQ) method for O ( N ) Kohn–Sham Density Functional Theory (DFT) calculations at high temperature. Specifically, we develop an efficient and scalable finite-difference implementation of the infinite-cell Clenshaw–Curtis SQ approach, in which results for the infinite crystal are obtained by expressing quantities of interest as bilinear forms or sums of bilinear forms, that are then approximated by spatially localized Clenshaw–Curtis quadrature rules. We demonstrate the accuracy of SQDFT by showing systematic convergence of energies and atomic forces with respect to SQ parameters to reference diagonalization results, and convergence with discretization to established planewave results, for both metallic and insulating systems. We further demonstrate that SQDFT achieves excellent strong and weak parallel scaling on computer systems consisting of tens of thousands of processors, with near perfect O ( N ) scaling with system size and wall times as low as a few seconds per self-consistent field iteration. Finally, we verify the accuracy of SQDFT in large-scale quantum molecular dynamics simulations of aluminum at high temperature. Program summary Program Title: SQDFT Program Files doi: http://dx.doi.org/10.17632/h4jnrmh9v3.1 Licensing provisions: GNU General Public License 3 (GPL) Programming language: C/C++ External routines/libraries: MVAPICH2 2.1 ( http://mvapich.cse.ohio-state.edu/ ) or later. Nature of problem: Quantum mechanical calculation of static and dynamic properties of condensed matter at high temperature in the framework of Kohn–Sham Density Functional Theory (DFT). Solution method: High-order finite-difference discretization. Calculation of the electronic ground state using the self-consistent field (SCF) iteration in conjunction with Periodic Pulay extrapolation/mixing. Local reformulation of the electrostatics in terms of the electrostatic potential and pseudocharge densities. Solution of the Poisson equation using the Alternating Anderson–Richardson (AAR) method. Calculation of the electron density, band structure energy, electronic entropy energy, and atomic forces using the infinite-cell Clenshaw–Curtis Spectral Quadrature (SQ) approach, in which results for the infinite crystal are obtained by expressing quantities as bilinear forms or sums of bilinear forms, that are then approximated by spatially localized Clenshaw–Curtis quadrature rules. NVE molecular dynamics using the leapfrog method and NVT molecular dynamics using the Verlet algorithm with the Nose–Hoover thermostat. Extrapolation of electron density between molecular dynamics steps. Parallelization via domain decomposition. Restrictions: Cubical domain. Local Density Approximation (LDA). Troullier–Martins pseudopotentials without relativistic or non-linear core corrections. [ABSTRACT FROM AUTHOR]