GÀconvergence a b s t r a c tWe present a real-space, non-periodic, finite-element formulation for Kohn-Sham density functional theory (KS-DFT). We transform the original variational problem into a local saddle-point problem, and show its well-posedness by proving the existence of minimizers. Further, we prove the convergence of finite-element approximations including numerical quadratures. Based on domain decomposition, we develop a parallel finite-element implementation of this formulation capable of performing both all-electron and pseudopotential calculations. We assess the accuracy of the formulation through selected test cases and demonstrate good agreement with the literature. We also evaluate the numerical performance of the implementation with regard to its scalability and convergence rates. We view this work as a step towards developing a method that can accurately study defects like vacancies, dislocations and crack tips using density functional theory (DFT) at reasonable computational cost by retaining electronic resolution where it is necessary and seamlessly coarse-graining far away.
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...
As the second 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 extended systems. Specifically, employing a local formulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local force component, we develop a finite-difference framework wherein both the energy and atomic forces can be efficiently calculated to within desired accuracies in DFT. We demonstrate using a wide variety of materials systems that SPARC achieves high convergence rates in energy and forces with respect to spatial discretization to reference plane-wave result; exponential convergence in energies and forces with respect to vacuum size for slabs and wires; energies and forces that are consistent and display negligible 'egg-box' effect; accurate properties of crystals, slabs, and wires; and negligible drift in molecular dynamics simulations. We also demonstrate that the weak and strong scaling behavior of SPARC is similar to well-established and optimized plane-wave implementations for systems consisting up to thousands of electrons, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for performing DFT simulations of extended systems.where the summation index J runs over all atoms in Ω, and the summation index J ′ runs over the J th atom and its periodic images. In addition, the coefficients γ Jl and projection functions χ Jlm can be expressed as(6) Above, u J ′ lm are the isolated atom pseudowavefunctions, V J ′ l are the angular momentum dependent pseudopotentials, and V J ′ are the local components of the pseudopotentials, with l and m signifying the azimuthal and magnetic quantum numbers, respectively.Above, the summation index J runs over all atoms in Ω, and the summation index J ′ runs over the J th atom and its periodic images. The electronic ground-state is determined by solving the non-linear eigenvalue problem in Eqn. 16 using the Self-Consistent Field (SCF) method [52]. Specifically, a fixed-point iteration is performed with respect to the potential V ef f = V xc + φ, which is further accelerated using mixing/extrapolation schemes [53,54,55,56]. In each iteration of the SCF method, the electrostatic potential is calculated by solving the Poisson equation, and the electron density is determined by computing the eigenfunctions of the linearized Hamiltonian. The orthogonality requirement amongst the Kohn-Sham orbitals makes such a procedure scale asymptotically as O(N 3 ) with respect to the number of atoms, which severely limits the size of systems that can be studied. To overcome this restrictive scaling, O(N ) approaches [20,21,57,58] will be subsequently developed and implemented into SPARC. are summed to obtain f h J,loc . Algorithm 2: Calculation of the local component of the atomic force. Input: R, φ (i,j,k) , V J , and r b J f h,p J...
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.
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