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We introduce featom, an open source code that implements a high-order finite element solver for the radial Schrödinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the κ = ± 1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10 − 8 Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schrödinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines. Program Title: featom CPC Library link to program files: https://doi.org/10.17632/962fzmm7f7.1 Licensing provisions: MIT Programming language: Fortran with command-line interfaces Nature of problem: Solution of the Schrödinger, Dirac, and Kohn–Sham equations of density functional theory for isolated atoms. Solution method: A high-order finite element method is used to assemble a generalized eigenproblem that is then solved for eigenvalues and orbitals. The Poisson equation is solved using the finite element method also. Self-consistent field equations are solved using Pulay mixing. Additional comments including restrictions and unusual features: Our solution methodology for the Dirac equation does not suffer from spurious states. We maintain high accuracy even for κ = ± 1 states. Unlike other methods, our approach is not limited to Coulombic or self-consistent potentials and can handle non-uniform meshes, including exponential meshes. Restrictions: Spherical symmetry. [ABSTRACT FROM AUTHOR]

Ab initio pseudopotentials are a linchpin of modern molecular and condensed matter electronic structure calculations. In this work, we employ multi-objective optimization to maximize pseudopotential softness while maintaining high accuracy and transferability. To accomplish this, we develop a formulation in which softness and accuracy are simultaneously maximized, with accuracy determined by the ability to reproduce all-electron energy differences between Bravais lattice structures, whereupon the resulting Pareto frontier is scanned for the softest pseudopotential that provides the desired accuracy in established transferability tests. We employ an evolutionary algorithm to solve the multi-objective optimization problem and apply it to generate a comprehensive table of optimized norm-conserving Vanderbilt (ONCV) pseudopotentials (https://github.com/SPARC-X/SPMS-psps). We show that the resulting table is softer than existing tables of comparable accuracy, while more accurate than tables of comparable softness. The potentials thus afford the possibility to speed up calculations in a broad range of applications areas while maintaining high accuracy. [ABSTRACT FROM AUTHOR]

A robust and general solver for the radial Schrödinger, Dirac, and Kohn–Sham equations is presented. The formulation admits general potentials and meshes: uniform, exponential, or other defined by nodal distribution and derivative functions. For a given mesh type, convergence can be controlled systematically by increasing the number of grid points. Radial integrations are carried out using a combination of asymptotic forms, Runge–Kutta, and implicit Adams methods. Eigenfunctions are determined by a combination of bisection and perturbation methods for robustness and speed. An outward Poisson integration is employed to increase accuracy in the core region, allowing absolute accuracies of 10−8 Hartree to be attained for total energies of heavy atoms such as uranium. Detailed convergence studies are presented and computational parameters are provided to achieve accuracies commonly required in practice. Comparisons to analytic and current-benchmark density-functional results for atomic number are presented, verifying and providing a refinement to current benchmarks. An efficient, modular Fortran 95 implementation, dftatom, is provided as open source, including examples, tests, and wrappers for interface to other languages; wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines. Program summary: Program title: dftatom Catalogue identifier: AEPA_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEPA_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: MIT license No. of lines in distributed program, including test data, etc.: 14122 No. of bytes in distributed program, including test data, etc.: 157453 Distribution format: tar.gz Programming language: Fortran 95 with interfaces to Python and C. Computer: Any computer with a Fortran 95 compiler. Operating system: Any OS with a Fortran 95 compiler. RAM: 500 MB Classification: 2.1. External routines: Numpy (http://www.numpy.org/) and Cython (http://cython.org/) Nature of problem: Solution of the Schrödinger, Dirac, and Kohn–Sham equations of Density Functional Theory for isolated atoms. Solution method: Radial integrations are carried out using a combination of asymptotic forms, Runge–Kutta, and implicit Adams methods. Eigenfunctions are determined by a combination of bisection and perturbation methods. An outward Poisson integration is employed to increase accuracy in the core region. Self-consistent field equations are solved by adaptive linear mixing. Restrictions: Spherical symmetry Unusual features: Radial integrators work for general potentials and meshes. No restriction to Coulombic or self-consistent potentials; no restriction to uniform or exponential meshes. Outward Poisson integration. Fallback to bisection for robustness. Running time: For uranium, non-relativistic density functional calculation execution time is around 0.6 s for 10−6 a.u. accuracy in total energy on an Intel Core i7 1.46 GHz processor. [ABSTRACT FROM AUTHOR]

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]