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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]

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]