High-order finite element method for atomic structure calculations

TL;DR

This paper presents exttt{featom}, an open source high-order finite element code for atomic structure calculations that achieves high accuracy and efficiency in solving quantum equations for heavy atoms.

Contribution

The paper introduces a novel finite element solver with systematic convergence control, high accuracy, and significant speed improvements over existing methods, along with an open source implementation.

Findings

Achieves $10^{-8}$ Hartree accuracy for heavy atoms like uranium.

Demonstrates systematic convergence with mesh and polynomial order.

Provides benchmark results for atomic numbers 1 to 92.

Abstract

We introduce \texttt{featom}, an open source code that implements a high-order finite element solver for the radial Schr\"odinger, 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 $\kappa=\pm1$ 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 Schr\"odinger 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 \texttt{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.