Meta-GGA density functional calculations on atoms with spherically symmetric densities in the finite element formalism

TL;DR

This paper extends the implementation of density functional calculations to meta-GGA functionals within the finite element formalism for atoms with spherically symmetric densities, enabling more accurate atomic calculations for use in polyatomic systems.

Contribution

The work introduces a new implementation of meta-GGA density functionals in the finite element formalism for spherically symmetric atoms, advancing atomic DFT calculations.

Findings

Many meta-GGA functionals are ill-behaved for Li and Na atoms.

Basis set truncation errors vary strongly with the functional used.

Total energies converge to 0.1 μEh when densities below 10^{-11} a0^{-3} are screened.

Abstract

Density functional calculations on atoms are often used for determining accurate initial guesses as well as generating various types of pseudopotential approximations and efficient atomic-orbital basis sets for polyatomic calculations. To reach the best accuracy for these purposes, the atomic calculations should employ the same density functional as the polyatomic calculation. Atomic density functional calculations are typically carried out employing spherically symmetric densities, corresponding to the use of fractional orbital occupations. We have described their implementation for density functional approximations (DFAs) belonging to the local density approximation (LDA) and generalized gradient approximation (GGA) levels of theory as well as Hartree-Fock (HF) and range-separated exact exchange [S. Lehtola, Phys. Rev. A 2020, 101, 012516]. In this work, we describe the extension to meta-GGA functionals using the generalized Kohn-Sham scheme, in which the energy is minimized with respect to the orbitals, which in turn are expanded in the finite element formalism with high-order numerical basis functions. Furnished with the new implementation, we continue our recent work on the numerical well-behavedness of recent meta-GGA functionals [S. Lehtola and M. A. L. Marques, J. Chem. Phys. 2022, 157, 174114]. We pursue complete basis set (CBS) limit energies for recent density functionals, and find many to be ill-behaved for the Li and Na atoms. We report basis set truncation errors (BSTEs) of some commonly used Gaussian basis sets for these density functionals and find the BSTEs to be strongly functional dependent. We also discuss the importance of density thresholding in DFAs and find that all of the functionals studied in this work yield total energies converged to $0.1\ \mu E_{h}$ when densities smaller than $10^{-11}a_{0}^{-3}$ are screened out.