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 [Lehtola, S.Phys.
Rev. A2020,101, 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 [Lehtola, S.; Marques,
M. A. L.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 0.1μEhwhen
densities smaller than 10–11a0–3are screened
out.