  Sun et al. [J. Chem. Phys. 144, 191101 (2016)] suggested that common density
functional approximations (DFAs) should exhibit large energy errors for excited
states as a necessary consequence of orbital nodality. Motivated by
self-interaction corrected density functional calculations on many-electron
systems, we continue their study with the exactly solvable $1s$, $2p$, and $3d$
states of 36 hydrogenic one-electron ions (H-Kr$^{35+}$) and demonstrate with
self-consistent calculations that state-of-the-art DFAs indeed exhibit large
errors for the $2p$ and $3d$ excited states. We consider 56 functionals at the
local density approximation (LDA), generalized gradient approximation (GGA) as
well as meta-GGA levels, also including several hybrid functionals like the
recently proposed machine-learned DM21 local hybrid functional. The best
non-hybrid functional for the $1s$ ground state is revTPSS. The $2p$ and $3d$
excited states are more difficult for DFAs as Sun et al. predicted, and LDA
functionals turn out to yield the most systematic accuracy for these states
amongst non-hybrid functionals. The best performance for the three states
overall is observed with the BHandH global hybrid GGA functional, which
contains 50% Hartree-Fock exchange and 50% LDA exchange. The performance of
DM21 is found to be inconsistent, yielding good accuracy for some states and
systems and poor accuracy for others. Based on these results, we recommend
including a variety of one-electron cations in future training of
machine-learned density functionals.
