How good are recent density functionals for ground and excited states of one-electron systems?

Abstract: 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.