ΔSCF Excitation Energies up a Ladder of Ground-State Density Functionals

Density functional theory (DFT) is a widespread and effective tool in electronic structure calculations for ground-state electron systems. Its success has prompted exploration into the use of DFT for noncollective excited states. The delta self-consistent field (ΔSCF) method allows for the extension of DFT to excited-state energies by restricting the Kohn-Sham orbital occupations, producing an excited-state electron density, and then computing its energy. In this paper, we examine the performance of the LSDA, PBE generalized-gradient approximation (GGA), and SCAN/r²SCAN meta-GGA for the excitation energies of several important systems. We consider the energies of atoms with atomic numbers 1-18. For the hydrogen atom, where we use the exact electron density and have no multiplet splitting, we find significant improvement up the ladder from LSDA to PBE to SCAN. For the uniform gas, we find an effective mass different from the bare mass only with r²SCAN. We split the case of multielectron atoms into non-Aufbau excitations, where the highest-energy electron is excited to the lowest state in the next nl subshell (where accuracy is least limited by available basis sets), and spin-flip excitations, where the spin of an electron is flipped, leading to a higher energy state of the same nl configuration. We find reasonably accurate approximate excitation energies, except for the spin-flip cases where the auxiliary noninteracting wave function of a non-Hund's-rule spin state is not well described by a single determinant, but the single-determinant results are improved by spin purification.