Atomic Ionization: sd energy imbalance and Perdew-Zunger self-interaction correction energy penalty in 3d atoms

To accurately describe the energetics of transition metal systems, density functional approximations (DFAs) must provide a balanced description of s- and d- electrons. One measure of this is the sd transfer error, which has previously been defined as $E(\mathrm{3d}^{n-1}\mathrm{4s}^{1}) - E(\mathrm{3d}^{n-2}\mathrm{4s}^{2})$. Theoretical concerns have been raised about this definition due to its evaluation of excited-state energies using ground-state DFAs. A more serious concern appears to be strong correlation in the ${4s}^2$ configuration. Here we define a ground-state measure of the sd energy imbalance, based on the errors of s- and d-electron second ionization energies of the 3d atoms, that effectively circumvents the aforementioned problems. We find an improved performance as we move from LSDA to PBE to r$^2$SCAN for first-row transition metal atoms. However, we find large (~ 2 eV) ground-state sd energy imbalances when applying a Perdew-Zunger 1981 self-interaction correction. This is attributed to an "energy penalty" associated with the noded 3d orbitals. A local scaling of the self-interaction correction to LSDA results in a balance of s- and d-errors.