Removal of residual nonspherical self-interaction error in LDA+$U$

In the established LDA+$U$ method, the electron self-interaction, which is generally nonspherical and orbital-dependent, is removed in a mean-field way. This results in residual self-interaction errors, particularly pronounced for $f$-electrons. An alternative double counting scheme that modifies the exchange, not Hartree, energy of LDA is proposed as a remedy. We show that LDA+$U$ with our approach preserves the expected degeneracy of $f^1$ and $f^2$ states in free ions and the correct ground states in the PrO$_2$ solid.