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In approximate density functional theory (DFT), the self-interaction error isan electron delocalization anomaly associated with underestimated insulatinggaps. It exhibits a predominantly quadratic energy-density curve that isamenable to correction using efficient, constraint-resembling methods such asDFT + Hubbard $U$ (DFT+$U$). Constrained DFT (cDFT) enforces conditions on DFTexactly, by means of self-consistently optimized Lagrange multipliers, andwhile its use to automate error corrections is a compelling possibility, weshow that it is limited by a fundamental incompatibility with constraintsbeyond linear order. We circumvent this problem by utilizing separate linearand quadratic correction terms, which may be interpreted either as distinctconstraints, each with its own Hubbard $U$ type Lagrange multiplier, or as thecomponents of a generalized DFT+$U$ functional. The latter approach prevails inour tests on a model one-electron system, $H_2^+$, in that it readily recoversthe exact total-energy while symmetry-preserving pure constraints fail to doso. The generalized DFT+$U$ functional moreover enables the simultaneouscorrection of the total-energy and ionization potential or the correction ofeither together with the enforcement of Koopmans condition. For the lattercase, we outline a practical, approximate scheme by which the required pair ofHubbard parameters, denoted as U1 and U2, may be calculated fromfirst-principles.

Inapplicability of exact constraints and a minimal two-parameter generalization to the DFT+$U$ based correction of self-interaction error