In electronic structure methods based on the correction of approximate density-functional theory (DFT) for systematic inaccuracies, Hubbard U parameters may be used to quantify and amend the self-interaction errors ascribed to selected subspaces. Here, in order to enable the accurate, computationally convenient calculation of U by means of DFT algorithms that locate the ground-state by direct total-energy minimization, we introduce a reformulation of the successful linear-response method for U in terms of the fully-relaxed constrained ground-state density. Defining U as an implicit functional of the ground-state density implies the comparability of DFT + Hubbard U (DFT+U) total-energies, and related properties, as external parameters such as ionic positions are varied together with their corresponding first-principles U values. Our approach provides a framework in which to address the partially unresolved question of self-consistency over U, for which plausible schemes have been proposed, and to precisely define the energy associated with subspace many-body self-interaction error. We demonstrate that DFT+U precisely corrects the total energy for self-interaction error under ideal conditions, but only if a simple self-consistency condition is applied. Such parameters also promote to first-principles a recently proposed DFT+U based method for enforcing Koopmans' theorem
