  While density functional theory (DFT) is widely applied for its combination
of cost and accuracy, corrections (e.g., DFT+U) that improve it are often
needed to tackle correlated transition-metal chemistry. In principle, the
functional form of DFT+U, consisting of a set of localized atomic orbitals (AO)
and a quadratic energy penalty for deviation from integer occupations of those
AOs, enables the recovery of the exact conditions of piecewise linearity and
the derivative discontinuity. Nevertheless, for practical transition-metal
complexes, where both atomic states and ligand orbitals participate in bonding,
standard DFT+U can fail to eliminate delocalization error (DE). Here, we show
that by introducing an alternative valence-state (i.e., molecular orbital or
MO) basis to the DFT+U approach, we recover exact conditions in cases where
standard DFT+U corrections have no error-reducing effect. This MO-based DFT+U
also eliminates DE where standard AO-based DFT+U is already successful. We
demonstrate the transferability of our approach on a range of ligand field
strengths (i.e., from H_2O to CO), electron configurations (i.e., from Sc to Fe
to Zn), and spin states (i.e., low-spin and high-spin) in representative
transition-metal complexes.
