Molecular DFT+U: A Transferable, Low-Cost Approach to Eliminate Delocalization Error

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 (AOs) 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 for which 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 representative transition-metal complexes with a range of ligand field strengths, electron configurations (i.e., from Sc to Zn), and spin states.尽管密度泛函理论（DFT）因兼顾计算成本与精度而得到广泛应用，但为处理关联型过渡金属化学体系，往往需要引入改进其性能的修正方法（如DFT+U）。从原理上看，DFT+U的泛函形式由一组定域原子轨道（AOs）以及针对这些轨道非整数占据情况的二次能量惩罚项构成，能够恢复分段线性性与导数不连续性的严格条件。然而，对于同时涉及原子态与配体轨道成键的实际过渡金属配合物而言，标准DFT+U往往无法消除离域误差（DE）。本文研究表明，为DFT+U方法引入替代价态基组（即分子轨道（MO）基组）后，可在标准DFT+U无法起到误差抑制作用的场景中恢复严格条件；同时该基于分子轨道的DFT+U还可在标准基于原子轨道的DFT+U已能有效消除误差的场景中，进一步去除离域误差。我们通过涵盖不同配体场强度、电子构型（即从钪到锌）与自旋态的典型过渡金属配合物，验证了所提方法的可迁移性。