Where Does the Density Localize? Convergent Behavior for Global Hybrids, Range Separation, and DFT+U

Approximate density functional theory (DFT) suffers from many-electron self-interaction error, otherwise known as delocalization error, that may be diagnosed and then corrected through elimination of the deviation from exact piecewise linear behavior between integer electron numbers. Although paths to correction of energetic delocalization error are well-established, the impact of these corrections on the electron density is less well-studied. Here, we compare the effect on density delocalization of DFT+U (i.e., semilocal DFT augmented with a Hubbard U correction), global hybrid tuning, and range-separated hybrid tuning on a diverse test set of 32 transition metal complexes and observe the three methods to have qualitatively equivalent effects on the ground state density. Regardless of valence orbital diffuseness (i.e., from 2p to 5p), ligand electronegativity (i.e., from Al to O), basis set (i.e., plane wave versus localized basis set), metal (i.e., Ti, Fe, Ni), and spin state, or tuning method, we consistently observe substantial charge loss at the metal and gain at ligand atoms (∼0.3–0.5 e or more). This charge loss at the metal is preferentially from the minority spin, leading to increasing magnetic moment as well. Using accurate wave function theory references, we observe that a minimum error in partial charges and magnetic moments occurs at higher tuning parameters than typically employed to eliminate energetic delocalization error. These observations motivate the need to develop multifaceted approximate-DFT error correction approaches that separately treat density delocalization and energetic errors to recover both correct density and orbital energy-derived properties