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.