Where Does the Density Localize in the Solid State? Divergent Behavior for Hybrids and DFT+U

Approximate
density functional theory (DFT) is widely used in chemistry
and physics, despite delocalization errors that affect energetic and
density properties. DFT+U (i.e., semilocal DFT augmented with a Hubbard
U correction) and global hybrid functionals are two commonly employed
practical methods to address delocalization error. Recent work demonstrated
that in transition-metal complexes both methods localize density away
from the metal and onto surrounding ligands, regardless of metal or
ligand identity. In this work, we compare density localization trends
with DFT+U and global hybrids on a diverse set of 34 transition-metal-containing
solids with varying magnetic state, electron configuration and valence
shell, and coordinating-atom orbital diffuseness (i.e., O, S, Se).
We also study open-framework solids in which the metal is coordinated
by molecular ligands, i.e., MCO3, M­(OH)2, M­(NCNH)2, K3M­(CN)6 (M = V–Ni). As in
transition-metal complexes, incorporation of Hartree–Fock exchange
consistently localizes density away from the metal, but DFT+U exhibits
diverging behavior, localizing density (i) onto the metal in low-spin
and late transition metals and (ii) away from the metal in other cases
in agreement with hybrids. To isolate the effect of the crystal environment,
we extract molecular analogues from open-framework transition-metal
solids and observe consistent localization of the density away from
the metal in all cases with both DFT+U and hybrid exchange. These
observations highlight the limited applicability of trends established
for functional tuning on transition-metal complexes even to equivalent
coordination environments in the solid state.