Impact of the Kohn-Sham delocalization error on the 4f shell localization and population in lanthanide complexes

The extent of ligand to metal donation bonding and mixing of 4f (and 5d) orbitals with ligand orbitals is studied by Kohn-Sham (KS) calculations for LaX₃ (X = F, Cl, Br, I), GdX₃, and LuX₃ model complexes, CeCl₆^2-, YbCp₃, and selected lanthanide complexes with larger ligands. The KS delocalization error (DE) is quantified via the curvature of the energy for noninteger electron numbers. The extent of donation bonding and 4f-ligand mixing correlates well with the DE. For Lu complexes, the DE also correlates with the extent of mixing of ligand and 4f orbitals in the canonical molecular orbitals (MOs). However, the localized set of MOs and population analyses indicate that the closed 4f shell is localized. Attempts to create situations where mixing of 4f and ligand orbitals occurs due to a degeneracy of fragment orbitals were unsuccessful. For La(III) and, in particular, for Ce(IV), Hartree-Fock, KS, and coupled cluster singles and doubles calculations are in agreement in that excess 4f populations arise from ligand donation, along with donation into the 5d shell. Likewise, KS calculations for all systems with incompletely filled 4f shells, even those with "optimally tuned" functionals affording a small DE, produce varying degrees of excess 4f populations which may be only partially attributed to 5d polarization.