Ligand Local Rigidity as a Quantitative Descriptor of Lanthanide Size Selectivity

Separation of the rare-earth elements relies on the small, monotonic contraction in the ionic radius across the trivalent lanthanide series. Ligand preorganization is a powerful strategy for amplifying this size difference, but within the N,O-tetradentate ligand family examined here, it approaches a structural ceiling as the most preorganized scaffolds become fully fused. Here, using density functional theory, we introduce cavity rigidity as a quantitative framework for lanthanide size selectivity and define local rigidity (LR) as the resistance of the ligand cavity to the compression required for metal binding. The LR reflects both the intrinsic stiffness of the ligand framework and the extent to which the cavity must deform upon coordination. Across the N,O-tetradentate ligands examined, the computed La/Lu size selectivity increases linearly with LR. The same relationship is observed for five additional lanthanide pairs with larger ionic radius differences producing steeper slopes. Within this ligand family, the results point to two routes for increasing thermodynamic size selectivity: increasing the framework stiffness or increasing the equilibrium donor separation.