Ligand-Field-Dependent Behavior of Meta-GGA Exchange in Transition-Metal Complex Spin-State Ordering

Prediction of spin-state ordering in transition metal complexes is essential for understanding catalytic activity and designing functional materials. Semilocal approximations in density functional theory, such as the generalized-gradient approximation (GGA), suffer from several errors including delocalization error that give rise to systematic bias for more covalently bound low-spin electronic states. Incorporation of exact exchange is known to counteract this bias, instead favoring high-spin states, in a manner that has recently been identified to be strongly ligand-field dependent. In this work, we introduce a tuning strategy to identify the effect of incorporating the Laplacian of the density (i.e., a meta-GGA) in exchange on spin-state ordering. We employ a diverse test set of M(II) and M(III) first-row transition metal ions from Ti to Cu as well as octahedral complexes of these ions with ligands of increasing field strength (i.e., H₂O, NH₃, and CO). We show that the sensitivity of spin-state ordering to meta-GGA exchange is highly ligand-field dependent, stabilizing high-spin states in strong-field (i.e., CO) cases and stabilizing low-spin states in weak-field (i.e., H₂O, NH₃, and isolated ions) cases. This diverging behavior leads to generally improved treatment of isolated ions and strong field complexes over a standard GGA but worsened treatment for the hexa-aqua or hexa-ammine complexes. These observations highlight the sensitivity of functional performance to subtle changes in chemical bonding.