Exact ligand cone angles.

Many properties of transition-metal complexes depend on the steric bulk of bound ligands, usually quantified by the Tolman (θ) and solid (Θ) cone angles, which have proven utility but suffer from various limitations and coarse approximations. Here, we present an improved, mathematically rigorous method to determine an exact cone angle (θ°) by solving for the most acute right circular cone that contains the entire ligand. The procedure is applicable to any ligand, planar or nonplanar, monodentate or polydentate, bound to any metal center in any environment, and it is ideal for analyzing structures from quantum chemical computations as well as X-ray crystallography experiments. Exact cone angles were evaluated for a wide array of phosphine and amine ligands bound to palladium, nickel, or platinum by optimizing structures using B3LYP/6-31G* density functional theory with effective core potentials for the transition metals. The mean absolute deviations of the standard θ and Θ parameters from the exact cone angles were 15-25°, mostly caused by distortions from the assumed idealized structures.