Analytical algorithms for ligand cone angles calculations. Application to triphenylphosphine palladium complexes

We defined the smallest enclosing cone angle as the Tolman cone angle for null atomic spheres radii. Then we provide a simple analytical algorithm to compute the smallest enclosing cone at fixed apex, which works in the case of unsymmetrical ligand. We applied it to compute ligand cones for a family of triphenylphosphine palladium complexes, and we showed that both the angle of the cone and its resulting solid angle strongly correlate with the Tolman cone angle, thus suggesting that there is no more need for atomic radii. We also defined the best cone of fixed apex fitting a population of unit vectors. We proposed a simple analytical algorithm to compute it, which is proved to work in any d -dimensional Euclidean space. We defined the conicity index κ to evaluate quantitavely the pertinence of the best fitting cone. We used this best fit cone to define a mean ligand cone, and thus a mean cone angle and a mean cone axis. We applied it to our family of triphenylphosphine palladium complexes and we observed that the axis of the individual cones deviated from the mean cone axis by at most 13.2°. The observed conicity index was small κ = 0.0177 , indicating a very good fit for the whole family of complexes.