Initial hardness response and hardness profiles in the study of Woodward-Hoffmann rules for electrocyclizations

The fundamental principles of pericyclic reactions are governed by the Woodward-Hoffmann rules, which state that these reactions can only take place if the symmetries of the reactants' molecular orbitals and the products' molecular orbitals are the same. As such, these rules rely on the nodal structure of either the wave function or the frontier molecular orbitals, so it is unclear how these rules can be recovered in the density functional reactivity theory (or "conceptual DFT"), where the basic quantity is the strictly positive electron density. A third, nonsymmetry based approach to predict the outcome of pericyclic reactions is due to Zimmerman which uses the concept of the aromatic transition states: allowed reactions possess aromatic transition states, while forbidden reactions possess antiaromatic transition states. Based on our recent work on cycloadditions, we investigate the initial response of the chemical hardness, a central DFT based reactivity index, along the reaction profiles of a series of electrocyclizations. For a number of cases, we also compute complete initial reaction coordinate (IRC) paths and hardness profiles. We find that the hardness response is always higher for the allowed modes than for the forbidden modes. This suggests that the initial hardness response along the IRC is the key for casting the Woodward-Hoffmann rules into conceptual DFT