Revisiting the Maximum Hardness Principle: A Quantitative Analysis on Reaction Datasets

Chemical hardness is one of the fundamental concepts in chemical reactivity theory, rigorously defined within the framework of Conceptual Density Functional Theory (CDFT). The associated maximum hardness principle (MHP), which postulates that a favorable direction of reaction is toward the state of maximum hardness, has been widely applied as a guiding rule to govern the direction of chemical reactions. However, several studies have questioned its validity. In this work, we investigated the quantitative applicability of the MHP using two reaction datasets, viz, a dipolar cycloaddition dataset of 5269 reaction profiles and the more comprehensive BH9 dataset of 449 reactions. We adopted different approximations to compute the hardness, including Kohn-Sham orbital based frontier molecular orbital (FMO) and the finite difference approximation (FDA). The effect of using different definitions to compute the average hardness of bimolecular reactions is analyzed on the validity of MHP. Our analysis revealed dependence of hardness values on the level of theory, definitions and approximations used which should be kept in mind before criticizing the MHP. Among 5269 cycloaddition reactions, approximately 80% of the reactions are found to obey the MHP. For BH9 dataset, MHP is found to better supported for unimolecular reactions, while remaining strongly dependent on the definition used to compute the average hardness for bimolecular reactions. Reactions in which the individual hardness of the reactants differs significantly are more likely to disobey MHP. Nonetheless, since these electronic structure principles are qualitative in nature, their validity cannot be expected to be universal.