Hardness maximization or equalization? New insights and quantitative relations between hardness increase and bond dissociation energy.

It has been overlooked that the change of hardness, η, upon bonding is intimately connected to thermochemical cycles, which determine whether hardness is increased according to Pearson's "maximum hardness principle" (MHP) or equalized, as expected by Datta's "hardness equalization principle" (HEP). So far the performances of these likely incompatible "structural principles" have not been compared. Computational validations have been inconclusive because the hardness values and even their qualitative trends change drastically and unsystematically at different levels of theory. Here I elucidate the physical basis of both rules, and shed new light on them from an elementary experimental source. The difference, Δη = ηmol - <ηat >, of the molecular hardness, ηmol , and the averaged atomic hardness, <ηat >, is determined by thermochemical cycles involving the bond dissociation energies D of the molecule, D+ of its cation, and D- of its anion. Whether the hardness is increased, equalized or even reduced is strongly influenced by ΔD = 2D - D+  - D- . Quantitative expressions for Δη are obtained, and the principles are tested on 90 molecules and the association reactions forming them. The Wigner-Witmer symmetry constraints on bonding require the valence state (VS) hardness, ηVS , instead of the conventional ground state (GS) hardness, ηGS . Many intriguingly "unpredictable" failures and systematic shortcomings of said "principles" are understood and overcome for the first time, including failures involving exotic and/or challenging molecules, such as Be2, B2 , O3 , and transition metal compounds. New linear relationships are discovered between the MHP hardness increase ΔηVS and the intrinsic bond dissociation energy Di . For bond formations, MHP and HEP are not compatible, and HEP does not qualify as an ordering rule.