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Bond orders are attributed a new role in rationalizing the electronegativity equalization (ENE) and maximum hardness (MH) rules. The following rules and theorems are formulated for chemical species (atoms, groups, molecules), X, Y, XY, their ionization energies, I, electron affinities, A, electronegativity, χ = ½(I + A), and chemical hardness, η = ½ (I − A). Rule 1 Sanderson’s principle of electronegativity equalization is supported (individual deviations < 10%) by association reactions, X + Y → XY, if the ionic bond dissociation energies are equal, D (XY+) = D (XY−), or, equivalently, if the relative bond orders are equal, BOrel (XY+) = BOrel (XY−). Rule 2 Sanderson’s principle of electronegativity equalization is supported (individual deviations < 10%) by association reactions, X + Y → XY, if the formal bond orders, FBO, of the ions are equal, FBO (XY+) = FBO (XY−). Theorem 1 The electronegativity is not equalized by association reactions, X + Y → XY, if the formal bond orders of the ions differ, FBO (XY+) − FBO (XY−) ≠ 0. Theorem 2 The chemical hardness is increased by nonpolar bond formation, 2X → X2, if (and for atomic X: if and only if) the sum BOrel (X2+) + BOrel (X2−) < 2. Rule 3 The chemical hardness is decreased, thus the “maximum hardness principle” violated by association reactions, X + Y → XY, if (but not only if) BOrel (XY+) + BOrel (XY−) > 2. The theorems are proved, and the rules corroborated with the help of elementary thermochemical cycles according to the first law of thermodynamics. They place new conditions on the “structural principles” ENE and MH. The performances of different electronegativities and hardness scales are compared with respect to ENE and MH. The scales based on valence-state energies perform consistently better than scales based on ground-state energies. The present work provides the explanation for the order of magnitude better performance of valence-state ENE compared to that of the ground-state ENE. We here show by a new approach that the combination of several fuzzy concepts clarifies the situation and helps in defining the range of validity of rules and principles derived from such concepts.

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, Δ η  =  η mol – < η at >, of the molecular hardness, η mol , and the averaged atomic hardness, < η at >, 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 Δ D =  2 D – D +   − D − . Quantitative expressions for Δ η 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, η VS , instead of the conventional ground state (GS) hardness, η GS . 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 Be 2, B 2 , O 3 , and transition metal compounds. New linear relationships are discovered between the MHP hardness increase Δ η VS and the intrinsic bond dissociation energy D i . For bond formations, MHP and HEP are not compatible, and HEP does not qualify as an ordering rule.