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

Abstract

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 = η -<η >, of the molecular hardness, η , and the averaged atomic hardness,<η >, 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, η , instead of the conventional ground state (GS) hardness, η. 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 B, O, and transition metal compounds. New linear relationships are discovered between the MHP hardness increase Δη and the intrinsic bond dissociation energy D . For bond formations, MHP and HEP are not compatible, and HEP does not qualify as an ordering rule.

Eliminating symmetry problems in electronegativity equalization and correcting self‐interaction errors in conceptual DFT

“…Two excited states, 3 E and 1 E , are formed by the removal of an electron from the degenerate ( 1e ) MO of the CH 3 molecule . This is the main bonding orbital and spans all three CH bonds.…”

“…[1][2][3][4]19,23,[30][31][32][33]49,75,76] Ruedenberg's theory is semi-empirically accounted for in the VS-Atoms-in-Molecules (VSAM) model of bonding. [19][20][21]23,32,55,[66][67][68][69][70][71]83,88,92] Consider a constant total number of electrons in the molecule, N 0 , and continuous orbital occupation numbers, 0 ≤ n i ≤ 2. In the VSAM model, we bypass the calculations of ρ VS and π VS , and model the sharing-penetration energy proportional to ¼ n 2 i J, just as in the RHF method.…”

“…The VPA function is thus a specific charge dependent generalization of Mulliken's χ VS . The VPA further depends on the VS orbital hardness [19][20][21]23,32,55,[66][67][68][69][70][71]83,88] η VS ðX, iÞ = 1=2 ½I VS ðX,iÞ − A VS ðX, iÞ = 1=2 J VS ðX,iÞ ð27Þ…”

“…Note that different bonds may have different values of α VP , in accordance with the photoelectron spectra, where different peaks may be attributed to specific bonds. [52][53][54][55] Because of the proportionality [18,86] between the χ VS (X, i) and η VS (X, i), the CT-average <α VP (at)> CT of eqs. 32 and 34 is very similar to the geometric mean, <α VP (at) > GM .…”

“…The updated I and A data are listed in Supporting Information Table S1, for their references, see Refs. [52][53][54][55][56]. Figure 7 gives an overview of the results.…”