Chemical potential of molecules contrasted to averaged atomic electronegativities: alarming differences and their theoretical rationalization

Electronegativity Equilibration

Controlling the distribution of electrons in materials is the holy grail of chemistry and material science. Practical attempts at this feat are common but are often reliant on simplistic arguments based on electronegativity. One challenge is knowing when such arguments work, and which other factors may play a role. Ultimately, electrons move to equalize chemical potentials. In this work, we outline a theory in which chemical potentials of atoms and molecules are expressed in terms of reinterpretations of common chemical concepts and some physical quantities: electronegativity, chemical hardness, and the sensitivity of electronic repulsion and core levels with respect to changes in the electron density. At the zero-temperature limit, an expression of the Fermi level emerges that helps to connect several of these quantities to a plethora of material properties, theories and phenomena predominantly explored in condensed matter physics. Our theory runs counter to Sanderson’s postulate of electronegativity equalization and allows a perspective in which electronegativities of bonded atoms need not be equal. As chemical potentials equalize in this framework, electronegativities equilibrate.

Temperature-Dependent Approach to Electronic Charge Transfer

A charge transfer model is developed within the framework of the grand canonical ensemble through the analysis of the behavior of the fractional charge as a function of the chemical potential of the bath when the temperature and the external chemical potential are kept fixed. Departing from the fact that, before the interaction between two species, each one has a zero fractional charge, one can identify two situations after the interaction occurs where the fractional charge of at least one of the species is different from zero, indicating that there has been charge transference. One of them corresponds to the case when one of the species is immersed in a bath conformed by the other one, while the other is related to the case in which both species are present in equal amounts (stoichiometric proportion). Correlations between the fractional charges and average energies, thus obtained with experimental equilibrium constants, kinetic rate constants, hydration constants, and bond enthalpies, indicate that, although at the experimental temperatures, they are very small quantities, they have chemically meaningful information. Additionally, in the stoichiometric case, one also finds a rather good correlation between the equalized chemical potential and the one obtained from experimental information for a test set of diatomic and triatomic molecules.

Theorems and rules connecting bond energy and bond order with electronegativity equalization and hardness maximization

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.

Electronegativity-a perspective

Electronegativity is a very useful concept but it is not a physical observable; it cannot be determined experimentally. Most practicing chemists view it as the electron-attracting power of an atom in a molecule. Various formulations of electronegativity have been proposed on this basis, and predictions made using different formulations generally agree reasonably well with each other and with chemical experience. A quite different approach, loosely linked to density functional theory, is based on a ground-state free atom or molecule, and equates electronegativity to the negative of an electronic chemical potential. A problem that is encountered with this approach is the differentiation of a noncontinuous function. We show that this approach leads to some results that are not chemically valid. A formulation of atomic electronegativity that does prove to be effective is to express it as the average local ionization energy on an outer contour of the atom's electronic density.

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 = 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 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 _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, B₂, O₃, and transition metal compounds. New linear relationships are discovered between the MHP hardness increase Δη _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.

Symmetry laws improve electronegativity equalization by orders of magnitude and call for a paradigm shift in conceptual density functional theory

The strict Wigner-Witmer symmetry constraints on chemical bonding are shown to determine the accuracy of electronegativity equalization (ENE) to a high degree. Bonding models employing the electronic chemical potential, μ, as the negative of the ground-state electronegativity, χ(GS), frequently collide with the Wigner-Witmer laws in molecule formation. The violations are presented as the root of the substantially disturbing lack of chemical potential equalization (CPE) in diatomic molecules. For the operational chemical potential, μ(op), the relative deviations from CPE fall between -31% ≤ δμ(op) ≤ +70%. Conceptual density functional theory (cDFT) cannot claim to have operationally (not to mention, rigorously) proven and unified the CPE and ENE principles. The solution to this limitation of cDFT and the symmetry violations is found in substituting μ(op) (i) by Mulliken's valence-state electronegativity, χ(M), for atoms and (ii) its new generalization, the valence-pair-affinity, α(VP), for diatomic molecules. Mulliken's χ(M) is equalized into the α(VP) of the bond, and the accuracy of ENE is orders of magnitude better than that of CPE using μ(op). A paradigm shift replacing the dominance of ground states by emphasizing valence states seems to be in order for conceptual DFT.