Comments on Electronegativity Equalization with Pauling Units

The chemical hardness concept and its realization within the conceptual density functional theory is approached with innovative perspectives, such as the electronegativity and hardness equalization of atoms in molecules connected with the softness kernel, in order to examine the structure-reactivity equalization ansatz between the electronic sharing index and the charge transfer either in the additive or geometrical mean picture of bonding. On the other hand, the maximum hardness principle presents a relation with the chemical stability of the hardness concept. In light of the inverse relation between hardness and polarizability, the minimum polarizability principle has been proposed. Additionally, this review includes important applications of the chemical hardness concept to solid-state chemistry. The mentioned applications support the validity of the electronic structure principles regarding chemical hardness and polarizability in solid-state chemistry.

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.

In analogy with Sanderson's electronegativity equalization principle, it is possible to postulate a principle of spin potential equalization in the E[N(alpha), N(beta)] representation of the spin polarized density functional theory, where N(alpha) and N(beta) refer to the number of electrons with spins alpha and beta, respectively. The principle provides simple expressions to evaluate the energy changes DeltaE between two interacting molecules, A and B, together with the electron transfer, DeltaN(alpha) and DeltaN(beta). The model is illustrated for a series of addition reactions of electrophilic, nucleophilic, and ambiphilic carbenes to alkenes in their singlet and triplet multiplicities. The results are in a consistent qualitative agreement with the experimental reactivity established for these systems.

Conceptual density functional theory provides descriptors (such as chemical hardness, the Fukui function, and the dual descriptor) that enable valuable computational insights into the reactivity of battery electrolyte components.

In this report, we have attempted to explore whether the hardness equalization principle can be conceived analogous to the well‐established electronegativity equalization principle. We have relied upon the fact that the hardness, like the electronegativity, is a qualitative property, and there is commonality in the basic philosophy of the origin and the operational significance of these two fundamental descriptors—the electronegativity and the hardness of atoms in physics and chemistry. Starting from the empirical radial dependent formula of computing the hardness of atoms suggested by us, we have derived an ansatz of the molecular hardness assuming that hardness equalization principle is operative and justifiably valid. For a validity test, we have applied the suggested ansatz to compute the hardness of as many as four sets of compounds with widely divergent physical and chemical nature. As hardness is not an experimentally measurable property, there is no benchmark to perform any validity test of our computed data. We have, therefore, computed the hardness data of these four different sets of the compound invoking the approximate and operational formula of Parr and Pearson, (I − A)/2, to evaluate hardness values and to compute I and A, we have invoked Koopmans' theorem and an ab initio quantum chemical method. We have observed that there is a close correlation between the four sets of hardness data computed through the semi‐empirical ansatz of this work and the quantum mechanical method. Thus, it appears that the ansatz of computing molecular hardness derived on the basis of the hardness equalization principle is efficacious in computing molecular hardness. The detailed comparative study suggests that the paradigm of the hardness equalization principle may be another law of nature like the established electronegativity equalization principle. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

We extend the framework of conceptual density functional theory (DFT) to include fourth-order energy derivatives. We present the series of quantum reactivity descriptors at this order for the canonical and the grand canonical ensembles. After introducing a direct computational methodology, we investigate and describe the quartic curvature λ and the third-order Fukui function f(3)(r). These higher-order descriptors capture nonlinear aspects of electronic reactivity that go beyond the conventional concepts of chemical potential, hardness, or Fukui functions. The global descriptor λ, obtained via finite-difference approximations of frontier orbital energies, quantifies the curvature of the chemical hardness and offers insight into the electronic (in)stability of molecular systems under charge perturbations. The local descriptor f(3)(r), derived as the third-order response of the electron density, reveals spatially resolved regions of charge-transfer sensitivity. We apply these descriptors to a series of push-pull organic molecules and open-shell systems, demonstrating their ability to highlight differences in internal charge transfer character and electronic delocalization. The results support the use of fourth-order conceptual DFT tools as chemically meaningful indicators of reactivity beyond the harmonic regime.

The Xα method with a self-consistent parameter α has been used to calculate the electronegativity of several diatomic molecules. The electronegativities have been determined on the principle of electronegativity equalization using the results of atomic calculations.

Comment on “A new equation based on ionization energies and electron affinities of atoms for calculating of group electronegativity” by S. Kaya and C. Kaya [Comput. Theoret. Chem. 1052 (2015) 42–46

Conceptual Density Functional Theory and Some Recent Developments

A formulation of the chemical potential (electronegativity) equalization principle is presented from the perspective of density-functional theory. The resulting equations provide a linear-response framework for describing the redistribution of electrons upon perturbation by an applied field. The method has two main advantages over existing electronegativity equalization and charge equilibration methods that allow extension to accurate molecular dynamics simulations. Firstly, the expansion of the energy is taken about the molecular ground state instead of the neutral atom ground states; hence, in the absence of an external field, the molecular charge distribution can be represented by static point charges and dipoles obtained from fitting to high-level ab initio calculations without modification. Secondly, in the presence of applied fields or interactions with other molecules, the density response can be modeled accurately using basis functions. Inclusion of basis functions with dipolar or higher order multipolar character allows molecules or chemical groups to have correct local anisotropic polarizabilities. A modified semiempirical form of the hardness matrix has been introduced that can be evaluated efficiently using Gaussians, and requires only one parameter per basis function. Applications at two basis-set levels demonstrate the method can accurately reproduce induced dipole moments and estimated chemical potentials obtained from density-functional calculations for a variety of molecules. Inclusion of basis functions beyond the conventional spherical-atom type is essential in some instances. The present formulation provides the foundation for a promising semi-empirical model for polarization and charge transfer in molecular simulations.

We report the synthesis and characterization, colorimetric anion sensors properties, density functional theory (DFT) calculation studies of a Schiff base (E)-2-[(3,5-bis(trifluoromethyl)phenylimino)methyl]-4-chlorophenol. The molecular structure of the title compound was experimentally determined using X-ray single-crystal data and was compared to the structure predicted by theoretical calculations using DFT. The experimental and calculated data (using DFT) of the title compound have been compared. To investigate the tautomeric stability, some properties such as total energy, HOMO and LUMO energies, the chemical hardness (η), the ionization potential, the electron affinity, the absolute electronegativity (χ), the absolute softness (σ) of the compound were obtained at B3LYP/6–311++G(d,p) level in the gas phase and solvent phase. The calculated results showed that the enol-imine form of the compound was more favorite than keto-amine form. The colorimetric response of the Schiff base receptors in DMSO was investigated.

A new method for evaluating the antioxidant activity of organic compounds based on quantum chemical calculations of their electronic structure using the DFT B3LYP/6-31G (d, p) method has been proposed. The geometric parameters of antioxidant molecules were optimized and the reactivity indices were determined from the energy values of the highest occupied and the lowest unoccupied molecular orbitals: absolute electronegativity; electronic chemical potential; absolute "chemical hardness". The found indicators allow us to quantify the antioxidant activity by building their dependence on the energy of the lower vacant molecular orbitals. The correlation of the obtained parameters with the standard, such as trolox, allows determining the relative antioxidant activity of the test substance.

Metal hexafluorides hydrolyze at ambient temperature to deposit compounds having fluorine-to-oxygen ratios that depend upon the identity of the metal. Uranium-hexafluoride hydrolysis, for example, deposits uranyl fluoride (UO2F2), whereas molybdenum hexafluoride (MoF6) and tungsten hexafluoride deposit trioxides. Here, we pursue general strategies enabling the prediction of depositing compounds resulting from multi-step gas-phase reactions. To compare among the three metal-hexafluoride hydrolyses, we first investigate the mechanism of MoF6 hydrolysis using hybrid density functional theory (DFT). Intermediates are then validated by performing anharmonic vibrational simulations and comparing with infrared spectra [McNamara et al., Phys. Chem. Chem. Phys. 25, 2990 (2023)]. Conceptual DFT, which is leveraged here to quantitatively evaluate site-specific electrophilicity and nucleophilicity metrics, is found to reliably predict qualitative deposition propensities for each intermediate. In addition to the nucleophilic potential of the oxygen ligands, several other contributing characteristics are discussed, including amphoterism, polyvalency, fluxionality, steric hindrance, dipolar strength, and solubility. To investigate the structure and composition of pre-nucleation clusters, an automated workflow is presented for the simulation of particle growth. The workflow entails a conformer search at the density functional tight-binding level, structural refinement at the hybrid DFT level, and computation of a composite free-energy profile. Such profiles can be used to estimate particle nucleation kinetics. Droplet formation is also considered, which helps to rationalize the different UO2F2 particle morphologies observed under varying levels of humidity. Development of predictive methods for simulating physical and chemical deposition processes is important for the advancement of material manufacturing involving coatings and thin films.

Highly Hazardous Pesticides (HHPs) pose severe risks to human health and the environment, making it essential to understand their molecular stability and degradation pathways. In this study, the Quantum Theory of Atoms in Molecules (QTAIM) was applied to four representative organophosphate pesticides, allowing the identification of electronically weak bonds as intrinsic sites of lability. These findings are consistent with reported hydrolytic, oxidative, enzymatic, and microbial degradation routes. Importantly, QTAIM descriptors proved largely insensitive to solvation, confirming their intrinsic character within the molecular electronic structure. To complement QTAIM, conceptual DFT (Density Functional Theory) reactivity indices were analyzed, revealing that solvent effects induce more noticeable variations in global and local descriptors than in topological parameters. In addition, a Topological Analysis of the Fukui Function (TAFF) was performed, which mapped nucleophilic, electrophilic, and radical susceptibilities directly onto QTAIM basins. The TAFF analysis confirmed that bonds identified as weak by QTAIM (notably P–O, P–S, and P–N linkages) also coincide with the most reactive sites, thereby reinforcing their mechanistic role in degradation pathways. This integrated framework highlights the robustness of QTAIM, the sensitivity of global and local reactivity descriptors to solvation revealed by conceptual DFT, and the complementary insights provided by TAFF, contributing to risk assessment, remediation strategies, and the rational design of safer pesticides.

Conceptual Density Functional Theory