Beyond electronegativity and local hardness: Higher-order equalization criteria for determination of a ground-state electron density

Higher-order global softnesses, local softnesses, and softness kernels are defined along with their hardness inverses. The local hardness equalization principle recently derived by the authors is extended to arbitrary order. The resulting hierarchy of equalization principles indicates that the electronegativity/chemical potential, local hardness, and local hyperhardnesses all are constant when evaluated for the ground-state electron density. The new equalization principles can be used to test whether a trial electron density is an accurate approximation to the true ground-state density and to discover molecules with desired reactive properties, as encapsulated by their chemical reactivity indicators.     

On the foundations of chemical reactivity theory

In formulating chemical-reactivity theory (CRT) so as to give it a deep foundation in density-functional theory (DFT), Parr, his collaborators, and subsequent workers have introduced reactivity indices as properties of isolated reactants, some of which are in apparent conflict with the underlying DFT. Indices which are first derivatives with respect to electron number are staircase functions of number, making electronegativity equalization problematic. Second derivative indices such as hardness vanish, putting hardness-based principles out of reach. By reformulating CRT within our partition theory, which provides an exact decomposition of a system into its component species, we resolve the conflict. We show that the reactivity of a species depends on its chemical context and define that context. We establish when electronegativity equalization holds and when it fails. We define a generalization of hardness, a hardness matrix containing the self-hardness of the individual species and the mutual hardnesses of the pairs of species of the system, and identify the physical origin of hardness. We introduce a corresponding generalization of the Fukui function as well as of the local and global softnesses and the softness kernel of the earlier formulation. We augment our previous formulation of the partition theory by introducing a model energy function and express the difference between the exact and the model forces on the nuclei in terms of the new reactivity indices. For simplicity, our presentation is limited to time-reversal invariant systems with vanishing spin density; it is straightforward to generalize the theory to finite spin density.     

Phenomenological chemical reactivity theory for mobile electrons

We present herein a model to deal with the chemical reactivity, selectivity and site activation concepts of π electron systems derived by merging the classical Coulson–Longuet-Higgins response function theory based on the Hückel molecular orbital (HMO) theory and the conceptual density functional theory. HMO-like expressions for the electronic chemical potential, chemical hardness and softness, including their local counterparts, atomic and bond Fukui functions and non-local response functions are derived. It is shown that sophisticated non-local concepts as site activation may be cast into deeper physical grounds by introducing a simplified version of static response functions. In this way, useful quantities such as self and mutual polarizabilities originally defined through the HMO parameters can be redefined as self and mutual softnesses. The model is illustrated by discussing the classical Hammett free energy relationship describing inductive substituent effects on the reactivity of benzoic acids.     

Systematic formulations for electronegativity and hardness and their atomic scales within density functional softness theory

A unified Mulliken valence with Parr ground‐state electronegativity picture is presented. It provides a useful analytical tool on which the absolute hardness as well ionization potential and electron affinity functionals are based. For all these chemical reactivity indices, systematic approximate density functionals are formulated within density functional softness theory and are applied to atomic systems. For the absolute hardness, a special relationship with the new electronegativity ansatz and a particular atomic trend paralleling the absolute electron affinity are established that should complement and augment the earlier finite‐difference energetic approach. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Frontier orbital and density functional study of the variation of the hard–soft behavior of monoborane (BH3) and boron trifluoride (BF3) as a function of angles of reorganization from planar (D3h) to pyramidal (C3v) shape

In chemical response the BH₃ and BF₃ molecules undergo the physical process of planar (D_3h) to pyramidal (C_3v) reorganization in shape as the condition precedent to the event of chemical reaction under the requirement of symmetry. A frontier orbital and density functional study of the variation of the stability of electronic structures and chemical reactivity of associated with the physical process of D_3h to C_3v geometry reorganization has been performed. The theoretical parameters viz. eigenvalues of HOMO and LUMO, the HOMO and LUMO energy gap, the global hardness and global softness, the chemical potential, the condensed Fukui function, and local softness of B atom, the reaction site, have been computed over a wide range of ∠XBX angles. The nature of variation in the intrinsic chemical reactivity, global and local, of the molecules associated with their geometry reorganization during the chemical event of charge transfer interaction involving their frontier molecular orbitals has been quantitatively explored. The hardness profiles as a function of reaction coordinates are consistent with the principle of maximum hardness (PMH). Results demonstrate that the hardness and softness are not a static and invariable property of molecules but a dynamic and variable function of molecular structure. The hardness parameters and the HOMO–LUMO gap of the molecules are so modified with the distortion of molecular geometry that, after a certain stage of molecular deformation, the profiles of such parameters of the molecules intersect and cross each other, signifying that the relative order of the intrinsic hardness of their equilibrium geometry is reversed. The intrinsically hard molecule BF₃ becomes softer than the intrinsically soft molecule BH₃ as a consequence of structural distortion. The increase in chemical reactivity computed in terms of density functional parameters are transparent and justified in terms of the profiles of the eigenvalues of the frontier orbitals. The profiles of chemical potential reveal the inherent difference in the tendency of backdonation from two molecules. The computed values of Fukui functions and local softness parameters of the B atom site demonstrate that the concept of local softness can be exploited for a theoretical analysis and understanding of the characteristic chemical events of the molecules under consideration. The profiles of the Fukui functions and local softness parameters of the two molecules seem to reflect and reveal their intrinsic difference in the tendency of receiving donation in the LUMO (electrophilicity) and that of backdonation from the HOMO (nucleophilicity) and the inherent difference of overall reactivity of the two molecules by a simultaneous operation of two opposing processes of charge transfer. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Local and nonlocal density functional calculations of the molecular structure of isomeric thiadiazoles

The chemistry of thiadiazoles and their derivatives is of considerable interest in chemistry owing to their pharmacological and potential industrial applications. In this context, a detailed study of isomeric thiadiazole molecules has been done using local (SVWN; Slater, and Vosko, Wilk and Nusair) and nonlocal (BLYP; Becke, and Lee, Yang and Parr) density functionals and optimizing the molecular geometries by means of the gradient technique. A charge sensitivity analysis of the studied molecule has been performed by resorting to density functional theory, obtaining several sensitivity coefficients such as the molecular energy, net atomic charges, global and local hardness, global and local softness and Fukui functions. With these results and the analysis of the dipole moments, the molecular electrostatic potentials and the total electron density maps, several conclusions have been inferred about the preferred sites of chemical reaction of the studied compounds. The condensed Fukui functions are shown to be one of the best criteria for predicting chemical reactivity.     

The unconstrained local hardness: an intriguing quantity, beset by problems

Developing a mathematical approach to the local hard/soft acid/base principle requires an unambiguous definition for the local hardness. One such quantity, which has aroused significant interest in recent years, is the unconstrained local hardness. Key identities are derived for the unconstrained local hardness, δμ/δρ(r). Several identities are presented which allow one to determine the unconstrained local hardness either explicitly using the hardness kernel and the inverse-linear response function, or implicitly by solving a system of linear equations. One result of this analysis is that the problem of determining the unconstrained local hardness is infinitely ill-conditioned because arbitrarily small changes in electron density can cause enormous changes in the chemical potential. This is manifest in the exponential divergence of the unconstrained local hardness as one moves away from the system. This suggests that one should be very careful when using the unconstrained local hardness for chemical interpretation.     

Polarization justified Fukui functions: the theory and applications for molecules

The Fukui functions based on the computable local polarizability vector have been presented for a group of simple molecules. The necessary approximation for the density functional theory softness kernel has been supported by a theoretical analysis unifying and generalizing early concepts produced by the several authors. The exact relation between local polarizability vector and the derivative of the nonlocal part of the electronic potential over the electric field has been demonstrated. The resulting Fukui functions are unique and represent a reasonable refinement when compared to the classical ones that are calculated as the finite difference of the density in molecular ions. The new Fukui functions are strongly validated by their direct link to electron dipole polarizabilities that are reported experimentally and by other computational methods.     

On the redistribution of electrons for chemical reaction systems

A regional density-functional theory is formulated and applied to the study of ground-state electron redistributions during the course of a chemical reaction. If for a given increment of the reaction process, accumulation of electrons occurs in a certain region of space, then it is called the dynamic acceptor region, denoted by P. The complement is called the dynamic donor region, denoted by Q. The regional energy itself is determined as a unique functional of the electron density of the total system. The regional transfer potentials are defined in such a way that they add to give the total chemical potential, and their values along the reaction coordinate are found to be different between P and Q. The difference between the regional transfer potentials is shown to provide the driving force for electron transfer from Q to P. A characteristic coordinate for following electron transfer and an associated excitation potential are introduced. The excitation potential is a measure of regional virtual excitation due to regional interactions. The regional transfer potential gives the local character of electron transferability, while the excitation potential gives the global character. The theory encompasses the concepts of regional hardness and softness and sheds light on the HSAB principle.     

New link between conceptual density functional theory and electron delocalization

In this paper we give a new definition of the softness kernel based on the exchange-correlation density. This new kernel is shown to correspond to the change of electron fluctuation upon external perturbation, thus helping to bridge the gap between conceptual density functional theory and some tools describing electron localization in molecules. With the aid of a few computational calculations on diatomics we illustrate the performance of this new computational tool.     

On the importance of the "density per particle" (shape function) in the density functional theory

The central role of the shape function sigma(r) from the density functional theory (DFT), the ratio of the electron density rho(r) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential micro of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined.     

Stability conditions for density functional reactivity theory: an interpretation of the total local hardness

The second-order Taylor series expansions commonly used in the density functional chemical reactivity theory are used to define local stability conditions for electronic states. Systems which satisfy these conditions are stable to infinitesimal perturbations due to approaching chemical reagents. The basic formalism considered here supersedes previous variational approaches to chemical reactivity theory like the electrophilicity, potentialphilicity, and chargephilicity. The total local hardness emerges naturally in this analysis, and can be clearly interpreted. When the total local hardness is small, the system is relatively insensitive to perturbations. Furthermore, minus the total local hardness is an energetically favorable perturbation of the external potential.     

Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory

The conventional analysis of Perdew and Levy, and Sham and Schlu?ter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, ((?ρ(s)(r('),r))/?N)(v(s) ), at integers. (3) Consistent with the Perdew-Levy-Sham-Schlu?ter conclusion that the exact Kohn-Sham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchange-correlation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electrons.     

Local-density approximation for orbital densities applied to the self-interaction correction

A simple approximation to the functional derivative of Perdew-Zunger-type self-interaction-corrected local-spin density functional is suggested. In this approach, the orbital density |phi(isigma)(r)|(2) is regarded as a functional of the local electron density |phi(isigma)(r)|(2)=n(isigma)(n(sigma)(r)) so as to enable a functional derivative of n(isigma)(n(sigma)(r)) with respect to n(sigma)(r). Our computational results show that this approximation gives fairly good estimates of the total energy, the ionization potential, and the electron affinity for atoms. Comparative studies of this method with the averaged-density approximation and the global averaging method for the self-interaction correction are made.