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.     

A new approach to local hardness

The applicability of the local hardness as defined by the derivative of the chemical potential with respect to the electron density is undermined by an essential ambiguity arising from this definition. Further, the local quantity defined in this way does not integrate to the (global) hardness-in contrast with the local softness, which integrates to the softness. It has also been shown recently that with the conventional formulae, the largest values of local hardness do not necessarily correspond to the hardest regions of a molecule. Here, in an attempt to fix these drawbacks, we propose a new approach to define and evaluate the local hardness. We define a local chemical potential, utilizing the fact that the chemical potential emerges as the additive constant term in the number-conserving functional derivative of the energy density functional. Then, differentiation of this local chemical potential with respect to the number of electrons leads to a local hardness that integrates to the hardness, and possesses a favourable property; namely, within any given electron system, it is in a local inverse relation with the Fukui function, which is known to be a proper indicator of local softness in the case of soft systems. Numerical tests for a few selected molecules and a detailed analysis, comparing the new definition of local hardness with the previous ones, show promising results.     

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.     

Quantum dot with N interacting electrons confined in a power-law external potential

The ground-state physical properties, such as electron density, chemical potential, and total energy, of a two-dimensional quantum dot with N interacting electrons confined in a power-law external potential are numerically determined by the Thomas-Fermi approximation. The effect of the confining potential on properties such as electron density and chemical potential is examined for both interacting and non-interacting systems. It is shown that the results of the calculations are in excellent agreement with those given in the literature. The results indicate that interactions and the shape of the confinement affect the density and thus the ground-state properties of the electrons significantly.     

The Fukui function of an atom in a molecule: A criterion to characterize the reactive sites of chemical species

The principle of hard and soft acids and bases is interpreted as the result of two opposing tendencies, one related to the charge transfer process (chemical potential equalization principle), and the other one related to the reshuffling of the electronic density (maximum hardness or minimum softness principle). A local version of the principle is elucidated by assuming that these tendencies are dominated by the local properties rather than by the global properties of the molecule. This principle is used together with the Fukui function of the atoms in the molecule to characterize the reactive sites. The results presented for the nucleophilic addition to the pyridinium ion, and for the electrophilic substitution on pyridine oxide show the usefulness of these concepts in describing the inherent reactivity of chemical species.     

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.     

Spin-state splittings, highest-occupied-molecular-orbital and lowest-unoccupied-molecular-orbital energies, and chemical hardness

It is known that the exact density functional must give ground-state energies that are piecewise linear as a function of electron number. In this work we prove that this is also true for the lowest-energy excited states of different spin or spatial symmetry. This has three important consequences for chemical applications: the ground state of a molecule must correspond to the state with the maximum highest-occupied-molecular-orbital energy, minimum lowest-unoccupied-molecular-orbital energy, and maximum chemical hardness. The beryllium, carbon, and vanadium atoms, as well as the CH(2) and C(3)H(3) molecules are considered as illustrative examples. Our result also directly and rigorously connects the ionization potential and electron affinity to the stability of spin states.     

Ruling out any electrophilicity equalization principle

Two gas-phase electrophilicity indices, ω(1) and ω(2), introduced by Parr, von Szentpa?ly, and Liu are tested with respect to the recently proposed "principle of electrophilicity equalization." Although electronegativity is equalized in many cases, there is no functioning "hardness equalization principle" nor are the electrophilicity indices principally equalized during molecule formation: they cannot be generally expressed as the mean of the corresponding atomic indices. For large metal clusters and [n]fullerenes, both electrophilicity indices increase proportional to n(1/3) and n(1/2), respectively, as the hardness values converge to zero. Two "principles" are shown to be obsolete: the "geometric mean principle for hardness equalization" and the "principle of electrophilicity equalization", with the latter somewhat relying on the former. An appeal is made to exercise careful judgment before proposing and publishing new structural principles.     

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.     

Polarizability as a local functional of the electron density

Expressions for the multipole polarizability and shielding factor for an atom are obtained using the model where the total electron energy is assumed to be a local functional of the electron density. This simple model correctly predicts the leading term in the 1/Z expansion of the polarizability. Further, the simple local density functional for the polarizability, when evaluated with ground-state Hartree-Fock densities, yields numerical values for atoms which are, in general, in reasonable agreement with those obtained from coupled Hartree-Fock theory.     

Orbitally resolved charge sensitivity analysis: Basic concepts and relations

Following the recent developments of the charge sensitivity analysis (CSA ) in the atoms-in-molecules (AIM ) resolution, the corresponding CSA quantities in the orbital (or shell) resolution (OR ) are defined. The OR electron population variables, in the ordinary closed-shell SCF problem, are the elements of the bond-order matrix P , and their conjugates, “chemical potentials,” F ^T = ?E/? P , are the respective Fock matrix elements, appropriate for the representation in question; here E is the SCF energy. The second derivatives ?²E/? P ? P define the OR hardness tensor from which all related OR CS s, e.g., the hardness, softnesses, Fukui function (FF ) indices, etc., can be determined. The rigid potentials and hardness tensor, corresponding to the “frozen” orbital approximation, are examined in more detail, and the decoupled representation of the normal orbitals (N oO ) is introduced, in which the rigid hardness tensor becomes diagonal. Illustrative valence-shell N oO contours for the water molecule are given and discussed. The new approximation for the OR FF indices, as the orbital occupation probabilities, is proposed on the basis of the density matrix functional development of Donnely and Parr for natural orbitals, and the relevant expressions for the molecular fragment (collection of orbitals) quantities are summarized.     

Modeling of the mutual molecular polarization with an electronegativity equalization approach

A model for the mutual polarization of two approaching molecules is proposed, exploiting the principle of electronegativity equalization. The deformation of the electronic density of one molecule is the response to the perturbation of its chemical potential due to the electrostatic potential of the other molecule. The electronic densities, the density deformations, and the electrostatic potentials of both molecules are described with a previously developed asymptotic density model (ADM ). The ADM model allows a partitioning of all relevant properties in terms of atomic quantities. The perturbation of the chemical potential is given in atomic resolution, and the change of the electronic density is represented in terms of atomic charges. A hardness tensor, which determines the changes of the atomic chemical potentials due to the changes of the atomic charges, is modeled consistently with the ADM and earlier approaches. The results of the model, the changes of atomic charges within the molecules due to their mutual interaction, are compared with the changes of atomic charges obtained from population analysis of ab initio calculations. © 1995 John Wiley & Sons, Inc.     

Density functional theory of complex transition densities

We present an extension of Hohenberg-Kohn-Sham density functional theory to the domain of complex local potentials and complex electron densities. The approach is applicable to resonance (Siegert) [Phys. Rev. 56, 750 (1939)] states and other scattering and transport problems that can be described by a normalized state of a Hamiltonian containing a complex local potential. Such Hamiltonians are non-Hermitian and their eigenvalues are in general complex, the imaginary part being inversely proportional to the lifetime of the system. The one-to-one correspondence between complex local potentials nu and complex electron densities rho is established provided that the complex variables are sufficiently close to real local potentials and densities of nondegenerate ground states. We show that the exchange-correlation functionals, contributing to the complex energy, are determined through analytic continuation of their ground-state-theory counterparts. This implies that the exchange-correlation effects on the lifetime of a resonance are, under appropriate conditions, already determined by the functionals of the ground-state theory.     

On the electronic structure and chemical bonding in the tantalum trimer cluster

The electronic structure and chemical bonding in the Ta 3 (-) cluster are investigated using photoelectron spectroscopy and density functional theory calculations. Photoelectron spectra are obtained for Ta 3 (-) at four photon energies: 532, 355, 266, and 193 nm. While congested spectra are observed at high electron binding energies, several low-lying electronic transitions are well resolved and compared with the theoretical calculations. The electron affinity of Ta 3 is determined to be 1.35 +/- 0.03 eV. Extensive density functional calculations are performed at the B3LYP/Stuttgart +2f1g level to locate the ground-state and low-lying isomers for Ta 3 and Ta 3 (-). The ground-state for the Ta 3 (-) anion is shown to be a quintet ( (5)A 1') with D 3 h symmetry, whereas two nearly isoenergetic states, C 2 v ( (4)A 1) and D 3 h ( (6)A 1'), are found to compete for the ground-state for neutral Ta 3. A detailed molecular orbital analysis is performed to elucidate the chemical boding in Ta 3 (-), which is found to possess multiple d-orbital aromaticity, commensurate with its highly symmetric D 3 h structure.