Bond fukui indices: Comparison of frozen molecular orbital and finite differences through mulliken populations

Bond Fukui functions and matrices are introduced for ab initio levels of theory using a Mulliken atoms in molecules model. It is shown how these indices may be obtained from first‐order density matrix derivatives without need for going to second‐order density matrices as in a previous work. The importance of taking into account the nonorthogonality of the basis in ab initio calculations is shown, contrasting the present results with previous work based on Hückel theory. It is shown how the extension of Fukui functions to Fukui matrices allows getting more insight into the nature of bond Fukui functions. All presently introduced indices respect the necessary normalization conditions and include the classical single atom condensed Fukui functions. © 2013 Wiley Periodicals, Inc.     

Critical thoughts on computing atom condensed Fukui functions

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory.     

Atom and Bond Fukui Functions and Matrices: A Hirshfeld‐I Atoms‐in‐Molecule Approach

The Fukui function is often used in its atom‐condensed form by isolating it from the molecular Fukui function using a chosen weight function for the atom in the molecule. Recently, Fukui functions and matrices for both atoms and bonds separately were introduced for semiempirical and ab initio levels of theory using Hückel and Mulliken atoms‐in‐molecule models. In this work, a double partitioning method of the Fukui matrix is proposed within the Hirshfeld‐I atoms‐in‐molecule framework. Diagonalizing the resulting atomic and bond matrices gives eigenvalues and eigenvectors (Fukui orbitals) describing the reactivity of atoms and bonds. The Fukui function is the diagonal element of the Fukui matrix and may be resolved in atom and bond contributions. The extra information contained in the atom and bond resolution of the Fukui matrices and functions is highlighted. The effect of the choice of weight function arising from the Hirshfeld‐I approach to obtain atom‐ and bond‐condensed Fukui functions is studied. A comparison of the results with those generated by using the Mulliken atoms‐in‐molecule approach shows low correlation between the two partitioning schemes.     

Removing electrons can increase the electron density: a computational study of negative Fukui functions

Ab initio and density-functional theory calculations for a family of substituted acetylenes show that removing electrons from these molecules causes the electron density along the C-C bond to increase. This result contradicts the predictions of simple frontier molecular orbital theory, but it is easily explained using the nucleophilic Fukui function-provided that one is willing to allow for the Fukui function to be negative. Negative Fukui functions emerge as key indicators of redox-induced electron rearrangements, where oxidation of an entire molecule (acetylene) leads to reduction of a specific region of the molecule (along the bond axis, between the carbon atoms). Remarkably, further oxidization of these substituted acetylenes (one can remove as many as four electrons!) causes the electron density along the C-C bond to increase even more. This work provides substantial evidence that the molecular Fukui function is sometimes negative and reveals that this is due to orbital relaxation.     

Can one oxidize an atom by reducing the molecule that contains it?

Negative values for the condensed Fukui function are identified as the key to designing molecules in which reduction of the molecule is associated with oxidation of one of the atomic centers, or vice versa. Sufficient conditions for negative condensed Fukui functions are derived, and metal complexes are identified as likely candidates for this exotic redox chemistry. Based on our theoretical understanding of where negative values of the Fukui function occur [P. W. Ayers, R. C. Morrison and R. K. Roy. J. Chem. Phys., 2002, 116, 8731], molecular-orbital diagrams for molecules where molecular oxidation is coupled to atomic reduction (or vice versa) are sketched. Whether one could design a metal complex with these properties is an open question but, if one could, then that compound would have fascinating redox chemistry and interesting magnetic properties. Candidate molecules for this property include metal complexes with small metal-to-ligand and/or ligand-to-metal charge transfer excitation energies.     

Quantum chemical study of the inhibitive properties of 2-pyridyl-azoles

Four molecules that have been proven to act as corrosion inhibitors of mild steel in acidic media are studied. The inhibitive efficiency of these molecules is explained by means of electronic structure calculations of the protonated species that seem to represent better the actual situation of the experimental conditions. By assuming that the interaction between the inhibitor and the metallic surface occurs through donation and back-donation, it is shown, with a simple charge transfer model, that the interaction energy is favored when hardness increases, in agreement with the experimentally observed inhibition efficiencies. A local analysis with Hirshfeld condensed Fukui functions, and local Fukui functions, provides further support to the donation and back-donation mechanism.     

Application of density functional theory concepts to the study of the chemical reactivity of thiadiazoles

Several useful concepts derived from Density Functional Theory have been applied to study the chemical reactivity of 1,2,5- and 1,3,4-thiadiazoles. Total hardness of the molecules in terms of the HOMO-LUMO gap as a measure of aromaticity and the condensed Fukui functions related to the variations of the net charges of the atoms resulting from a Mulliken population analysis were calculated in order to determine the reactivity of different sites within the molecules studied. The net charges have been obtained from calculations made in the context of the Hartree-Fock-LCAO approximation and the results compared with the existing experimental evidence on thiadiazoles and related compounds.     

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.     

Atomic charges,dipole moments,and Fukui functions using the Hirshfeld partitioning of the electron density

In the Hirshfeld partitioning of the electron density, the molecular electron density is decomposed in atomic contributions, proportional to the weight of the isolated atom density in the promolecule density, constructed by superimposing the isolated atom electron densities placed on the positions the atoms have in the molecule. A maximal conservation of the information of the isolated atoms in the atoms-in-molecules is thereby secured. Atomic charges, atomic dipole moments, and Fukui functions resulting from the Hirshfeld partitioning of the electron density are computed for a large series of molecules. In a representative set of organic and hypervalent molecules, they are compared with other commonly used population analysis methods. The expected bond polarities are recovered, but the charges are much smaller compared to other methods. Condensed Fukui functions for a large number of molecules, undergoing an electrophilic or a nucleophilic attack, are computed and compared with the HOMO and LUMO densities, integrated over the Hirshfeld atoms in molecules.     

Theoretical Definition of a Functional Group and the Molecular Orbital Paradigm

It is the purpose of this review to demonstrate that the empirical classification of the observations of chemistry in terms of the properties assigned to functional groups is a consequence of and is predicted by physics. This is accomplished by showing that the atoms and functional groups of chemistry can be identified with bounded space-filling objects whose properties are defined by quantum mechanics. The quantum mechanical definition of a group is combined with a new pictorial representation of its form to obtain a unified picture which should make it eminently recognizable to chemists. This picture, when combined with the demonstrated ability of these groups to recover the measured properties of atoms in molecules, is offered as one which meets the expectations a chemist associates with the concept of a functional group. The manner in which this physical definition of a group differs fundamentally from models of functional groups based upon molecular orbital theory is discussed.     

Negative and Infinite Fukui Functions: The Role of Diagonal Dominance in the Hardness Matrix

The possible genesis of negative atom condensed Fukui functions is discussed based on hardness kernel matrix relationships. The recent hypothesis that diagonal dominance of the hardness matrix is a requirement for positive Fukui functions is proven, and general considerations also predict the possibility of regions with numerically unstable Fukui functions, including discontinuities.     

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.     

Chemical reactivity and selectivity using Fukui functions: basis set and population scheme dependence in the framework of B3LYP theory

The use of Fukui functions for the site selectivity of the formaldehyde molecule for nucleophilic, electrophilic and radical attacks has been made with special emphasis to the dependence of Fukui values on the basis sets as well as population schemes in the framework of B3LYP theory. Out of the five population schemes selected viz., Mulliken population analysis, natural population analysis, CHELP, CHELPG and atoms in molecules (AIM), it is found that the CHELPG and AIM schemes predict precise reactive site with less dependency on the basis sets. Charges derived from Hirshfeld partitioning, calculated using the BLYP/dnd method (implemented in the DMOL³ package), provide non-negative Fukui values for all the molecular systems considered in this study. Supporting results have been obtained for acetaldehyde and acetone molecules at the 6-31+G** basis set level. These results support the fact that high Fukui values correspond to soft–soft interaction sites. On the other hand, the correlation of the low Fukui value to the hard–hard interaction site merits further investigation. Received: 10 November 2001 / Accepted: 6 March 2002 / Published online: 13 June 2002     

Quantum similarity study of atoms: a bridge between hardness and similarity indices

A hardness based similarity index for studying the quantum similarity for atoms is analyzed. The investigation of hardness and Fukui functions of atoms leads to the construction of a quantum similarity measure, which can be interpreted as a quantified comparison of chemical reactivity of atoms. Evaluation of the new measure reveals periodic tendencies throughout Mendeleev's table. Moreover on the diagonal the global hardness was recovered. Considering a corresponding quantum similarity index reveals that renormalization of the measure can mask periodic patterns. The hardness was calculated for atoms with nuclear charge 3相似文献   

Influence of electron correlation and degeneracy on the Fukui matrix and extension of frontier molecular orbital theory to correlated quantum chemical methods

The Fukui function is considered as the diagonal element of the Fukui matrix in position space, where the Fukui matrix is the derivative of the one particle density matrix (1DM) with respect to the number of electrons. Diagonalization of the Fukui matrix, expressed in an orthogonal orbital basis, explains why regions in space with negative Fukui functions exist. Using a test set of molecules, electron correlation is found to have a remarkable effect on the eigenvalues of the Fukui matrix. The Fukui matrices at the independent electron model level are mathematically proven to always have an eigenvalue equal to exactly unity while the rest of the eigenvalues possibly differ from zero but sum to zero. The loss of idempotency of the 1DM at correlated levels of theory causes the loss of these properties. The influence of electron correlation is examined in detail and the frontier molecular orbital concept is extended to correlated levels of theory by defining it as the eigenvector of the Fukui matrix with the largest eigenvalue. The effect of degeneracy on the Fukui matrix is examined in detail, revealing that this is another way by which the unity eigenvalue and perfect pairing of eigenvalues can disappear.     

How big are atoms in crystals?

Atoms and bonds are central concepts in structural chemistry, but neither are concepts that arise naturally from the physics of condensed phases. It is ironic that the internuclear distances in crystals that are readily measured depend on the sizes of atoms, but since atoms in crystals can be defined in many different ways, all of them arbitrary and often incompatible, there is no natural way to express atomic size. I propose a simple coherent picture of Atoms-in-Crystals which combines properties selected from three different physically sound definitions of atoms and bonds. The charge density of the free atom that is used to construct the procrystal is represented by a sphere of constant charge density having the quantum theory of atoms in molecules (QTAIM) bonded radius. The sum of these radii is equal to the bond length that correlates with the bond flux (bond valence) in the flux theory of the bond. The use of this model is illustrated by answering the question: How big are atoms in crystals? The QTAIM bonded radii are shown to be simple functions of two properties, the number of quantum shells in the atomic core and the flux of the bond that links neighbouring atoms. Various radii can be defined. The univalent bonded radius measures the intrinsic size of the atom and is the same for all cations in a given row of the periodic table, but the observed bonded radius depends also on the bond flux that reflects the chemical environment.     

Evaluation of methods to predict reactivity of gold nanoparticles

Several methods have appeared in the literature for predicting reactivity on metallic surfaces and on the surface of metallic nanoparticles. All of these methods have some relationship to the concept of frontier molecular orbital theory. The d-band theory of Hammer and N?rskov is perhaps the most widely used predictor of reactivity on metallic surfaces, and it has been successfully applied in many cases. Use of the Fukui function and the condensed Fukui function is well established in organic chemistry, but has not been so widely applied in predicting the reactivity of metallic nanoparticles. In this article, we will evaluate the usefulness of the condensed Fukui function in predicting the reactivity of a family of cubo-octahedral gold nanoparticles and make comparison with the d-band method.