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.     

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.     

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.     

Relation between the Fukui function and the Coulomb hole

By using a coarse-grain representation of the molecular electronic density, we demonstrate that the value of the condensed Fukui function at an atomic site is directly related to the polarization charge (Coulomb hole) induced by a test electron removed (or added) from (at) the atom. The link between the formation of an electron-hole pair and the condensed Fukui function provides insights on the possible negativity of the Fukui function which is interpreted in terms of two phenomena: overscreening and over-strengthening.     

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.     

Computing Fukui functions without differentiating with respect to electron number. I. Fundamentals

By using perturbations in the molecular external potential, the authors deduce the Fukui function from the change in Kohn-Sham orbital energies, avoiding the troublesome differentiation of the density with respect to electron number. Though this paper focuses on the Fukui function, the same general technique can be used to compute the functional derivative of any observable with respect to the external potential. In this paper, the method is used to compute the Fukui function for the beryllium atom and the formaldehyde molecule. The follow-up paper (part II) addresses the problem of computing condensed reactivity indicators.     

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.     

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.     

Revisiting the calculation of condensed Fukui functions using the quantum theory of atoms in molecules

The analysis of previously reported shortcomings of the condensed Fukui functions obtained making use of the quantum theory of atoms in molecules indicates these drawbacks are due to the inadequacy of the definition employed to compute them and not to the partitioning. A new procedure, which respects the mathematical definition and solves these problems, is presented for the calculation of condensed Fukui functions for atomic basins defined according to the quantum theory of atoms in molecules. It is tested in a set of 18 molecules, which includes the most controversial reported cases.     

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.     

Preferential site of attack on fullerene cations: frontier orbitals and rate coefficients

An analysis of reaction efficiency is presented for reactions of carbonaceous ions and molecules. Our results show that the combination of experimental rate-coefficient measurements and computations of the condensed Fukui functions of frontier molecular orbitals and pyramidal angles of pi orbitals is very useful for elucidating the reactive sites on fullerene carbon clusters in the gas phase.     

Comparison among four different ways to condense the Fukui function

Four different ways to condense the Fukui function are compared. Three of them perform a numerical integration over different basins to define the condensed Fukui function, and the other one is the most traditional Fukui function using Mulliken population analysis. The basins are chosen to be the basins of the electron density (AIM), the basins of the electron localization function (ELF), and the basins of the Fukui function itself. The use of the last two basins is new and presented for the first time here. It is found that the last three methods yield results which are stable against a change in the basis set. The condensed Fukui function using the basins of the ELF is not able to give information on the reactivity of an acceptor molecule. In general, the condensed Fukui function using the basins of the density or the basins of the Fukui function describe the reactivity trends well. The latter is preferred, because it only contains information about the Fukui function itself and it gives the right information for donor as well as acceptor centers.     

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     

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.     

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.     

The Fukui matrix: a simple approach to the analysis of the Fukui function and its positive character

The Fukui matrix is introduced as the derivative of the one-electron reduced density matrix with respect to a change in the number of electrons under constant external potential. The Fukui matrix extends the Fukui function concept: the diagonal of the Fukui matrix is the Fukui function. Diagonalizing the Fukui matrix gives a set of eigenvectors, the Fukui orbitals, and accompanying eigenvalues. At the level of theory used, there is always one dominant eigenvector, with an eigenvalue equal to 1. The remaining eigenvalues are either zero or come in pairs with eigenvalues of the same magnitude but opposite sign. Analysis of the frontier molecular orbital coefficient in the eigenvector with eigenvalue 1 gives information on the quality of the frontier molecular orbital picture. The occurrence of negative Fukui functions can be easily interpreted in terms of the nodal character of the dominant eigenvector versus the characteristics of the remaining eigenvectors and eigenvalues.     

Studies on predicting reactive sites of 3,9-diazatetraasteranes by conceptual density functional theory and experiment

Theoretical studies on predicting potential reactive sites of attractive compounds are of significance, which are capable of providing deeper insight into their chemical behaviors as well as foundations for discovering new drug molecules. The 3,9-diazatetraasterane derivatives are a kind of cage compounds equipping with potent biological activities. In this paper, predication of reactive sites of 3,9-diazatetraasteranes bearing phenyl and ester groups (6,12-diphenyl-3,9-diazahexacyclo[6.4.0.0^2.7.0^4.11.0^5.10]-dodecane-1,5,7,11-tetracarboxylate) was carried out by the conceptual density functional theory. Firstly, a conformational search of 3,9-diazatetraasterane was performed by molecular dynamics simulations associating with Quantum Mechanics calculations in order to obtain relative stable conformers. And then the reactive site prediction of 3,9-diazatetraasterane was carried out by the Fukui function and condensed Fukui function. The results exhibited that the potentially chemical reactive site is the N atom of 3,9-diazatetraasterane scaffold, which is readily attacked by electrophilic agent. Meanwhile, the calculated results were validated by experimental outcomes in the derivation of 3,9-diazatetraasteranes.

     

Benzene fused five-membered heterocycles. A theoretical approach

The Nuclear Independent Chemical Shift of each ring, as a criterion of aromaticity, is used to explain the stability order of benzopyrrole, benzofuran and benzothiophene, and their isomers. The results indicate that the benzene ring is aromatic in all the systems. The five-membered rings of benzopyrrole, benzofuran and benzothiophene are also aromatic, whereas those of isobenzopyrrole, isobenzofuran and isobenzothiophene are non-aromatic. This could be an explanation of the stability of the former molecules. The molecular orbitals and the condensed Fukui functions derived from the electronic structure calculations are also reported. These reactivity indices explain the expected electrophilic substitution of these compounds. The theoretical structure, ionization energies, order of aromaticity, stability and reactivity are in good agreement with the experimental results. The usefulness of this approach to determine the reactivity is discussed since their stability and reactivity may be understood. The reactivity indices are useful to explain and confirm the experimental information, and for molecules with unknown reactive behavior, this approach could help to predict some of the reactions.     

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.