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.     

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.     

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.     

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.     

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.     

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.     

Natural orbital Fukui function and application in understanding cycloaddition reaction mechanisms

A new condensed form of the Fukui function, the natural orbital Fukui function (NOFF), is proposed and derived from natural bond orbital occupancy. It is defined as the change in natural bond orbital occupancy upon electronic perturbation (electron addition to, or depletion from, a molecular system). Applying NOFF to a series of cycloaddition reactions (e.g., [4 + 2] and [2 + 1] cycloadditions) illustrates the effectiveness of the concept in interpreting bond breakage and formation mechanisms.     

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.     

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.     

Derivation of the linear relationship between SWCNTs functionalization energies and sidewall curvature

A simple linear relationship between the functionalization reaction energies for the exohedral monovalent addition on the surface of an ideal, infinitely long, single-walled carbon nanotube (SWCNT) and the reciprocal SWCNT radius has been derived employing the hard?Csoft acid basis principle and the tight binding model. The slope of the derived linear relationship is a function of the effective number of valence electrons involved in the functionalization reaction. The intercept of the derived linear relationship, equal to the reaction energies on a planar graphite surface, is a function of the electrophilicity of the monovalent addend and of the condensed Fukui function of its reacting atom. The theoretical predictions of this simple formula are coherent with the computational density functional theory data reported in the literature.     

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.     

Theoretical study of the interaction of molecular oxygen with copper clusters

A new method based on frontier orbital theory has been used to investigate the binding site of molecular oxygen to neutral and anion copper clusters. It has been shown that one can make useful predictions of the binding sites based on the knowledge of the donor local reactivity of the cluster using the condensed Fukui function, f(-)(Ff). In this way, it was found that Cu(3), Cu(5), and Cu(5)(-) have the highest reactivity toward molecular oxygen.     

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.     

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.     

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.     

Face-integrated Fukui function: understanding wettability anisotropy of molecular crystals from density functional theory

Wettability is one of the anisotropic surface properties of molecular crystals that exhibit the structural variance of chemical moieties on various growth faces. The divergence in liquid-solid interactions at different faces is thought to be related to the inherent responding capacity or sensitivity of a solid surface to the perturbation in electronic structures and atomic positions as a result of the contact by a liquid. Since the Fukui function, according to density functional theory (DFT), is a local function for describing such sensitivity to the structural perturbation and is directly related to local softness, it has been proposed and tested to use an integrated Fukui function over a crystallographic plane for describing the anisotropy of solid-liquid interactions. It is found that the contact angle of a polar solvent, such as water, on a crystal surface shows an intimate connection to the integrated Fukui functions of the surface, illustrating an extension of Pearson's HSAB (hard and soft acids and bases) to crystal systems. The concept of face-integrated Fukui function and the approach to apply the HSAB with the DFT-based concepts may provide a powerful means for describing anisotropic properties, including wettability of organic crystals.