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.     

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.     

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.     

Density functional theory,chemical reactivity,and the Fukui functions

Foundations of Chemistry - We review the early works which were precursors of the Conceptual Density Functional Theory. Starting from Thomas–Fermi approximation and from the exact formulation...     

Efficient representation of the linear density-density response function

We present a thorough derivation of the mathematical foundations of the representation of the molecular linear electronic density-density response function in terms of a computationally highly efficient moment expansion. Our new representation avoids the necessities of computing and storing numerous eigenfunctions of the response kernel by means of a considerable dimensionality reduction about from 10³ to 10¹. As the scheme is applicable to any compact, self-adjoint, and positive definite linear operator, we present a general formulation, which can be transferred to other applications with little effort. We also present an explicit application, which illustrates the actual procedure for applying the moment expansion of the linear density-density response function to a water molecule that is subject to a varying external perturbation potential. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.     

Reduced eigensystem representation of the linear density-density response function

The linear density-density response function represents a formulation of the generalized density response of a molecular (or extended) system to arbitrary perturbing potentials. We have recently established an approach for reducing the dimension of the (in principle infinite) eigenspace representation (the moment expansion) and generalized it to arbitrary self-adjoint, positive-definite, and compact linear operators. Here, we present a modified representation—the reduced eigensystem representation—which allows to define a trivial criterion for the convergence of the approximation to the density response. By means of this novel eigensystem-like structure, the remarkable reduction of the dimensionality becomes apparent for the calculation of the density-density response function.     

Analytical green's functions for interacting bands: Coupled linear exciton bands

Analytical Green's functions for interacting bands were calculated for two coupled linear exciton bands. The total, partial and overlap density-of-states functions and their corresponding F functions were also obtained in an analytical form. This approach enables us to discuss analytically the nature of van Hove singularities and the impurity states involving the interacting bands.     

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.     

Optical linear response function with linear and diagonal quadratic electron-vibration coupling in mixed quantum-classical systems

Optical linear response function of linearly and quadratically coupled mixed quantum-classical condensed phase systems is derived. The linear response function is derived using Kapral's formalism of statistical mechanics in mixed quantum-classical systems. Our mixed quantum-classical linear dipole moment correlation function J(t) is compared with the full quantum J(t) [Y. J. Yan and S. Mukamel, J. Chem. Phys. 85, 5908 (1986)] in the high temperature limit. Model calculations and discussion of our results are presented. Various formulas of Franck-Condon factors for both linear and quadratic coupling are discussed.     

Moment expansion of the linear density‐density response function

We present a low rank moment expansion of the linear density‐density response function. The general interacting (fully nonlocal) density‐density response function is calculated by means of its spectral decomposition via an iterative Lanczos diagonalization technique within linear density functional perturbation theory. We derive a unitary transformation in the space of the eigenfunctions yielding subspaces with well‐defined moments. This transformation generates the irreducible representations of the density‐density response function with respect to rotations within SO(3). This allows to separate the contributions to the electronic response density from different multipole moments of the perturbation. Our representation maximally condenses the physically relevant information of the density‐density response function required for intermolecular interactions, yielding a considerable reduction in dimensionality. We illustrate the performance and accuracy of our scheme by computing the electronic response density of a water molecule to a complex interaction potential. © 2015 Wiley Periodicals, Inc.     

Group electronegativity and Fukui function studies of the substituent effects in aromatic and inorganic systems

The group electronegativities (GE ) of molecular fragments, including the environmental contributions due to both the electrostatic interactions and electron distribution relaxation, and the Fukui function (FF ) indices of the charge sensitivity analysis (CSA ) are correlated with the known substituent effects in molecular systems. The semiempirical CSA in the atoms-in-molecules (AIM ) resolution has been applied to substituted benzenes and square platinum complexes treated as illustrative examples. The calculated FF indices and GE are both shown to constitute adequate reactivity criteria that qualitatively reproduce the known substituent effects. The FF index (second-order property) is found to be a more sensitive detector of the substituent influence than is the corresponding GE parameter (first-order property). © 1992 John Wiley & Sons, 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.     

Radial behavior of gradient expansion approximation to atomic Fukui function and shell structure of atoms

An approximation to the Fukui function in atoms recently proposed in the form of a gradient correction to the local density approximation expression is here investigated. The spatial behavior of this function is analyzed, focusing on the gradient correction term. Physical information on the shell structure of atoms is shown to be conveyed by the radial distribution of that term. The analytically modeled densities (AMD) procedure is also implemented, and global atomic hardnesses are calculated with Hartree-Fock and AMD representations of atomic electron densities. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 488–503, 1998     

Modeling temporary anions in density functional theory: calculation of the Fukui function

Two approaches are investigated for modeling electron densities of temporary anions in density functional theory (DFT). Both rely on an artificial binding of the excess electron, in one case by a compact basis set and in the other by a potential wall. The key feature of the calculations is that the degree of binding is controlled in both cases by knowledge of the negative electron affinity of the corresponding neutral, approximated in terms of DFT local functional frontier orbital eigenvalues and vertical ionization potential, A=-(epsilon(LUMO)+epsilon(HOMO))-I. To illustrate the two approaches, Fukui functions for nucleophilic attack are determined in four molecules with increasingly negative electron affinities. They yield very similar results, which are notably different to those determined without artificial electron binding. The use of a potential wall has the attractive feature that large, diffuse basis sets can be used, avoiding the need for a compact basis, tailored to a particular molecule.