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

Does the radical GPRI strongly depend on the population scheme? A comparative study to predict radical attack on unsaturated molecules with the radical general-purpose reactivity indicator

The reactivity of 22 unsaturated molecules undergoing attack by a methyl radical (⋅CH3) have been elucidated using the condensed radical general-purpose reactivity indicator (condensed radical GPRI) appropriate for relatively nucleophilic or electrophilic molecules. Using the appropriate radical GPRI equation for electrophilic attack or nucleophilic radical attack, seven different population schemes were used to assign the most reactive atoms in each of the 22 molecules. The results show that the condensed radical GPRI is sensitive to the population scheme chosen, but less sensitive than the radical Fukui function. Therefore, the reliability of these methods depends on the population scheme. Our investigation indicates that the condensed radical GPRI is most accurate in predicting the dominant products of the methyl radical addition reactions on a variety of unsaturated molecules when the Hirshfeld, Merz-Singh-Kollman, or Voronoi deformation density population schemes are used. Furthermore, for all populations schemes in the majority of instances where the radical Fukui function failed the radical GPRI was able to identify the most reactive atom under certain reactivity conditions.

Peculiar Differences between Two Copper Complexes Containing Similar Redox-Active Ligands: Density Functional and Multiconfigurational Calculations

Transition metal complexes featuring redox-active ligands often exhibit multiple redox states, influenced by the interplay between the metal center and the ligand. This study delves into the electronic structures of two mononuclear complexes of copper with two similar redox-active urea azine ligands. The ligands differ by the replacement of an NCH3 moiety by an S atom in the ligand backbone. Experimental analysis yields pronounced electronic structural disparities between these complexes, observable in both the solution and solid phases. Conventional quantum chemical methods, such as density functional theory using different functionals (B3LYP, TPSSh, and CAM-B3LYP), remain inadequate to rationalize the observed spectroscopic anomalies. However, a multiconfigurational approach elucidates the disparate behaviors of these complexes. Multireference perturbation theory, based on complete active space self-consistent field computations, identifies Cu(I) in the case of the complex with the NCH3 containing ligands and a state with substantial Cu(II) contributions in the case of the complex with the S atom containing ligands. In contrast, DFT indicates Cu(I) in both scenarios.

Extending Conceptual Density Functional Theory toward First-Order Reduced Density Matrices: An Open Subsystems Viewpoint on the Fukui Matrix

As a matrix extension of the Fukui function, a reactivity descriptor grounded within Conceptual Density Functional Theory, the Fukui matrix extends Frontier Molecular Orbital Theory to correlated regimes with its eigendecomposition in Fukui occupations and Fukui naturals. Despite successful applications, the questions remain as to whether replacing a quantity derived from a purely density-based framework by its matrix extension is theoretically well-founded and what chemical information is contained in the corresponding eigendecomposition. In this study, we show that the matrix extension of the Fukui function is only well-defined if one also generalizes the external potential to become nonlocal, leading to the introduction of Conceptual First-Order Reduced Density Matrix Functional Theory. By interpreting the Anderson impurity model from an interacting open subsystem perspective, we show how Fukui occupations and Fukui naturals reflect the influence of an increasing (static) correlation and which characteristic patterns we should expect within a molecular context. This study represents a step in generalizing Conceptual Density Functional Theory beyond its density-based perspective.

Properties of the density functional response kernels and its implications on chemistry

An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato's theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed.

Conceptual density functional theory based electronic structure principles

In this review article, we intend to highlight the basic electronic structure principles and various reactivity descriptors as defined within the premise of conceptual density functional theory (CDFT). Over the past several decades, CDFT has proven its worth in providing valuable insights into various static as well as time-dependent physicochemical problems. Herein, having briefly outlined the basics of CDFT, we describe various situations where CDFT based reactivity theory could be employed in order to gain insights into the underlying mechanism of several chemical processes.

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.

Revisiting electroaccepting and electrodonating powers: proposals for local electrophilicity and local nucleophilicity descriptors

It is shown that the electrophilicity index is also a rational choice for measuring nucleophilicity.

An intermediate level of approximation for computing the dual descriptor

At present, there are two levels of approximation to compute the dual descriptor (DD). The first uses the total electronic density of the original system along with the electronic densities of the system with one more electron and one less electron, but this procedure is time consuming and normal termination of computation of total electronic densities is not guaranteed. The second level of approximation uses only the electronic densities of frontier molecular orbitals, HOMO and LUMO, to avoid the former approximation; however, the orbital relaxation implicitly included in the first level of approximation is absent in the second, thus risking an incorrect interpretation of local reactivity. Between the lowest occupied molecular orbital (LOMO) and the highest unoccupied molecular orbital (HUMO), a framework to provide an expression of the DD in terms of the electronic densities of all molecular orbitals (except HUMO and LOMO) has been proposed to be implemented by programmers as a computational code. This methodology implies another level of approximation located between the conventional approximation methods mentioned above. In this study, working equations have been oriented toward molecular closed- and open-shell systems. In addition, the mathematical expression for a closed-shell system was applied to acetylene in order to assess the capability of this approach to generate the DD.

In pursuit of negative Fukui functions: examples where the highest occupied molecular orbital fails to dominate the chemical reactivity

In our quest to explore molecules with chemically significant regions where the Fukui function is negative, we explored reactions where the frontier orbital that indicates the sites for electrophilic attack is not the highest occupied molecular orbital. The highest occupied molecular orbital (HOMO) controls the location of the regions where the Fukui function is negative, supporting the postulate that negative values of the Fukui function are associated with orbital relaxation effects and nodal surfaces of the frontier orbitals. Significant negative values for the condensed Fukui function, however, were not observed.

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.

Manifestations of noninnocent ligand behavior

The potential of redox-active ligands to behave "noninnocently" in transition-metal coordination compounds is reflected with respect to various aspects and situations. These include the question of establishing "correct" oxidation states, the identification and characterization of differently charged radical ligands, the listing of structural and other consequences of ligand redox reactions, and the distinction between barrierless delocalized "resonance" cases M(n)/L(n) ↔ M(n+1)L(n-1) versus separated valence tautomer equilibrium situations M(n)/L(n) ⇌ M(n+1)L(n-1). Further ambivalence arises for dinuclear systems with radical bridge M(n)(μ-L(•))M(n) versus mixed-valent alternatives M(n+1)(μ-L(-))M(n), for noninnocent ligand-bridged coordination compounds of higher nuclearity such as (μ(3)-L)M(3), (μ(4)-L)M(4), (μ-L)(4)M(4), or coordination polymers. Conversely, the presence of more than one noninnocently behaving ligand at a single transition-metal site in situations such as L(n)-M-L(n-1) or L(•)-M-L(•) may give rise to corresponding ligand-to-ligand interaction phenomena (charge transfer, electron hopping, and spin-spin coupling) and to redox-induced electron transfer with counterintuitive oxidation-state changes. The relationships of noninnocent ligand behavior with excited-state descriptions and perspectives regarding material properties and single-electron or multielectron reactivity are also illustrated briefly.

The Utility of the HSAB Principle via the Fukui Function in Biological Systems

The hard/soft acid-base principle has long been known to be an excellent predictor of chemical reactivity. The Fukui function, a reactivity descriptor from conceptual density functional theory, has been shown to be related to the local softness of a system. The usefulness of the Fukui function is explored and demonstrated herein for three common biological problems: ligand docking, active site detection, and protein folding. In each type of study, a scoring function is developed based on the local HSAB principle using atomic Fukui indices. Even with necessary approximations for its use in large systems, the Fukui function remains a useful descriptor for predicting chemical reactivity and understanding chemical systems.

Do the local softness and hardness indicate the softest and hardest regions of a molecule?

In this work, we will show that the largest values of the local softness and hardness do not necessarily correspond to the softest and hardest regions of the molecule, respectively. Based on our results, we will argue that it is more useful to interpret the local softness and the local hardness as functions that measure the "local abundance" or "concentration" of the corresponding global properties. This new point of view helps reveal how and when these local reactivity indices are most useful.

Conceptual DFT: the chemical relevance of higher response functions

In recent years conceptual density functional theory offered a perspective for the interpretation/prediction of experimental/theoretical reactivity data on the basis of a series of response functions to perturbations in the number of electrons and/or external potential. This approach has enabled the sharp definition and computation, from first principles, of a series of well-known but sometimes vaguely defined chemical concepts such as electronegativity and hardness. In this contribution, a short overview of the shortcomings of the simplest, first order response functions is illustrated leading to a description of chemical bonding in a covalent interaction in terms of interacting atoms or groups, governed by electrostatics with the tendency to polarize bonds on the basis of electronegativity differences. The second order approach, well known until now, introduces the hardness/softness and Fukui function concepts related to polarizability and frontier MO theory, respectively. The introduction of polarizability/softness is also considered in a historical perspective in which polarizability was, with some exceptions, mainly put forward in non covalent interactions. A particular series of response functions, arising when the changes in the external potential are solely provoked by changes in nuclear configurations (the "R-analogues") are also systematically considered. The main part of the contribution is devoted to third order response functions which, at first sight, may be expected not to yield chemically significant information, as turns out to be for the hyperhardness. A counterexample is the dual descriptor and its R analogue, the initial hardness response, which turns out to yield a firm basis to regain the Woodward-Hoffmann rules for pericyclic reactions based on a density-only basis, i.e. without involving the phase, sign, symmetry of the wavefunction. Even the second order nonlinear response functions are shown possibly to bear interesting information, e.g. on the local and global polarizability. Its derivatives may govern the influence of charge on the polarizability, the R-analogues being the nuclear Fukui function and the quadratic and cubic force constants. Although some of the higher order derivatives may be difficult to evaluate a comparison with the energy expansion used in spectroscopy in terms of nuclear displacements, nuclear magnetic moments, electric and magnetic fields leads to the conjecture that, certainly cross terms may contain new, intricate information for understanding chemical reactivity.

Understanding the Woodward–Hoffmann Rules by Using Changes in Electron Density

The Woodward-Hoffmann rules for pericyclic reactions are explained entirely in terms of directly observable physical properties of molecules (specifically changes in electron density) without any recourse to model-dependent concepts, such as orbitals and aromaticity. This results in a fundamental explanation of how the physics of molecular interactions gives rise to the chemistry of pericyclic reactions. This construction removes one of the key outstanding problems in the qualitative density-functional theory of chemical reactivity (the so-called conceptual DFT). One innovation in this paper is that the link between molecular-orbital theory and conceptual DFT is treated very explicitly, revealing how molecular-orbital theory can be used to provide "back-of-the-envelope" approximations to the reactivity indicators of conceptual DFT.

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.