How to Compute the Fukui Matrix and Function for Systems with (Quasi-)Degenerate States

Spin-Polarized Conceptual Density Functional Theory from the Convex Hull

We present a new, nonarbitrary, internally consistent, and unambiguous framework for spin-polarized conceptual density-functional theory (SP-DFT). We explicitly characterize the convex hull of energy, as a function of the number of electrons and their spin, as the only accessible ground states in spin-polarized density functional theory. Then, we construct continuous linear and quadratic models for the energy. The nondifferentiable linear model exactly captures the simplicial geometry of the complex hull about the point of interest and gives exact representations for the conceptual DFT reactivity indicators. The continuous quadratic energy model is the paraboloid of maximum curvature, which most tightly encloses the point of interest and neighboring vertices. The quadratic model is invariant to the choice of coordinate system (i.e., {N, S} vs {Nα, Nβ}) and reduces to a sensible formulation of spin-free conceptual DFT in the appropriate limit. Using the quadratic model, we generalize the Parr functions {P+(r), P-(r)} (and their derivatives with respect to number of electrons) to this new spin-polarized framework, integrating the Parr function concept into the context of (spin-polarized) conceptual DFT, and extending it to include higher-order effects.

Extending the Scope of Conceptual Density Functional Theory with Second Order Analytical Methodologies

In the context of the growing impact of conceptual density functional theory (DFT) as one of the most successful chemical reactivity theories, response functions up to second order have now been widely applied; in recent years, among others, particular attention has been focused on the linear response function and also extensions to higher order have been put forward. As the larger part of these studies have been carried using a finite difference approach to compute these concepts, we now embarked on (an extension of) an analytical approach to conceptual DFT. With the ultimate aim of providing a complete set of analytically computable second order properties, including the softness and hardness kernels, the hardness as the simplest second order response function is scrutinized again with numerical results highlighting the difference in nature between the analytical hardness (referred to as hardness condition) and the Parr-Pearson absolute chemical hardness. The hardness condition is investigated for its capability to gauge the (de)localization error of density functional approximations (DFAs). The analytical Fukui function, besides overcoming the difficulties in the finite difference approach in treating negatively charged systems, also showcases the errors of deviating from the straight-line behavior using fractional occupation number calculations. Subsequently, the softness kernel and its atom-condensed inverse, the hardness matrix, are accessed through the Berkowitz-Parr relation. Revisiting the softness kernel confirms and extends previous discussions on how Kohn's Nearsightedness of Electronic Matter principle can be retrieved and identified as the physicist's version of the chemist's "transferability of functional groups" concept. The accurate, analytical hardness matrix evaluation on the other hand provides further support for the basics of Nalewajski's charge sensitivity analysis. Based on Parr and Liu's functional expansion of the energy functional, a new energy decomposition is introduced with an order of magnitude analysis of the different terms for a series of simple molecules both at their equilibrium geometry and upon variation in bond length and dihedral angle. Finally, for the first time, the perturbation expansion of the energy functional is studied numerically up to second order now that all response functions and integration techniques are at hand. The perturbation expansion energies are in excellent agreement with those obtained directly from DFA calculations giving confidence in the convergence of the perturbation series and its use in judging the importance of the different terms in reactivity investigations.

Connection between nuclear and electronic Fukui functions beyond frontier molecular orbitals

Based on the relationship between average local ionization energy Ī(r) and average local electron affinity Ā(r) with the electronic Fukui functions, i.e., f-(r) and f+(r), respectively, in this paper, we establish a connection between nuclear and electronic Fukui functions beyond frontier molecular orbitals. As a consequence of this connection, we obtain expressions of average nuclear Fukui functions interpreted as a variation of average nucleophilicity or electrophilicity (weighted by the electronic orbital Fukui functions) with respect to nuclear displacements, which goes beyond the highest occupied molecular orbital/or lowest unoccupied molecular orbital consideration. Furthermore, from this connection and considering the frontier molecular orbital approximation, we derive expressions of nuclear Fukui functions in terms of the atom-condensed electronic Fukui functions, which imply a locality in the chemical reactivity and could be used to study the variation of local nucleophilicity or electrophilicity with respect to nuclear displacements. Finally, this new way to interpret the nuclear Fukui function could be useful in the future to study the chemical reactivity related to molecular vibrations, internal rotations, bond dissociation, chemical reaction along the model of reaction coordinate, and so on.

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.

Molecular interactions from the density functional theory for chemical reactivity: Interaction chemical potential, hardness, and reactivity principles

In the first paper of this series, the authors derived an expression for the interaction energy between two reagents in terms of the chemical reactivity indicators that can be derived from density functional perturbation theory. While negative interaction energies can explain reactivity, reactivity is often more simply explained using the “|dμ| big is good” rule or the maximum hardness principle. Expressions for the change in chemical potential (μ) and hardness when two reagents interact are derived. A partial justification for the maximum hardness principle is that the terms that appear in the interaction energy expression often reappear in the expression for the interaction hardness, but with opposite sign.

Molecular Interactions From the Density Functional Theory for Chemical Reactivity: The Interaction Energy Between Two-Reagents

Reactivity descriptors indicate where a reagent is most reactive and how it is most likely to react. However, a reaction will only occur when the reagent encounters a suitable reaction partner. Determining whether a pair of reagents is well-matched requires developing reactivity rules that depend on both reagents. This can be achieved using the expression for the minimum-interaction-energy obtained from the density functional reactivity theory. Different terms in this expression will be dominant in different circumstances; depending on which terms control the reactivity, different reactivity indicators will be preferred.

DFT Study for Adsorbing of Bromine Monochloride onto BNNT (5,5), BNNT (7,0), BC2NNT (5,5), and BC2NNT (7,0)

The study of intermolecular interactions is of great importance. This study attempted to quantitatively examine the interactions between bromine monochloride (BrCl) with pristine boron nitride nanotube (BNNT) armchair (5,5) and zigzag (7,0) as well as armchair (5,5) BC2NNT and zigzag (7,0) BC2NNT in vacuum. Quantum mechanical studies of such systems are possible in the density functional theory (DFT) framework. For this purpose, various functionals, such as B3LYP-D3, [Formula: see text]B97XD, and M062X, have been used. One of the most suitable basis functionals for the systems studied in this research is 6-311G (d), which has been used in both optimization calculations and calculations related to wave function analyses. The main part of this work is the study of various analyses that reveal the nature of the intermolecular interactions between the two components introduced above. The results of conceptual DFT, natural bond orbital, non-covalent interactions, and quantum theory of atoms in molecules (QTAIM) were consistent and in favor of physical adsorption in all systems. Gallium had more adsorption energy than other dopants. The HOMO–LUMO energy gaps were as follows: BNNT (5,5): 10.296, BNNT (7,0): 9.015, BC2NNT (5,5): 7.022, and BC2NNT (7,0): 5.979[Formula: see text]eV at B3LYP-D3/6-311G (d) model chemistry. The strongest interaction is related to the BC2NNT (7,0)/BrCl cluster: [Formula: see text][Formula: see text]eV. The results of QTAIM and NCI analysis identified the intermolecular interactions of the type of strong van der Waals interaction for these nanotubes. The sensitivity of the adsorption increased when a gas molecule interacted with carbon-doped BNNT, and the change in the frontier orbital gap could be used to design nanosensors to detect BrCl gas.

Proposal of a Fermi-Dirac-Derived Reactivity Descriptor: Beyond the Frontier MO Model

In this paper, we derive a reactivity descriptor stemming from the Fermi-Dirac population scheme, applied to density functional calculations on molecular systems. Assuming that molecular orbitals only marginally change when temperature is slightly increased from 0 K, we study the response of electron density to a change in temperature. Connection with usual conceptual density functional theory descriptors is made, and the T-variation of electron density for some representative examples is given and discussed.

Unifying Conceptual Density Functional and Valence Bond Theory: The Hardness-Softness Conundrum Associated with Protonation Reactions and Uncovering Complementary Reactivity Modes

In this study, we address the long-standing issue-arising prominently from conceptual density functional theory (CDFT)-of the relative importance of electrostatic, i.e., "hard-hard", versus spin-pairing, i.e., "soft-soft", interactions in determining regiochemical preferences. We do so from a valence bond (VB) perspective and demonstrate that VB theory readily enables a clear-cut resolution of both of these contributions to the bond formation/breaking process. Our calculations indicate that appropriate local reactivity descriptors can be used to gauge the magnitude of both interactions individually, e.g., Fukui functions or HOMO/LUMO orbitals for the spin-pairing/(frontier) orbital interactions and molecular electrostatic potentials (and/or partial charges) for the electrostatic interactions. In contrast to previous reports, we find that protonation reactions cannot generally be classified as either charge- or frontier orbital-controlled; instead, our results indicate that these two bonding contributions generally interplay in more subtle patterns, only giving the impression of a clear-cut dichotomy. Finally, we demonstrate that important covalent, i.e., spin pairing, reactivity modes can be missed when only a single spin-pairing/orbital interaction descriptor is considered. This study constitutes an important step in the unification of CDFT and VB theory.

Predicting Deprotonation Sites Using Alchemical Derivatives

An alchemical transformation is any process, physical or fictitious, that connects two points in the chemical space. A particularly important transformation is the vanishing of a proton, whose energy can be linked to the proton dissociation enthalpy of acids. In this work we assess the reliability of alchemical derivatives in predicting the proton dissociation enthalpy of a diverse series of mono- and polyprotic molecules. Alchemical derivatives perform remarkably well in ranking the proton affinity of all molecules. Additionally, alchemical derivatives could be use also as a predictive tool because their predictions correlate quite well with calculations based on energy differences and experimental values. Although second-order alchemical derivatives underestimate the dissociation enthalpy, the deviation seems to be almost constant. This makes alchemical derivatives extremely accurate to evaluate the difference in proton affinity between two acid sites of polyprotic molecule. Finally, we show that the reason for the underestimation of the dissociation enthalpy is most likely the contribution of higher-order derivatives.

The Role of the Density Response Kernel in the Protonation Process

The interaction between a Lewis base and a proton produces large changes in the electron distribution of the proton-accepting species. The quantum description of the change in the electronic properties can be predicted by perturbative models. Density functional theory-based reactivity indices are useful quantities to predict the changes in the energy and the electron density of electronic systems. However, the perturbation produced by the proton, namely, the electric field of a bare nucleus, usually leads to the use of perturbative terms beyond the first order. In this work, we analyze the effect of the protonation on the electronic structure of different Lewis bases. We also identify the leading term in the perturbative expansion that determines the protonation site and try to relate it with the chemical nature of the Lewis base. Even when the interaction is initially dominated by the electrostatic forces, we found that the electron redistribution effects can play a more relevant role for highly polarizable species.

Orbital-Weighted Dual Descriptor for the Study of Local Reactivity of Systems with (Quasi-) Degenerate States

An alternative response function, based on the dual descriptor in terms of Koopmans' approximation, is hereby proposed for the description of chemical reactivity in systems with (quasi-) degenerate frontier molecular orbitals. This descriptor is constructed from Fukui functions that include contributions from different orbitals, i.e., orbital-weighted Fukui functions. The methodology is applied to three case studies: the first case consists of a series of benchmark organic and inorganic molecules from which the dual descriptor, based only on frontier orbitals, is not appropriate to describe their reactivity. The second case deals with the proper description of chemical reactivity in Diels-Alder reactions between fullerene C₆₀ and cyclopentadiene (CP), revealing the importance of considering secondary orbital interactions for an adequate regioselectivity description. The third, and last case, consists of a series of polycyclic aromatic hydrocarbons (PAHs) possessing molecular orbital degeneracy. By means of analyzing of this descriptor, an alternative approach to the description of aromaticity is proposed. In all cases, the proposed index called "orbital-weighted dual descriptor" has proven to accurately describe the chemical reactivity and aromaticity of the studied systems.

New advances in conceptual-DFT: an alternative way to calculate the Fukui function and dual descriptor

An alternative way of calculating the Fukui function and the partial derivative of second order of the electronic density with respect to the number of electrons N is presented, the new formulas agree with the usual ones but only in cases without degeneracy. The new operative formulas are more general than the previous ones and are the right ones for those problematic cases where one or both of the frontier molecular orbitals are degenerate. Finally, we present a new way of applying the finite difference approximation that leads to more realistic results than the usual formulas. Graphical abstract A new way of calculating the Fukui function is presented that results in a new operative formula of the function. It has also been obtained the partial derivative of second order of the electronic density with respect to the number of electrons N, and it agree with the usual formula of the dual descriptor function but only in cases without degeneration.

How Biochemical Environments Fine-Tune a Redox Process: From Theoretical Models to Practical Applications

In this study, we give a new physical insight into how enzymatic environments influence a redox process. This is particularly important in a biochemical context, in which oxidoreductase enzymes and low-molecular-weight cofactors create a microenvironment, fine-tuning their specific redox potential. We present a new theoretical model, quantitatively backed up by quantum chemically calculated data obtained for key biological sulfur-based model reactions involved in preserving the cellular redox homeostasis during oxidative stress. We show that environmental effects can be quantitatively predicted from the thermodynamic cycle linking ΔΔ G(OX/RED)_ref-ligand values to the differential interaction energy ΔΔ G_int of the reduced and oxidized species with the environment. Our obtained data can be linked to hydrogen-bond patterns found in protein active sites. The thermodynamic model is further understood in the framework of molecular orbital theory. The key insight of this work is that the intrinsic properties of neither a redox couple nor the interacting environment (e.g., ligand) are enough by themselves to uniquely predict reduction potentials. Instead, system-environment interactions need to be considered. This study is of general interest as redox processes are pivotal to empower, protect, or damage organisms. Our presented thermodynamic model allows a pragmatically evaluation on the expected influence of a particular environment on a redox process, necessary to fully understand how redox processes take place in living organisms.

Analysis of molecular and (di)atomic dual-descriptor functions and matrices

In this work, the dual-descriptor is studied in matrix form [Formula: see text] and both coordinates condensed to atoms, resulting in atomic and diatomic (or where applicable, bond) condensed single values. This double partitioning method of the dual-descriptor matrix is proposed within the Hirshfeld-I atoms-in-molecule framework although it is easily extended to other atoms-in-molecules methods. Diagonalizing the resulting atomic and bond dual-descriptor matrices gives eigenvalues and eigenvectors describing the reactivity of atoms and bonds. The dual-descriptor function is the diagonal element of the underlying matrix. The extra information contained in the atom and bond resolution is highlighted and the effect of choosing either the fragment of molecular response or response of molecular fragment approach is quantified. Graphical Abstract Atom and bond condensed dual descriptor matrices and functions are derived from molecular ones using Hirshfeld-I atoms in molecules weight functions.

Going beyond the three-state ensemble model: the electronic chemical potential and Fukui function for the general case

The analytical working equations for the chemical potential and the Fukui function for the case of any number of ground and excited states is presented.

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.

The influence of zero-flux surface motion on chemical reactivity

Motion of zero-flux surfaces when an electron is removed from an ICCF molecule.

Towards the rationalization of catalytic activity values by means of local hyper-softness on the catalytic site: a criticism about the use of net electric charges

By means of the Spin-Polarized Conceptual Density Functional Theory (SP-CDFT), three 2,6-bis(imino)pyridine catalysts based on iron(II), used for polymerization of ethylene, were studied. The catalysts differed by the substituent group, bearing either -H, -NO2 or -OCH3. To date, catalytic activity, a purely experimental parameter measuring the mass of polyethylene produced per millimole of iron per time and pressure unit at a fixed temperature, has not been explained in terms of local hyper-softness. The latter is a purely theoretical parameter designed for quantifying electronic effects; it is measured using the metal atom responsible for the coordination process with the monomer (ethylene). Because steric effects are not relevant in these kinds of catalysts and only electronic effects drive the catalytic process, an interesting link is found between catalytic activity and the local hyper-softness condensed on the iron atom by means of four functionals (B3LYP, BP86, B97D, and VSXC). This work demonstrates that the use of local hyper-softness, predicted by the SP-CDFT, is a suitable parameter for explaining order relationships among catalytic activity values, thus quantifying the electronic influence of the substituent group inducing this difference; the use of only net electric charges does not lead to clear conclusions. This finding can aid in estimating catalytic activities leading to a more rational design of new catalysts via computational chemistry.

Philicity and Fugality Scales for Organic Reactions

Theoretical scales of reactivity and selectivity are important tools to explain and to predict reactivity patterns, including reaction mechanisms. The main achievement of these efforts has been the incorporation of such concepts in advanced texts of organic chemistry. In this way, the modern organic chemistry language has become more quantitative, making the classification of organic reactions an easier task. The reactivity scales are also useful to set up a number of empirical rules that help in rationalizing and in some cases anticipating the possible reaction mechanisms that can be operative in a given organic reaction. In this review, we intend to give a brief but complete account on this matter, introducing the conceptual basis that leads to the definition of reactivity indices amenable to build up quantitative models of reactivity in organic reactions. The emphasis is put on two basic concepts describing electron-rich and electron-deficient systems, namely, nucleophile and electrophiles. We then show that the regional nucleophilicity and electrophilicity become the natural descriptors of electrofugality and nucleofugality, respectively. In this way, we obtain a closed body of concepts that suffices to describe electron releasing and electron accepting molecules together with the description of permanent and leaving groups in addition, nucleophilic substitution and elimination reactions.