Can we predict ambident regioselectivity using the chemical hardness?

The hard/soft acid/base (HSAB) principle is a cornerstone in our understanding of chemical reactivity preferences. Motivated by the success of the original ("global") version of this rule, a "local" counterpart was readily proposed to account for regioselectivity preferences, in particular, in ambident reactions. However, ample experimental evidence indicates that the local HSAB principle often fails to provide meaningful predictions. Here we examine the assumptions behind the standard proof of the local HSAB rule, showing that it is based on a flawed premise. By solving this issue, we show that it is critical to consider not only the charge transferred between the different reacting centers but also the charge reorganization within the non-reacting parts of the molecule. We propose different reorganization models and derive the corresponding regioselectivity rules for each.

Hardness of molecules and bandgap of solids from a generalized gradient approximation exchange energy functional.

The deviations from linearity of the energy as a function of the number of electrons that arise with current approximations to the exchange-correlation (XC) energy functional have important consequences for the frontier eigenvalues of molecules and the corresponding valence-band maxima for solids. In this work, we present an analysis of the exact theory that allows one to infer the effects of such approximations on the highest occupied and lowest unoccupied molecular orbital eigenvalues. Then, we show the importance of the asymptotic behavior of the XC potential in the generalized gradient approximation (GGA) in the case of the NCAPR functional (nearly correct asymptotic potential revised) for determining the shift of the frontier orbital eigenvalues toward the exact values. Thereby we establish a procedure at the GGA level of refinement that allows one to make a single calculation to determine the ionization potential, the electron affinity, and the hardness of molecules (and its solid counterpart, the bandgap) with an accuracy equivalent to that obtained for those properties through energy differences, a procedure that requires three calculations. For solids, the accuracy achieved for the bandgap lies rather close to that which is obtained through hybrid XC energy functionals, but those also demand much greater computational effort than what is required with the simple NCAPR GGA calculation.

Analysis of the kinetic energy functional in the generalized gradient approximation.

A new density functional for the total kinetic energy in the generalized gradient approximation is developed through an enhancement factor that leads to the correct behavior in the limits when the reduced density gradient tends to 0 and to infinity and by making use of the conjoint conjecture for the interpolation between these two limits, through the incorporation, in the intermediate region of constraints that are associated with the exchange energy functional. The resulting functional leads to a reasonable description of the kinetic energies of atoms and molecules when it is used in combination with Hartree-Fock densities. Additionally, in order to improve the behavior of the kinetic energy density, a new enhancement factor for the Pauli kinetic energy is proposed by incorporating the correct behavior into the limits when the reduced density gradient tends to 0 and to infinity, together with the positivity condition, and imposing through the interpolation function that the sum of its integral over the whole space and the Weiszacker energy must be equal to the value obtained with the enhancement factor developed for the total kinetic energy.

Temperature-Dependent Approach to Electronic Charge Transfer.

A charge transfer model is developed within the framework of the grand canonical ensemble through the analysis of the behavior of the fractional charge as a function of the chemical potential of the bath when the temperature and the external chemical potential are kept fixed. Departing from the fact that, before the interaction between two species, each one has a zero fractional charge, one can identify two situations after the interaction occurs where the fractional charge of at least one of the species is different from zero, indicating that there has been charge transference. One of them corresponds to the case when one of the species is immersed in a bath conformed by the other one, while the other is related to the case in which both species are present in equal amounts (stoichiometric proportion). Correlations between the fractional charges and average energies, thus obtained with experimental equilibrium constants, kinetic rate constants, hydration constants, and bond enthalpies, indicate that, although at the experimental temperatures, they are very small quantities, they have chemically meaningful information. Additionally, in the stoichiometric case, one also finds a rather good correlation between the equalized chemical potential and the one obtained from experimental information for a test set of diatomic and triatomic molecules.

Constrained dipole moment density functional theory for charge distributions in force fields for the study of molecular fluids.

A new procedure, based on electronic structure calculations that only requires a dipole moment value for a given molecule as input and, from which the charges for all the atoms in it are uniquely determined, is developed and applied to the study of molecular fluids with classical dynamics. The dipole moment value considered for the isolated molecule is the one that reproduces the dielectric constant of its corresponding fluid. Following previous work, the Lennard-Jones parameters are determined to reproduce the liquid density and the surface tension at the liquid-vapor interface. The force field thus obtained leads to a reasonable description of several properties such as heats of vaporization, self-diffusion coefficients, shear viscosities, isothermal compressibilities, and volumetric expansion coefficients of pure substances.

Negative Electron Affinities and Derivative Discontinuity Contribution from a Generalized Gradient Approximation Exchange Functional.

Two methods to calculate negative electron affinities systematically from ground-state density functional methods are presented. One makes use of the lowest unoccupied molecular orbital energy shift provided by approximate inclusion of derivative discontinuity in the nearly correct asymptotic potential (NCAP) nonempirical, constraint-based generalized gradient approximation exchange functional. The other uses a second-order perturbation calculation of the derivative discontinuity based on the NCAP exchange-correlation potential. On a set of thirty-eight molecules, NCAP leads to a rather accurate description that is improved further through the perturbation correction. The results presented show the importance of the asymptotic behavior of the exchange-correlation potential in the calculation of negative electron affinities as well as demonstrating the versatility of the NCAP functional.

Electronegativities of Pauling and Mulliken in Density Functional Theory.

Electronegativity is a fundamental concept in chemistry that allows one to infer important aspects about the interactions between chemical species. In the present work we make use of the framework provided by the density functional theory of chemical reactivity, to discuss in a unified way the approaches to the concept of electronegativity developed by Pauling and by Mulliken. Our analysis starts by making use of the identification of the electronegativity of Mulliken with the chemical potential of density functional theory, and continues to show that the ionic correction proposed by Pauling can be derived, with certain approximations, from the quadratic smooth interpolation of the energy as a function of the number of electrons in terms of the chemical potentials and the hardnesses of the interacting species, from which one can infer the close qualitative relationship between Pauling's electronegativity and the electrophilicity concept.

Generalized Gradient Approximation Exchange Energy Functional with Near-Best Semilocal Performance.

We develop and validate a nonempirical generalized gradient approximation (GGA) exchange (X) density functional that performs as well as the SCAN (strongly constrained and appropriately normed) meta-GGA on standard thermochemistry tests. Additionally, the new functional (NCAP, nearly correct asymptotic potential) yields Kohn-Sham eigenvalues that are useful approximations of the density functional theory (DFT) ionization potential theorem values by inclusion of a systematic derivative discontinuity shift of the X potential. NCAP also enables time-dependent DFT (TD-DFT) calculations of good-quality polarizabilities, hyper-polarizabilities, and one-Fermion excited states without modification (calculated or ad hoc) of the long-range behavior of the exchange potential or other patches. NCAP is constructed by reconsidering the imposition of the asymptotic correctness of the X potential (-1/ r) as a constraint. Inclusion of derivative discontinuity and approximate integer self-interaction correction treatments along with first-principles determination of the effective second-order gradient expansion coefficient yields a major advance over our earlier correct asymptotic potential functional [CAP; J. Chem. Phys. 2015 , 142 , 054105 ]. The new functional reduces a spurious bump in the CAP atomic exchange potential and moves it to distances irrelevantly far from the nucleus (outside the tail of essentially all practical basis functions). It therefore has nearly correct atomic exchange-potential behavior out to rather large finite distances r from the nucleus but eventually goes as - c/ r with an estimated value for the constant c of around 0.3, so as to achieve other important properties of exact DFT exchange within the restrictions of the GGA form. We illustrate the results with the Ne atom optimized effective potentials and with standard molecular benchmark test data sets for thermochemical, structural, and response properties.

Chemical hardness: Temperature dependent definitions and reactivity principles.

In this brief report, we show that the three different chemical hardness definitions developed in the framework of the temperature-dependent density functional theory-namely, the electronic, the thermodynamic, and the Helmholtz hardnesses-imply both the hard and soft acids and bases (HSAB) principle and the maximum hardness (MH) principle. These hardnesses are identified as the second derivative of a thermodynamic state function and avoid the somewhat arbitrary approach, based on the parabolic interpolation of the energy versus electron number, that is normally used to justify these principles. This not only leads to a more mathematically sound justification of the HSAB and MH principles in the low-temperature limit but also establishes that the HSAB and the MH principles hold at any temperature of chemical relevance.

Local and nonlocal counterparts of global descriptors: the cases of chemical softness and hardness.

A new strategy, recently reported by us to develop local and linear (nonlocal) counterparts of global response functions, is applied to study the local behavior of the global softness and hardness reactivity descriptors. Within this approach a local counterpart is designed to identify the most important molecular fragments for a given chemical response. The local counterpart of the global softness obtained through our methodology corresponds to the well-known definition of local softness and, in agreement with what standard conceptual chemical reactivity in density functional theory dictates, it simply reveals the softest sites in a molecule. For the case of the local hardness, we obtain two expressions that lead to different information regarding the values of the hardness at the different sites within a chemical species. The performance of these two proposal were tested by comparing their corresponding atom-condensed values to experimentally observed reactivity trends for electrophilic attack on benzene and ethene derivatives.

Local electrophilicity.

In this work some possibilities for deriving a local electrophilicity are studied. First, we consider the original definition proposed by Chattaraj, Maiti, and Sarkar (J Phys Chem A 107:4973, 2003), in which the local electrophilicity is given by the product of the global electrophilicity, and the Fukui function for charge acceptance is derived by two different approaches, making use of the chain rule for functional derivatives. We also modify the proposals based on the electron density so as to have a definition with the same units of the original definition, which also introduces a dependence in the Fukui function for charge donation. Additionally, we also explore other possibilities using the tools of information theory and the temperature dependent reactivity indices of the density functional theory of chemical reactivity. The poor results obtained from the last two approaches lead us to conjecture that this is due to the fact that the global electrophilicity is not a derivative, like most of the other reactivity indices. The conclusion is that Chattaraj's suggestion seems to be the simplest, but at the same time a very reliable approach to this important property.

Global and local charge transfer in electron donor-acceptor complexes.

The formation of electron donor-acceptor complexes is studied with global and local charge transfer partitionings. The 1-parabola model is applied to the bromination reaction of alkenes and the correlations found between the global and local charge transferred with the transition energy of the charge transfer bands and the kinetic rate constants indicate that the nucleophilic attack of alkenes to bromine is the electronic process controlling the reactivity in the formation of the electron donor-acceptor complexes in this reaction. The 2-parabolas model is used in studying the nitrosation of aromatic compounds where colorful electron donor-acceptor complexes are formed. In this case, and like previous applications of the 2-parabolas model, the consistent usage of the model mandates the explicit consideration of reaction conditions in preparing the reactants to have a direction of electron transfer that is consistent with the chemical potential differences. For the nitrosation reaction this implies considering the nitrosonium cation as the charge acceptor. Both applications support that the charge transferred predicted from chemical reactivity models can be used as a scale to measure the nucleophilicity in reactivity trends. Graphical Abstract ᅟ.

Reply to the 'Comment on "Revisiting the definition of local hardness and hardness kernel"' by C. Morell, F. Guégan, W. Lamine, and H. Chermette, Phys. Chem. Chem. Phys., 2018, 20, DOI.

This reply complements the comment of Guégan et al. about our recent work on the revision of the local hardness and the hardness kernel concepts. Guegan et al. analyze our work using a Taylor series expansion of the energy as a functional of the electron density, to show that our procedure opens a new way to define local descriptors. In this contribution we show that the strategy we followed for the local hardness and the hardness kernel is even more general, and that it can be used to derive from a global response function its corresponding local and non-local counterparts by: (1) requiring that the integral over one of the two variables that characterizes the non-local function leads to the local function, and that the integral over the local function leads to the global response index, and (2) assuming that the global and local functions are related through the electronic density, by making use of the chain rule for functional derivatives.

Role of Reaction Conditions in the Global and Local Two Parabolas Charge Transfer Model.

The local and global charge transfer approach based on the two parabolas model is applied to several problems aiming to show the importance of incorporating the reaction conditions to evaluate the global and local chemical descriptors. It is shown that, by preparation of the reactants, the chemical potentials of the reacting species determined by the two parabolas model satisfy the condition for the transfer of electrons in the direction dictated by the chemical potential difference. The model is applied to the hydration of alkenes, showing that it recovers Markovnikov's rule, to aromatic nitration, and to the interaction of nitrobenzenes with 1,3-diethylurea, an electrochemically controlled hydrogen-bonding problem. The applications presented show that to satisfy the charge transfer directionality established by the chemical potential differences obtained from the two parabolas model, one has to incorporate the reaction conditions in the evaluation of the global and local chemical descriptors. The global and local charge transfer predicted along these lines allows one to determine the direction of electron transfer prevailing in the reaction and also the most relevant atoms participating in the interactions between the reactants, aiding in the unraveling of the chemical interactions present in the system under investigation.

Thermodynamic Justification for the Parabolic Model for Reactivity Indicators with Respect to Electron Number and a Rigorous Definition for the Electrophilicity: The Essential Role Played by the Electronic Entropy.

The temperature-dependence of the Helmholtz free energy with respect to the number of electrons is analyzed within the framework of the Grand Canonical Ensemble. At the zero-temperature limit, the Helmholtz free energy behaves as a Heaviside function of the number of electrons; however, as the temperature increases, the profile smoothens and exhibits a minimum value at noninteger positive values of the fractional electronic charge. We show that the exact average electronic energy as a function of the number of electrons does not display this feature at any temperature, since this behavior is solely due to the electronic entropy. Our mathematical analysis thus indicates that the widely used parabolic interpolation model should not be viewed as an approximation for the average electronic energy, but for the dependence of the Helmholtz free energy upon the number of electrons, and this analysis is corroborated by numerical results. Finally, an electrophilicity index is defined for the Helmholtz free energy showing that, for a given chemical species, there exists a temperature value for which this quantity is equivalent to the electrophilicity index defined within the parabolic interpolation of the electronic energy as a function of the number of electrons. Our formulation suggests that the convexity property of the energy versus the number of electrons together with the entropic contribution does not allow for an analogous nucleophilicity index to be defined.

Thermodynamic responses of electronic systems.

We present how the framework of the temperature-dependent chemical reactivity theory can describe the panorama of different types of interactions between an electronic system and external reagents. The key reactivity indicators are responses of an appropriate state function (like the energy or grand potential) to the variables that determine the state of the system (like the number of electrons/chemical potential, external potential, and temperature). We also consider the response of the average electron density to appropriate perturbations. We present computable formulas for these reactivity indicators and discuss their chemical utility for describing electronic, electrostatic, and thermal changes associated with chemical processes.

Thermodynamic hardness and the maximum hardness principle.

An alternative definition of hardness (called the thermodynamic hardness) within the grand canonical ensemble formalism is proposed in terms of the partial derivative of the electronic chemical potential with respect to the thermodynamic chemical potential of the reservoir, keeping the temperature and the external potential constant. This temperature dependent definition may be interpreted as a measure of the propensity of a system to go through a charge transfer process when it interacts with other species, and thus it keeps the philosophy of the original definition. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures and up to temperatures of chemical interest, one finds that for zero fractional charge, the thermodynamic hardness is proportional to T-1(I-A), where I is the first ionization potential, A is the electron affinity, and T is the temperature. However, the thermodynamic hardness is nearly zero when the fractional charge is different from zero. Thus, through the present definition, one avoids the presence of the Dirac delta function. We show that the chemical hardness defined in this way provides meaningful and discernible information about the hardness properties of a chemical species exhibiting integer or a fractional average number of electrons, and this analysis allowed us to establish a link between the maximum possible value of the hardness here defined, with the minimum softness principle, showing that both principles are related to minimum fractional charge and maximum stability conditions.

New Fukui, dual and hyper-dual kernels as bond reactivity descriptors.

We define three new linear response indices with promising applications for bond reactivity using the mathematical framework of τ-CRT (finite temperature chemical reactivity theory). The τ-Fukui kernel is defined as the ratio between the fluctuations of the average electron density at two different points in the space and the fluctuations in the average electron number and is designed to integrate to the finite-temperature definition of the electronic Fukui function. When this kernel is condensed, it can be interpreted as a site-reactivity descriptor of the boundary region between two atoms. The τ-dual kernel corresponds to the first order response of the Fukui kernel and is designed to integrate to the finite temperature definition of the dual descriptor; it indicates the ambiphilic reactivity of a specific bond and enriches the traditional dual descriptor by allowing one to distinguish between the electron-accepting and electron-donating processes. Finally, the τ-hyper dual kernel is defined as the second-order derivative of the Fukui kernel and is proposed as a measure of the strength of ambiphilic bonding interactions. Although these quantities have never been proposed, our results for the τ-Fukui kernel and for τ-dual kernel can be derived in zero-temperature formulation of the chemical reactivity theory with, among other things, the widely-used parabolic interpolation model.

Donation and back-donation analyzed through a charge transfer model based on density functional theory.

The net charge transfer process that occurs between two species, A and B, interacting with each other, may be decomposed into two processes: one in which A receives charge from B, which can be identified as the electrophilic channel for A or the nucleophilic channel for B, and a second in which A donates charge to B, which can be identified as the nucleophilic channel for A or the electrophilic channel for B. By determining the amount of charge associated with both processes through the minimization of the interaction energy associated with each case, the expressions for the amount of charge involved in each case can be expressed in terms of the directional chemical potentials and the hardnesses of the interacting species. The correlation between the charges obtained for the interaction between phosphine ligands of the type PRR'R'' and Ni, and the A1 carbonyl stretching frequency provides support for their interpretation as measures of the electrophilicity and nucleophilicity of a chemical species, and, at the same time, allows one to describe the donation and back-donation processes in terms of the density functional theory of chemical reactivity.