ADCHα-I population analysis and constrained dipole moment density functional theory in force fields for molecular simulations.

A new population analysis, ADCHα-I, based on the interpolation between the Hirshfeld (H) and the iterative Hirshfeld (H-I) methods through a parameter α and on the atomic dipole moment corrected Hirshfeld (ADCH) methodology is proposed, in combination with the constrained dipole moment density functional theory (CD-DFT) previously developed, to determine the charge distributions of force fields. Following this approach, the electronic density of the isolated molecule is determined for the value of the dipole moment that reproduces the experimental dielectric constant, in order to incorporate through this property the effects of the surrounding molecules in the liquid, and to carry on this information to the molecular simulation, the new population analysis is built to obtain the set of charges that reproduces this dipole moment. By selecting α = 1/2, one is led to charges that are larger than the ones obtained through H and ADCH and smaller than those of H-I and that incorporate, at the local level, information about the response of isolated atoms to donate or to accept charge, which is not considered in ADCH. The results obtained for several liquid properties indicate that the combination of CD-DFT with this population analysis leads to a good description of the charge distributions in force fields used in molecular simulations.

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.

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.

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.

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.

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.

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 chemical potential, local hardness, and dual descriptors in temperature dependent chemical reactivity theory.

In this work we establish a new temperature dependent procedure within the grand canonical ensemble, to avoid the Dirac delta function exhibited by some of the second order chemical reactivity descriptors based on density functional theory, at a temperature of 0 K. Through the definition of a local chemical potential designed to integrate to the global temperature dependent electronic chemical potential, the local chemical hardness is expressed in terms of the derivative of this local chemical potential with respect to the average number of electrons. For the three-ground-states ensemble model, this local hardness contains a term that is equal to the one intuitively proposed by Meneses, Tiznado, Contreras and Fuentealba, which integrates to the global hardness given by the difference in the first ionization potential, I, and the electron affinity, A, at any temperature. However, in the present approach one finds an additional temperature-dependent term that introduces changes at the local level and integrates to zero. Additionally, a τ-hard dual descriptor and a τ-soft dual descriptor given in terms of the product of the global hardness and the global softness multiplied by the dual descriptor, respectively, are derived. Since all these reactivity indices are given by expressions composed of terms that correspond to products of the global properties multiplied by the electrophilic or nucleophilic Fukui functions, they may be useful for studying and comparing equivalent sites in different chemical environments.

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

Making use of the grand canonical ensemble the derivation of the analytical equations for the chemical potential and the Fukui function in the general case of any number of ground and excited states is presented. The expressions thus obtained allow one to establish that the ensemble of three consecutive ground states that has been usually used to analyze the effects of temperature in these quantities provides a satisfactory description for them at temperatures of chemical interest. Nevertheless, some situations must be considered cautiously, as for example, when the N + k and N + k + 1 (N is the electron number) ground states are (nearly) quasidegenerate or when the first excited state of both the anion and the cation (with respect to the reference state) is very low in energy. Results for the copper atom (with the ground state of Cu+ as the reference state), using some selected ensemble models constituted by several ground and excited states, are presented to show that the very low-lying excited states of some of the copper species are able to contribute to chemical reactivity at relatively low temperatures (∼2000 K). A relevant aspect is that due to its generality, the present approach provides a new way to study the reactivity of the chemical species under extreme conditions.

Revisiting the definition of local hardness and hardness kernel.

An analysis of the hardness kernel and local hardness is performed to propose new definitions for these quantities that follow a similar pattern to the one that characterizes the quantities associated with softness, that is, we have derived new definitions for which the integral of the hardness kernel over the whole space of one of the variables leads to local hardness, and the integral of local hardness over the whole space leads to global hardness. A basic aspect of the present approach is that global hardness keeps its identity as the second derivative of energy with respect to the number of electrons. Local hardness thus obtained depends on the first and second derivatives of energy and electron density with respect to the number of electrons. When these derivatives are approximated by a smooth quadratic interpolation of energy, the expression for local hardness reduces to the one intuitively proposed by Meneses, Tiznado, Contreras and Fuentealba. However, when one combines the first directional derivatives with smooth second derivatives one finds additional terms that allow one to differentiate local hardness for electrophilic attack from the one for nucleophilic attack. Numerical results related to electrophilic attacks on substituted pyridines, substituted benzenes and substituted ethenes are presented to show the overall performance of the new definition.

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.

Global and Local Partitioning of the Charge Transferred in the Parr-Pearson Model.

Through a simple proposal, the charge transfer obtained from the cornerstone theory of Parr and Pearson is partitioned, for each reactant, in two channels: an electrophilic, through which the species accepts electrons, and the other, a nucleophilic, where the species donates electrons. It is shown that this global model allows us to determine unambiguously the charge-transfer mechanism prevailing in a given reaction. The partitioning is extended to include local effects through the Fukui functions of the reactants. This local model is applied to several emblematic reactions in organic and inorganic chemistry, and we show that besides improving the correlations obtained with the global model it provides valuable information concerning the atoms in the reactants playing the most important roles in the reaction and thus improving our understanding of the reaction under study.

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.

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.

Electronic chemical response indexes at finite temperature in the canonical ensemble.

Assuming that the electronic energy is given by a smooth function of the number of electrons and within the extension of density functional theory to finite temperature, the first and second order chemical reactivity response functions of the Helmholtz free energy with respect to the temperature, the number of electrons, and the external potential are derived. It is found that in all cases related to the first or second derivatives with respect to the number of electrons or the external potential, there is a term given by the average of the corresponding derivative of the electronic energy of each state (ground and excited). For the second derivatives, including those related with the temperature, there is a thermal fluctuation contribution that is zero at zero temperature. Thus, all expressions reduce correctly to their corresponding chemical reactivity expressions at zero temperature and show that, at room temperature, the corrections are very small. When the assumption that the electronic energy is given by a smooth function of the number of electrons is replaced by the straight lines behavior connecting integer values, as required by the ensemble theorem, one needs to introduce directional derivatives in most cases, so that the temperature dependent expressions reduce correctly to their zero temperature counterparts. However, the main result holds, namely, at finite temperature the thermal corrections to the chemical reactivity response functions are very small. Consequently, the present work validates the usage of reactivity indexes calculated at zero temperature to infer chemical behavior at room and even higher temperatures.