The constrained dipole moment density functional theory allows one to control the magnitude and the components of the molecular dipole moment. In this work, we present a methodology which can control the dipole, quadrupole, and octupole moments in a variational and nonempirical way. This development allows us to estimate the individual (dipole, quadrupole, and octupole) and the combined (dipole-quadrupole, dipole-octupole, quadrupole-octupole, and dipole-quadrupole-octupole) multipole contributions in the formation of the ground state of the complex, taking as reference the frozen state of the complex. These contributions allow us to introduce an approximation to the variational frozen state and the corresponding frozen contribution. To test the reliability of the theoretical development, we study four sets of noncovalent complexes from the literature with a total of 24 systems. The individual and the combined multipole contributions results reveal the nature of the interaction between fragments according to these multipole moments, and the rather fast convergence of the multipole expansion, which, according to the results obtained, indicates that by including just the dipole, quadrupole, and octupole moments, one can describe the frozen state reasonably well.

Because the gradient expansion for the kinetic energy functionals around the uniform electron gas limit diverges for molecular systems, hyperasymptotic resummation techniques seem appealing. We discuss some renormalization approaches including rational Pad & eacute; and Meijer-G resummation methods that are well-adapted to strongly divergent expansions and evaluate their performance. In general, the problem of kinetic energy functionals appears extremely challenging. Imposing constraints on the kinetic energy functional may be beneficial and potentially provide guidance for future investigations.

The perspective presented in this work shows explicitly that the first order perturbation based on the changes that occur in the chemical potential and the hardness of each of the interacting species, due to the change in their external potential that arises when they are in the presence of each other, can be translated into the language of conceptual density functional theory (CDFT). Through this approach, one is led to important insights about the relevant role played by the chemical potential, the hardness, the Fukui function and the dual descriptor to understand the chemical reactivity at the global level by means of the first two, and of the different sites within a molecule by means of the last two. In particular, it is shown that the covalent contribution to the interaction energy can be expressed in terms of these four indicators, and because of the close relation between them and their corresponding approximations in terms of the frontier orbitals, the present work highlights the fundamental role of frontier orbital theory in the study of chemical reactivity. The analysis presented of the "|Δμ| big is good" rule reinforces the validation of this theory from first principles, as Parr and Yang had anticipated in the work that gave rise to the Fukui function. Additionally, the simplified approach developed to study specific cases, together with the systems considered, shows that the perturbed descriptors provide relevant insights into the way in which the reactants approach each other and about the possible dominant effects present in a reaction.

Recently, Miranda-Quintana, Heidar-Zadeh, and Ayers have elaborated a simple proof of the "Δμ big is good" rule for reactions that are dominated by charge transfer, making use of the smooth quadratic interpolation for fractional electron number and the analysis of several derivatives. In this work, we extend this approach, showing that one can derive a direct relationship of the change in the energy as a function of the change in the absolute value of the chemical potential that implies the "Δμ big is good" rule. In addition, we show that this rule plays an important role in the fulfillment of the hard and soft acids and bases (HSAB), the maximum hardness, and the minimum electrophilicity principles, as anticipated intuitively by Parr.

CONTEXT:The formulation of conceptual density functional theory in the grand canonical ensemble provides a theoretical framework that allows one to establish additional insights about the response functions that characterize this approach. In particular, through this procedure, one can establish the local counterpart of the chemical potential which, when integrated over all the space, leads to the global quantity and the local counterpart of the hardness that not only provides a function free of ambiguities, but also generates through its integration over all the space the well-defined value of the global quantity given by the difference of the vertical first ionization potential and electron affinity. In the present work, the non-local counterpart of these local reactivity descriptors is derived making use of the Fukui kernel descriptor previously developed by us. Then, the local and non-local chemical potential and hardness, thus obtained, are applied to study site and bond reactivities of several systems, to rationalize the behavior of kinetic and thermodynamic properties, through the chemical information that these indexes provide. METHODS:The electronic structure calculations required to evaluate the reactivity indexes analyzed in this work were done with the PBE0 exchange-correlation energy functional. The geometry optimization was done in all cases in a modified version of the NWChem program, while the Hirshfeld population analysis was done in a modified version of the demon2k program. For the electrophilic addition of hydrogen halides (HX) to several substituted ethenes and the hydration reaction of aldehydes and ketones, the 6-311G** basis set was used, while for the bond enthalpies of chemical reactions where there is a homolytic bond break and the trans influence in which the lability of the leaving ligand is modified by the ligand opposite to it, the Def2-TZVP was used.

A comparative analysis between the procedure originally followed to establish the local hardness concept and a recent one based on a local chemical potential, defined within the grand canonical ensemble formalism, is done to get a better understanding of the main aspects involved in both procedures and to show that the local and non-local counterparts of global reactivity descriptors recently developed, constitute an excellent complement to analyze chemical reactivity within the framework of conceptual density functional theory. A comparative analysis between the procedure originally followed to establish the local hardness and the hardness kernel concepts and a recent one based on a local chemical potential, defined within the grand canonical ensemble formalism, is done. The original local softness and softness kernel remain unchanged.

The original constrained dipole moment density functional theory allows one to control the magnitude of the molecular dipole moment in a variational and non-empirical way. In this work, we extend this methodology to control the three Cartesian components of the molecular dipole moment. The new theoretical development is suitable for the calculation of the charge-transfer energy contributions to the total interaction energies in non-covalent complexes. To test the reliability of the theoretical development, we form three sets of non-covalent complexes from the literature with a total of fifty-one systems. The former set of complexes includes many different types of non-covalent interactions, the second set consists of prototypical non-covalent complexes and three biologically relevant interactions between DNA base pairs, and the third set comprises halogen bonding complexes. We determined the charge-transfer energy contributions and the total interaction energies of all these complexes. The calculated charge-transfer energies are in very good agreement with the ones calculated using the fragment-based Hirshfeld methodology, which has been proven to be reliable. Nevertheless, the new procedure relies on the molecular dipole moment, which is observable, while the fragment-based Hirshfeld methodology relies on a definition of a population analysis.

This work compiles almost a decade of theoretical progress in temperature-dependent chemical reactivity theory to introduce the first finite-temperature charge transfer model, predicting fractional electron transfers during chemical interactions. The key insight is that electronic heat drives charge transfer. By analyzing thermodynamic parameters like electronic heat capacity, softness, and chemical potential, the framework explains how species transition from inert to reactive states, where electrons are decorrelated enough to enable charge transfer. A crucial aspect of this model is the role of thermal fluctuations, which governs molecular response functions and facilitates the simultaneous exchange of energy and charge. This model is reduced to a simple linear equation in the chemical potential of the reservoir. When extrapolated, it supports the electrophilicity index, adding a correction term and providing a working formula more influenced by electron affinity. These findings offer new pathways to analyze and predict chemical interactions under the finite temperature regime.

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.

Until quite recently, Conceptual DFT (CDFT) was mainly based on the energy functional, E[N,v], where the number of electrons N and the external potential v are state variables. One of the strengths of CDFT, however, is the ease with which additional and/or different state variables can be incorporated. Here, the incorporation of new variables—namely temperature and external fields—is discussed, outlining the motivation for these extensions, sketching their theoretical/computational context, and presenting some elucidative examples. Using the Grand Canonical Ensemble, finite temperature can be introduced, ameliorating the N-differentiability problem and leading to new well-behaved chemical reactivity concepts. Incorporating temperature as an additional fundamental variable enables us to formulate a novel suite of “thermodynamic” reactivity descriptors, including important concepts like the heat capacity of electronic systems. The mathematical structure underpinning the set of (well-behaved) finite-temperature reactivity indices can guide the formulation of plausible definitions for local analogs of global descriptors. This endeavor is especially significant in the case of local hardness, a concept that has remained elusive since the inception of CDFT. The ever-increasing portfolio of experimental reaction conditions to creatively synthesize new molecules, needs the introduction of various external “fields” like electric and magnetic fields (ε and B), mechanical forces (F) and pressure (P) to describe the state of the chemical system. The conventional energy functional can be expressed in a general form, E[N,v,X], where X denotes the “field”. Response functions to changes in the field can then be defined in analogy to classical thermodynamics. The electric field results display a case of a field-induced selectivity in a reaction channel of the Fukui function. Remarkably, atomic electronegativity and hardness in magnetic fields display a piecewise behavior in magnetic fields, associated to configurational jumps upon increasing field strength. The overall compression of their ranges for stronger fields may be insightful when investigating chemistry in extremely high fields. The electronegativity and hardness of diatomics under mechanical force can be traced back to changes in equilibrium distances in the neutral, cationic and anionic state, parallel with the evolution of an intrinsic atomic volume under pressure.

The charge-transfer energy contribution is one of the most controversial components of the total interaction energy. Commonly, the energy associate to a charge-transfer process depends on population analysis. Therefore, the results further depend on how the population analysis is defined, and certainly, the results may be arbitrary. Moreover, another important feature of the current methods is the basis sets dependency. The results of methodologies that depend on orbital-based population analyses tend to have a strong dependency on the size of the basis set utilized. This basis set dependency is eliminated by using spatial partitioning population analyses. However, these methodologies still rely on the arbitrary choice of how to divide the space. In this work, we study the use of the molecular dipole moment as a reference to describe the charge transfer-free system, i.e., a system in which the charge-transfer process is avoided. We use the recently developed constrained dipole moment density functional theory methodology to constrain the dipole moment of several systems according to reference values. These dipole moment references do not present charge transfer nor polarization contributions. In this manner, we have calculated the charge-transfer energy contributions and the total interaction energies of 13 non-covalent complexes. In addition, we determined two long range charge-transfer excitations considering the dipole moment as a reference. The calculated charge-transfer energy contributions and excitation energies are in a very good agreement with the fragment-based Hirshfeld methodology. Nevertheless, the constrained dipole moments results do not depend on population analysis. Moreover, the method is robust with respect to the strength of the charge transfer and the basis set size.

The theoretical framework of conceptual density functional theory is applied to the study of molecular excitations by considering that the differences between the ground and excited state electronic densities may be treated as if they were the consequence of intramolecular charge transfer processes. Through this approach, it is found that the hardness is directly related with the absorption and emission energies, and that the dual descriptor provides a description of the regions of the molecule where charge is added or removed, that in its condensed form leads to an estimation of the effective charge involved in the excitation. All this chemically significative information is then used to analyze the effects of the substituent in the fluorescence of naphthalimide derivatives.

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.

A smooth interpolation between the Hirshfeld and the iterative Hirshfeld isolated atomic densities used to determine the weight function that distributes the mo-lecular electronic density among the atoms is proposed to analyze the effect of the Fukui function on the atomic charges obtained. The interpolation makes use of a parameter alpha, that leads to the Hirshfeld weight function when alpha = 0, and to the iterative Hirshfeld weight function when alpha = 1. Through the calculation of the correlation coefficient R2 between these charges and those corresponding to fits of the electrostatic potential, it is found that R2 presents a maximum when alpha is equal to 0.95, and is almost constant for values of alpha between 0.85 and 1.0. The second main correlation was found with the atomic dipole moment corrected Hirshfeld charges that leads to a maximum when alpha lies between 0.5 and 0.6. Thus, the additional parameter that results from the interpolation may be used to obtain charges similar to those found through schemes that only assign charges, to obtain at the same time the atomic densities that add up to the molecular electron density, or the additional parameter may be used to find a set of charges in a molecule aimed to give a better description of a specific property, delivering at the same time a set of atomic densities that may be very important to derive other properties of the atoms in a molecule.

A new procedure based on the two parabolas model of the energy and the electronic density for fractional electron number is used with the assumption that the changes to the isolated values of these two quantities due to the presence of another interacting species can be incorporated through a multiplicative constant in the second order term. The expressions thus obtained for the chemical potential, hardness, Fukui function and dual descriptor reactivity indexes of conceptual density functional theory have the same form of those obtained through a first order perturbation approach within the grand canonical ensemble. The perturbation parameters are then evaluated by imposing the chemical potential and hardness equalization principles for the interaction between species A and B to form AB, and it is applied to show for a group of substituted ethenes that the condensed to atom perturbed local chemical potential and local hardness evaluated at the carbon atom that follows the Markovnikov’s rule lead to better correlation with the activation energy of their reaction with HCl than the unperturbed descriptors. A similar situation is found for the correlation of the condensed to atom local chemical potential evaluated at N in the aniline molecules with the experimental pK_a values. The results obtained indicate that through the perturbed descriptors, that introduce information of the electronic structure on each species of the other one with which it interacts allow one to obtain an improved description of their chemical reactivity.

Departing from the framework provided by the grand canonical ensemble, the conceptual density functional theory reactivity indexes for isolated systems are modified to incorporate through a first order perturbation approach the effects on them induced by the presence of another interacting species. The general procedure established leads to the same perturbed expressions for the chemical potential and the hardness previously developed and allows one to derive the perturbed expressions for the Fukui function and the dual descriptor. The expressions obtained include the effects on the reactivity indexes of a given species when it interacts with another one, through additional parameters that can be determined to satisfy known specific conditions associated with the interacting systems studied, to improve the description of their chemical reactivity.

Making use of the initial and final Fukui functions approximated by the charge density of the initial (occupied) orbital and the final (unoccupied in the ground state) orbital to identify the charge transfer excitations in a time-dependent density functional theory calculation, and the acceptance and donation regions, we analyze the performance of constrained density functional theory to predict the excitation energies by considering several alternatives to fix the amount of charge transferred. It is shown that charge transfer excitations energies may be accurately determined through this approach when one fixes the final charge in the acceptance region to a value that complements the net charge already present to minus one.