On the importance of the “density per particle” (shape function) in the density functional theory

Shannon entropy variation as a global indicator of electron density contraction at interatomic regions in chemical reactions

CONTEXT

The negative of the Shannon entropy derivative is proposed to account for electron density contraction as the chemical bonds are breaking and forming during a chemical reaction. We called this property the electron density contraction index, EDC, which allows identifying stages in a reaction that are dominated by electron contraction or expansion. Four different reactions were analyzed to show how the EDC index changes along the reaction coordinate. The results indicate that the rate of change of Shannon entropy is directly related to the rate of change of the electron density at the bond critical points between all the atomic pairs in the molecular systems. It is expected that EDC will complement the detailed analysis of reaction mechanisms that can be performed with the theoretical tools available to date.

METHODS

Density functional theory calculations at the B3LYP/6-31G(d,p) level of theory were carried out using Gaussian 16 to analyze the reaction mechanisms of the four reactions studied. The reaction paths were obtained via the intrinsic reaction coordinate method, which served as the reaction coordinate to obtain the reaction force and the EDC profiles in each case. Shannon entropy and electron density at the bond critical points were calculated using the Multiwfn 3.7 package.

Energetic Information from Information-Theoretic Approach in Density Functional Theory as Quantitative Measures of Physicochemical Properties

The Hohenberg-Kohn theorem of density functional theory (DFT) stipulates that energy is a universal functional of electron density in the ground state, so energy can be thought of having encoded essential information for the density. Based on this, we recently proposed to quantify energetic information within the framework of information-theoretic approach (ITA) of DFT (J. Chem. Phys. 2022, 157, 101103). In this study, we systematically apply energetic information to a variety of chemical phenomena to validate the use of energetic information as quantitative measures of physicochemical properties. To that end, we employed six ITA quantities such as Shannon entropy and Fisher information for five energetic densities, yielding twenty-six viable energetic information quantities. Then, they are applied to correlate with physicochemical properties of molecular systems, including chemical bonding, conformational stability, intermolecular interactions, acidity, aromaticity, cooperativity, electrophilicity, nucleophilicity, and reactivity. Our results show that different quantities of energetic information often behave differently for different properties but a few of them, such as Shannon entropy of the total kinetic energy density and information gain of the Pauli energy density, stand out and strongly correlate with several properties across different categories of molecular systems. These results suggest that they can be employed as quantitative measures of physicochemical properties. This work not only enriches the body of our knowledge about the relationship between energy and information, but also provides scores of newly introduced explicit density functionals to quantify physicochemical properties, which can serve as robust features for building machine learning models in future studies.

On the Periodicity of the Information Theory and Conceptual DFT-Based Reactivity Descriptors

The periodic trends in conceptual density functional and information theory-based reactivity descriptors are reported for the atoms H to Ba (Z = 1 to 56). Ionization potential, electron affinity, electronegativity, and hardness show periodic behavior following the Aufbau principle and popular electronic structure principles. They are in agreement with those reported in standard chemistry textbooks. The trend in the electrophilicity index, however, shows an interesting behavior, where it contradicts earlier reports. Our calculation reveals that the noble gas elements correspond to minimum ω values in each period which obey the minimum electrophilicity principle as well as reflect their low reactivity. Periodic trends in electroaccepting and electrodonating powers, along with that of net electrophilicity, are as expected. The behavior of information theory-based Shannon and GBP entropies, along with the Shannon entropy of shape function are also explored across the periodic table.

Constrained iterative Hirshfeld charges: A variational approach

We develop a variational procedure for the iterative Hirshfeld (HI) partitioning scheme. The main practical advantage of having a variational framework is that it provides a formal and straightforward approach for imposing constraints (e.g., fixed charges on certain atoms or molecular fragments) when computing HI atoms and their properties. Unlike many other variants of the Hirshfeld partitioning scheme, HI charges do not arise naturally from the information-theoretic framework, but only as a reverse-engineered construction of the objective function. However, the procedure we use is quite general and could be applied to other problems as well. We also prove that there is always at least one solution to the HI equations, but we could not prove that its self-consistent equations would always converge for any given initial pro-atom charges. Our numerical assessment of the constrained iterative Hirshfeld method shows that it satisfies many desirable traits of atoms in molecules and has the potential to surpass existing approaches for adding constraints when computing atomic properties.

N-Derivatives of Shannon entropy density as response functions

The exact first and second order partial derivatives of Shannon entropy density with respect to the number of electrons at constant external potential are introduced as new descriptors for prediction of the active sites of a molecule. The derivatives, which are a measure of the inhomogeneity of electron density, are calculated both exactly (from analytical forms) and approximately (using the finite difference method) for some molecular systems. According to the maximum entropy principle, the extreme value of the first order derivative on the surface of a given molecule should determine the active sites of the molecule in electrophilic and nucleophilic attack. The second order derivative indicates where the Shannon entropy is more concentrated or depleted during the electron exchange. Although these derivatives on the surfaces of helium and neon atoms are uniform, the corresponding values for argon, krypton and xenon atoms are not. This could explain the greater tendency of heavy noble gas atoms to form stable compounds. A dual descriptor is also defined as the difference between the left and right first order derivatives of Shannon entropy density, which allows one to simultaneously predict the preferable sites for electrophilic and nucleophilic attack over the system at point r. Therefore, the reactivity of an atom in a molecule requires the non-uniformity of the first and second order derivatives of Shannon entropy density on the surface of that atom.

Shannon, Rényi, Tsallis Entropies and Onicescu Information Energy for Low-Lying Singly Excited States of Helium

Knowledge of the electronic structures of atomic and molecular systems deepens our understanding of the desired system. In particular, several information-theoretic quantities, such as Shannon entropy, have been applied to quantify the extent of electron delocalization for the ground state of various systems. To explore excited states, we calculated Shannon entropy and two of its one-parameter generalizations, Rényi entropy of order α and Tsallis entropy of order α , and Onicescu Information Energy of order α for four low-lying singly excited states (1s2s 1 S e , 1s2s 3 S e , 1s3s 1 S e , and 1s3s 3 S e states) of helium. This paper compares the behavior of these three quantities of order 0.5 to 9 for the ground and four excited states. We found that, generally, a higher excited state had a larger Rényi entropy, larger Tsallis entropy, and smaller Onicescu information energy. However, this trend was not definite and the singlet–triplet reversal occurred for Rényi entropy, Tsallis entropy and Onicescu information energy at a certain range of order α .

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

From the definition of a local chemical potential, well-behaved expressions for the local hardness and the dual descriptors are derived.

The electron localization as the information content of the conditional pair density

In the present work, the information gained by an electron for "knowing" about the position of another electron with the same spin is calculated using the Kullback-Leibler divergence (DKL) between the same-spin conditional pair probability density and the marginal probability. DKL is proposed as an electron localization measurement, based on the observation that regions of the space with high information gain can be associated with strong correlated localized electrons. Taking into consideration the scaling of DKL with the number of σ-spin electrons of a system (N(σ)), the quantity χ = (N(σ) - 1) DKLfcut is introduced as a general descriptor that allows the quantification of the electron localization in the space. fcut is defined such that it goes smoothly to zero for negligible densities. χ is computed for a selection of atomic and molecular systems in order to test its capability to determine the region in space where electrons are localized. As a general conclusion, χ is able to explain the electron structure of molecules on the basis of chemical grounds with a high degree of success and to produce a clear differentiation of the localization of electrons that can be traced to the fluctuation in the average number of electrons in these regions.

From information theory to quantitative description of steric effects

In this study, further evidences of the efficiency of informational theory for a quantitative description of steric effects are showcased.

Scaling properties of information-theoretic quantities in density functional reactivity theory

A number of strong linear correlations between information-theoretic quantities and electron populations for atoms, molecules, and atoms-in-molecules have been disclosed.

Fuzzy electron density fragments in macromolecular quantum chemistry, combinatorial quantum chemistry, functional group analysis, and shape-activity relations

Conspectus Just as complete molecules have no boundaries and have "fuzzy" electron density clouds approaching zero density exponentially at large distances from the nearest nucleus, a physically justified choice for electron density fragments exhibits similar behavior. Whereas fuzzy electron densities, just as any fuzzy object, such as a thicker cloud on a foggy day, do not lend themselves to easy visualization, one may partially overcome this by using isocontours. Whereas a faithful representation of the complete fuzzy density would need infinitely many such isocontours, nevertheless, by choosing a selected few, one can still obtain a limited pictorial representation. Clearly, such images are of limited value, and one better relies on more complete mathematical representations, using, for example, density matrices of fuzzy fragment densities. A fuzzy density fragmentation can be obtained in an exactly additive way, using the output from any of the common quantum chemical computational techniques, such as Hartree-Fock, MP2, and various density functional approaches. Such "fuzzy" electron density fragments properly represented have proven to be useful in a rather wide range of applications, for example, (a) using them as additive building blocks leading to efficient linear scaling macromolecular quantum chemistry computational techniques, (b) the study of quantum chemical functional groups, (c) using approximate fuzzy fragment information as allowed by the holographic electron density theorem, (d) the study of correlations between local shape and activity, including through-bond and through-space components of interactions between parts of molecules and relations between local molecular shape and substituent effects, (e) using them as tools of density matrix extrapolation in conformational changes, (f) physically valid averaging and statistical distribution of several local electron densities of common stoichiometry, useful in electron density databank mining, for example, in medicinal drug design, and (g) tools for combinatorial quantum chemistry approaches using fuzzy fragment databanks and rapid construction of a large number of approximate electron densities for large sets of related molecules, relevant in theoretical molecular and nanostructure design.

An information-theoretic resolution of the ambiguity in the local hardness

A definition of the local hardness, suitable for application in the local hard/soft acid/base principle, is derived by applying information theory.

Electron density shape analysis of a family of through-space and through-bond interactions

A family of styrene derivatives has been used to study the effects of through-space and through-bond interactions on the local and global shapes of electron densities of complete molecules and a set of substituents on their central rings. Shape analysis methods which have been used extensively in the past for the study of molecular property-molecular shape correlations have shown that in these molecules a complementary role is played by the through-space and through-bond interactions. For each specific example, the dominance of either one of the two interactions can be identified and interpreted in terms of local shapes and the typical reactivities of the various substituents. Three levels of quantum chemical computational methods have been applied for these structures, including the B3LYP/cc-pVTZ level of density functional methodology, and the essential conclusions are the same for all three levels. The general approach is suggested as a tool for the identification of specific interaction types which are able to modify molecular electron densities. By separately influencing the through-space and through-bond components using polar groups and groups capable of conjugation, some fine-tuning of the overall effects becomes possible. The method described may contribute to an improved understanding and control of molecular properties involving complex interactions with a possible role in the emerging field of molecular design.

Information carriers and (reading them through) information theory in quantum chemistry

This Perspective discusses the reduction of the electronic wave function via the second-order reduced density matrix to the electron density ρ(r), which is the key ingredient in density functional theory (DFT) as a basic carrier of information. Simplifying further, the 1-normalized density function turns out to contain essentially the same information as ρ(r) and is even of preferred use as an information carrier when discussing the periodic properties along Mendeleev's table where essentially the valence electrons are at stake. The Kullback-Leibler information deficiency turns out to be the most interesting choice to obtain information on the differences in ρ(r) or σ(r) between two systems. To put it otherwise: when looking for the construction of a functional F(AB) = F[ζ(A)(r),ζ(B)(r)] for extracting differences in information from an information carrier ζ(r) (i.e. ρ(r), σ(r)) for two systems A and B the Kullback-Leibler information measure ΔS is a particularly adequate choice. Examples are given, varying from atoms, to molecules and molecular interactions. Quantum similarity of atoms indicates that the shape function based KL information deficiency is the most appropriate tool to retrieve periodicity in the Periodic Table. The dissimilarity of enantiomers for which different information measures are presented at global and local (i.e. molecular and atomic) level leads to an extension of Mezey's holographic density theorem and shows numerical evidence that in a chiral molecule the whole molecule is pervaded by chirality. Finally Kullback-Leibler information profiles are discussed for intra- and intermolecular proton transfer reactions and a simple S(N)2 reaction indicating that the theoretical information profile can be used as a companion to the energy based Hammond postulate to discuss the early or late transition state character of a reaction. All in all this Perspective's answer is positive to the question of whether an even simpler carrier of information than the electron density function ρ(r) can be envisaged: the shape function, integrating to 1 by construction fulfils this role. On the other hand obtaining the information (or information difference) contained in one (or two) systems from ρ(r) or σ(r) can be most efficiently done by using information theory, the Kulback-Leibler information deficiency being at the moment (one of) the most advisable functionals.

Particle number and probability density functional theory and A-representability

In Hohenberg-Kohn density functional theory, the energy E is expressed as a unique functional of the ground state density rho(r): E = E[rho] with the internal energy component F(HK)[rho] being universal. Knowledge of the functional F(HK)[rho] by itself, however, is insufficient to obtain the energy: the particle number N is primary. By emphasizing this primacy, the energy E is written as a nonuniversal functional of N and probability density p(r): E = E[N,p]. The set of functions p(r) satisfies the constraints of normalization to unity and non-negativity, exists for each N; N = 1, ..., infinity, and defines the probability density or p-space. A particle number N and probability density p(r) functional theory is constructed. Two examples for which the exact energy functionals E[N,p] are known are provided. The concept of A-representability is introduced, by which it is meant the set of functions Psi(p) that leads to probability densities p(r) obtained as the quantum-mechanical expectation of the probability density operator, and which satisfies the above constraints. We show that the set of functions p(r) of p-space is equivalent to the A-representable probability density set. We also show via the Harriman and Gilbert constructions that the A-representable and N-representable probability density p(r) sets are equivalent.

Fisher and Jensen-Shannon divergences: Quantitative comparisons among distributions. Application to position and momentum atomic densities

The Fisher divergence (FD) and Jensen-Shannon divergence (JSD) are used in this work with the aim of providing quantitative measures of the discrepancies between two arbitrary D-dimensional distribution functions, the FD being of local character and the JSD of global one. In doing so, the concepts of Fisher information and Shannon entropy associated to a distribution are the essential quantities for building up these comparative functionals. This kind of relative measures are here applied to the study of the one-particle densities in both conjugated spaces (position and momentum) of neutral atoms, discussing the results as compared to those provided by other previous functional measures. It is clearly shown how these divergences provide relevant information on the atomic shell structure, up to a level which depends on the considered space and measure.

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

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

Conceptual DFT: the chemical relevance of higher response functions

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

Condensed-to-atoms electronic Fukui functions within the framework of spin-polarized density-functional theory

A simple formalism devised to calculate the condensed-to-atoms Fukui function [R. R. Contreras, P. Fuentealba, M. Galván, and P. Pérez, Chem. Phys. Lett. 304, 405 (1999)] has been further analyzed within a spin-polarized density-functional theory framework. The model is based on a frozen-core approximation to these local reactivity indices [M. Galván, A. Vela, and J. L. Gázquez, J. Phys. Chem. 92, 6470 (1988)], giving us an extended local reactivity description of systems based on the frontier spin-up and spin-down molecular orbitals. Degenerate molecular spin orbitals have been explicitly included in our model equations. Computational results for the nitric oxide (NO) and some simple carbene systems are presented in order to test the model. These quantities have been discussed in the context of changes both in charge density and spin density within the context of electron charge transfer or spin-polarization processes.

Critical analysis and extension of the Hirshfeld atoms in molecules

The computational approach to the Hirshfeld [Theor. Chim. Acta 44, 129 (1977)] atom in a molecule is critically investigated, and several difficulties are highlighted. It is shown that these difficulties are mitigated by an alternative, iterative version, of the Hirshfeld partitioning procedure. The iterative scheme ensures that the Hirshfeld definition represents a mathematically proper information entropy, allows the Hirshfeld approach to be used for charged molecules, eliminates arbitrariness in the choice of the promolecule, and increases the magnitudes of the charges. The resulting "Hirshfeld-I charges" correlate well with electrostatic potential derived atomic charges.

Alternatives to the electron density for describing Coulomb systems

Stimulated by the difficulty of deriving effective kinetic energy functionals of the electron density, the authors consider using the local kinetic energy as the fundamental descriptor for molecular systems. In this ansatz, the electron density must be expressed as a functional of the local kinetic energy. There are similar results for other quantities, including the local temperature and the Kohn-Sham potential. One potential advantage of these approaches--and especially the approach based on the local temperature--is the chemical relevance of the fundamental descriptor.