The Fukui Potential and the Capacity of Charge and the Global Hardness of Atoms

Extending the definition of atomic basis sets to atoms with fractional nuclear charge

Alchemical transformations showed that perturbation theory can be applied also to changes in the atomic nuclear charges of a molecule. The alchemical path that connects two different chemical species involves the conceptualization of a non-physical system in which an atom possess a non-integer nuclear charge. A correct quantum mechanical treatment of these systems is limited by the fact that finite size atomic basis sets do not define exponents and contraction coefficients for fractional charge atoms. This paper proposes a solution to this problem and shows that a smooth interpolation of the atomic orbital coefficients and exponents across the periodic table is a convenient way to produce accurate alchemical predictions, even using small size basis sets.

Atoms-In-Molecules' Faces of Chemical Hardness by Conceptual Density Functional Theory

The chemical hardness concept and its realization within the conceptual density functional theory is approached with innovative perspectives, such as the electronegativity and hardness equalization of atoms in molecules connected with the softness kernel, in order to examine the structure-reactivity equalization ansatz between the electronic sharing index and the charge transfer either in the additive or geometrical mean picture of bonding. On the other hand, the maximum hardness principle presents a relation with the chemical stability of the hardness concept. In light of the inverse relation between hardness and polarizability, the minimum polarizability principle has been proposed. Additionally, this review includes important applications of the chemical hardness concept to solid-state chemistry. The mentioned applications support the validity of the electronic structure principles regarding chemical hardness and polarizability in solid-state chemistry.

On the Prediction of Lattice Energy with the Fukui Potential: Some Supports on Hardness Maximization in Inorganic Solids

Using perturbation theory within the framework of conceptual density functional theory, we derive a lower bound for the lattice energy of the ionic solids. The main element of the lower bound is the Fukui potential in the nuclei of the molecule corresponding to the unit formula of the solid. Thus, we propose a model to calculate the lattice energy in terms of the Fukui potential. Our method, which is extremely simple, performs well as other methods using the crystal structure information of alkali halide solids. The method proposed here correlates surprisingly well with the experimental data on the lattice energy of a diverse series of solids having even a non-negligible covalent characteristic. Finally, the validity of the maximum hardness principle (MHP) is assessed, showing that in this case, the MHP is limited.

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

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

Arbitrarily accurate quantum alchemy

Doping compounds can be considered a perturbation to the nuclear charges in a molecular Hamiltonian. Expansions of this perturbation in a Taylor series, i.e., quantum alchemy, have been used in the literature to assess millions of derivative compounds at once rather than enumerating them in costly quantum chemistry calculations. So far, it was unclear whether this series even converges for small molecules, whether it can be used for geometry relaxation, and how strong this perturbation may be to still obtain convergent numbers. This work provides numerical evidence that this expansion converges and recovers the self-consistent energy of Hartree-Fock calculations. The convergence radius of this expansion is quantified for dimer examples and systematically evaluated for different basis sets, allowing for estimates of the chemical space that can be covered by perturbing one reference calculation alone. Besides electronic energy, convergence is shown for density matrix elements, molecular orbital energies, and density profiles, even for large changes in electronic structure, e.g., transforming He₃ into H₆. Subsequently, mixed alchemical and spatial derivatives are used to relax H₂ from the electronic structure of He alone, highlighting a path to spatially relaxed quantum alchemy. Finally, the underlying code that allows for arbitrarily accurate evaluation of restricted Hartree-Fock energies and arbitrary order derivatives is made available to support future method development.

Analytic Alchemical Derivatives for the Analysis of Differential Acidity Assisted by the h Function

Analytical calculation of alchemical derivatives based on auxiliary density perturbation theory is described, coded, and validated. For the case where the nucleus is a hydrogen atom and the nuclear charge is changed from 1 to 0, it turns out that a good estimate of the proton binding energies can be obtained very efficiently. First-order results correspond exactly to the molecular electrostatic potential evaluated at the hydrogen nucleus location (removing self-repulsion), in agreement with previously reported extensive studies. Therefore, the second-order results reported here are refinements in accuracy that finally allow a quantitative exploration of differential acidity. Furthermore, the recently reported h function is produced in its analytical form as a byproduct and local descriptor associated with the proton binding energy values found with this approach. In an example application, proton binding energies are computed for a family of imidazolium derivatives to demonstrate the capabilities and the stability of the method with respect to changes in basis set or exchange-correlation functional.

Links among the Fukui potential, the alchemical hardness and the local hardness of an atom in a molecule

This paper presents a brief summary of the difficulty that resides in the definition of the elusive concept of local chemical hardness. We argue that a definition of local hardness should be useful to a reactivity principle and not just as a mere definition. We then continue with a formal discussion about the benefits and difficulties of using the Fukui potential, which is interpreted as an alchemical derivative (alchemical hardness), as descriptor of local hardness of molecules. Computational evidence shows that the alchemical hardness is at least as good a descriptor as the combination of other two well-stabilized descriptors of local hardness, such as the Fukui function and grand canonical local hardness. Although our results are auspicious for the alchemical hardness as descriptor of local hardness, we finish by calling the attention of the community on the importance of discussing the raison d'être of a local hardness function and its main characteristics. We suggest that an axiomatic construction of local hardness could be they way of constructing a local hardness which is both useful and free of arbitrariness.

Predicting Deprotonation Sites Using Alchemical Derivatives

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

Elucidating Enzymatic Catalysis Using Fast Quantum Chemical Descriptors

In general, computational simulations of enzymatic catalysis processes are thermodynamic and structural surveys to complement experimental studies, requiring high level computational methods to match accurate energy values. In the present work, we propose the usage of reactivity descriptors, theoretical quantities calculated from the electronic structure, to characterize enzymatic catalysis outlining its reaction profile using low-level computational methods, such as semiempirical Hamiltonians. We simulate three enzymatic reactions paths, one containing two reaction coordinates and without prior computational study performed, and calculate the reactivity descriptors for all obtained structures. We observed that the active site local hardness does not change substantially, even more so for the amino-acid residues that are said to stabilize the reaction structures. This corroborates with the theory that activation energy lowering is caused by the electrostatic environment of the active sites. Also, for the quantities describing the atom electrophilicity and nucleophilicity, we observed abrupt changes along the reaction coordinates, which also shows the enzyme participation as a reactant in the catalyzed reaction. We expect that such electronic structure analysis allows the expedient proposition and/or prediction of new mechanisms, providing chemical characterization of the enzyme active sites, thus hastening the process of transforming the resolved protein three-dimensional structures in catalytic information.

The Pauli principle and the confinement of electron pairs in a double well: Aspects of electronic bonding under pressure

It has become recently clear that chemical bonding under pressure is still lacking guiding principles for understanding the way electrons reorganize when their volume is constrained. As an example, it has recently been shown that simple metals can become insulators (aka electrides) when submitted to high enough pressures. This has lead to the general believe that "a fundamental yet empirically useful understanding of how pressure alters the chemistry of the elements is lacking" [R. J. Hemley, High Pressure Res. 30, 581 (2010)]. In this paper, we are interested in studying the role that the Pauli principle plays on the localization/delocalization of confined noninteracting electrons. To this end, we have considered the simple case of a 1-dimensional (1-D) double well as a confining potential, and the Electron Localization Function (ELF) has been used to characterize the degree localization/delocalization of the systems of noninteracting electrons. Then, we have systematically studied the topology of the ELF as a function of the double well parameters (barrier eight and wells distance) and of the number of electrons. We have found that the evolution of the ELF distributions has a good correspondence with the evolution of chemical bonding of atomic solids under pressure.

Atomic/Ionic Radius as Mathematical Limit of System Energy Evolution

The classical, in its nature, concept of atomic or ionic radii, although profitable in many fields, is represented by an ambiguous choice of formulations. In this work, we propose a definition of atomic and ionic radii rooted in chemical principles and conceptual density functional theories. The estimation based on electron density fundamental response functions has been successfully tested. The generalized approach has been shown to be applicable to atoms in any oxidation state. The radii display good correlation with classical estimates, such as Shannon. The atomic and ionic radii obtained according to this scheme are directly comparable between different elements, without any adjustment procedures requiring fitting constants. The definition also has a clear physical interpretation, which supports understanding of size-related phenomena and trends.

Exploring chemical space with alchemical derivatives: alchemical transformations of H through Ar and their ions as a proof of concept

Alchemical derivatives have been used previously to obtain information about transformations in which the number of electrons is unchanged. Here an approach for combining changes in both the number of electrons and the nuclear charge is presented.

Benchmark values of chemical potential and chemical hardness for atoms and atomic ions (including unstable anions) from the energies of isoelectronic series

We present benchmark values for the electronic chemical potential and chemical hardness from reference data for ionization potentials and electron affinities. In cases where the energies needed to compute these quantities are not available, we estimate the ionization potential of the metastable (di)anions by extrapolation along the isoelectronic series, taking care to ensure that the extrapolated data satisfy reasonable intuitive rules to the maximum possible extent. We also propose suitable values for the chemical potential and chemical hardness of zero-electron species. Because the values we report are faithful to the trends in accurate data on atomic energies, we believe that our proposed values for the chemical potential and chemical hardness are ideally suited to conceptual studies of chemical trends across the periodic table. The critical nuclear charge (Z critical) of the isoelectronic series with 2 < N < 96 has also been reported for the first time.

On understanding the chemical origin of band gaps

Conceptual DFT and quantum chemical topology provide two different approaches based on the electron density to grasp chemical concepts. We present a model merging both approaches, in order to obtain physical properties from chemically meaningful fragments (bonds, lone pairs) in the solid. One way to do so is to use an energetic model that includes chemical quantities explicitly, so that the properties provided by conceptual DFT are directly related to the inherent organization of electrons within the regions provided by topological analysis. An example of such energy model is the bond charge model (BCM) by Parr and collaborators. Bonds within an ELF-BCM coupled approach present very stable chemical features, with a bond length of ca. 1 Å and 2[Formula: see text]. Whereas the 2[Formula: see text] corroborate classical views of chemical bonding, the fact that bonds always expand along 1 Å introduces the concept of geometrical transferability and enables estimating crystalline cell parameters. Moreover, combining these results with conceptual DFT enables deriving a model for the band gap where the chemical hardness of a solid is given by the bond properties, charge, length, and a Madelung factor, where the latter plays the major role. In short, the fundamental gap of zinc-blende solids can be understood as given by a 2[Formula: see text] bond particle asymmetrically located on a 1 Å length box and electrostatically interacting with other bonds and with a core matrix. This description is able to provide semi-quantitative insight into the band gap of zinc-blende semiconductors and insulators on equal footing, as well as a relationship between band gap and compressibility. In other words, merging these different approaches to bonding enables to connect measurable macroscopic behavior with microscopic electronic structure properties and to obtain microscopic insight into the chemical origin of band gaps, whose prediction is still nowadays a difficult task. Graphical Abstract Conceptual DFT couples to quatum chemcial topology to explain the band gap of zinc-blende solids.

On the trends of Fukui potential and hardness potential derivatives in isolated atoms vs. atoms in molecules

In the present study, trends of electronic contribution to molecular electrostatic potential [Vel(r¯)(r=0)], Fukui potential [v(+)f|(r=0) and v(-)f|(r=0)] and hardness potential derivatives [Δ(+)h(k) and Δ(-)h(k)] for isolated atoms as well as atoms in molecules are investigated. The generated numerical values of these three reactivity descriptors in these two electronically different situations are critically analyzed through the relevant formalism. Values of Vel(r¯) (when r → 0, i.e., on the nucleus) are higher for atoms in molecules than that of isolated atoms. In contrast, higher values of v(+)|(r=0) and v(-)|(r=0) are observed for isolated atoms compared to the values for atoms in a molecule. However, no such regular trend is observed for the Δ(+)h(k) and Δ(-)h(k) values, which is attributed to the uncertainty in the Fukui function values of atoms in molecules. The sum of Fukui potential and the sum of hardness potential derivatives in molecules are also critically analyzed, which shows the efficacy of orbital relaxation effects in quantifying the values of these parameters. The chemical consequence of the observed trends of these descriptors in interpreting electron delocalization, electronic relaxation and non-negativity of atomic Fukui function indices is also touched upon. Several commonly used molecules containing carbon as well as heteroatoms are chosen to make the investigation more insightful.

QSAR study of the DPPH· radical scavenging activity of coumarin derivatives and xanthine oxidase inhibition by molecular docking

A Quantitative Structure-Activity Relationship (QSAR) of coumarins by genetic algorithms employing physicochemical, topological, lipophilic and electronic descriptors was performed. We have used experimental antioxidant activities of specific coumarin derivatives against the DPPH· radical molecule. Molecular descriptors such as Randic Path/Walk, hydrophilic factor and chemical hardness were selected to propose a mathematical model. We obtained a linear correlation with R2 = 96.65 and Q LOO2 = 93.14 values. The evaluation of the predictive ability of the model was performed by applying the Q ASYM2, $\hat r^2 $ and Δr m2 methods. Fukui functions were calculated here for coumarin derivatives in order to delve into the mechanics by which they work as primary antioxidants. We also investigated xanthine oxidase inhibition with these coumarins by molecular docking. Our results show that hydrophobic, electrostatic and hydrogen bond interactions are crucial in the inhibition of xanthine oxidase by coumarins.

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.