Some Recent Advances in Density-Based Reactivity Theory.

Establishing a chemical reactivity theory in density functional theory (DFT) language has been our intense research interest in the past two decades, exemplified by the determination of steric effect and stereoselectivity, evaluation of electrophilicity and nucleophilicity, identification of strong and weak interactions, and formulation of cooperativity, frustration, and principle of chirality hierarchy. In this Featured Article, we first overview the four density-based frameworks in DFT to appreciate chemical understanding, including conceptual DFT, use of density associated quantities, information-theoretic approach, and orbital-free DFT, and then present a few recent advances of these frameworks as well as new applications from our studies. To that end, we will introduce the relationship among these frameworks, determining the entire spectrum of interactions with Pauli energy derivatives, performing topological analyses with information-theoretic quantities, and extending the density-based frameworks to excited states. Applications to examine physiochemical properties in external electric fields and to evaluate polarizability for proteins and crystals are discussed. A few possible directions for future development are followed, with the special emphasis on its merger with machine learning.

Mechanistic Study and Conceptual Chemical Reactivity Analysis of Hydroboration of Carbon Dioxide Catalyzed by a Manganese(I)-PNP-Pincer Complex.

Designing efficient and selective catalysts for carbon dioxide reduction is an intensive research area in the recent literature on homogeneous catalysis. In this work, we study the catalytic activity of a newly reported Mn(I)-PNP-pincer catalyst with an embedded aromatic ring. First, we systematically examine its capability to yield different products and highlight the importance of ligand aromaticity and steric effects on metal-ligand cooperativity. We then further conceptually probe its reactivity with descriptors from both conceptual density functional theory and an information-theoretic approach, thereby proposing a novel partitioning of the reaction coordinate into three relevant regions. Our results show that the reactivity in these different regions is governed by different properties such as steric effects, electrophilicity/nucleophilicity, or aromaticity. We anticipate that this methodology, with the analytical tools employed in this study, can be generalized and extended to other catalytic systems and find applications in designing better catalysts.

Impacts of external fields on aromaticity and acidity of benzoic acid: a density functional theory, conceptual density functional theory and information-theoretic approach study.

The impact of external fields on the molecular structure and reactivity properties has been of considerable interest in the recent literature. Benzoic acid as one of the most widely used compounds in medicinal and materials sciences is known for its dual propensity in aromaticity and acidity. In this work, we systematically investigate the impact of a uniform external electric field on these properties. We apply density functional theory, conceptual density functional theory, and an information-theoretic approach to appreciate the change pattern of aromaticity and acidity properties in external fields with different strengths. Our results show that they possess different change patterns under external fields, which can be satisfactorily rationalized by variations in reactivity descriptors and partial charges. The surprising yet novel results from this study should enrich the body of our knowledge about the impact of external fields for different kinds of electronic properties and provide guidance and foundation for future studies of this phenomenon in other molecular systems.

Directionality and additivity effects of molecular acidity and aromaticity for substituted benzoic acids under external electric fields.

Our recent study [M. Li et al.Phys. Chem. Chem. Phys., 2023, 25, 2595-2605] unveiled that the impact of an external electric field on molecular acidity and aromaticity for benzoic acid is directional, which can be understood using changes in frontier orbitals and partial charges. However, it is unclear if the effect will disappear when substituting groups are present and whether new patterns of changes will show up. In this work, as a continuation of our efforts to appreciate the impact of external electric fields on physiochemical properties, we find that the directionality effect is still in place for substituted benzoic acid derivatives and that there exists the additivity effect with respect to the number of substituent groups, regardless of the direction of the applied field and the type of substituting groups. We confirm the findings using electron-donating and electron-accepting groups with the electric field applied either parallelly or perpendicularly to the carboxyl group along the benzene ring. The directionality and additivity effects uncovered from this work should enrich the body of our knowledge about the impact of external electric fields on physiochemical properties and could be applicable to other systems and properties as well.

Topological analysis of information-theoretic quantities in density functional theory.

We have witnessed considerable research interest in the recent literature about the development and applications of quantities from the information-theoretic approach (ITA) in density functional theory. These ITA quantities are explicit density functionals, whose local distributions in real space are continuous and well-behaved. In this work, we further develop ITA by systematically analyzing the topological behavior of its four representative quantities, Shannon entropy, two forms of Fisher information, and relative Shannon entropy (also called information gain or Kullback-Leibler divergence). Our results from their topological analyses for 103 molecular systems provide new insights into bonding interactions and physiochemical properties, such as electrophilicity, nucleophilicity, acidity, and aromaticity. We also compare our results with those from the electron density, electron localization function, localized orbital locator, and Laplacian functions. Our results offer a new methodological approach and practical tool for applications that are especially promising for elucidating chemical bonding and reactivity propensity.

Simultaneous identification of strong and weak interactions with Pauli energy, Pauli potential, Pauli force, and Pauli charge.

Strong and weak interatomic interactions in chemical and biological systems are ubiquitous, yet how to identify them on a unified theoretical foundation is still not well established. Recently, we proposed employing Pauli energy-based indexes, such as strong covalent interaction and bonding and noncovalent interaction indexes, in the framework of density functional theory for the purpose. In this work, we extend our previous theoretical work by directly employing Pauli energy, Pauli potential, Pauli force, and Pauli charge to simultaneously identify both strong covalent bonding and weak noncovalent interactions. Our results from this work elucidate that using their signature isosurfaces, we can identify different types of interactions, either strong or weak, including single, double, triple, and quadruple covalent bonds, ionic bond, metallic bond, hydrogen bonding, and van der Waals interaction. We also discovered strong linear correlations between Pauli energy derived quantities and different covalent bond orders. These qualitative and quantitative results from our present study solidify the viewpoint that a unified approach to simultaneously identify both strong and weak interactions is possible. In our view, this work signifies one step forward towards the goal of establishing a density-based theory of chemical reactivity in density functional theory.

Toward Density-Based and Simultaneous Description of Chemical Bonding and Noncovalent Interactions with Pauli Energy.

Chemical bonds and noncovalent interactions are extraordinarily important concepts in chemistry and beyond. Using density-based quantities to describe them has a long history in the literature, yet none can satisfactorily describe the entire spectrum of interactions from strong chemical bonds to weak van der Waals forces. In this work, employing Pauli energy as the theoretical foundation, we fill in that knowledge gap. Our results show that the newly established density-based index can describe single and multiple covalent bonds, ionic bonds, metallic bonds, and different kinds of noncovalent interactions, all with unique and readily identifiable signature shapes. Two new descriptors, BNI (bonding and noncovalent interaction) index and USI (ultra-strong interaction) index, have been introduced in this work. Together with NCI (noncovalent interaction) and SCI (strong covalent interaction) indexes already available in the literature, a density-based description of both chemical bonds and noncovalent interactions is accomplished.

Density functional theory studies of boron clusters with exotic properties in bonding, aromaticity and reactivity.

Atomic clusters are unique in many perspectives because of their size and structure features and are continuously being applied for different purposes. To unveil their unconventional properties, in this work, using neutral tetraboron clusters as illustrative examples, we study their exotic behaviors in bonding, aromaticity, and reactivity. We show that both double and triple bonds can be formed, ring current patterns can be totally different, and both electrophilic and nucleophilic reactivities can coexist simultaneously. These features are often in contrast with our conventional chemical wisdom and could enrich the possibility for their potential applications. The methodologies employed in this work can be readily applied to other systems. Our studies should help us better appreciate atomic clusters with many atypical properties and henceforth yield novel applications.

Quantifying Frustrations for Molecular Complexes with Noncovalent Interactions.

Molecular systems bound together through noncovalent interactions are involved in a lot of life-essential processes such as molecular recognition, signal transduction, and allosteric regulation. While cooperation as an important effect discovered in these systems focuses on the behavior of system's entirety, we need also examine the behavior of individual parts. In this work, using the distortion energy as the descriptor, we quantify frustration as the energetic loss of individual parts due to the formation of nonadditive molecular complexes. The applicability of our approach has been illustrated by a few simple clusters. Our results show that the frustration effect is smaller than the cooperation effect, but same as cooperativity, it can be both positive and negative. The ultimate benefit of a system made of multiple parts is dictated by the balance between the cooperative behavior among parts and the sacrifice from its individuals. This conflicting yet complementary conceptual pair of cooperation and frustration provides us with a different perspective from the systems' viewpoint for molecular complexes. This new angle of appreciating molecular complexes can be applied in conformational changes, enzymatic catalysis, and many more.

Quantifications and Applications of Relative Fisher Information in Density Functional Theory.

Though density functional theory is widely accepted as one of the most successful developments in theoretical chemistry in the past few decades, the knowledge of how to apply this new electronic structure theory, to help us better understand chemical processes and transformations, is still an unaccomplished task. The information-theoretic approach is emerging as a viable option for that purpose in the recent literature, providing new insights about steric effect, cooperativity, electrophilicity, nucleophilicity, stereoselectivity, homochirality, etc. In this work, based on the result from a recent paper by one of us [ J. Chem. Phys, 2019, 151, 141103], we present two quantifications of the relative Fisher information and discuss their physiochemical properties and possible applications. To that end, their analytical properties have been elucidated. They have also been applied to six categories of systems to illustrate their applicability. A better descriptor to quantify the single bond rotation barrier has been obtained. The relative Fisher information can also simultaneously determine electrophilicity and nucleophilicity, and effectively describe helical structures with different homochiral and heterochiral propensities. As integral parts of the information-theoretic approach, these newly introduced quantities will provide us with more analytical tools toward the long-term goal of crafting a chemical reactivity theory in the density-based language.

Density Functional Theory and Information-Theoretic Approach Study on the Origin of Homochirality in Helical Structures.

Homochirality of macromolecules such as proteins and DNA is one of the most striking features in nature; yet, there is still no convincing theory to explain its origin. In a recent work by one of the present authors (J. Phys. Chem. Lett. 2020, 11, 8690-8696), a general proposal from the viewpoint of thermodynamics has been put forward. It proposes that it is the handedness of helices ubiquitous in biological macromolecules that plays the decisive role. It also unveiled that there exist strong cooperativity effects dominated by favorable electrostatic interactions in the homochiral conformer. In this work, making use of analytical tools, we recently developed a density functional theory and an information-theoretic approach and through four sets of helical structures we designed for the present study, we examine these systems to provide new insights about these properties. We found that the 310-helix and the α-helix are markedly different in cooperativity from the viewpoint of both the total energy and its three components. The electrostatic dominance of homochiral species is manifested by both the electron charge distribution and information gain. At the atomic level, different elements behave significantly differently because they play different roles in the systems. Our results from this work validate that these analytical tools can be applied to homochiral systems, which can be further extended to others with potential interest in asymmetric synthesis and macromolecular assembly where the Principle of Homochirality Hierarchy comes into play.

Switching between Hückel and Möbius aromaticity: a density functional theory and information-theoretic approach study.

Benziporphyrins are versatile macrocycles exhibiting aromaticity switching behaviors. The existence of both Hückel and Möbius (anti)aromaticity has been reported in these systems, whose validity is respectively governed by the [4n + 2] and [4n] π-electron rule on the macrocyclic pathway. Despite the experimental evidence on the floppiness of benziporphyrins, the switching mechanism between Hückel and Möbius structures is still not clear, as well as the factors influencing the stability of the different π-conjugation topologies. For these reasons, we performed a systematic study on A,D-di-p-benzihexaphyrins(1.1.1.1.1.1) with two redox states corresponding to [28] and [30] π-electron conjugation pathways. Whereas benzi[28]hexaphyrin obeys Möbius aromaticity, benzi[30]hexaphyrin follows Hückel aromaticity. The dynamic interconversion between Möbius and Hückel aromaticity is investigated through the rotation of a phenylene ring, which acts as the topology selector. Further analyses of the energy profiles using energy decomposition and information-theoretic approaches provide new insights into conformational stability, aromaticity and antiaromaticity for these species. Strong and opposite cross correlations between aromaticity indexes and information-theoretic quantities were found for the two macrocyclic systems with opposite global aromaticity and antiaromaticity behaviors. These results indicate that Hückel and Möbius aromaticity and antiaromaticity, though qualitatively different, are closely related and can be interchanged, and information-theoretic quantities provide a novel understanding about their relevance. Our present results should provide in-depth insights to appreciate the nature and origin about Möbius (anti)aromaticity and its close relationship with Hückel (anti)aromaticity.

Global and local aromaticity of acenes from the information-theoretic approach in density functional reactivity theory.

In this work, we report a systematic study on the global and local aromaticity of acenes using a series of model structures from 2-acene to 11-acene. A recently developed ansatz, an information-theoretic approach coached into density functional reactivity theory has been employed, which essentially provides different density functionals characterizing the molecular electron density distribution. Based on the correlation analysis of six conventional aromaticity indices with eight information-theoretic quantities, we examined the aromaticity of acenes from both global and local perspectives. From the global aromaticity viewpoint, our results suggest that different descriptors based on various physicochemical properties are intrinsically dependent. A novel laminated feature ruling local aromaticity of acenes has been unveiled, from which we found that the distance from the terminal rings plays the critical role. Based on the shape of the correlation plots between the conventional aromaticity indices and information-theoretic quantities, the latter could be separated into three subgroups. The seemingly contradictory results from global and local aromaticity perspectives not only present us the uniqueness of the acene systems but all demonstrate the effectiveness of the information-theoretic approach from density functional reactivity theory. Besides strengthening the validity of a series of new aromaticity descriptors, our results should lead to more clear insights into the chemical significance of the information-theoretic quantities.

Is It Possible To Determine Oxidation States for Atoms in Molecules Using Density-Based Quantities? An Information-Theoretic Approach and Conceptual Density Functional Theory Study.

The oxidation state, also called oxidation number, of atoms in molecules is a fundamental chemical concept. It is defined as the charge of an atom in a molecule after the ionic approximation of its heteronuclear bonds is applied. Even though for simple molecules the assignment of oxidation states is straightforward, redundancy and ambiguity do exist for others. In this work, we present a density-based framework to determine the oxidation state using the quantities from the information-theoretic approach. As a proof of concept, we examined six elements for a total of 49 molecules. Strong linear correlations were obtained with Shannon entropy, Ghosh-Berkowitz-Parr entropy, information gain, relative Rényi entropy of orders 2 and 3, and Hirshfeld charge. We also discovered that the crystal radius of elements plays the key role in rationalizing the system dependent nature of these strong linear correlations. The validity and effectiveness of our results were demonstrated by the examples of molecules containing elements with two or more oxidation states. Our results should be applicable to more complicated systems in assigning different oxidation states.

Using Pauli energy to appraise the quality of approximate semilocal non-interacting kinetic energy density functionals.

It is well-known that the kinetic energy density (KED) functional is the most difficult to approximate in density functional theory (DFT), yet to take full advantage of DFT with its density-based descriptive capability of molecular properties, an accurate account of KED is a must. To have a better idea of how an approximate KED formula behaves and where we should focus in the future development of better approximate KEDs, in this work we propose to employ the Pauli energy to assess their quality. We tested the performance of a total of 22 approximate semilocal noninteracting KED functionals from the literature for 18 neutral atoms and 20 small molecules. We found that generalized gradient approximation formulas of the KED functional can often reasonably accurately predict the total kinetic energy value for atoms and molecules but failed miserably to forecast the integrated values for Pauli energy related properties. The reason behind this is that presently available approximate KED functionals are unable to accurately account for the kinetic energy distribution in the medium range away from nuclei, where the Pauli energy plays a crucial role. Our results strongly suggest that the key information missing in approximate KED functionals comes from the medium regions, not nuclear cusps nor asymptotic areas, and the Pauli energy is a reliable measure of the quality of approximate KED functionals. Future efforts in developing better KED approximations should be invested in the regions of molecules where chemical bonds are formed in order to accurately account for the Pauli energy.

Molecular acidity: An accurate description with information-theoretic approach in density functional reactivity theory.

Molecular acidity is one of the important physiochemical properties of a molecular system, yet its accurate calculation and prediction are still an unresolved problem in the literature. In this work, we propose to make use of the quantities from the information-theoretic (IT) approach in density functional reactivity theory and provide an accurate description of molecular acidity from a completely new perspective. To illustrate our point, five different categories of acidic series, singly and doubly substituted benzoic acids, singly substituted benzenesulfinic acids, benzeneseleninic acids, phenols, and alkyl carboxylic acids, have been thoroughly examined. We show that using IT quantities such as Shannon entropy, Fisher information, Ghosh-Berkowitz-Parr entropy, information gain, Onicescu information energy, and relative Rényi entropy, one is able to simultaneously predict experimental pKa values of these different categories of compounds. Because of the universality of the quantities employed in this work, which are all density dependent, our approach should be general and be applicable to other systems as well. © 2017 Wiley Periodicals, Inc.

Steric charge.

The steric effect is one of the most widely used chemical concepts in chemistry, yet a generally accepted implementation of its formulation and quantification from a theoretical viewpoint, if even possible, is still an unaccomplished task. Based on the energetics viewpoint of our earlier proposal using the Weizsäcker kinetic energy as a quantification of the energy contribution from the steric effect as well as our recent work using steric force as a reliable local descriptor to account for the origin of the stereoselective propensity in chemical reactions, in this work, we systematically examine the local behavior and general applicability of another closely related quantity, steric charge. To this end, its physical origin and physiochemical properties are formalized and highlighted, and are shown with numerical illustrations and verifications. To showcase its usefulness in appreciating chemical reactivity, we present three case studies of steric charge in this work. In the case of ethane flexible rotation, we found that the eclipsed conformer is more sterically charged than the staggered conformer, leading to the reduced stability and higher energy of the former. For the case of SN2 reactions, a remarkable linear correlation has been obtained between the steric charge on the central atom in the transition state and the reaction barrier height of SN2 reactions, in good agreement with experimental findings. In the case of stereoselectivity properties for the nucleophilic addition of carbonyl compounds, we found that steric charge is equally applicable as steric force to justify the stereoselective origin for the nucleophilic attack to the carbonyl carbon atom with different substituent groups. Put together, our results from the present study should pave the way towards the general use of steric charge, together with steric energy and steric force, as an insightful global and local descriptor to appreciate and quantify chirality and stereoselectivity related phenomena in chemical processes and transformations.

Quantification and origin of cooperativity: insights from density functional reactivity theory.

Cooperativity is a widely used chemical concept whose existence is ubiquitous in chemical and biological systems but whose quantification is still controversial and origin much less appreciated. In this work, using the interaction energy of a molecular system, which is composed of multiple copies of a building block, we propose a quantitative measurement to evaluate the cooperativity effect. This quantification approach is then applied to six molecular systems, i.e., water cluster, argon cluster, protonated water cluster, zinc atom cluster, water cluster on top of a graphene sheet, and alpha helix of glycine amino acids, each with up to 20 copies of the building block. Cooperativity is seen in all these systems. Both positive and negative cooperativity effects are observed. Employing the two energy partition schemes in density functional theory and the information-theoretic quantities such as Shannon entropy, Fisher information, information gain, etc., we then examine the origin of the cooperativity effect for these systems. Strong linear correlations between the cooperativity measure and some of these theoretical quantities have been unveiled. With these correlations, we are able to quantitatively account for their origin of cooperativity. Our results show that the interactions governing the existence and validity of the cooperativity effect are complicated. An opposite mechanism in enthalpy-entropy compensation for positive and negative cooperativity has been unveiled. These results should provide new insights and understandings from a different viewpoint about the nature and origin of cooperativity to appreciate this vastly important chemical concept.

Identifying Strong Covalent Interactions with Pauli Energy.

As one of the most widely used chemical concepts whose origin can be traced back to Lewis theory of bonding a century ago, a covalent bond involves sharing one or more pairs of electrons between atoms. A strong covalent interaction (SCI) is such a covalent bond that two or more electron pairs are shared, yielding a double, triple, quadruple, or even higher bond order. Despite its ubiquity and usefulness, a robust and generally applicable approach to accurately identify strong covalent interactions and determine their bond orders is still lacking. In this work, an SCI index is proposed from density functional theory using the Pauli energy, which is the contribution of the Pauli exclusion principle to the kinetic energy. Illustrative examples from organic, inorganic, and organometallic systems were provided. Its close relationship with the electron localization function (ELF) was elucidated. Both ELF and SCI generate similar results. Two complexes with a quintuple metal-metal bond have been confirmed. A stronger than quintuple bond has been showcased. This work should provide a robust approach to determine bond orders for strong covalent interactions in complex systems, pinpoint the physiochemical origin of strong covalent interactions, and rationalize the usefulness of both SCI and ELF. These tools should be able to be applied to other systems in different fields to effectively appreciate strong covalent interactions.

Theoretical Study of pKa Values for Trivalent Rare-Earth Metal Cations in Aqueous Solution.

Molecular acidity of trivalent rare-earth metal cations in aqueous solution is an important factor dedicated to the efficiency of their extraction and separation processes. In this work, the aqueous acidity of these metal ions has been quantitatively investigated using a few theoretical approaches. Our computational results expressed in terms of pKa values agree well with the tetrad effect of trivalent rare-earth ions extensively reported in the extraction and separation of these elements. Strong linear relationships have been observed between the acidity and quantum electronic descriptors such as the molecular electrostatic potential on the acidic nucleus and the sum of the valence natural atomic orbitals energies of the dissociating proton. Making use of the predicted pKa values, we have also predicted the major ionic forms of these species in the aqueous environment with different pH values, which can be employed to rationalize the behavior difference of different rare-earth metal cations during the extraction process. Our present results should provide needed insights not only for the qualitatively understanding about the extraction and separation between yttrium and lanthanide elements but also for the prediction of novel and more efficient rare-earth metal extractants in the future.