| It is one of the important tasks for current theoretic chemistry to predict and rationalize molecular reactivity qualitatively and quantitatively. Fukui’s frontier molecular orbital theory and Woodward-Hoffmann Rules through the conservation of the orbital symmetry, after the mature of wave function theory in the 1950 s, are two representative theories of reactivity. However, the correlation effect of electrons must be considered when one rationalizes the reactivity by the language of wave functions. Since the orbital itself is a single electron approximation, the more precise the calculation is, such as at the CCSD or MP2 level of calculations, the fuzzier concept for the orbital is, and therefore the description of reactivity with molecular orbital would no longer be applicable.The computational method of density functional theory（DFT） has become mature in the early 1990 s, whose basic idea is that the physicochemical properties of atoms, molecules and solid alike are functionals of the electron density. The energy of a system is the functional of the electron density as well. Because of this simplicity, to solve the wave function for a system with N particles, which should be a problem with 4N degrees of freedom, now it is a problem with 3 spacious degrees of freedom from the electron density, making the problem greatly simplified, with the calculation results at similar levels to the ab initio method. This change of computational methodology focus from ab initio theory to DFT is known as the second revolution of quantum chemistry. According to DFT, the electron density of a molecular system alone should suffice in determining all its properties in the ground state, including its structure and reactivity properties. Is it possible to recast the reactivity theory with DFT in a similar approach as in the wave function theory?In the 1980 s, Robert G. Parr of the University of North Carolina at Chapel Hill and coworkers had proved that chemical potential from the Euler equation in DFT is equivalent to the electronegativity concept in chemistry. Since then, a new research field has emerged and is still in continuous development, which is called conceptual density functional theory（CDFT）, also known as Chemical DFT or reactivity DFT. CDFT regards the chemical reactivity as the responses to the changes in the number of electrons,external potential, or others, which can be simulated by the perturbation expansion using a Taylor series of the total energy. CDFT conceptualizes the first and second order derivatives appearing in the Taylor expansion in terms of chemical insights. Accordingly, various reactivity descriptors such as electronegativity, hardness, softness, Fukui function, dual descriptor etc. have been introduced to relate with the traditional chemical terms and related chemical significances. This not only rediscovered Fukui’s FMO theory, but also explained the validity of the Woodward-Hoffmann rules without revoking orbitals and their symmetry.Although CDFT made great progresses in the last thirty years, there is still a distance between the theory and the reality of applications. CDFT, as chemical activity theory in chemistry, is far from becoming the mainstream. In the current literature there are a variety of ways and means to apply DFT to appreciate the molecular reactivity, both qualitatively and quantitatively. In the 1980 s, Bader and others categorized and characterized chemical bonds and weak interactions utilizing the density related quantities. Lately, a new effort, called density functional reactivity theory（DFRT）, had been undertaken to establish a chemical reactivity framework through directly using the electron density and its related quantities as descriptors. DFRT makes use of the functional of density to quantify the chemical reactivity, such as the charge distribution, the steric effect, electrophilicity, and nucleophilicity. Since the electron density alone should suffice in determining all properties in the ground state, including the chemical reactivity, in the strict sense, DFRT is indeed the chemical reactivity theory in DFT, aspart of DFT.In this dissertation, we call for new developments of this DFRT by expanding its scope, developing its connotations, and promoting its applications. We propose to incorporate the information theory into DFRT. Information theory developed by Shannon and others as a branch of applied mathematics, electrical engineering, and computer science is involved in the quantification of information, which is often a probability distribution function. Since the quantities in information theory employ only the electron density and its relative quantities, they are naturally correlated with DFT. The information theory quantities we examined in this dissertation to describe the molecular structure and reactivity are as follows:1 Shannon entropy, SS: measuring the spatial delocalization of the electron density.2 Fisher information, IF: describing the sharpness or concentration of the electron density distribution.3 Fisher information in terms of the Laplacian of the electron density, I’F:4 Ghosh-Berkowitz-Parr entropy, SGBP: transcribing the ground-state density functional theory into a local thermodynamics through the phase-space distribution function. SGBP can describe efficiently the chemical bond formation.5 Information gain（or Kullback-Leibler divergence, or relative entropy, or information divergence）, IG: a non-symmetric measure of the entropy difference between two probability distribution functions.6 Hirshfeld charge: under the first-order approximation for the information gain, and the atom in molecule partitioned by Hirshfeld stockholder partition, IG is almost the sum of Hirshfeld charge for atoms in molecule.7 Rényi entropy, Rn: when n approaches to 1, Rn reduces to the SS.8 Tsallis entropy, Tn: a generalization of the standard Boltzmann-Gibbs entropy.9 Onicescu information energy, En: the common term in Rn and Tn, a finer measure of dispersion distribution than that of SS.The above 9 information theoretic quantities are based on the electron density as the probability distribution. In DFRT, there is another probability distribution function, shape function, σ（r）. It is related to the electron density ρ（r） through the following relationship:r sr =)（）( r N. The 9 above information theoretic quantities can be redefined with the shape function.（Chapter 1.4 for details） In this dissertation, we systematically investigated these information theoretic quantities and apply them to describe the molecular structure and chemical reactivity properties.Shubin Liu proposed in 2007 to quantify the steric effect in the framework of DFRT with the Weizs?cker kinetic energy TW:This quantity is related to the Fisher information by only a factor of 8:which as can be seen is closely related to quantities in theinformation theoretic approach. This dissertation also utilized this quantity to understand and explain some chemical phenomena and properties.Next, we will briefly summarize the main points for each of chaptersincluded this dissertation, Chapters 2 to 10.I. Dissecting molecular descriptors into atomic contributions in density functional reactivity theory（Chapter 2）In this Chapter, a new basin-based integration algorithm has been implemented, whose reliability and effectiveness have been extensively examined. Applying this new analysis tool to a list of simple hydrocarbon systems and different bonding processes, including stretching, bending, and rotating, interesting patterns for the atomic and molecular values of these quantities have been observed. Based on what we havepresented in this work, following conclusions are in order.（i） For homonuclear diatomic molecules, as the two atoms get closer, both Fisher information and Shannon entropy decrease, at least until they reach the equilibrium distance.（ii） For heteronuclear diatomic molecules, the molecular value of both quantities decreases, so does the atomic value of the Fisher information, but for Shannon entropy, its atomic value for the two atoms behave differently, going to opposite directions.（iii） For angle bending in water molecule, the same trend is observed as the heteronuclear diatomic case, where all quantities decrease as the bended angle increases, except for the atomic value of Shannon entropy for oxygen atom.（iv） For the dihedral angle rotation in ethane, molecular values of the two quantities share the same trend, but their atomic counterparts behave differently, going opposite directions for both quantities. We now know that the extra steric repulsion in ethane in its eclipsed conformation originates from the carbon atoms, not from hydrogen atoms.（v） For hydrocarbon molecule studied in this work, as more methyl groups are attached to the central carbon atom, its atomic values of Shannon entropy and Fisher information both decrease, while those of the attached hydrogen atoms often increase.II. On the relationship among Ghosh-Berkowitz-Parr entropy, Shannon entropy and Fisher information（Chapter 3）In their seminal work thirty years ago, Ghosh, Berkowitz, and Parr reformulated the ground-state density-functional theory into a local version of thermodynamics, where a new concept, called Ghosh-Berkowitz-Parr entropy, was proposed. Employing the noninteracting kinetic energy density and Thomas-Fermi kinetic energy density, this entropy was formulated like the Sackur-Tetrode equation in classical thermodynamics. Not much has been known about its properties. In this contribution, we investigate its relationship with Shannon entropy and Fisher information from the numerical perspective. To that end, we have examined 36 neutral atoms and 42 molecular systems in this Chapter. We have considered both molecular and atomic contributions with Bader’s zero-flux criterion to partition atoms in molecules. Our results show that these quantities are closely correlated, and yet their correlations might be complicated since no universal relationship among them has been observed.III. Scaling Properties of Information-Theoretic Quantities in Density Functional Reactivity Theory（Chapter 4）Density functional reactivity theory（DFRT） employs the electron density and its related quantities to describe reactivity properties of a molecular system. Quantities from information theory such as Shannon entropy, Fisher information, and Ghosh-BerkowitzParr entropy are natural descriptors within the DFRT framework. They have been previously employed to quantify electrophilicity, nucleophilicity and the steric effect. In this Chapter, we examine their scaling properties with respect to the total number of electrons. To that end, we considered their representations in terms of both the electron density and the shape function for isolated atoms and neutral molecules. We also investigated their atomic behaviors in different molecules with three distinct partitioning schemes: Bader’s zero-flux, Becke’s fuzzy atom, and Hirshfeld’s stockholder partitioning.Strong linear relationships of these quantities as a function of the total electron population are reported for atoms, molecules, and atoms in molecules. These relationships reveal how these information-theoretic quantities depend on the molecular environment and the electron population. These trends also indicate how these quantities can be used to explore chemical reactivity for real chemical processes.IV. Rényi Entropy, Tsallis Entropy, and Onicescu Information Energy in Density Functional Reactivity Theory（Chapter 5）Quantities from the information-theoretic approach such as Shannon entropy and Fisher information are simple density functionals, which have shown great potentials as reactivity descriptors. In this Chapter, we introduce three closely related quantities, Rényi entropy, Tsallis entropy, and Onicescu information energy. We evaluated these quantities for a number of neutral atoms and molecules. Their scaling properties with respect to the total number of electrons and electronic energy have also been unveiled. In addition, using the second-order Onicescu information energy as an example, we examine its changing patterns as a function of the dihedral angle rotation for the ethane molecule at both molecular and atoms-in-molecules levels. The new quantities introduced in thisChapter should serve as additional reactivity descriptors to provide in-depth insights for the accurate prediction of structure and reactivity properties for molecular systems.V. Information Functional Theory: Electronic Properties as Functionals of Information for Atoms and Molecules（Chapter 6）How to accurately predict electronic properties of a Columbic system with the electron density obtained from experiments such as X-ray crystallography is still an unresolved problem. The information-theoretic approach recently developed in the framework of density functional reactivity theory is one of the efforts to address the issue. In this Chapter, using 27 atoms and 41 molecules as illustrative examples, we present a case study to demonstrate that one is able to satisfactorily describe such electronic properties as the total energy and its components with information-theoretic quantities like Shannon entropy, Fisher information, Ghosh-Berkowitz-Parr entropy, and Onicescu information energy. Closely related to the earlier attempt of expanding density functionals using simple homogeneous functionals, this work not only confirms Nagy’s proof that Shannon entropy alone should contain all the information needed to adequately describe an electronic system, but also provides a feasible pathway to map the relationship between the experimentally available electron density and various electronic properties for Columbic systems such as atoms and molecules. Extensions to other electronic properties are straightforward.VI. Information Conservation Principle Determines Electrophilicity, Nucleophilicity, and Regioselectivity（Chapter 7）Electrophilic and nucleophilic reactions are important chemical transformations involving charge acceptance and donation, so chemical intuition suggests that atomic charges should be a reliable descriptor to determine electrophilicity, nucleophilicity, and regioselectivity. Nevertheless, no such theoretical framework has been established as of yet. In the Chapter, we report that the Hirshfeld charge can be used for such purposes. We justify this usage by showing that it results from the Information Conservation Principle. This principle not only decides where electrophilic and nucleophilic attacks will preferably occur but also dictates the amount of the Hirshfeld charge distribution,which, as we have shown, remarkably strongly correlates with experimental scales of both electrophilicity and nucleophilicity.VII. Computational Study of Chemical Reactivity using Information-Theoretic Quantities from Density Functional Reactivity Theory for Electrophilic Aromatic Substitution Reactions（Chapter 8）The electrophilic aromatic substitution for nitration, halogenation, sulfonation, and acylation is a vastly important category of chemical transformation. Its reactivity and regioselectivity is predominantly determined by nucleophilicity of carbon atoms on the aromatic ring, which in return is immensely influenced by the group that is attached to the aromatic ring a prior. In this Chapter, taking advantage of recent developments in quantifying nucleophilicity（electrophilicity） with descriptors from the informationtheoretic approach in density functional reactivity theory, we examine the reactivity properties of this reaction system from three perspectives. These include scaling patterns of information-theoretic quantities such as Shannon entropy, Fisher information, GhoshBerkowitz-Parr entropy and information gain at both molecular and atomic levels, quantitative predictions of the barrier height with both Hirshfeld charge and information gain, and energetic decomposition analyses of the barrier height for the reactions. To that end, we focused, in this Chapter, on the identity reaction of the mono-substituted-benzene molecule reacting with hydrogen fluoride using boron trifluoride as the catalyst in the gas phase. We also considered 19 substituting groups, 9 of which are ortho/para directing and the other 9 meta directing, besides the case of R=-H. Similar scaling patterns for these information-theoretic quantities found for stable species elsewhere were disclosed for these reactions systems. We also unveiled novel scaling patterns for information gain at the atomic level. The barrier height of the reactions can reliably be predicted by using both the Hirshfeld charge and information gain at the regioselective carbon atom. The energy decomposition analysis ensued yields an unambiguous picture about the origin of the barrier height, where we showed that it is the electrostatic interaction that plays the dominant role, while the roles played by exchange-correlation and steric effects are minor but indispensable. Results obtained in this work should shed new light for better understanding the factors governing the reactivity for this class of reactions and assistingongoing efforts for the design of new and more efficient catalysts for such kind of transformations.VIII. Density Functional Reactivity Theory Study of SN2 Reactions from the Information-Theoretic Perspective（Chapter 9）As a continuation of our recent efforts to quantify chemical reactivity with quantities from the information-theoretic approach from the framework of density functional reactivity theory, the effectiveness of applying these quantities to quantify electrophilicity for the bimolecular nucleophilic substitution（SN2） reactions in both gas phase and water solvent is presented in this chapter. We examined a total of 21 self-exchange SN2 reactions for the compound with the general chemical formula of R1R2R3C-F, where R1, R2, and R3 represent substituting alkyl groups such as-H,-CH3,-C2H5,-C3H7, and-C4H9 in both gas and solvent phases. Our findings confirm that scaling properties for information-theoretic quantities found elsewhere are still valid. It has also been verified that the barrier height has the strongest correlation with the electrostatic interaction, but the contributions from the exchange-correlation and steric effects, though less significant, are indispensable. We additionally unveiled that the barrier height of these SN2 reactions can reliably be predicted not only by the Hirshfeld charge and information gain at the regioselective carbon atom, as have been previously reported by us for other systems, but also by other information-theoretic descriptors such as Shannon entropy, Fisher information, and Ghosh-Berkowitz-Parr entropy on the same atom. These new findings provide further insights for the better understanding of the factors impacting the chemical reactivity of this vastly important category of chemical transformations.To wrap up, in this dissertation, a set of information theoretic quantities are introduced into DFRT. It is proved both theoretically and numerically not only that there are close relationships among the quantities, but also that these quantities can describe qualitatively and quantitatively classical chemical reactivity, such as electrophilicity, nucleophilicity, regioselectivity, energy barriers for important chemical reactions such as electrophilic aromatic substitution reactions and SN2 reactions. These successes confirm the feasibility and general applicability of employing information theory to developDFRT. Since the electron density determines everything in the ground state, it is not surprising that they are intrinsically correlated. How to expand our understanding on other reactivity properties, such as molecular acidity or basicity, redox potentials, stereoselectivity, and diasteroselectivity is something we should address in the future. It is our cautiously optimistic belief that DFRT brings us to a brand new field with doors wide open in appreciating molecular reactivity with information theoretic approach. These works should fill in blanks in our knowledge about molecular reactivity and provide valuable contributions for the maturity and perfection of DFRT and DFT as a whole. |
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