密度泛函活性理論中的信息論方法

劉述斌

（Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, USA）

密度泛函活性理論中的信息論方法

劉述斌*

（Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, USA）

密度泛函活性理論（DFRT）運用簡單的密度泛函探討和定量化分子的反應(yīng)活性，是近來發(fā)展起來的一個關(guān)于分子活性理論的新方法。在新近的文獻中，這樣的簡單密度泛函的例子包括香農(nóng)熵，費舍爾信息以及其它來自信息論中的密度泛函。本文綜述了DFRT信息論方法的原理，包括物理信息極小原理、最小信息增益原理和信息守恒原理?？偨Y(jié)了DFRT信息論方法在電子密度、形態(tài)密度和分子中的原子三種表述下的理論框架。此外，還介紹了運用信息論方法在定量描述空間位阻效應(yīng)、親電性、親核性和區(qū)域選擇性中的突出應(yīng)用，以及對親電芳香取代反應(yīng)的鄰對間位取代效應(yīng)的起源和本質(zhì)提供的一個全新詮釋。最后簡要地展望了該領(lǐng)域的幾個可能的未來發(fā)展方向。

香農(nóng)熵；費舍爾信息；密度泛函理論反應(yīng)；立體效應(yīng)；親電性；親核性；區(qū)域選擇性；鄰對間位取代效應(yīng)

1 Introduction

Density functional theory （DFT）1as an alternative approach to solve the Schrodinger equation for the calculation of electronic structures and properties for molecules and solids has enjoyed tremendous success in the past few decades. It is theoretically rigorous, conceptually attractive, and computationally efficient. Its complete discard of orbitals in the mind of its original inventors is even more enthralling. This is not only becauseof the complex nature of the single-particle and total wave functions but also the fact that the concept of atomic and molecular orbitals introduced as artifacts is the most crude approximation in the wave function theory. Although Kohn-Sham orbitals were later employed for the practical implementation of DFT, the rigor and effort of developing such a theoretical framework without the use of orbitals have been lingering over people's minds for quite some time, and are still driving a number of people moving towards that direction. Such a practice in DFT is called the orbital-free DFT （OF-DFT） in the literature2–4.

Closely related to this endeavor is the application of DFT to chemical reactivity theory in chemistry, where the mainstream for over several decades has been to make use of molecular orbitals and their related properties to predict and rationalize molecular reactivity5–7. Examples include Fukui's frontier molecular orbital （FMO）8theory and the Woodward-Hoffmann's rules through the conservation of the orbital symmetry9. Nevertheless, 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 properties1. Is it possible to recast the reactivity theory with molecular orbitals in the language of the electron density without scarifying the rigor and generality More importantly, is it possible to expand the scope of our present orbital-based understanding about molecular reactivity with the introduction of descriptors from the electron density and its associated quantities

The answer to both of the above questions is yes. At least, we believe they should be. Pioneered by its inventor, Robert G. Parr of the University of North Carolina at Chapel Hill, the first such a theoretical framework is the conceptual density functional theory （CDFT）5–7, also called chemical DFT, reactivity DFT, among others. It regards the chemical reactivity of a molecular system 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 interprets these processes by conceptualizing the first and second order derivatives appearing in the Taylor expansion in terms of chemical insights. For that, various reactivity descriptors such as electronegativity, hardness/softness, Fukui function, dual descriptor, etc.5–7, have been introduced and linked to relevant chemical meanings in conventional chemical language. They not only rediscovered Fukui's FMO theory1,5–7, but also explained the validity of the Woodward-Hoffmann rules without revoking orbitals and their symmetry.10

The approach of employing the first and second order derivatives in explaining molecular reactivity, however, does not really grasp the essence of DFT. That is, it does not directly employing electron density and its related quantities for the purpose. In retrospect, density related quantities have been extensively utilized by Bader11and others12in categorizing and characterizing chemical bonds and weak interactions. Nevertheless, no systematic effort is seen until recently. There is a recent effort in the literature, trying to establish a chemical reactivity framework using functionals based purely on the electron density and its related quantities. Such a practice is now called DFRT（density functional reactivity theory）, to differentiate from the original work by Parr and others on CDFT （conceptual density functional theory）. To the best knowledge of the present author, the first work along this line is the seminal paper in 2000 by Nalewajski and Parr13, who unveiled that minimizing the information gain （also called Kullback-Leibler divergence） leads to the “stockholder partition” of the electron density, i.e., the so-called Hirshfeld charge14. In 2007, we proposed to use the Weizs?cker kinetic energy functional as a quantification for the steric effect15. Numerous applications have followed since then. Very recently, based on the work by Nalewajski and Parr, reliable predictions of electrophilicity, nucleophilicity and regioselectivity using the Hirshfeld charge and information gain have been reported16–22. These novel developments share a couple of common points. On the one hand, they all make use of density functionals, not just density, its gradient or Laplacian used in Bader's AIM （atoms in molecules） theory11as reactivity descriptors. These reactivity related density functionals, on the other hand, all fall into the same umbrella of the informationtheoretic approach, which coincidently has seen tremendous developments in recent years from the physics viewpoint23–28.

In this review, we outline principles of the information-theoretic approach as well as its recent developments from the chemical reactivity perspective. This is an emerging field so it is impossible to be comprehensive in the literature review. Rather, we focus more on the recent progresses by the present author and coworkers, and, for the benefit of readers and potential researchers, provide personal outlooks for its possible future development directions.

2 Conceptual DFT and DFRT

In Conceptual DFT1,5–7, the starting point is the conjecture that any ground-state property of an electronic system such as the total electronic energy, E, can be sufficiently described by two variables, the total electron number N and the external potential v（r）, E ≡ E［N, ν（r）］. When an attacking agent comes across leading to the changes in both N and v（r）, ΔN and Δv（r）, the subsequent change in the total energy, ΔE, can be approximated by the Taylor expansion up to the second order of the following formula1,5–7

where （?E/?N）vand ［δE/δv（r）］Nare the first-order partial derivatives of the total energy with respect to N and v（r） with v（r）and N fixed, respectively, （?2E/?N2）vand ［δ2E/δv2（r）］Nare thecorresponding second-order terms, and ［?/?N（δE/δv）N］vis the second-order cross term.

What conceptual DFT has accomplished over the last few decades was to make sense of these derivatives in terms of chemical concepts. For instance, （?E/?N）vis nothing but the widely used concept of electronegativity in chemistry. One of the second-order term, （?2E/?N2）vwas defined as global hardness, η, because it is closely related to the principle of hard and soft acid and base （HSAB）. More importantly, the second-order cross term, ［?/?N（δE/δv）N］vwas named as the Fukui function, f（r）, by Parr and Yang29, because it recovers the frontier orbital results from Fukui's FMO theory as a special case. A few reviews and treatises on the subject are available in the literature1,5–7. Readers are referred to those references for more details. Recent developments include examinations of third-order terms of the above Tayler series. The most prominent work is the dual descriptor by Morell, Grand, and Toro-Labbé in 200530,

This quantity can be used to explain the validity of the Woodward-Hoffmann rules, which are often believed that molecular orbitals and their symmetry play the essential role and thus densitybased indices are unable to interpret their origin. The recent work using the dual descriptor clearly demonstrated that orbital and symmetry are not mandatory to interpret the validity of these rules10. We recently applied these ideas in molecular acidity/basicity study31–35and in predicting mechanisms for protoncoupled electron transfer reactions36,37.

While CDFT has been with us for about thirty years with lots of insights provided, establishments accomplished, and progresses and developments still continuing1,5–7, there are apparent discrepancies between theory and reality. It is still far from being accepted by the chemical community as the mainstream of chemical reactivity theory. There are ongoing controversies and debates about the general validity of the principles from this theory such as maximum hardness principle and minimum electrophilicity principle38–42. Its general applicability and prediction robustness are often questioned.

There is a recent effort in the literature in applying DFT to examine and rationalize molecular reactivity in a vastly different manner. In this case, instead of employing the energy derivatives introduced above, density functionals themselves are directly used to quantify reactivity properties for molecules such as the charge distribution13,43–45, steric effect15,46–58, and molecular electrophilicity/ nucleophilicity16–22. This effort is similar to the conventional undertaking in DFT, where one tries to find more accurate and better behaved density functionals to approximate energy components such as the exchange-correlation energy and non-interacting kinetic energy. What the new approach is trying to establish is to use density functionals to provide new insights for reactivity properties and quantify them if possible. This very idea of employing density functionals to quantify molecular reactivity is indeed akin to and consistent with the original spirit of the DFT inventors. That is, since the electron density alone should suffice in determining all properties in the ground state, any property of a molecule could be expressed as a functional of the electron density. Since this approach deviate so much from the original idea of CDFT, we call this new approach DFRT, abbreviated from density functional reactivity theory. To be more specific, DFRT stands for a chemical reactivity theory in DFT where molecular reactivity properties are appreciated and quantified by density functionals.

3 Information-theoretic approach

3.1 Electron density representation

Information theory developed by Shannon59and 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 electron density is a continuous probability distribution function, as early as in the year of 1980, information theory has been applied to DFT to study atoms and molecules60–62. We call this particular category of work the information-theoretic approach.

A key measure of information is entropy, quantifying the uncertainty involved in predicting the value of the distribution function. Shannon entropy is the first such a measure widely used in the literature, which reads59

where sS（r） is the Shannon entropy density and ρ（r） is the total electron density, satisfying the following condition in relation to the total number of electrons, N, of the system,

Shannon entropy measures the spatial delocalization of the electronic density. The second important measure in information theory is the Fisher information, IF, defined as follows63

which is a gauge of the sharpness or concentration of the electron density distribution. In Eq.（5）, iF（r） is the Fisher information density and ρ（r） is the density gradient. Earlier, we have proved that there is an equivalent expression for the Fisher information in terms of the Laplacian of the electron density2ρ47

Equations （5） and （6） are equal in the sense that they can be derived by partial integration from one to the other, and that the two integrals have the same value. As have been shown, local behaviors of the two integrals, iF（r） and i'F（r）, are markedly different. More importantly, we have proved the existence of the following rigorous relationship among the three quantities, sS（r）, iF（r）, and i'F（r）47,

whose validity has subsequently been verified by numerical results.

The third quantity in the same spirit is the Ghosh-Berkowitz-Parr （GBP） entropy64,

where t（r, ρ） is the kinetic energy density, which is related to the total kinetic energy TSvia

and tTF（r； ρ） is the Thomas-Fermi kinetic energy density,

with k as the Boltzmann constant （set to be unity for convenience in this work

The GBP entropy originates from the effort to transcribe the ground-state density functional theory into a local thermodynamics through the phase-space distribution function f（r, p）, which is a function of both the electron position r and momentum p as its two basic variables. The conditions of such a recast of DFT into thermodynamics are that the phase-space distribution function is associated with the ground state electron density ρ（r） and kinetic energy density t（r； ρ） through the following relationships

The specific form of the local kinetic energy t（r； ρ） used is the following,

Very recently, three information-theoretic quantities, Rényi entropy, Tsallis entropy, and Onicescu information energy, are introduced as new reactivity descriptors in DFRT65. The Rényi entropy of order n, where n ≥ 0 and n ≠ 1, is defined as66

When n approaches to 1, the Rényi entropy, Eq.（14）, reduces to the Shannon entropy, Eq.（3）. The Tsallis entropy of order n is defined as follows,67

It is a generalization of the standard Boltzmann-Gibbs entropy. The common term in Eqs.（14） and （15） is the integral of the nth power of the electron density, which is called the Onicescu information energy of order n68:

Onicescu introduced this quantity in an attempt to define a finer measure of dispersion distribution than that of Shannon entropy in information theory.

Closely related to the concept of entropy in information theory is the relative entropy, which is a non-symmetric measure of the entropy difference between two probability distribution functions. Well known examples in the literature are the relative Shannon entropy, also called information gain, Kullback-Leibler divergence, or information divergence, defined by69

and the relative Rényi entropy of order n24,

3.2 Shape function representation

Information-theoretic quantities defined in Eqs.（3）–（18） employ the electron density as the probability distribution function. There is another distribution function in DFRT, the shape function σ（r）1,19,70–72, which is related to the electron density ρ（r）and the total number of electrons N through the following relationship,

with the following normalization condition

Information-theoretic quantities defined in Eqs.（3）–（18） can similarly be redefined with the shape function, yielding19,65

Because of Eq.（19）, quantities in these two representations are correlated, except for the GBP entropy, which does not have ananalytical expression between the two representations. As can be readily proved, we have19,65

These rigorous relationships between the two representations of information-theoretic quantities enable us to obtain them interchangeably from one representation to the other.

3.3 Atoms-in-molecules representation

Another important aspect of the information-theoretic approach is to reevaluate the above quantities from the perspective of atoms in molecules19,65,73,74. To consider atomic contributions of an information-theoretic quantity in a molecular system, three approaches are available to perform atom partitions in molecules. They are Becke's fuzzy atom approach75, Bader's zero-flux AIM approach11, and Hirshfeld's stockholder approach14. The total electron population N of the system is the summation of electron density in each atomic contribution, NA,

and the quantities in Eqs.（3）–（18） can be rewritten as73,74where ρAis the electron density on atom （or group） A in a molecule, whose total molecular electron density is ρ（ris the counterpart of atom （or group） A in the reference state, which can be neutral atom, or ion, or group, etc., and ΩAis the atomic basin of atom A in the molecule. The counterpart in terms of the shape function can be derived similarly.

All these formulas shown above in the information-theoretic approach as well as the three partition schemes for atoms in molecules have been successfully implemented in the Multiwfn package76, whose reliability and applicability have extensively been tested and confirmed.

All the quantities introduced above from the informationtheoretic approach share the same characteristics: they are all simple density functionals！ Different from the density functionals for energy components such as the noninteracting kinetic energy TSand the exchange-correlation energy Exc, where many different levels of approximations have been introduced1, the form of these density functionals is not only analytical but also much simpler. Still, they all have their sound origin from information theory. Do they provide any insight from the perspective of chemical reactivity Also, except the known relationship in Eq.（7）, are they correlated in any manner

We will address these and other questions in the following sections.

4 Principle of extreme physical information

The Euler equation of DFT for a real interacting systemcan be recast in the model, non-interacting system and becomes1

where TS, vKS, and μ are the noninteracting kinetic energy, Kohn-Sham potential, and chemical potential arising from the constraint that the total number of electrons must be fixed to N, Eq.（4）. Furthermore, the noninteracting kinetic energy can be divided into two pieces,

where TWis the Weizs?cker kinetic energy77

and TPis the Pauli energy due to the Pauli Exclusion Principle78,79. With Eq.（45）, the Euler equation becomes

where vPis the Pauli potential defined as the functional of derivative of TPwith respect to the electron density. With the help that

Eq.（47） reads

Equation （49） is similar to the Kohn-Sham equation, but it doesnot require the introduction of any orbitals. It is the working formula for the orbital-free DFT （OF DFT） approach in DFT2–4.

This OF-DFT equation can, however, be derived from the principle of extreme physical information, which states that the“physical information” K of a system should be extreme:26

where IFis the Fisher information, J is the set of constraints that are to be imposed, and K is the difference between IFand J, called the “physical information”. According to the literature80,81, these constraints are three-fold. （a） The total wave function of the system is antisymmetric. As shown by the Nagy and others80,81, this condition generates the Pauli potential vP（r）.（b） The total density of the noninteracting system should be the same as that of the interacting system. Same as the Kohn-Sham equation, this constraints yields the Kohn-Sham potential vKS（r）.（c） Lastly, the total electron density is normalized to N, Eq.（4）, which results in a Lagrange multiplier μ, the chemical potential of the system.

Put together, that is to say, minimizing the Fisher information subject to the above three constraints should give us the same Euler equation in Eq.（49） as what we have obtained from the OF-DFT approach. What the principle of extreme physical information tells us is that one does not have to use the electronic energy when deriving Euler equation in DFT. It could be substituted by information. In other words, information alone is adequate in understanding all properties of the system.

5 Scaling properties of informationtheoretic quantities

Quantities from the information-theoretic approach are simple density functionals. One of the first properties of these functionals one wishes to examine is their scaling properties82–84. Scaling properties of energy density functionals in DFT have been extensively studied in the literature85,86. There are two distinct categories of the scaling, one via scaling the coordinate variable and the other through scaling the electron density. One can also scale both coordinate and density variables at the same time, leading to the so-called hybrid scaling82–86.

Homogeneity of a functional Q［ρ］ of degree m in （or “with respect to” ） coordinate scaling is defined as

where

Equation （52） is imposed as the condition of the coordinate scaling because it will always keep the scaled densitynormalized to N. It has been previously proved that for a wellbehaved functional, Eq.（51） is equivalent to

Examples of homogeneous functionals in coordinate scaling include the kinetic and exchange energy functionals. There are numerous studies in the literature to express the correlation energy functionals in terms of homogeneous functionals in coordinate scaling.

A functional Q［ρ］ is homogeneous of degree n in density scaling if it satisfies the following condition:

Another definition of the homogeneity with respect to density scaling is as follows,

Examples of homogeneous functionals in density scaling include the Thomas-Fermi formula for the kinetic energy and the Dirac formula for the exchange energy. The density scaling is related to a quantity's extensive/intensive nature. For example, Fisher information is homogeneous of degree one in density scaling because if one scales the density by ζ times, IFis amplified by ζ times as well, that is, IF［ζρ］ = ζIF［ρ］. However, for the Shannon entropy, no such relationship can be obtained when the density is scaled by ζ times. Instead, we get

Therefore, Shannon entropy is not a homogeneous functional in density scaling. This is also true for the GBP entropy functional, which is not known to satisfy any homogeneity relations in both density and coordinate scaling. For the alternative definition of the Fisher information defined in Eq.（6）, we have the similar density scaling property as the Shannon entropy. So, even though the two forms of Fisher information yield the same molecular value, not only their local behaviors are vastly different, but their scaling properties are also well. Similarly, we can prove that Rényi entropy and Tsallis entropy are not homogeneous functionals in density scaling, but the Onicescu information energy of order n is homogeneous of degree n in density scaling. For counterpart quantities in the shape function representation, similar conclusions can be obtained.

While these analyses of scaling properties are helpful in understanding analytical properties of these quantities, it is not straightforward to verify their validity with numerical results. To overcome this problem, in our recent work19, we proposed to approach it in a little different manner. Instead of scaling the electron density for the same molecule, we consider the scaling property of these quantities in different molecular systems with respect to the total electron population. At the meanwhile, we also examine the scaling property at the atoms-in-molecules level using three different ways of atomic partition scheme from the literature, Becke's fuzzy atom approach, Bader's zeroflux AIM approach, and Hirshfeld's stockholder approach. As an example, shown in Table 1 are the numerical values for a few quantities for the first and second row neutral atoms, with the correlation coefficient with respect to the total number of electrons shown in the last row. As can be seen from Table 1, linear dependences of Iσare remarkable, indicating that there exist strong scaling properties for these quantities19. Table 2 displays the atomic values for Rényi entropy, Tsallis entropy, Onicescu information energy of orders 2and 3, respectively, with R2values with respect to the total atomic number shown at the end as well. Because they are not linear functionals with respect to the total number of electrons, their R2values are not close to unit65.

Table 3 exhibits the numerical results for a total of 42 neutral molecules, with the correlation coefficients also shown at the end.19Although the R2values are not as good as the atomic case for Iσandthey are still pretty decent. Plus, for SGBP, andsame as the atomic case, the linear correlations are still extremely strong.

Table 1 Information-theoretic quantities （in atomic units） for first and second-row neutral atoms

At the atoms-in-molecule level of theory, the three atom partition schemes yield similar, though not identical, results, as shown in Table 4 by their correlation coefficients with different information-theoretic quantities19. Fig.1 demonstrates three of these strong correlations for carbon and hydrogen atoms in the 42 systems studied, using Becke's fuzzy atom partitioning approach as an example. We also studied the behavior of these quantities as a function of molecular conformation change at both molecular and atoms-in-molecules levels. Shown in Fig.2, as an example, is the profiles of the second-order Onicescu information energy as a function of the ∠HCCH dihedral angle for the ethane molecule, which is change from 0° （staggered isomer） to 360° at both levels65.

These scaling properties with respect to the total number of electron populations at both the molecular and atoms in molecules levels have also been investigated and confirmed in a few reaction systems20–22.

6 Quantification of steric effect

Steric effect is one of the most widely used concepts in chemistry. It originates from the fact that each atom in a molecule occupies a certain amount of space. When atoms are brought together, hindrance will be induced in the expense of shape, energy, reactivity, etc. According to Weisskopf87, steric effect is resulted from the “kinetic energy pressure” in atoms and molecules. Description of the effect employing Pauli exchange interactions from the wave function theory is available88, but a satisfactory quantification is still lacking. Earlier, we proposed to quantify the steric effect within the framework of density functional reactivity theory with the Weizs?cker kinetic energy TW15,

which is related to the Fisher information by a factor of 8,

This approach of quantifying the steric effect was based on the assumption that the total electronic energy of a system comes from the independent contributions of three physiochemical effects, steric, electrostatic, and quantum due to the exchangecorrelation effect15:

If the above assumption is valid, we can prove that using the Weizs?cker kinetic energy （i.e., also the Fisher information） to express the steric effect, Eq.（57）, should be true.

The proof is straightforward, shown below. Since in DFT, we know that1

where TS, Vne, J, Vnn, and Excstand for the noninteracting kinetic, nuclear-electron attraction, classical electron-electron Cou-lomb repulsion, nuclear-nuclear repulsion, and exchange-correlation energy density functionals, respectively. Three components, Vne, Vnn, and J, are of the electrostatic nature, Ee= Vne+ J + Vnn. For the quantum part, we have

where the Pauli energy is78,79,

which denotes the portion of the kinetic energy that embodies the effect from the antisymmetric requirement of the total wave function by the Pauli Exclusion Principle. Put together, we have the density-based formulation of the steric energy15,

So, the validity of Eq.（57） solely depends on the assumption of Eq.（59）. If Eq.（59） is true, so is Eq.（57）.

Table 2 Atomic values of new information-theoretic quantities （in atomic unit） obtained with the total electron density

The physical meaning of the above quantification of the steric effect is that it represents a hypothetical state where electrons are assumed to be bosons and Esis simply the total energy of this state excluding contributions from other two effects, electrostatic and quantum. If the density of the hypothetical boson state were the same as that of the fermionic state, ρ（r）, the total wave function of the hypothetical state would just bewhere N is the number of electrons. The total kinetic energy of the hypothetical state, from which Weisskopf's “kinetic energy pressure”87for the steric effect is calculated, is simply Eq.（57）.

From the three-dimensional space point of view, in the above hypothetical state, when all electrons in the ground state are squeezed into the lowest orbital, the space occupied by electrons should be minimal. Still, different atoms with different electrons would occupy different amount of space. Henceforth, the space withheld by the hypothetical state and represented by Esshould be an intrinsic property of the system15.

From the Fisher information point of view, which is a measure of the localization of the electron density distribution, this new definition of the steric effect, Eq.（57）, quantifies spikes of electron densities around atomic nuclei. According to the principle of extreme physical information discussed above, the Fisher information should be minimized subject to the validityof the three constraints26,80,81. This means that when a molecular system is formed, atoms in molecules tend to occupy the minimal amount of space, if the three conditions mentioned in Section 4 are satisfied.

Table 3 Information-theoretic quantities （in atomic units） for 42 molecular systems

There are a few nice properties associated with this quantification of the steric effect in DFRT.15For instance, the integrand of Eq.（57） is non-negative everywhere, and thus repulsive in nature. It vanishes for the case of a homogeneous electron gas. It is extensive because it is homogeneous of degree one in density scaling, as shown in the above section, so the larger the system, the larger the steric repulsion. If Bader's atoms-in-molecules approach or any of other atom partition methods is ad-opted, the steric energy can be partitioned at both the atomic and functional group levels. Its corresponding steric potential, steric charge, and steric force have been defined and evaluated for both molecules and solids46,55.

Table 4 Correlation coefficient values （R2） of all least square fitts with respect to the total number of electron populations for the informationtheoretic quantities presented in this work at atoms-in-molecules level with AIM， Becke， and Hirshfeld partitionings

Fig.1 Scaling results for （a-c） carbon and （d-f） hydrogen atoms in molecules results with Becke's fuzzy partitioning approach for 192 carbon atoms and 252 hydrogen atoms in 42 molecular systems studied

This new quantification has been applied to a number of systems and chemical phenomena46–58, such as conformational changes of small molecules, SN2 reactions, chained and branched alkanes, water clusters, anomeric effect, weak interactions, orbital interactions, and other systems. Reasonably good trends and linear relationships between theoretical result and experimental scales by Taft have recently been observed at both group and entire molecular levels.

In particular, a clear picture has emerged from these studies in conformation changes57, showing that for all systems considered thus far, the electrostatic interaction plays the predominant role, whereas other effects such as steric and exchangecorrelation interactions play minor but indispensable roles. As an example, Fig.3 shows the energetic profiles of the （a） total energy and （b） exchange-correlation energy differences with respect to the 360° flexible rotation with the step size of 5° for each of the six different single bonds, C－C, C－N, C－O, N－N, N－O, and O－O for six simple molecules CH3CH3, CH3NH2, CH3OH, NH2NH2, NH2OH, and H2O2, respectively. As can be seen from the figure, these energy profiles are vastly different from one another in both barrier heights and curve shapes57.

From the energy component analysis viewpoint, as shown in Fig.4, however, a much simpler picture is available. Our resultsunambiguously show that the electrostatic interaction is the dominant contributor to the rotation barrier height. The higher the barrier, the larger the electrostatic interaction. On the other hand, our results also confirm and consolidate the earlier explanations in the literature, where both steric repulsion and hyperconjugation （quantum） effects were employed to validate the rotation barriers. As can be seen from Figs.4b and 4c, though with a smaller coefficient, both steric effect and exchangecorrelation interactions positively contribute to the rotation barrier, suggesting that the higher the bond rotation barrier, the stronger the steric repulsion and the smaller the quantum effect due to exchange-correlation interactions57.

Fig.2 Profiles of information-theoretic quantities as a function of the HCCH dihedral angle for the ethane molecule

Fig.3 Profiles of （a） total energy and （b） exchange-correlation energy differences of six molecules

7 Minimum information gain principle

In 2000, Nalewajski and Parr13started the era of employing a simple density functional to quantify properties related to mo-lecular structure and reactivity. In their original work, they employed the relative Shannon entropy from the information-theoretic approach, Eq.（43）, which is sometimes also called the information gain, Kullback-Leibler divergence, information divergence, missing information, or entropy deficiency. From it, they derived the Hirshfeld partition of atoms in molecules, and thus the Hirshfeld charge, with the so-called minimum information gain principle.

Reprinted with permission from Ref.57. Copyright 2013 American Chemical Society. 1 cal = 4.1868 J

Rewriting Eq.（43） as the information gain using atomic density where ρAis the electron density on atom （or group） A in a molecule, whose total electron density is ρ and the total number of electrons is N, andis the counterpart of atom （or group） A in the reference state, which can be neutral atom, or ion, or group, etc.

Minimizing the information gain, subject to the condition that atomic densities are always normalized to the total electron N

leads to

where λ is the Lagrange multiplier. Functional differentiation of Eq.（66） with respect to all ρA, after some algebraic manipulations, yields that for all atoms A, we have14This is the well-known “stockholder partition” of the electron density for atoms in molecules first proposed by Hirshfeld.

That is to say, if one employs the Hirshfeld scheme to partition atoms in a molecule, the information gain due to the formation of the molecule from the composing pieces will be minimal. In other words, atoms in molecules partitioned in this manner will preserve their identity （e.g. electrophilic and nucleophilic properties, etc.） of the reference state as much as possible. This nature of minimal information deficiency is the essence of the minimum information gain principle.

The milestone of this seminal work by Nalewajski and Parr is the fact that it serves as the first example to quantify a reactivity property of molecules with a density functional from the information-theoretic approach. Moreover, since charge is not associated with any physical observable, it has no unique definition. There are many ways to quantify charge in the literature. This work provides the first example to derive charges from a physiochemical principle.

In the next sections, besides rationalizing the validity of the Hirshfeld charge, we will demonstrate how this principle can be utilized to quantify other reactivity properties such as regioselectivity, electrophilicity, and nucleophilicity.16

8 Information conservation principle

Very recently, we expanded the context of the above principle, providing an in-depth understanding on the origin and nature of a few reactivity properties such as regioselectivity, electrophilicity, and nucleophilicity.16The expansion was based on a special case of the above principle, which was called the information conservation principle.

According to the above principle, atoms in molecules tend to keep their identity of their reference state as much as possible, indicating that ρAandshould be similar and the differencebetween the two densities should be simulated by a perturbation expansion using the Taylor series. We can define a new variable, x =so the information gain in Eq.（64） becomes

Since x is expected to be small, usingas the firstorder approximation, we have

where qAis the Hirshfeld charge on atom （or group） A. This result shows that under the first-order approximation, the information gain simply gives rise to the Hirshfeld charge distribution.

At the meanwhile, since ρAandsatisfy the same normalization condition, Eq.（65）, the total information gain in Eq.（69）must vanish16, suggesting that under the first-order approximation, the information before and after a system is formed should be conserved. We call this result the information conservation principle. This principle, stemmed from the above first-order approximation, is a special case of the minimum information gain principle, Eq.（64）, with no information gained at all. The actual value of information gain in Eq.（64） should come from the second and other higher order terms in the Taylor expansion in Eq.（68）, with

To verify the validity of the first-order approximation introduced in Eq.（68）, shown in Fig.5 is the comparison of the two quantities without （Fig.5（a）） and with （Fig.5（b）） the approximation for the nitrobenzene molecule. As one can see from the two plots, no discernible difference can be seen from the two, and thus it confirms its validity16.

Fig.5 Validity check of the first-order approximation for information gain: （a） information gain in Eq.（64） and （b） deformation density in Eq.（69） for the nitrobenzene molecule

According to this principle, when a new molecular system is formed from its components, the identity of its ingredients will be preserved, at least to the first-order approximation. That is to say, if a component is electrophilic or nucleophilic in nature, it will still be so in the newly formed system as well. Moreover, to preserve the identity of the components in the new system, they will have to adjust themselves in such a manner that each of the components becomes charged according to its stockholder contribution in the electron density, as shown in Eq.（67）. Put together, the new understanding from the information conservation principle proposed in this work will provide a novel approach in quantifying reactivity properties, such as electrophilicity, nucleophilicity, and regioselectivity, as will be illustrated in the following section by a number of different molecular systems16–22.

9 Nucleophilicity and electrophilicity

As two of the most widely used concepts in chemistry89–91, electrophilicity and nucleophilicity measure the ability of an electrophile and nucleophile to accept and donate electrons, respectively. Related to these reactivity properties is the concept of regioselectivity, on which atom or atoms these electrophilic or nucleophilic transformations are most likely to take place. A unified effort towards a general scale of nucleophilicity and electrophilicity using experimental rate constants has recently been introduced by Mayr and coworkers92–94. Theoretically, with the Fukui function from conceptual DFT one can predict the reactive site of electrophilic and nucleophilic attacks1,29. Neverthe-less, any effort to provide a unified theoretical framework to simultaneously quantify electrophilicity and nucleophilicity and at the same time determine their regioselectivity has been unsuccessful until recently.

In principle, the property of electrophilicity and nucleophilicity should be closely related to the lack and excess of electrons, respectively, at different atoms in molecules. Therefore, the conventional chemical wisdom that information of the atomic charge distribution in a molecule should suffice in quantifying these properties must still be valid. That is, a negatively charged atom in molecules is able to donate electrons so it should be nucleophilic in nature, whereas a positively charge atom is capable of accepting electrons so it be electrophilic. Our recent work has demonstrated that it is indeed the case16. Based on the principles introduced above, we employed the Hirshfeld charge for the purposes and demonstrated that regioselectivity, electrophilicity and nucleophilicity can all simultaneously be determined.

As illustrative examples, we studied 21 electrophilic systems（Scheme 1） and 22 nucleophilic molecules （Scheme 2）, and compared their experimental scale of electrophilicity/nucleophilicity with both information gain and Hirshfeld charge. Figs.6 and 7 display those comparisons, from which we find that remarkable agreement between Mayr's experimental scales and our theoretical measures can be achieved16.

Scheme 2 Twenty-two nucleophilic molecules studied

In another recent work, we applied the idea to nitrogen-containing systems17. Five different categories of compounds were studied, including benzenediazonium, azodicarboxylate, diazo,primary and secondary amines, with a total of 40 molecules. Our results show that there still exist strong linear correlations between the Hirshfeld charge and their experimental scales of electrophilicity and nucleophilicity. However, these correlations depend on the valence state and bonding environment of the nitrogen element. The linear relationship only holds true within the same category of compounds.

Fig.6 Comparison of the experimental electrophilicity scale of Mayr et al. with （a） information gain and （b） Hirshfeld charge for 21 systems listed in Scheme 1

Fig.7 Comparison of the experimental nucleophilicity scale by Mayr et al. with （a） information gain and （b） Hirshfeld charge for 22 systems listed in Scheme 2

In addition, we separately studied electrophilic aromatic substitution18,20,21and bimolecular nucleophilic substitution （SN2） reactions22to quantitatively examine their nucleophilicity and electrophilicity, respectively, by comparing information gain or Hirshfeld charge at the reactive center with the reaction barrier height obtained from transition states. Again, remarkably strong correlations between these quantities have been obtained. These and other results unambiguously show that information gain and Hirshfeld charge are reliable descriptors to accurately determine nucleophilicity and electrophilicity at the same time for different systems.

Closely related to nucleophilicity and electrophilicity is the chemical concept of regioselectivity, which is simply the preferred reactive center for nucleophilic or electrophilic reactions. A nucleophilic attach should take place on the atom or atoms with the most negative charge, whereas an electrophilic attach is most likely to occur at the atom or atoms with the most positive charge. The validity of our prediction with the regioselectivity has been demonstrated in the studies present above. A more interesting example on regioselectivity is the para/ortho and meta group directing effects in electrophilic aromatic substitu-tion reactions, as will be discussed below18,20.

10 Regioselectivity: nature of ortho/para and meta group directing

An electrophilic aromatic substitution is a chemical transformation in which the hydrogen atom of an aromatic ring is replaced by an electrophile as a result of an electrophilic attack on the aromatic ring. Important examples of such reactions include aromatic nitration, halogenation, sulfonation, acylation, and alkylation. Electrophilic aromatic substitution reactions as one of the most fundamental chemical processes are affected by atoms or groups already attached to the aromatic ring95. The groups that promote substitution at the ortho/para or meta positions are, respectively, called ortho/para and meta directing groups, which are often characterized by their capability to donate electrons to or withdraw electrons from the ring. Though resonance and inductive effects have been employed in textbooks and in the literature to justify this phenomenon96,97, no satisfactory quantitative interpretation is available until now.

The logic to apply the information-theoretic approach to this problem is the following. When a functional group, either ortho/ para or meta directing, is placed on the benzene ring, the nucleophilic property of the benzene ring will subsequently be altered, leading to the nonhomogeneous behavior of the six carbon atoms of the substituted benzene ring in electron donation. This change of nucleophilicity, the capability of donating electrons to the incoming electrophile, of the substituted benzene ring should be the root cause of the above ortho/para and meta directing phenomenon.18

Table 5 Hirshfeld charges at ortho-， meta-， and para-positions for mono-substituted benzene derivatives Ar-R， with both ortho/para （left four columns） and meta （right four columns） directing groups R

Table 5 shows the Hirshfeld charge at ortho-, meta- and para-positions for mono-substituted benzene derivatives, Ar-R, with either ortho/para or meta directing group, R. We found that （i） for ortho/para directing groups （left 4 columns in the Table）, in all the cases, the most negative charged atom is on either ortho or para position, and there is no single incidence where the meta-position possesses the most negative charge. For the meta directing groups （the right 4 columns）, however, for a vast majority of the systems such as －CX3, －CHO,－CN, －COR, －NO2, －NO, and －SO3H, the meta position possesses the largest Hirshfeld charge, consistent with experimental evidence. The outliers aresystems, where the meta position has the second largest negative charge18.

Table 6 Prediction of ortho/para and meta group directing for 23 mono-substituted benzene derivatives， Ar-R， in electrophilic aromatic substitution

Using the Hirshfeld charge as an indication of the regioselectivity, we can predict the reaction centers for unknown systems. Some results are shown in Table 618.

To provide quantitative understanding about the role of the Hirshfeld charge on the impact of ortho/para and meta group directing effect20,21, we continued the study by examining the transition state structure and activation energy of an identity reaction for a series of mono-substituted-benzene molecules reacting with hydrogen fluoride using BF3as the catalyst in the gas phase （Scheme 3）. A total of 18 substitution groups were considered, nine of which were ortho/para directing and the other nine groups meta-directing. From this study20,21, we found that the barrier height of these reactions strongly correlates with the Hirshfeld charge on the regioselective site for both ortho/para and meta-directing groups, with the correlation coefficient R2both better than 0.96. We also discovered a less accurate correlation between the barrier height and HOMO energy. These results reconfirm the validity and effectiveness of employing the Hirshfeld charge as a reliable descriptor of both reactivity and regioselectivity for this vastly important category of chemical transformations. Fig.8 exhibits some of the results from the study, confirming the close relationship between Hirshfeld charge and regioselectivity. This relationship has also recently been confirmed by SN2 reaction systems22.

Scheme 3 Electrophilic aromatic substitution reaction studied using the identity reaction of mono-substituted-benzene molecules reacting with hydrogen fluoride using BF3as the catalyst in the gas phase

11 Outlooks: reactivity as density functionals

What is reactivity Reactivity of a molecular system is the tendency of the system to undergo a chemical transformation, which should be an intrinsic property of the system determined a priori by its composition and structure. This tendency could include changing from one conformation of the molecule to an-other one, losing or accepting one or more electrons or protons, responding to attacks by external reagents such as catalysts or external fields such as photoexcitation, among other possible changes or perturbations. Molecular orbital theory does provide us with insights using frontier orbitals. However, its main obstacle is the fact that the more accurate the computational results are, the more ambiguous their physical meanings are. This is because orbitals are human invention and thus their applicability is limited. On the other hand, one does not have to use orbitals to understand and predict molecular reactivity. DFT provides with us such a rigorous and robust alternative, where one employs electron density and its associated quantities instead of orbitals for the purpose. Earlier success in conceptual DFT with the introduction of concepts such as electronegativity, hardness, Fukui function, and dual descriptors has evidently witnessed the validity and effectiveness of such alternative tools. There exist, however, many challenges. We recently adventured a new pathway by appreciating and quantifying reactivity properties with density functionals. Key questions remaining are now the following: How should this tendency of molecular changes be described Can it be represented by one or more density functionals Or is it possible that each tendency of the above changes be appropriately attributed by a density functional Keeping in mind that according to DFT, the electron density alone is adequate to determine all these properties in the ground state, we are confident that such a proposition is not completely unjustified.

Fig.8 Strong linear correlations between Hirshfeld charge and reaction barrier height for electrophilic aromatic substation reactions with both ortho/para and meta directing groups

Indeed, above results from the recent literature demonstrated the validity of such an effort to describe structure and reactivity properties using density functionals. Steric effect quantified using the Weizs?cker kinetic energy density functional is one example. Electrophilicity, nucleophilicity, and regioselectivity in terms of information gain or Kullback-Leibler divergence is another one, which enables us to provide a completely new understanding about the ortho/para/meta group directing phenomena for electrophilic aromatic substitution reactions. The density functionals we discussed in this review not only are simple in their forms, but also originate from the information theory. They possess amiable scaling properties, and because of the nature of the electron density, many of these quantities are inter-correlated. In addition, numerous principles have been established in the literature, such as the principle of extreme physical information, minimum information gain principle, and information conservation principle. These works should serve as the foundation of the information-theoretic approach in density functional reactivity theory.

So, what is next We can answer this question by proceeding in two pathways. From the theoretical development point of view, there are a number of topics that we can quickly pick up and pursue in the near future. At first, the Kullback-Leibler divergence is the relative Shannon entropy. How does it differ from the relative Rényi entropy Are they somehow related Secondly, what are the local behaviors of steric/quantum potential, steric/quantum charge, and other local functions and quantities stemmed from the information-theoretic approach What are their roles in determining reactivity and selectivity for chemical reactions More importantly, how can we go beyond the information-theoretic approach and systematically develop different density functionals to address different needs in predicting and understanding different molecular structure and reactivity properties

From the viewpoint of structure and reactivity properties, acidity/basicity and redox potentials are the two extremely important intrinsic physiochemical properties of molecules that require a lot of our attentions. Earlier, we proposed to employ the molecular electrostatic potential on the nucleus of the acidic orbasic atom or its valence natural atomic orbital energy as an accurate descriptor of the acidity or basicity31–35. This work however has its limitations. That is, different systems tend to have different correlations so any effort to put different categories of molecules into one scaling system has failed miserably. We observed the same issue with electrophilicity and nucleophilicity scales. Are these phenomena related Is it ever possible to come up with a universal scale for acidity/basicity and electrophilicity/nucleophilicity Finally, with reliable tools dealing with regioselectivity and electrophilicity/nucleophilicity available in the literature, we are in a better position than ever to tackle the problems related to stereoselectivity, enantioselectivity, diasteroselectivity, and other chiral or asymmetric chemistry related topics.

To wrap up, it is our humble but cautiously optimistic belief that a brand new door to appreciate and entertain molecular reactivity from the perspective of density functional reactivity theory has just opened. A bright future lies ahead. Opportunities are abundant.

Acknowledgment: The author is indebted immensely to many of his former and present students and worldwide collaborators in the last few years. In particular, he is grateful to Dr. LU Tian of Beijing Kein Research Center for Natural Sciences for coding all the formulas in the text in the MultiWFN package, and to Professor PARR Robert G of University of North Carolina at Chapel Hill for constant attention, discussion, and encouragement. The author also wishes to express his sincere gratitude to Professor Emeritus LI Run-E of Hunan Normal University, Changsha, for her stimulation of the author's interest in molecular reactivity when he took the introductory organic chemistry 32 years ago taught by Professor LI. The author acknowledges support from the Research Computing Center, University of North Carolina at Chapel Hill for accessing computing facilities.

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Information-Theoretic Approach in Density Functional Reactivity Theory

LIU Shu-Bin*
（Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, USA）

Density functional reactiνity theory （DFRT） is a recent endeaνor to appreciate and quantify molecular reactiνity with simple density functionals. Examples of such density functionals recently inνestigated in the literature included Shannon entropy, Fisher information, and other quantities from information theory. This reνiew presents an oνerνiew on the principles of the information-theoretic approach in DFRT, including the extreme physical information principle, minimum information gain principle, and information conserνation principle. Three representations of this approach with electron density, shape function, and atoms-in-molecules are also summarized. Moreoνer, their applications in quantifying steric effect, electrophilicity, nucleophilicity, and regioselectiνity are highlighted, so are the recent results in a completely new understanding about the nature and origin of ortho/para and meta group directing phenomena in electrophilic aromatic substitution reactions. A brief outlook of a few possible future deνelopments is discussed at the end.

Shannon entropy; Fisher information; Density functional reactiνity theory; Steric effect; Electrophilicity; Nucleophilicity; Regioselectiνity; Ortho/para and meta group directing effect

?r） is the

tate density satisfying the same normalization condition as ρ（r）. This reference density can be from the same molecule with different conformation or from the reactant of a chemical reaction when the transition state is investigated.

O641

10.3866/PKU.WHXB201510302

Received: October 2, 2015； Revised: October 29, 2015； Published on Web: October 30, 2015.

*Corresponding author. Email: shubin@email.unc.edu； Tel: +1-919-962-4032

?Editorial office of Acta Physico-Chimica Sinica