## Fukui函数和局域软度应用于亲电加成反应的区位选择性的研究

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## Fukui Function and Local Softness Related to the Regioselectivity of Electrophilic Addition Reactions

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 基金资助: 国家自然科学基金.  21483083国家自然科学基金.  21483083, 21133005辽宁省自然科学基金.  2014020150

Corresponding authors: Email: jane_xu@dicp.ac.cn (X.Z.)Email: zhaodxchem@lnnu.edu.cn; Tel.: +86-411-8215-8977 (Z.D.)

 Fund supported: theNationalNaturalScienceFoundationofChina.  21483083the National Natural Science Foundation of China.  21483083, 21133005the Natural Science Foundation of Liaoning Province, China.  2014020150

Fukui函数、局域软度、广义Fukui函数以及广义软度通常被称为反应描述符。使用它们研究和探讨了HCl与不对称烯烃以及溴苯硒与不对称苯乙烯的亲电加成反应的区位选择性。在MP2/6-311++G(d, p)理论水平下，采用有限差分方法计算这些反应描述符，同时也使用ABEEMσπ方法进行了计算。ABEEMσπ模型下的局域软度和广义局域软度，分别结合局域硬-软酸碱(HSAB)原理，得出亲电试剂氯化氢与溴苯硒，更容易进攻不对称乙烯和苯乙烯中的马氏碳原子，符合马氏规则。而有限差分方法不能完全地解释该系列反应的区位选择性。此外，主要产物所对应的马氏碳原子的广义局域软度值，就能够预测出此类反应的活性序列，所得结果与速率常数有很好的关联。

Abstract

Regioselectivities of electrophilic addition reactions of hydrogen chloride to asymmetric alkenes and benzeneselenenyl bromide to substituted styrenes have been investigated by using reactivity descriptors, including Fukui function f(r), local softness s(r), generalized Fukui function fG(r), and generalized local softness sG(r). All of them are obtained from the finite difference approximation method calculated by ab initio method at MP2/6-311++G(d, p) level of theory and our ABEEMσπ model, respectively. According to the generalized version of the local hard-soft and acid-base (HSAB) principle, the forecasted regioselectivities of our investigated additions using the ABEEMσπ model are in fair agreement with the experimental values. In particular, we can also rationalize their reaction rate constants by the generalized local softness, i.e., the softest the site is, the easiest the reaction is. Hence, the generalized reactivity descriptors work quite well.

Keywords： Fukui function ; Local softness ; Local hard-soft and acids-bases (HSAB) principle ; Electrophilic addition reactions ; Finite difference approximation ; ABEEMσπ model

ZHU Zunwei, YANG Qiaofeng, XU Zhenzhen, ZHAO Dongxia, FAN Hongjun, YANG Zhongzhi. Fukui Function and Local Softness Related to the Regioselectivity of Electrophilic Addition Reactions. Acta Physico-Chimica Sinica[J], 2018, 34(5): 519-527 doi:10.3866/PKU.WHXB201710126

## 1 Introduction

Electrophilic addition of an electrophile to alkenes is one of the most widely studied electrophilic reactions 1-9, as shown in Fig. 1. The analysis of this sort of reactions has attracted great concern of both experimental and theoretical studies. Regioselectivity for the electrophilic addition has been shown to follow the empirical Markovnikov's rules 10, the addition of an acidic proton to a double bond of an alkene yields a product where the proton is bound to the carbon atom bearing the largest number of hydrogen atoms when the substituent of alkene is electron-donating group. And when the substituent is electron-accepting group, the proton of acid favors to attack the carbon atom bearing the smallest number of hydrogen atoms, which calls the anti-Markovnikov's rule. Many theoretical 1-4, 8 and experimental 5-7, 9 studies have focused on the regioselectivity of electrophilic addition to alkene all the while, such as, frontier molecular orbital (FMO) theory is sometimes used for explaining the regioselectivity of reaction 8. Suresh and his coworkers have employed the molecular electrostatic potential to confirm the regioselectivity of Markovnikov reaction 2. Recently, Yang, Ding and Zhao 1 have performed to use the frontier electron density of initial-state carbon atoms in molecular face theory (MFT) to estimate its regioselectivity: if the frontier electron density encoded on the Markovnikov carbon atom is larger than that of anti-Markovnikov, the reaction may be predicted to proceed on the Markovnikov route, and otherwise it may prefer the anti-Markovnikov route.

### Fig 1

Fig 1   The regioselectivities of electrophilic additions of hydrogen chloride to the substituted ethenes (R3 = H) and benzeneselenenyl bromide to substituted styrenes (R1 = R2 = H), including Markovnikov and anti-Markovnikov products.

The local hard-soft and acids-bases (HSAB) principle is an efficient method in conceptual density functional theory (CDFT) to predict the regio-and stereoselectivities of reactions, especially the corresponding softness matching in a local approach 11-17 for two or four or much more reactive points between two reactants. Li-Evans 18 proposed that for a hard reaction the site of minimal Fukui function (FF) is preferred and for a soft reaction the site of maximal Fukui function is preferred. Gazquez and Mendez 19 stated that the reaction between two chemical species will not necessarily occur through their softest atoms, but through those sites whose local softness are close to each other. In this respect, Geerlings, Proft and Langenacker 20 suggested that local softness should be used as an intermolecular reactivity descriptor, whereas the FF is as an intramolecular one. Thus, the comparisons of the FF or condensed FF values for different systems are meaningless because they represent only the relative reactivity among different sites within a molecule. Therefore, in order to rationalize the intermolecular reactivity, we have proposed a series of generalized reactivity descriptors 21, including generalized Fukui function (GFF) and generalized local softness (GLS) 22.

Recently, based on the generalized reactivity descriptors we have been successful to predict and to explain the regio-/stereoselectivities of Diels-Alder reactions 22 and the enzymatic catalyzed reactions of biological system 21 and to correlate their intermolecular reactivities of all reactions in terms of the atom-bond electronegativity equalization method (ABEEMσπ) model with the local HSAB principle at its generalized version. And we have obtained the results in good agreement with the experimentally observed outcomes.

In this paper, we will use the usual reactivity descriptors and the generalized one combined with the local HSAB principle to investigate the regioselectivities of the electrophilic additions of alkene including the hydrogen chloride and benzeneselenyl with unsymmetrical alkene and to rationalize their order of reaction rate constants by the ab initio method at the level of MP2/6-311++G(d, p) with the finite difference approximation (FDA) method and the ABEEMσπ model. It should be noted that the FDA method involves the three systems of N, N + 1, and N - 1 electrons, but ABEEMσπ model only involves one system of N electron.

### 2.1 The reactivity descriptors

Fukui function (FF) is one of the important reactivity descriptors in predicting the intramolecular reactivity in CDFT 23, 24. Parr and Yang defined the FF $f\text{(}\vec{r}\text{)}$ and local softness $s(\vec{r})$ 25, 26 as:

$f{\text{(}}\vec r{\text{)}} = {\left( {\frac{{\partial \rho {\text{(}}\vec r{\text{)}}}}{{\partial N}}} \right)_{\nu \left( {\vec r} \right)}}$

$s(\vec r) = {\left( {\frac{{\partial \rho (\vec r)}}{{\partial \mu }}} \right)_{\nu \left( {\vec r} \right)}} = {\left( {\frac{{\partial \rho (\vec r)}}{{\partial N}}} \right)_{\nu \left( {\vec r} \right)}}{\left( {\frac{{\partial N}}{{\partial \mu }}} \right)_{\nu \left( {\vec r} \right)}} = f(\vec r)S$

where $\rho \text{(}\vec{r}\text{)}$ is the electron density at $\vec{r}$, N is the number of electrons for a molecular system, μ is the chemical potential, the negative of the electronegativity, $v\text{(}\vec{r}\text{)}$ is the external potential generated by the nuclei, S is the global softness.

### 2.2 The finite difference approximation (FDA)

In the FDA method, according to the Eq.(1), the condensed FF of nucleophilic attack for systems with electron gain can be written as

$f_k^ + = {q_k}{\text{(}}N + {\text{1)}} - {q_k}{\text{(}}N{\text{)}}$

and the condensed FF of electrophilic attack for systems with electron donation can be expressed as

$f_k^ - = {q_k}{\text{(}}N{\text{)}} - {q_k}{\text{(}}N - {\text{1)}}$

where qk(N + 1), qk(N), and qk(N - 1) stand for the partial charges on atom k in a molecule with N + 1, N, and N - 1 electrons at the same geometry structure, respectively 27.

A local softness descriptor $s(\vec{r})$ is related to the FF via Eq.(2), so the condensed local softness is related to the condensed FF 26 through

$s_k^ + = Sf_k^ + = S\left[{{q_k}{\text{(}}N {\text{ + 1)}}-{q_k}{\text{(}}N{\text{)}}} \right]$

$s_k^- = Sf_k^- = S\left[{{q_k}{\text{(}}N{\text{)}}-{q_k}(N-{\text{1)}}} \right]$

where, sk+ and sk- imply how global softness is redistributed among various atoms of the molecule by the condensed Fukui function. The global softness, S, can be given as 24 S = 1/I -A where I and A are the ionization potential and electron affinity, respectively. The first ionization potential I can be obtained by I= EN-1 - EN and the electron affinity A by A = EN+1 - EN with EN-1, EN, and EN+1 denoting the total energies of the systems with N - 1, N, and N + 1 electrons, respectively. The quantities involved can be calculated by an ab initio method at high level of theory.

### 2.3 ABEEMσπ model

Based on the DFT and electronegativity equalization method (EEM) 28-34, Yang and his coworkers have developed ABEEMσπ model 35-45, which explicitly partitions a molecule into atom, chemical bond, and lone pair (lp) regions. In this model, the single bond consists of one σ region, where the center of the charge for σ bond is located on the position of the ratio of the covalent atomic radii of two bonded atom; the double bond consists of one σ region and four π regions, where center of the σ bond charge is similar with the σ region of single bond and the π bond partial charges are placed above and below the double-bonded atoms at the covalent radii of the this double-bonded atoms perpendicular to the plane formed by the σ bond; and the center of charge and its orientation for the lp region is determined in terms of the chemical surrounding.

In terms of the definition of electronegativity based on DFT, the effective electronegativity of a region a, χa, can be expressed as:

${{\chi }_{a}}={{\chi }_{a}}^{*}+2{{\eta }_{a}}^{*}{{q}_{a}}+k\left[\sum\limits_{b?a}{\frac{{{q}_{b}}}{{{R}_{a, b}}}} \right]$

where χa* and 2ηa* are valence-state electronegativity and hardness of the region a, respectively. a and b denote two regions, including the atom or single bond σ or double bond σ and π or lone pairs regions. qa and qb are the partial charges of regions a and b, Ra, b denotes the distance between regions a and b, and k, 0.57, is an overall correction coefficient in this formalism 22, 35-47. The electronegativity equalization principle demands that the effective electronegativity of every region is equal to the overall electronegativity of the molecule, χmol:

${{\chi }_{a}}={{\chi }_{b}}=...={{\chi }_{m}}=...={{\chi }_\text{mol}}$

For an arbitrary molecule partitioned into m regions, solving the Eq.(8) with the constraint Eq.(9) on its net charge, qmol, if the parameters χa* and 2ηa* are known, we can obtain the charge of every region.

$\sum\limits_{a}{{{q}_{a}}}+\sum\limits_{b}{{{q}_{b}}}+...+\sum\limits_{m}{{{q}_{m}}}={{q}_{\text{mol}}}$

On the basis of the definition of the FF, we can express the FF of region a in our ABEEMσπ model as:

${{f}_{a}}=-{{\left( \frac{\text{d}{{q}_{a}}}{\text{d}N} \right)}_{\mu }}$

So, the global hardness 2ηmol can be expressed as:

$2{{\eta }_{\text{mol}}}=-{{\left( \frac{\partial {{\chi }_{a}}}{\partial N} \right)}_{\nu }}\text{=2}{{\eta }_{a}}^{\text{*}}{{f}_{a}}\text{+ }k\left[\sum\limits_{b\ne a}{\frac{{{f}_{b}}}{{{R}_{a\text{, }b}}}} \right]$

Hardness expressions for all the regions in a molecule like Eq.(11), altogether with the normalization condition of the FF, $\int{f\text{(}r\text{)d}r \text{= 1}}$, can be also solved to directly and quickly give the molecular hardness, 2ηmol, and in particular, the condensed FF fa of each region in the molecule if all parameters 2η* in Eq.(11) have been calibrated.

### 2.4 The generalized reactivity descriptors

The generalized Fukui function (GFF) ${{f}^{\text{G}}}\text{(}\vec{r}\text{)}$ and the generalized local softness (GLS) ${{s}^{\text{G}}}(\vec{r})$ have been proposed and their definitions 22 are expressed as Eqs.(12) and (13):

${{f}^{\text{G}}}\text{(}\vec{r}\text{) = }{{N}_{M}}f\text{(}\vec{r}\text{)}$

${{s}^{\text{G}}}\text{(}\vec{r}\text{) = }{{f}^{\text{G}}}\text{(}\vec{r}\text{)}S={{N}_{M}}f\text{(}\vec{r}\text{)}S={{N}_{M}}s\text{(}\vec{r}\text{)}$

where, the $f\text{(}\vec{r}\text{)}$ and $s(\vec{r})$ are local Fukui function (FF) and local softness, the usual reactivity descriptors, NM is the number of atoms for a molecular system, and S is the global softness. Obviously, according to Eq.(12), the GFF ${{f}^{\text{G}}}\text{(}\vec{r}\text{)}$ is normalized to NM because $f\text{(}\vec{r}\text{)}$ is normalized to 1 for a molecule. And ${{s}^{G}}\text{(}\vec{r}\text{)}$ is normalized to NMS rather than S, which means that the global softness is the average of the generalized local softness ${{s}^{G}}\text{(}\vec{r}\text{)}$. Based on the definition of ${{f}^{\text{G}}}\text{(}\vec{r}\text{)}$, the reactivity descriptor of the site is not only dependant on its charge and $f\text{(}\vec{r}\text{)}$, but also related to the number of the atoms in the molecule, NM, where the detailed ratiocination has been represented in Ref. 22.

### 3 Computational details

We investigated the electrophilic additions of hydrogen chloride to asymmetric alkenes and benzeneselenenyl bromide to substituted styrenes, as shown in Fig. 1. The geometries of all reactants were optimized and obtained by the B3LYP/6-311+ G(d, p) level of theory in Gaussian-03 48. All optimized reactants were stationary points of potential energy surface after checking the frequencies at the same level of theory.

### 3.1 Calibration of parameters χ* and 2η* for ABEEMσπ Model

According to Eq.(7) and Eq.(11), we have calibrated the parameters χ* and 2η*, through a regression and least-squares optimization procedure by dealing with some model molecules 35. For all model molecules, ab initio Hartree-Fock MO calculations were performed with STO-3G basis sets in Gaussian 03 48 and then the partial charges of all the model molecules were obtained by the Mulliken population analysis. Then the charge distributions obtained for the model molecules were brought into Eqs.(7–9) in order to determine the parameters χ* and 2η* through a regression and least-squares optimization procedure 22, 35-38, 46, 47. The old types of parameters were obtained from our previous work 37, and the new added types and parameters of χ* and 2η* are listed in Table 1.

Table 1   Parameters χ* and 2η* in ABEEMσπ Model.

 Type of ABEEMσπ parameter χ* 2η* Cl17―Ph―C626＝C62 C66 in ―Ph 2.500 6.800 π66 3.850 94.150 C626 2.500 7.200 π626 3.790 94.150 Cl17 3.010 20.099 C62＝C64―R C62 2.500 10.200 π62 3.500 80.319 C64 2.600 6.200 π64 3.580 88.150 Se― 1.800 34.970 Br― 2.880 70.019 σC-Se 4.900 35.000 σSe-Br 6.950 75.000 lpCl- 5.536 46.900 lpBr- 5.965 70.000 lpBe- 3.700 7.164

For the calibration, the reason why we use the minimum STO-3G basis set is not due to its time-consuming but more importantly due to its physical significance. Ab initio calculation with a higher level of basis set can give more accurate prediction of energy and geometry, but can not give more suitable partial charges than lower level of basis set for practical use. This phenomenon comes from the fact that a diffuse basis function located on an atom may to some extent cover the regions of the other adjacent atoms leading to a somewhat overestimating population of this atom in the Mulliken population analysis. Derouane and coworkers 49 showed that the formal charges calculated with the 6-21 basis set are higher than those computed with the STO-3G basis set, and thus suggested STO-3G charges may be more reliable. Wilson and Ichikawa 50 and Torrent-Sucarrat and their coworkers 51 pointed out that the charge transfer between atoms in a molecule is overestimated when the polarization basis sets are used. Huzinaka et al. 52 and Jakalian et al. 53, as well as our group47, had experienced that the use of higher level of basis sets overestimates the overlapping between their respective basis functions belonging to two atoms in a molecule. For example, if the 6-31G* basis set is used, the polarity of a molecule calculated by the partial charges is overestimated by 10%–15% than if the STO-3G basis set is used 53. Therefore, in the calibration process of the parameters, STO-3G basis set has been used in the ab initio calculations for all the model molecules to obtain the partial charges from Mulliken population analysis.

### 3.2 Calculation of Fukui function and local softness

Geerlings, Proft and Langenacker 20 suggested that local softness should be used as an intermolecular reactivity descriptor, whereas the FF is as an intramolecular one. Under the FDA method, via S = 1/(I - A), the global softness were obtained, where the first ionization potential I and the electron affinity A were calculated by ab initio method at MP2/6-311++G(d, p) level of theory. In terms of Eqs.(3) and (4), the condensed FFs of center atoms were calculated using the natural population analysis (NPA) at the MP2/6-311++G(d, p) level of theory, then obtained their local softness via Eqs.(5) and (6).

In the ABEEMσπ model, the FFs of center atoms were calculated by Eq.(11), then their GFFs were calculated by Eq.(12). Their global softness was obtained by Eq.(11) because of it being the inverse of the hardness, hence the local softness and the GLS were calculated by Eqs.(2) and (13), respectively.

### 3.3 Expression of the local HSAB principle under the finite difference approach and ABEEMσπ model

The local HSAB principle claimed 19 that the interaction between two molecules will occur not necessarily through their the softest atoms but rather through those atoms of two systems, and their Fukui functions of which are close. Based on this principle, the softness-matching criteria was proposed by Chandea, Nguyen, Geerlings and coworkers for understanding the regioselectivity of cycloaddition reactions 11-13, 15, 16. The softness-matching criterion at a local-local approach in the case of multiple sites of interaction has been cast in the form of the minimization of a quadratic form to articulate. In our investigated reactions, because there are two reaction center atoms in electrophilic additions, we will use the absolute values of differences between the local softness of the reaction center atoms of two reactants to express.

Hence, within the FDA method, the expression of the local HSAB principle is written as Eq.(14), where the i is the site of reactivity on molecule A, and the k is the site of reactivity on molecule B, as seen in Fig. 1, and the sAi+ is the condensed local softness of the ith atom in A, which represents that the electrophile H atom in HCl or [PhSe] group in PhSeBr acquires a electron sharing from the π-bond of substituted ethene and the sBk- is the condensed local softness of the kth atom in B, which represents that the reactant B will be attacked by H atom or [PhSe] group to donate an electron to be shared. And then, based on the proposed generalized reactivity descriptor, the local HSAB principle can be expressed as the Eq.(15). In this kind of the reaction, the superscript + denotes the reactivity descriptor of the electrophilic H atom, while the superscript – represents the reactivity index of double bond C atom in alkene.

$\Delta s=\left| s_{\text{A}i}^{+}-s_{\text{B}k}^{-} \right|$

$\Delta {{s}^{\text{G}}}=\left| s_{\text{A}i}^{\text{G}+}-s_{\text{B}k}^{\text{G}-} \right|$

In terms of ABEEMσπ model, the local HSAB principle can be expressed to the Eqs.(16) and (17), where the sAi, sBk, and $s_{\text{A}i}^{\text{G}}$, $s_{\text{B}k}^{\text{G}}$ are the local softness and GLS of ith atom in A and kth atom in B, respectively.

$\Delta s=\left| {{s}_{\text{A}i}}-{{s}_{\text{B}k}} \right|$

$\Delta {{s}^{\text{G}}}=\left| s_{\text{A}i}^{\text{G}}-s_{\text{B}k}^{\text{G}} \right|$

According to the local HSAB principle, the smaller the ∆s or ∆sG is, the easier the reaction is. In this paper, we only consider the state of single reactant, when ∆sMA < ∆sAM or ∆sMAG < ∆sAMG, the Markovnikov product should be the main; and when ∆sMA > ∆sAM or ∆sMAG > ∆sAMG, the anti-Markovnikov product should be favored. And the generalized reactivity descriptor can be further used to rationalize the reaction rate constants, i.e., the greater the sBkG is, the greater the reactivity is, and the easier the reaction is.

### 4.1 Regioselectivity of the addition of HCl to alkene

For the addition of HCl to unsymmetrical olefin CH2＝CR1R2, when the substituent is electron-donating group, such as alkyl, the reactions comply with the Markovnikov's rules to occur. When the substituent is electron-accepting group, such as ―CHO, ―COOH, the reactions comply with the anti-Markovnikov's rules. As seen in Fig. 1, these substituents belong to the alkyls, so the regioselectivities of these reactions abide by the Markovnikov's rules to produce the Markovnikov's products.

According to Eqs.(14)–(17), the difference values, $\Delta {{s}_{\text{MA}}}$, $\Delta {{s}_{\text{AM}}}$, $\Delta s_{\text{MA}}^{\text{G}}$ and $\Delta s_{\text{MA}}^{\text{G}}$, for the reactions of HCl with CH2＝CR1R2 were calculated by using the FDA method and the ABEEMσπ model, and listed in Table 2. Table 3 presents the detailed values of FF, GFF, local softness, and GLS of reactive center atoms obtained from these two methods.

Table 2   The difference values, ∆sMA, ∆sAM, ∆sMAG and ∆sAMG, for HCl with alkene in terms of MP2/6-311++G(d, p) level under the finite difference approximation, and our ABEEMσπ model.

 Finite difference approximation ABEEMσπ model ∆sMA ∆sAM ∆sAMG ∆sMAG 103∆sMA 103∆sAMG 10∆sMAG 10∆sMAG ethene 0.589 0.589 2.647 2.647 0.069 0.069 0.526 0.526 propene 0.533 0.581 6.017 5.579 2.099 9.250 0.739 1.760 1-butene 0.549 0.556 8.863 8.779 2.867 5.727 0.981 2.012 2-methylpropene 0.604 0.627 8.195 7.924 3.008 6.643 0.964 2.122

Table 3   The values of condensed f(r), fG(r), s(r), and sG(r) for the H atom of electrophile HCl and the CMA and CAM of unsymmetrical alkenes at the level of MP2/6-311++G(d, p) and the ABEEMσπ model.

 Finite difference approximation ABEEMσπ model f+ fG+ s+ sG+ f fG s sG HCl H 0.7503 1.5007 1.5446 3.0893 0.2855 0.5709 0.0133 0.0265 f- fG- s- sG- f fG s sG ethene CMA 0.4242 2.5451 0.9560 5.736 0.1196 0.7176 0.0132 0.0791 CAM 0.4242 2.5451 0.9560 5.736 0.1196 0.7176 0.0132 0.0791 propene CMA 0.4043 3.6388 1.0118 9.106 0.0796 0.7168 0.0112 0.1004 CAM 0.3849 3.4639 0.9632 8.668 0.1607 1.4464 0.0225 0.2025 1-butene CMA 0.3952 4.7424 0.9960 11.952 0.0623 0.7481 0.0104 0.1246 CAM 0.3924 4.7090 0.9890 11.868 0.1139 1.3672 0.0190 0.2277 2-methylpropene CMA 0.3842 4.6099 0.9404 11.285 0.0632 0.7580 0.0102 0.1229 CAM 0.3749 4.4989 0.9178 11.013 0.1227 1.4724 0.0199 0.2387

As shown in Table 2, following the FDA method, the values of ∆sMA (0.589), ∆sAM (0.589) and ∆sMAG (2.647), ∆sAMG (2.647) for the reaction between HCl with ethene are equal to each other, respectively, which, of course, indicates that there is no regioselectivity in this reaction. When the olefin is propene, the value of ∆sMA (0.533) is smaller than the relevant ∆sAM (0.581). According to the local HSAB principle, the H atom of HCl favors to attack the Markovnikov's carbon atom, so this result is in line with the experimental regioselectivity. However, the value of ∆sMAG (6.017) is greater than the ∆sAMG (5.579) by using the generalized local softness, then the regioselectivity of this reaction would be anticipated the anti-Markovnikov's attacking. But, this result is not in agreement with the experimental result. By using the same way to deal with the rest two additions of HCl to 1-butene and 2-methylpropene, we also obtain ∆sMA < ∆sAM and ∆sMAG > ∆sAMG. Hence, according to the local HSAB principle, within the FDA method, the predicted results obtained from the usual local softness are better than those from the generalized local softness (GLS).

And then, how about are the results in terms of the ABEEMσπ model? It is clearly seen from Table 3 that the both values of ∆sMA and ∆sAM are 0.069 × 10-3, and both ∆sMAG and ∆sAMG are 5.26 when the olefin is ethene, which indicates that the two double-bonded carbon atoms are identical. When R1 is H and R2 is ―CH3, $\Delta s_{\text{MA}}^{{}}$ is 2.099 × 10-3, $\Delta {{s}_{\text{AM}}}$ is 9.250 × 10-3 and $\Delta s_{\text{MA}}^{\text{G}}$ is 73.9, $\Delta s_{\text{AM}}^{\text{G}}$ is 17.60, i.e., $\Delta s_{\text{MA}}^{{}}$ < $\Delta {{s}_{\text{AM}}}$ and $\Delta s_{\text{MA}}^{\text{G}}$ < $\Delta s_{\text{AM}}^{\text{G}}$. Therefore, on basis of local HSAB principle, the Markovnikov's product should constitute the main product of this reaction, and this result is in agreement with the Markovnikov's rules. In the same way, when alkenes are 1-butene and 2-methylpropene, both $\Delta s_{\text{MA}}^{{}}$ < $\Delta {{s}_{\text{AM}}}$ and $\Delta s_{\text{MA}}^{\text{G}}$ < $\Delta s_{\text{AM}}^{\text{G}}$, so the results are consistent with the Markovnikov's rules. Consequently, both $\Delta s$ via usual local softness and $\text{ }\!\!\Delta\!\!\text{ }{{s}^{\text{G}}}$ via GLS from ABEEMσπ model can explain and forecast the regioselectivities of these reactions well, and the predicted results are better than those of the FDA method.

It was reported that the rate constants for the addition of HI to ethene, propene, and 2-methylpropene were in the ratio 1:90:700 5, 6, which indicated that with raising the substituents, the reaction rates gradually became greater and greater. Furthermore, experimental activation energies of the additions of HCl to ethene (166.105 kJ·mol-1), to propene (144.348 kJ·mol-1), and to 2-methylpropene (119.244 kJ·mol-1) were reported 2, 9, which implied when the substituents gradually become larger and larger, the additions of HCl to CH2CR1R2 are more and more easy to process. The generalized reactivity descriptors, GFF and GLS themselves, can rationalize the intermolecular reactivity, and especially forecast the order of reaction rate constants for a series of reactions. Hence, we applied the GLS, rather than $\Delta {{s}^{\text{G}}}$ to correlate the order of the reaction rate constants for these investigated additions.

The investigated electrophilic additions of HCl to alkenes, we have only considered the reactivity descriptors of reactants, which means that the values of GLS for the H atom of HCl are fixed and just the values of GLS for the CMA of alkenes are taken as variables, where the values of GLS for reaction centers by FDA method and ABEEMσπ model are listed in Table 3, i.e., we can disregard the GLS of H atom in HCl and only compare the GLS of CMA in alkenes of the main product. It is clear from our calculations that the values of $s_{\text{MA}}^{\text{G}}$ obtained from both FDA method and ABEEMσπ model (in Table 3) for alkenes increase with the substituents rising. Since the higher FF is, the higher reactivity is, i.e., the softest position of the molecule is, the easiest reactive site occurs. Hence, the addition for HCl to 2-methylpropene is the fastest reaction, the addition for HCl to ethene is the slowest one, and the addition of HCl to 1-butene is in the middle. These conclusions are just in agreement with experimental results and the results of usual local softness have not such regularity.

In a word, both the usual reactivity descriptors and the generalized ones from ABEEMσπ model in combination with the local HSAB principle can successfully interpret and forecast the regioselectivities of the additions of HCl to alkenes, which results are in agreement with the Markovnikov rule, but the results of FDA method are not good. And then, the values of $s_{\text{MA}}^{\text{G}}$ for the alkenes calculated by both the FDA method and the ABEEMσπ model can further correlate the reaction rate constants. When values of $s_{\text{MA}}^{\text{G}}$ gradually become big, the studied additions gradually become fast as substituents gradually increasing. But, the usual softness of CMA can not do this result. Therefore, we have chosen the other series of electrophilic additions to further check the validity and practicability of generalized descriptors again, as seen in Fig. 1.

### 4.2 The regioselectivity of addition for benzeneselenyl bromide with alkene

The additions of benzeneselenyl bromide to alkenes are usually considered to be the electrophilic addition reactions7. We have chosen four reactions between benzeneselenyl bromide and substituted styrene (XPhCH＝CH2), where the substituents PhX of alkenes are Ph, 3-ClPh, 4-ClPh and 4-CH3Ph, respectively. Here, [PhSe] group in electrophile is considered to be the H atom of HCl and it has the positive charge, which attacks one of the double-bonded carbon atoms of alkenes with enriched electron. The Br atomic charges obtained by HF/STO-3G level of theory and ABEEMσπ model are -0.100 and -0.140 a.u., respectively. Table 4 lists the charges of CMA and CAM, qMA and qAM, calculated by HF/STO-3G level of theory and ABEEMσπ model, respectively.

Table 4   The charges of CMA, CAM obtained from HF/STO-3G and ABEEMσπ Model.

 X―PhCH＝CH2 HF/STO-3G ABEEMσπ qMA qAM qMA qAM H -0.128 -0.057 -0.125 -0.054 3-Cl -0.122 -0.057 -0.116 -0.042 4-Cl -0.122 -0.058 -0.116 -0.042 4-CH3 -0.129 -0.057 -0.123 -0.051

These CMA possess much more negative charge and the CAM possess a little charge compared with CMA, hence, the electrophile [PhSe] group will favor to attack the CMA. Here, we could consider the substituents X together with benzene (X―Ph―) as a whole to be the electron-donating groups, i.e. these four reactions should obey the Markovnikov's rules according to the greater attraction between [PhSe] group and CMA, which are just in line with the experimental results7. We also can see that the qMA of 3-Cl and 4-Cl are less than that of the others, because the interaction between ―Cl, the electron-withdrawing group, and benzene, the electron-donating group, represents the character of the electron-donating group in nonpolar solution, so leading to that result, if the reactions react in polar solution, the result may be the opposite 7, however, the all calculations about these reactions were calculated at the gas state in vacuum. The charges of ABEEMσπ and ab initio method are in agreement by and large.

Then, we make use of the usual local softness and the generalized one from FDA method and the ABEEMσπ model combination with local HSAB principle to estimate the regioselectivities of above four additions, the values of ∆sMA, $\Delta {{s}_{\text{AM}}}$, $\Delta s_{\text{MA}}^{\text{G}}$ and $\Delta s_{\text{AM}}^{\text{G}}$ from these two methods are listed in Table 5, and the ratio of Markovnikov's product to anti-Markovnikov's from the experiment 7 are also listed in Table 5.

Table 5   The values of ∆sMA, ∆sAM, ∆sMAG and ∆sAMG for PhSeBr with substituted styrene (X-PhCH=CH2) at the level of MP2/6-311++G(d, p) with the finite difference approach and ABEEMσπ model.

 X―PhCH＝CH2 finite difference approach ABEEMσπ Model aMA: AM ∆sMA ∆sAM ∆sMAG ∆sAMG 104∆sMA 104∆sAM 102∆sMAG 102∆sAMG H 0.841 1.457 8.915 18.780 1.666 1.680 4.683 4.711 78:22 3-Cl 0.854 1.413 9.133 18.066 1.636 1.649 4.600 4.629 59:41 4-Cl 0.857 1.442 9.183 18.534 1.644 1.658 4.625 4.652 76:24 4-CH3 0.853 1.517 7.128 19.754 1.579 1.593 6.689 6.721 86:14

a Those reactions are reacting in benzene at 25 ℃.

It can be seen from that Table 5 the four $\Delta {{s}_{\text{MA}}}$ and $\Delta s_{\text{MA}}^{\text{G}}$ are all smaller than respective $\Delta {{s}_{\text{AM}}}$ and $\Delta s_{\text{AM}}^{\text{G}}$ under the FDA method. And the values of $\Delta {{s}_{\text{MA}}}$ and $\Delta s_{\text{MA}}^{\text{G}}$ from ABEEMσπ model are also smaller than their respective $\Delta {{s}_{\text{AM}}}$ and $\Delta s_{\text{AM}}^{\text{G}}$. Therefore, on basis of local HSAB principle, the Markovnikov's products should constitute the main products of these four reactions by means of FDA method and ABEEMσπ model, which results are in line with the experimental regioselectivities.

We can obtain two sequences: $\Delta s_{\text{MA}}^{\text{G}}(4\text{-C}{{\text{H}}_{3}})$ < $\Delta s_{\text{MA}}^{\text{G}}(\text{H})$ < $\Delta s_{\text{MA}}^{\text{G}}(3\text{-Cl})$ < $\Delta s_{\text{MA}}^{\text{G}}(4\text{-Cl})$ via FDA method and $\Delta s_{\text{MA}}^{\text{G}}(3\text{-Cl})$ < $\Delta s_{\text{MA}}^{\text{G}}(4\text{-Cl})$ < $\Delta s_{\text{MA}}^{\text{G}}(\text{H})$ < $\Delta s_{\text{MA}}^{\text{G}}(4\text{-C}{{\text{H}}_{3}})$ via ABEEMσπ model. According to the local HSAB principle, the smaller $\Delta {{s}^{\text{G}}}$ is, the easier reaction is. Luk and his coworkers 7 gave second-order rate constants, k, its unit being dm3∙mol-1∙s-1 of these reactions which are in order kH (2.58 ± 0.15) × 10-2, k3-Cl (1.57 ± 0.07) × 10-2, k4-Cl (2.20 ± 0.1) × 10-2, and k4-CH3 (2.77 ± 0.1) × 10-2, as shown in Fig. 2 (upper). The order of the experimental reaction rates is k3-Cl < k4-Cl < kH < k4-CH3. Therefore, the intermolecular reactivity predicted by ABEEMσπ model is just in a reverse order compared with the experimental rate constants.

### Fig 2

Fig 2   The line charts of the reaction rates (upper), the sG (middle) and sG- (lower) of CMA atoms in the four additions of benzeneselenyl bromide to the substituted styrenes.

Fig. 2 (middle) displays the line chart of the sG obtained from the ABEEMσπ model. And Fig. 2 (lower) represents the line charts of the sG- of the CMA atoms calculated by FDA method. It can be found from the Fig. 2 that the order of sMAG- is sMAG-(4-Cl) < sMAG-(3-Cl) < sMAG-(H) < sMAG-(4-CH3) (FDA method) and that of sMAG is sMAG(3-Cl) < sMAG(4-Cl) < sMAG(H) < sMAG(4-CH3) (ABEEMσπ model). The sequence via ABEEMσπ model is just in accord with that of the experimental reaction rates, but that of FDA method is not for the ―Cl substituted additions.

Therefore, the applications of the generalized reactivity descriptor combined with the local HSAB principle on this series of electrophilic additions demonstrate that both the values of $\Delta s$ and $\Delta {{s}^{\text{G}}}$ from ABEEMσπ model can forecast their regioselectivities and only the values of center atoms' generalized local softness of substituted ethenes calculated by ABEEMσπ model can rationalize the reaction rate constants rather than the difference of center atoms' generalized local softness of the two reactants. However, the results of the finite difference approximation are not well related to the experimental results.

## 5 Conclusions

For the addition reactions of HCl to the substituted ethenes and benzeneselenyl bromide to the substituted styrenes, according to the local HSAB principle, the values of the softness differences, $\Delta s$, in terms of ab initio method at the level of MP2/6-311++G(d, p) with the finite difference approximation (FDA) method and the ABEEMσπ model have been used to relate to their regioselectivities.

As the performance of the generalized reactivity descriptor, it is shown that the CMA atoms of all reactions prefer to be attacked in terms of ABEEMσπ model, which is in agreement with the experimental results. But, the results of FDA method can not obtain such good indication. However, it is shown that there are two inverse orders, compared with the orders of experimental rate constants for these two series of electrophilic additions by using the ΔsG from FDA and ABEEMσπ model. In fact, only generalized local softness (GLS) of center atoms can be related to the orders of the experimental reaction rate constants by both the FDA method and the ABEEMσπ model except the results of the FDA method for 3-Cl substituted addition with a little flaw.

Up to now, we have applied the generalized reactivity descriptors to study on several kinds of reactions, such as to predict the regio-and stereoselectivity of Diels-Alder reactions and to correlate their reaction rate constants, to rationalize the intermolecular reactivities and regioselectivities of enzymatic catalyzed nucleophilic reactions, etc. Moreover, we will continue to apply the generalized Fukui function and the generalized local softness to investigate other systems and to further check their rationality and validity.

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