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## Application of Natural Orbital Fukui Functions and Bonding Reactivity Descriptor in Understanding Bond Formation Mechanisms Underlying [2+4] and [4+2] Cycloadditions of o-Thioquinones with 1, 3-Dienes

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 基金资助: 国家自然科学基金.  21403097中央高校基本科研业务费专项资金.  lzujbky-2016-45

Corresponding authors: ZHOU Panpan, Email: zhoupp@lzu.edu.cn; Tel.: +86-931-8912862

 Fund supported: theNationalNaturalScienceFoundationofChina.  21403097the Fundamental Research Funds for the Central Universities, China.  lzujbky-2016-45

Abstract

o-Thioquinones can undergo either [2+4] or [4+2] cycloaddition reactions with acyclic dienes. To illustrate the bonding processes in these cycloadditions, the natural orbital Fukui function (NOFF) and bonding reactivity descriptor have been employed. The electrophilicity of a bond or an orbital in the o-thioquinone as well as in the acyclic diene has been found using the NOFF, which suggests that electron transfer takes place from an electron-donating bonding orbital to an electron-accepting antibonding/bonding orbital, leading to the cyclic product via the formation of a circular loop and two covalent bonds. The bonding reactivity descriptor shows that covalent bonds readily form between atom k1 of one molecule with a large fk1+ value and atom k2 of another molecule with a large fk2- value. Both the NOFF and the bonding reactivity descriptor are efficient tools for interpreting the mechanism underlying the [2+4] and [4+2] cycloaddition between o-thioquinones and acyclic dienes.

Keywords： Fukui function ; Natural orbital Fukui function (NOFF) ; Bonding reactivity descriptor ; Cycloaddition ; Nucleophilic/electrophilic attack

YAN Chaoxian, YANG Fan, WU Ruizhi, ZHOU Dagang, YANG Xing, ZHOU Panpan. Application of Natural Orbital Fukui Functions and Bonding Reactivity Descriptor in Understanding Bond Formation Mechanisms Underlying [2+4] and [4+2] Cycloadditions of o-Thioquinones with 1, 3-Dienes. Acta Physico-Chimica Sinica[J], 2018, 34(5): 497-502 doi:10.3866/PKU.WHXB201709222

## 1 Introduction

Mono-ortho-thioquinones 1-4 with the general formula 1 can serve as electron-poor heterodienes to react with many alkenes with the general formula 2 which act as electron-rich dienophiles via either a [2+4] 5 or a [4+2] 6-13 cycloaddition, leading to the formation of the spiro derivative (3) or the benzoxathiin cycloadduct (4) as the main product, as depicted in Scheme 1. Theoretical and experimental investigations of the [2+4] and [4+2] reaction mechanisms were also carried out by Menichetti and coworkers, and they suggested that the reactions of o-thioquinones with acyclic dienes undergo the [2+4] path and are kinetically favored while the reactions of o-thioquinones with cyclic dienes go through the [4+2] path and are thermodynamically favored 14, 15. These reactions stimulate our great interest in exploring the mechanisms of their different bonding processes.

### Scheme 1

Scheme 1   Reactions of mono-ortho-thioquinones and alkenes via either [2+4] or [4+2] cycloaddition.

Chemical reaction involves the bond breakage and formation, so using the chemical reactivity descriptor of a bond or an orbital to reveal the bonding process would be helpful for understanding the bonding process of the reaction. Recently, a new type of condensed Fukui function based on the natural bond orbital theory (NBO 16) for describing the chemical reactivity of a bond or an orbital, the so-called natural orbital Fukui function (NOFF) was proposed 17, which can effectively interpret the bond formation mechanism 17, 18. On the other hand, more recently, the bonding reactivity descriptor based on the condensed-to-atom Fukui function 19, 20 was proposed, and it is capable of evaluating the bonding trend between two atoms. Accordingly, in this work, we will apply NOFF and bonding reactivity descriptor to the [2+4] and [4+2] cycloadditions of o-thioquinones with 1, 3-dienes to elucidate and assess the different bonding processes.

## 2 Theoretical frameworks

For NOFF, the fnbo+ or fnbo- function indicates the electronic response of a natural bond orbital upon electron addition or removal (N is the total number of electrons), respectively 17. The positive fnbo+ or fnbo- suggests that the orbital can accept or donate electrons, respectively. While the negative fnbo+ or fnbo- may not necessarily imply the electron philicity of an orbital, meaning that the orbital does not accept or donate electrons, respectively.

$f_{{\rm{nbo}}}^ + = \left( {\frac{{\partial {n_{{\rm{nbo}}}}}}{{\partial N}}} \right)_{v{\rm{(r)}}}^ + = n_{{\rm{nbo}}}^{N + 1}-n_{{\rm{nbo}}}^N$

$f_{{\rm{nbo}}}^-= \left( {\frac{{\partial {n_{{\rm{nbo}}}}}}{{\partial N}}} \right)_{v{\rm{(r)}}}^-= n_{{\rm{nbo}}}^N-n_{{\rm{nbo}}}^{N - 1}$

The Fukui functions can be represented by the change in the chemical potential μ due to perturbations in the external potential ν(r) 21, 22,

$f(r) = {\left( {\frac{{\delta \mu }}{{\delta \nu (r)}}} \right)_N}$

For chemical reaction, favorable reactions are closely related to the stabilization of the frontier molecular orbitals 23, 24, which can be referred to the "|dμ| big is good" rule 25. Favorable electron-transfer interactions between reactants are related to the overlap between the Fukui functions of the reactive sites 26-28. In a chemical reaction, to stabilize the frontier electrons, the chemical potential μ should have the maximum change, so the reaction should occur between the charge-accepting site with the biggest fα+ (or fβ+) and the charge-donating site with the biggest fβ- (or fα-), meaning that:

$\Delta \mu \propto \max f_{\rm{ \mathsf{ α} }}^ + + \max f_{\rm{ \mathsf{ β} }}^{\rm{ - }}$

${\rm{or}}\;\Delta \mu \propto \max f_{\rm{ \mathsf{ β} }}^ + + \max f_{\rm{ \mathsf{ α} }}^ -$

For the reaction between two reactants (M1 and M2), their reaction sites (α and β) conform to the following energy expression 26-28:

$\int {\frac{{f_{{\rm{M}}1}^ + ({r_1})f_{{\rm{M}}2}^ - ({r_2})}}{{\left| {{r_1} - {r_2}} \right|}}{\rm{d}}{r_1}{\rm{d}}{r_2} \approx \sum\limits_{{\rm{ \mathsf{ α} }} \in {{\rm{M}}2}} {\sum\limits_{{\rm{ \mathsf{ β} }} \in {{\rm{M}}2}} {\frac{{f_\alpha ^ + f_\beta ^ - }}{{\left| {{r_{1;{\rm{ \mathsf{ α} }}}} - {r_{2;{\rm{ \mathsf{ β} }}}}} \right|}}} } } \le \frac{1}{4}\sum\limits_{{\rm{ \mathsf{ α} }} \in {{\rm{M}}2}} {\sum\limits_{{\rm{ \mathsf{ β} }} \in {{\rm{M}}2}} {\frac{{{{(f_{\rm{ \mathsf{ α} }}^ + + f_{\rm{ \mathsf{ β} }}^ - )}^2}}}{{\left| {{r_{1;{\rm{ \mathsf{ α} }}}} - {r_{2;{\rm{ \mathsf{ β} }}}}} \right|}}} }$

or

$\int {\frac{{f_{{\rm{M}}1}^ + ({r_1})f_{{\rm{M}}2}^ - ({r_2})}}{{\left| {{r_1} - {r_2}} \right|}}{\rm{d}}{r_1}{\rm{d}}{r_2} \approx \sum\limits_{{\rm{ \mathsf{ α} }} \in {{\rm{M}}2}} {\sum\limits_{{\rm{ \mathsf{ β} }} \in {{\rm{M}}2}} {\frac{{f_{\rm{ \mathsf{ α} }}^ + f_{\rm{ \mathsf{ β} }}^ - }}{{\left| {{r_{1;{\rm{ \mathsf{ α} }}}} - {r_{2;{\rm{ \mathsf{ β} }}}}} \right|}}} } } \le \frac{1}{4}\sum\limits_{{\rm{ \mathsf{ α} }} \in {{\rm{M}}2}} {\sum\limits_{{\rm{ \mathsf{ β} }} \in {{\rm{M}}2}} {\frac{{{{(f_{\rm{ \mathsf{ α} }}^ + + f_{\rm{ \mathsf{ β} }}^ - )}^2}}}{{\left| {{r_{1;{\rm{ \mathsf{ α} }}}} - {r_{2;{\rm{ \mathsf{ β} }}}}} \right|}}} }$

The reaction sites (α and β) are the bonding sites which occurs between M1 and M2, so the bonding between α and βtermed as fbonding can be written as the sum of their Fukui functions. The large value of fbonding is favorable because the arithmetic/geometric mean inequality indicates that ¼(fbonding)2 is an upper bound to the product of the Fukui functions. Therefore, the maximum change in chemical potential associated with electron transfer from site α (or β) to site β (or α) has the following relationship:

$f_{{\rm{bonding}}}^ \pm = {\rm{max}}f_{\rm{ \mathsf{ α} }}^ + + {\rm{max}}f_{\rm{ \mathsf{ β} }}^ - \sim \Delta \mu$

${\rm{or}}\;f_{{\rm{bonding}}}^ \pm = {\rm{max}}f_{\rm{ \mathsf{ β} }}^ - + {\rm{max}}f_{\rm{ \mathsf{ α} }}^ + \sim \Delta \mu$

The condensed-to-atom Fukui function for an atom k in a molecule can be derived from the difference in atomic charges 19, 20:

$f_k^ + = {q_k}(N)-{q_k}(N + 1), {\rm{governing}}\;{\rm{nucleophilic}}\;{\rm{attack}}$

$f_k^-= {q_k}(N-1)-{q_k}(N), {\rm{governing}}\;{\rm{electrophilic}}\;{\rm{attack}}$

where qk is the electronic charge of atom k and N is the number of electrons. These two condensed Fukui functions characterize the reactivity preferences for nucleophilic and electrophilic attacks on atom k, respectively.

In terms of the theoretical basis of fbonding mentioned above, it can be supposed that the bonding formation process can be represented using condensed-to-atom Fukui functions. The bonding would occur between the atom k1of M1 and the atom k2of M2 when the atoms k1and k2 possess large values of Fukui functions fk1+ (or fk1) and fk-2 (or fk2+). Thereby, the sum of their Fukui functions should be as large as possible. Consequently, the descriptor fbonding which reveals the bonding trend for nucleophilic/electrophilic reaction can be written as:

$f_{{\rm{bonding}}}^ \pm = f_{k1}^ + + f_{k2}^-$

It is termed as the bonding reactivity descriptor 29. Accordingly, the bonding formation would take place when the value of the fbonding+ quantity is as large as possible.

## 3 Computational methods

All calculations in this work were performed using the Gaussian 09 package 30. The molecular geometries of the studied systems were fully optimized using the B3LYP functional 31, 32 with 6-31+G(d, p) basis set. NBO 16, 33, 34 analysis was implemented at the same computational level, which can provide the natural orbital occupancy and natural atomic charge.

### 4 Results and discussion

The representative [2+4] and [4+2] cycloadditions of o-thioquinone (R1) with 1, 3-dienes (R2, R2') investigated previously 15 were selected for this study, as shown in Fig. 1. Different from the previous study, in this work, we analyzed the bonding formation mechanisms from the viewpoints of NOFF and bonding reactivity descriptor.

### Fig 1

Fig 1   The optimized geometries for o-thioquinone (R1) and 1, 3-dienes (R2, R2′) with some atoms numbered.

### 4.1 Perspective from NOFFs

For [2+4] cycloaddition of R1 with R2, one C＝S double bond in R1 and two C＝C double bonds in R2 directly participate in the reaction, they reorganize each other to form a six-membered ring containing one C＝C double bond and two new sigma bonds (i.e., C―C and C―S bonds). With regard to [4+2] cycloaddition of R1 with R2', the C＝S and C＝O double bonds in R1 react with one C=C double bond in R2' to give a six-membered ring containing one C＝C double bond and two new sigma bonds (i.e., C―S and C―O bonds). The nucleophilic or electrophilic nature of the double bond or its natural bond orbital involved in [2+4] and [4+2] cycloadditions determines the reaction process. Thereby, the reactivities of these double bonds will be assessed using the two NOFFs of fnbo+ and fnbo-.

The fnbo+ and fnbo- values for the C＝S and C＝O double bonds in R1 and for the C=C double bonds in R2 and R2' are summarized in Table 1. For the C＝S double bond in R1, its BD(1)C1―S1 (fnbo+ = 0.00004) is able to accept electrons while the BD(2)C1―S1 (fnbo- = 0.92991) is able to donate electrons, the BD*(1)C1–S1 is inactive and the BD*(2)C1―S1 (fnbo- = 0.02222) seems to be electron-donating. With respect to the C=O double bond in R1, both its BD(1)C2―O2 (fnbo- = 0.00008) and BD(2)C2―O2 (fnbo- = 0.01660) are able to donate electrons, the BD*(1)C2―O2 is inactive and the BD*(2) C2―O2 (fnbo- = 0.01291) seems to be electron-donating. A bonding orbital has the ability to accept electrons which will strengthen the bond, but it is energetically unfavorable for an antibonding orbital in donating electrons, so the electron-donating character of an antibonding orbital will be disregarded herein. For the C1＝C2 double bond in R2, the BD(1)C1―C2 is inactive and BD(2)C1―C2 (fnbo- = 0.96019) has the ability of donating electrons, but its BD*(1)C1―C2 is amphiphilic (fnbo+ = 0.00046, fnbo- = 0.00196) and the BD*(2)C1–C2 (fnbo- = 0.03795) seems to be electron-donating. The C3＝C4 double bond in R2 has the similar reactivity features to the C1＝C2 double bond (Table 1). The C3＝C4 double bond is located at the end of the R2, so it is more susceptible to being attacked by R1 than the C1＝C2 double bond. According to these values, in the [2+4] cycloaddition of R1 with R2 (Fig. 2), the BD(2)C3―C4 bonding orbital of R2 donates electrons to the BD(1)C1―S1 bonding orbital of R1, and the BD*(1)C1―C2 antibonding orbital of R2 accepts electrons from the BD(2)C1―S1 bonding orbital of R1. As a result, a circular loop forms and leads to the six-membered ring accompanied by the formations of two new covalent bonds (i.e., C―S and C―C bonds).

### Fig 2

Fig 2   Proposed mechanism of [2+4] cycloaddition of R1 with R2.

Table 1   NOFFs (unit in electrons) of the C1＝S1 and C2＝O2 double bonds in R1 and of the C1＝C2 and C3＝C4 double bonds in R2 and R2', and the reactivities of their natural bond orbitals based on the NOFF values a.

 Natural bond orbital fnbo+ fnbo- Reactivityb R1 BD(1)C1―S1 0.00004 -0.00119 nucleophilic BD(2)C1―S1 -0.96474 0.92991 electrophilic BD*(1)C1―S1 -0.00058 -0.00208 inactive BD*(2)C1―S1 -0.12195 0.02222 electrophilic BD(1)C2―O2 -0.00025 0.00008 electrophilic BD(2)C2―O2 -0.97636 0.01660 electrophilic BD*(1)C2―O2 -0.00021 -0.00034 inactive BD*(2)C2―O2 -0.10270 0.01291 electrophilic R2 BD(1)C1–C2 -0.00191 -0.00202 inactive BD(2)C1―C2 -0.96562 0.96019 electrophilic BD*(1)C1―C2 0.00046 0.00196 amphiphilic BD*(2)C1―C2 -0.04328 0.03795 electrophilic BD(1)C3―C4 -0.00141 -0.00348 inactive BD(2)C3―C4 -0.96961 0.96052 electrophilic BD*(1)C3–C4 0.00127 0.00198 amphiphilic BD*(2)C3―C4 -0.04842 0.03799 electrophilic R2' BD(1)C1―C2 -0.00209 -0.00273 inactive BD(2)C1―C2 -0.95650 0.95200 electrophilic BD*(1)C1―C2 0.00052 0.00206 amphiphilic BD*(2)C1―C2 -0.05322 0.05365 electrophilic BD(1)C3―C4 -0.00165 -0.00292 inactive BD(2)C3―C4 -0.95971 0.94557 electrophilic BD*(1)C3―C4 0.00113 0.00189 amphiphilic BD*(2)C3―C4 -0.05703 0.04618 electrophilic

aBD denotes bonding orbital; BD* denotes antibonding orbital. For BD and BD*, (1) denotes σ orbital, (2) denotes π orbital. b The reactivity of a bond or an orbital based on its fnbo+ and fnbo- functions can be derived from the definitions of NOFFs in Ref.17.

With regard to the C1＝C2 double bond in R2', the BD(1)C1―C2 is inactive and BD(2)C1―C2 (fnbo- = 0.95200) can donate electrons, the BD*(1)C1―C2 (fnbo+ = 0.00052, fnbo- = 0.00206) is amphiphilic and the BD*(2)C1―C2 (fnbo- = 0.05365) seems to be electron-donating. The C3＝C4 double bond has the similar reactivity features. Noticeably, different from the C1＝C2 double bond, the C3＝C4 double bond is located at the end of the R2', so it is more easily to be attacked by R1. Therefore, in the [4+2] cycloaddition of R1 with R2' (Fig. 3), the BD(2)C3―C4 bonding orbital of R2' donates electrons to the BD(1)C1–S1 bonding orbital of R1, and the BD*(1)C3―C4 antibonding orbital of R2' accepts electrons from the C2―O2 bonding orbital (e.g., BD(1)C2―O2 and BD(2)C2―O2) of R1. Consequently, the circular loop in leading to the six-membered ring forms which accomplishes the formations of two new covalent bonds (i.e., C―S and C―O bonds).

### Fig 3

Fig 3   Proposed mechanism of [4+2] cycloaddition of R1 with R2'.

### 4.2 Perspective from bonding reactivity descriptor

In this section, the negative Fukui function will not be considered because a negative Fukui function usually comes from the inability of a molecule to accommodate orbital relaxation caused by the changed number of electrons and/or improper charge partitioning techniques 35-37 or distorted molecular structures 38, 39. As summarized in Table 2, the C1 atom of R1 has the largest fk+ value (0.111) compared to other C atoms, indicating it is more susceptible to nucleophilic attack. The S1 and O2 atoms of R1 possessing large fk+ values (i.e., 0.123 and 0.311, respectively) mean their abilities to be susceptible to nucleophilic attack, while the O2 atom possessing the largest fk- value (0.543) suggests that it is more susceptible to electrophilic attack. For R2 or R2', the largest fk+ and fk- values are observed for its C4 atom, followed by the C1 atom possessing the larger fk+ and fk- values. Their C3 atoms also have large fk+ values. For the reaction between R1 and R2, according to the Fukui function values, it can be seen that the S1 atom of R1 and C4 atom of R2 readily forms C―S bond due to the large f bonding[±] value (i.e., fbonding± = fS1++fC4-), while the C1 atom of R1 and C1 atom of R2 forms C―C bond (fbonding± = fC1++fC1-). Although the O2 atom of R1 has the largest fk+ and fk- values, the C2 atom bonded with O2 has lower reactivity due to the smaller fk+ and fk- values. Both the C1 and S1 atoms of R1 with good reactivity render them to react with R2 via [2+4] cycloaddition. The C1＝C2 and C3＝C4 double bonds of R2' are different from those of R2, the steric effect makes the [4+2] cycloaddition more favorable, so only one C＝C double bond interacts with R1. The C4 atom of R2' has larger fk+ and fk- values than the C1 atom, and the C3 atom has larger fk+ value than the C2 atom, so the C3＝C4 double bond located at the end of R2' is more susceptible to being attacked by R1. Therefore, the bonding processes occur between S1 atom of R1 and C4 atom of R2' (fbonding± = fS1++fC4-), and between O2 atom of R1 and C3 of R2' (fbonding± = fC3++f O2-). The possible bonding mechanisms for [2+4] and [4+2] cycloadditions are shown in Figs. 4 and 5, respectively.

### Fig 4

Fig 4   Possible bonding mechanisms for [2+4] cycloaddition of R1 with R2.

### Fig 5

Fig 5   Possible bonding mechanisms for [4+2] cycloaddition of R1 with R2'.

Table 2   Fukui functions for atoms in the molecules R1, R2 and R2'.

 Molecule Atom (k) fk+ fk- R1 C1 0.111 0.011 C2 0.065 0.008 C3 0.009 0.028 C4 0.070 0.023 C5 0.030 0.041 C6 -0.018 -0.004 C7 0.060 0.048 C8 0.023 0.020 C9 0.075 0.073 C10 0.017 0.030 S1 0.123 0.066 O2 0.311 0.543 R2 C1 0.216 0.220 C2 0.088 0.112 C3 0.102 0.089 C4 0.338 0.295 C5 -0.012 -0.032 C6 -0.009 -0.022 R2′ C1 0.220 0.254 C2 0.065 0.098 C3 0.106 0.078 C4 0.339 0.305 C5 -0.010 -0.035 C6 -0.008 -0.025

## 5 Conclusions

In this work, natural orbital Fukui function (NOFF) which characterizes the electronic philicity of a bond or an orbital in a molecular system and bonding reactivity descriptor derived from Fukui functions which characterizes the bonding trend have been employed to explain the mechanisms of reactions between o-thioquinones and acyclic dienes via either [2+4] or [4+2] cycloaddition. NOFFs show that a bonding orbital with the electron-donating ability of one reactant interacts with an antibonding or bonding orbital with the electron-accepting ability of the other one, and vice versa. The electron transfer from an electron-donating bonding orbital to an electron-accepting antibonding/bonding orbital leads to the formation of a circular loop which is accompanied by the formations of two covalent bonds and thus the cyclic product. From bonding reactivity descriptor, the atom k1 of one molecule with a large value in fk1+ can readily form a covalent bond with atom k2of another molecule with a large value in fk2-, and the mechanism of [2+4] as well as [4+2] cycloaddition between o-thioquinone and acyclic diene is well interpreted. These results further suggest that NOFF and bonding reactivity descriptor are efficient Fukui functions within the framework of conceptual density functional theory (CDFT) 20, 40, 41 or density functional reactivity theory (DFRT) 25, 42-44 in explaining the bonding process of chemical reaction.

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Chapman O. L. ; McIntosh C. L. J. Chem. Soc. D 1971, 383.

Mayo P. D. ; Weedon A. C. ; Wong G. S. K. J.Org. Chem. 1979, 44, 1977.

Schulz R. ; Schweig A. Angew. Chem. Int. Ed. 1981, 20, 570.

Naghipur A. ; Reszka K. ; Sapse A.-M. ; Lown J. W. J.Am. Chem. Soc. 1989, 111, 258.

Capozzi G. ; Dios A. ; Franck R. W. ; Geer A. ; Marzabaldi C. ; Menichetti S. ; Nativi C. ; Tamarez M. Angew. Chem. Int. Ed. 1996, 35, 777.

Capozzi G. ; Falciani C. ; Menichetti S. ; Nativi C. J.Org. Chem. 1997, 62, 2611.

Capozzi G. ; Falciani C. ; Menichetti S. ; Nativi C. ; Raffaelli B. Chem. Eur. J. 1999, 5, 1748.

Capozzi G. ; Lo Nostro P. ; Menichetti S. ; Nativi C. ; Sarri P. Chem. Commun. 2001, 551.

Nair V. ; Mathew B. ; Radharkrishnan K. V. ; Rath N. P. Synlett 2000, 61.

Nair V. ; Mathew B. Tetrahedron Lett. 2000, 41, 6919.

Nair V. ; Mathew B. ; Rath N. P. ; Vairamani M. ; Prabhakar S. Tetrahedron 2001, 57, 8349.

Nair V. ; Mathew B. ; Menon R. S. ; Mathew S. ; Vairamani M. ; Prabhakar S. Tetrahedron 2002, 58, 3235.

Nair V. ; Mathew B. ; Thomas S. ; Vairamani M. ; Prabhakar S. J. Chem. Soc., Perkin Trans. 1 2001, 3020.

Menichetti S. ; Viglianisi C. Tetrahedron 2003, 59, 5523.

Contini A. ; Leone S. ; Menichetti S. ; Viglianisi C. ; Trimarco P. J.Org. Chem. 2006, 71, 5507.

Reed A. E. ; Curtiss L. A. ; Weinhold F. Chem. Rev. 1988, 88, 899.

Zhou P. ; Ayers P. W. ; Liu S. ; Li T. Phys. Chem. Chem. Phys. 2012, 14, 9890.

Priya A. M. ; Senthilkumar L. RSC Adv. 2014, 4, 23464.

Yang W. ; Mortier W. J. J.Am. Chem. Soc. 1986, 108, 5708.

Liu S. Acta Phys. -Chim. Sin. 2009, 25, 590.

Parr R. G. ; Yang W. J.Am. Chem. Soc. 1984, 106, 4049.

Yang W. ; Parr R. G. ; Pucci R. J.Chem. Phys. 1984, 81, 2862.

Albright T. A. ; Burdett J. K. ; Whangbo M. H.

Orbital Interactions in Chemistry

Wiley-Interscience: New York, NY, USA 1985.

Fujimoto H. ; Fukui K. ; Klopman G.

Chemical Reactivity and Reaction Paths

Wiley-Interscience: New York, NY, USA 1974.

Parr R. G. ; Yang W.

Density-Functional Theory of Atoms and Molecules

Oxford University Press: New York, NY, USA 1989.

Anderson J. S. M. ; Melin J. ; Ayers P. W. J.Chem. Theory Comput. 2007, 3, 358.

Ayers P. W. Faraday Discuss. 2007, 135, 161.

Berkowitz M. J.Am. Chem. Soc. 1987, 109, 4823.

Zhou P.-P. ; Liu S. ; Ayers P. W. ; Zhang R. -Q. J. Chem. Phys. 2017.

Frisch M. J. ; Trucks G. W. ; Schlegel H. B. ; Scuseria G. E. ; Robb M. A. ; Cheeseman J. R. ; Scalmani G. ; Barone V. ; Mennucci B. ; Petersson G. A. ; et al Gaussian 09, Revision D., 01, Gaussian, Inc.: Wallingford CT 2013.

Becke A. D. J.Chem. Phys. 1993, 98, 5648.

Lee C. ; Yang W. ; Parr R. G. Phys. Rev. B 1988, 37, 785.

Foster J. P. ; Weinhold F. J.Am. Chem. Soc. 1980, 102, 7211.

Reed A. E. ; Weinstock R. B. ; Weinhold F. J.Chem. Phys. 1985, 83, 735.

Roy R. K. ; Hirao K. ; Krishnamurty S. ; Pal S. J.Chem. Phys. 2001, 115, 2901.

Roy R. K. ; Hirao K. ; Pal S. J.Chem. Phys. 2000, 113, 1372.

Roy R. K. ; Pal S. ; Hirao K. J.Chem. Phys. 1999, 110, 8236.

Bultnick P. ; Carbó-Dorca R. J.Math. Chem. 2003, 34, 67.

Bultnick P. ; Carbó-Dorca R. ; Langenaeker W. J.Chem. Phys. 2003, 118, 4349.

Ayers P. W. ; Morell C. ; De Proft F. ; Geerlings P. Chem. Eur. J. 2007, 13, 8240.

Geerlings P. ; Proft F. D. ; Langenaeker W. Chem. Rev. 2003, 103, 1793.

Chattaraj P. K. ; Sarkar U. ; Roy D. R. Chem. Rev. 2006, 106, 2065.

Chermette H. J.Comput. Chem. 1999, 20, 129.

Parr R. G. ; Yang W. Annu. Rev. Phys. Chem. 1995, 46, 701.

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