DOI: 10.1002/ajoc.201200103

A Frontier Molecular Orbital Theory Approach to Understanding the MayrEquation and to Quantifying Nucleophilicity and Electrophilicity by Using

HOMO and LUMO Energies**

Lian-Gang Zhuo, Wei Liao, and Zhi-Xiang Yu*[a]

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Introduction

Understanding the reactivity of a chemical species, whichcould be either a stable or an active species (generated insitu or in a reaction as an active intermediate, or a theoreti-cally designed one), is one of the fundamental questions inchemistry. The development of theories to explain the fac-tors affecting reactivities is highly prized to understand whya reaction happens and how fast a reaction can be. Similar-ly, obtaining quantitative indexes of reactivities is also cru-cial in rational design, because these indexes can helpchemists calculate and predict the reaction rates of new ordesigned reactions. For these reasons, many efforts havebeen made by chemists to develop quantitative descriptorsof reactivities.[1]

The development of understanding reactivity can betraced back to the 1930s, when Ingold introduced the con-cepts of nucleophilicity and electrophilicity to quantify thestrength of nucleophiles and electrophiles.[2] Later, severalattempts to quantitatively describe molecules according tothis general concept have been proposed, for example bySwain and Scott,[3] Edwards,[4] and Ritchie.[5] In the 1990s,Mayr and co-workers developed an equation [logk=

s ACHTUNGTRENNUNG(E+N)] that can be used very broadly to measure nucleo-philicities (N) and electrophilicities (E).[6,7] Recently, theMayr equation, as it is now known, has been successfullyapplied to a broad range of reactions, thus greatly expand-ing the knowledge of the reactivities of molecules.[6]

Several theoretical and computational approaches havebeen devoted to understanding the Mayr equation and tocompute Mayr nucleophilicities and electrophilicities.[8]

Schindele, Houk, and Mayr conducted DFT calculations onthe affinities of benzhydryl cations (XC6H4)2CH+ for themethyl anion, hydroxide, and hydride anion and comparedthe computed values of rate and equilibrium constants withthose obtained experimentally in solution, revealing an ex-cellent linear correlation between the electrophilicity pa-rameters (E) and the calculated methyl anion affinities.[8a]

Contreras et al. proposed a series of models based onglobal reactivity indexes (w) defined by Parr et al. in termsof the electronic chemical potential and the chemical hard-ness.[8b–h] Some interesting features of the free-energy rela-tionships were found between N and w�, and between Eand w+ . Direct ab initio transition-state calculations wereperformed by Liu et al. to predict the nucleophilicity pa-rameters (N) for p nucleophiles in CH2Cl2.

[8i] In their study,the authors also used energies of the highest occupied mo-lecular orbitals (HOMOs) and lowest unoccupied molecu-lar orbitals (LUMOs) to describe the reactivities, and theyfound that LUMO and E are linearly correlated butHOMO and N are not. The reasons behind these resultshave not been well investigated, even though steric factorswere used to explain these effects. Despite these computa-tional efforts to rationalize the reactivity parameters, a theo-retical foundation and further understanding of the Mayrequation are needed to significantly advance our under-standing of reactivities.[6g]

[a] L.-G. Zhuo, W. Liao, Prof. Dr. Z.-X. YuBeijing National Laboratory for Molecular Sciences (BNLMS)Key Laboratory of Bioorganic Chemistry andMolecular Engineering of Ministry of EducationCollege of Chemistry, Peking UniversityBeijing 100871 (China)Fax: (+86) 10-6275-1708E-mail : yuzx@pku.edu.cn

[**] HOMO =highest occupied molecular orbital, LUMO = lowest unoc-cupied molecular orbital.

Supporting information for this article is available on the WWWunder http://dx.doi.org/10.1002/ajoc.201200103.

Abstract: Obtaining the reactivities(such as nucleophilicities and electro-philicities) of molecules is of funda-mental importance in chemistry. Mayrand co-workers have developed theMayr equation, which has been widelyused to quantify nucleophilicity andelectrophilicity. Herein we proposea theoretical understanding of theMayr equation based on frontier mo-lecular orbital (FMO) theory and theEyring equation of the transition statetheory, showing that the nucleophilici-ty of a molecule is related to theenergy of this molecule�s highest oc-cupied molecular orbital (HOMO),while the electrophilicity is related tothe energy of the lowest unoccupiedmolecular orbital (LUMO) of theelectrophile. Consequently, we pro-

pose a new approach by combiningthe FMO theory and the Mayr equa-tion to predict the reactivities of newmolecules. Ab initio calculation re-sults support these linear relationshipsbetween LUMO energies and theMayr electrophilicities (E) and theHOMO energies and the Mayr nucle-ophilicities (N) for sets of electro-philes and nucleophiles, respectively.For each set of nucleophiles or elec-trophiles, their different reactivitiesare mainly controlled by the electron-ic effects of the substituents. If othereffects, such as sterics, affect reactivityfor a set of electrophiles or nucleo-philes, the linear relationships be-tween HOMO levels and N valuesand LUMO levels and E valuescannot be secured. The present ap-

proach through combining Mayrequation and the quantitative FMOtheory suggests that the Mayr nucleo-philicity or electrophilicity of a newmolecule, which could be an inter-mediate of a reaction, unstable reac-tant, or a hypothetical reactive spe-cies, can be obtained through ab initiocalculations of the frontier molecularorbital energies, and this will greatlyexpand the data sets of Mayr nucleo-philicities and electrophilicities.

Keywords: density functionalcalculations · electrophilicity ·FMO theory · Mayr equation ·nucleophilicity

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Frontier molecular orbital (FMO) theory is widely andextensively used to explain reactivities and stereo- and re-gioselectivities.[9,10] The FMO theory assumes that the reac-tion rate of a bimolecular reaction is influenced by the en-ergies of HOMO and LUMO orbitals of the reactants. Ina substitution reaction, for example, FMO theory indicatesthat the reaction rate is proportional to the energy gap be-tween the HOMO of the nucleophile and LUMO of theelectrophile. The smaller this gap is, the greater the interac-tion between HOMOnu and LUMOel in stabilizing the tran-sition state, and consequently the faster the nucleophilic re-action is. The shape of these frontier molecular orbitals, re-flected by the orbital coefficients, should also be taken intoaccount when discussing the relative reactivities of differentreaction sites in the reactants (related to the regiochemis-try). Even though FMO theory is very successful in ad-dressing reactivity, it can only qualitatively explain or pre-dict a reaction rate. Only limited quantitative analysis usingFMO has been reported.[9h,i]

Considering that the Mayr equation can give quantitativeindexes of reactivities while FMO theory gives qualitativeones, we wondered whether there is an inherent relation-ship between the Mayr equation and the FMO theory, andconsequently if the Mayr equation and its nucleophilicityand electrophilicity can be understood by the FMO theory.We further wondered whether the Mayr equation and theFMO theory can be combined to provide a new approachto quantify or semi-quantify reactivities. If this could beachieved, we could get the Mayr nucleophilicities and elec-trophilicities of new species by just computing these spe-cies� frontier molecular orbital energies, and consequentlywe could greatly expand the data sets of Mayr nucleophilic-

ities and electrophilicities. Herein, we show that the Mayrequation and FMO theory are equivalent if electronic ef-fects dominate reactivities of the studied molecules. We fur-ther demonstrate that the nucleophilicity and electrophilici-ty in the Mayr equation are related to the HOMO andLUMO energies of a nucleophile and an electrophile, re-spectively. A general quantitative scheme of reaction ratewith regard to HOMO of a nucleophile and LUMO of anelectrophile is also proposed.

Results and Discussion

1. FMO Understanding of the Mayr Equation

Let�s begin by showing that the Mayr equation can be un-derstood by the FMO theory when the reactivities of a setof molecules are considered. These molecules could beeither electrophiles or nucleophiles with different reactivi-ties mainly dictated by the electronic effects of their differ-ent substituents.

log ki=j ¼ sj Ei þN j

� �ð1Þ

ln ki=j ¼ 2:303 sj Ei þN j

� �ð1aÞ

In the Mayr equation (1), ki/j is the experimentally mea-sured rate constant between electrophile i (i= a, b, c, …)and nucleophile j (j= A, B, C, …), sj is a nucleophile-specif-ic sensitivity parameter for the nucleophile j, Nj is the nu-cleophilicity of nucleophile j and Ei is the electrophilicity ofelectrophile i.

Before we derive the theoretical basis, let�s first reviewhow the Mayr equation was derived experimentally(Scheme 1).[6c] Initially, Mayr and co-workers measured thereaction rate of (p-MeOC6H4)2CH+ (a) with 2-methyl-1-pentene (A). By defining the nucleophile-specific parame-ter sA =1 for nucleophile A and defining the electrophilici-ty of electrophile a as Ea =0, they obtained the nucleophi-licity of A as NA =0.95 by entering the measured reactionrate into Equation (1). Then they measured the reactionrates of a series of electrophiles of b–v (see structures ofthese species in the Supporting Information) toward nucle-ophile A. By using these experimental rates in Equa-tion (1), they obtained the Ei values of all these electro-philes b–v. Mayr and co-workers then used these measuredelectrophilicities of the electrophiles a–v to obtain the nu-cleophilicities of other nucleophiles (B, C, D, …) by enter-ing the measured reaction rates into Equation (1). Conse-quently, the nucleophilicities (NB, NC, ND, …) and the nu-cleophile-specific parameters (sB, sC, sD, …) were obtained.

We now demonstrate herein that the Mayr equation canbe understood by using FMO theory assuming that elec-tronic effects are the dominant factor affecting reactivity.FMO theory suggests that the nucleophilic reaction be-tween A and the electrophile i should be faster if theHOMO of the nucleophile is higher while the LUMO ofelectrophile is lower.[10] Therefore, we propose that the acti-

Abstract in Chinese:

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vation enthalpies (DH 6¼i=A) of these reactions should approxi-

mately obey Equation (2):

DH 6¼i=A ¼ mALi � nAHA þ CA ð2Þ

where HA is the HOMO energy of nucleophile A and Li isthe LUMO energy of electrophile i, mA and nA are definedas LUMO and HOMO sensitivity factors for A, respective-ly, and CA is a constant for all these nucleophilic reactions

of A to i, which can be roughly regarded as a contributionfrom filled orbital interactions from reactants A and i, andthe distortion energies of the reactants.[11–13] Equation (2)indicates that the higher/lower the HOMO/LUMO gap is,the higher/lower the activation enthalpy of the reaction is.In traditional FMO theory, the contributions of HOMOand LUMO energies to the activation energy of the reac-tion are the same (based on perturbation theory),[9h] but wepropose that their contributions could be different, and weuse LUMO and HOMO sensitivity factors m and n toquantify those different contributions to the reactivity.

DFT calculations at the B3LYP/6-31G(d)[15,16] level in thegas phase supported the validity of the application of theFMO theory to the nucleophilic addition reactions betweenA and electrophiles a–v[6c] (Table 1). We found that these

quantitative relationships also hold for the nucleophilic re-actions between B–D and a–v (Table 1). It is interesting tonote that the computed rate constants (kcal) and experimen-tally measured ones (kexp) correlate very well.

After obtaining Equation (2) from FMO theory, we cansubmit this equation into the Eyring equation [Eq. (3)] ofthe transition state theory to get Equation (3 a):

ln ki=A ¼ �DH 6¼

i=A

RTþ

DS6¼i=A

Rþ ln

kBTh

ð3Þ

ln ki=A ¼ �mA

RTLi þ

nA

RTHA �

CA

RTþ

DS 6¼i=A

Rþ ln

kBTh

ð3aÞ

In Equation (3), kB is theBoltzman constant. The lastthree terms in Equa-tion (3 a) can be regarded asconstants, because all thesereactions are conducted atalmost the same tempera-ture, and the activation en-tropies for all of them arevery close (see Table 1).The activation entropies arevery close because thestructures of transitionstates are very similar. If

the last three terms can be divided arbitrarily into two con-

stants, namely, we can have C1A þ C2A ¼ �CA

RT þDS 6¼

i=A

R þ ln kBTh ,

Equation (3 a) can be written as Equation (3 b):

ln ki=A ¼ �mA

RTLi þ C1A

� �þ nA

RTHA þ C2A

� �ð3bÞ

This equation is similar to the Mayr equation. Therefore,we can define Equation (3 c) and Equation (3 d):

2:303 Ei ¼�mA

RTLi þ C1A ð3cÞ

2:303 NA ¼nA

RTHA þ C2A ð3dÞ

Then Equation (3 b) can be transformed to Equa-tion (3 e):

ln ki=A ¼ 2:303 Ei þNAð Þ ¼ 2:303 sA Ei þNAð Þ ð3eÞ

where sA = 1 is defined with respect to nucleophile A sothat Equation (3 e) and Equation (1 a) have the same form.

The nucleophilic reactions of nucleophile (B) towardelectrophiles a–v also follow Equation (4), based on theFMO theory, as supported by calculations shown inTable 1, entry 2. By following the previous procedure, wearrive at Equations (4b) and (4 c).

DH 6¼i=B ¼ mBLi � nBHB þ CB ð4Þ

ln ki=B ¼ �mB

RTLi þ C1B

� �þ nB

RTHB þ C2B

� �ð4bÞ

Table 1. The linear relationship between the activation energies andLUMO energies (Li) of electrophiles for the nucleophilic reactions ofA–D towards a–v (see Scheme 1).

j DH¼6 /Li DS¼6 [calmol�1 K�1] lnkcal/lnkexp

A DHi/A¼6 =8.99 Li + 70.3 �41.3 to �42.3 lnkcal =1.40 lnkexp�13.20

B DHi/B¼6 = 8.93 Li +76.2 �46.3 to �49.2 lnkcal =0.86 lnkexp�15.22

C DHi/C¼6 = 9.09 Li +49.7 �39.1 to �41.9 lnkcal =1.29 lnkexp�19.24

D DHi/D¼6 =10.8 Li + 81.3 �40.5 to �42.5 lnkcal =1.58 lnkexp�9.13

Scheme 1. Procedure of measuring Nj and Ei using the Mayr equation.

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Quantifying Nucleophilicity and Electrophilicity

ln ki=B ¼mB

mA�mA

RTLi þ C1A

� �þ mA

mB

nB

RTHB þ

mA

mBC2B

� ��

ð4cÞ

Equation (4b) has been written in another form, Equa-tion (4 c), in which, we tried to introduce2:303 Ei ¼ � mA

RT Li þ C1A so that the electrophile indexesfor the nucleophilic reactions of nucleophiles A and B to-wards electrophiles a–v are the same. In Equation (4c), we

have C1B = (mB/mA)C1A and C1B þ C2B ¼ �CB

RT þDS 6¼

i=B

R þ ln kBTh

Thus, we can define the nucleophilicity of B as2:303 NB ¼ mA

mB

nB

RT HB þ mA

mBC2B, and the nucleophile-specific

slope is sB =mB/mA.Consequently, we get Equation (4 d):

ln ki=B ¼ 2:303 sB Ei þNBð Þ ð4dÞ

Based on the above derivation, we can generalize theMayr nucleophilicity Nj of any nucleophile j, together withits nucleophile-specific slope sj (using nucleophile A as thereference nucleophile) as in Equation (5):

ln ki=j ¼mj

mA�mA

RTLi þ C1A

� �þ mA

mj

nj

RTHj þ

mA

mjC2j

� ��

ð5Þ

Equation (5) can be simplified to Equation (5 b):

ln ki=j ¼ sj fi Lið Þ þ fj Hj

� �� �¼ sj Ei þNj

� �ð5bÞ

where [Eqs. (6)–(8)]:

fi Lið Þ ¼ 2:303 Ei ¼ �mA

RTLi þ C1A ð6Þ

sj ¼mj

mAð7Þ

fj Hj

� �¼ 2:303 Nj ¼

mA

mj

nj

RTHj þ

mA

mjC2j ð8Þ

Thus, Equations (5)–(8) derived from the FMO theoryimplicate that the Mayr equation [Eq. (1)] can be well un-derstood by the FMO theory, and Equation (5) can be re-garded as a quantified FMO expression of the Mayr equa-tion (or this can be simplified as Equation (5b), which in-vokes two functions fi(Li) and fj(Hj) with Li and Hj as varia-bles).

Equation (7) suggests that the nucleophile-specific pa-rameter sj is related to the HOMO sensitivity factor mj ofnucleophile j. Nucleophiles j and A could usually have simi-lar HOMO sensitivity factors and sj is then close to 1, asfound by Mayr in the Mayr equation. The theoreticalvalues of sA, sB, sC and sD calculated on this basis are 1.0,1.0, 1.0, and 1.2 (mj/mA, mj are the slopes in Table 1). Thevalues measured by Mayr are 1.0, 1.62, 0.90, and 0.95, re-spectively. The difference between the theoretical and ex-

perimental values of s is due to the difference between thecomputed kcal and the experimentally measured kexp (thisrelationship is lnkcal = xj lnkexp +yj ; xj and yj are correctioncoefficients, Table 1). With this consideration, the nucleo-phile-specific parameter can be computed as sj ¼

mj

mA

xA

xj, and

the computed values of sA, sB, sC and sD are then 1.0, 1.64,1.03, and 1.06, close to the trend of Mayr�s experimentallymeasured values.

The above derivations [Eqs. (5)–(8)] indicate that theMayr nucleophilicity of a nucleophile is related to theHOMO energy of the nucleophile, while the Mayr electro-philicity is related to the LUMO energy of an electrophile.If both Ei/Li and Nj/Hj display linear relationships, we canthen greatly expand the data sets of Mayr nucleophilicitiesand electrophilicities by using the calculated Hj and Li ener-gies.

Here we want to point out that our derivation of Equa-tions (5)–(8) is based on FMO theory. It is well known thatFMO theory has its limitations. For example, FMO theoryis not quantitative and does not take into considerationsteric effects, solvent effects, distortion effects, and soforth.[10] Also for diffusion-controlled reactions,[10h] ourFMO approach has problems describing the reactivities ofreactants appropriately. In principle, we can include someadditional terms in Equation (2) to take account of allthese non-electronic effects, which will be a subject offuture study. At present, we will only consider sets of spe-cies that have different nucleophilicities or electrophilici-ties, mainly due to their different frontier molecular orbitalsinduced by their different substituents remote to the react-ing sites. In Scheme 2, we use the hypothetical reactivities

of substituted benzenes to depict the scope of using Equa-tions (5)–(8). For the set of species in Scheme 2 a, R groupsaffect the reactivities mainly through electronic effects, theFMO theory can be applied, and consequently, Equa-tions (5)–(8) can be used. For those species whose reactivi-ties are dominated mainly by both steric and electronic ef-fects, the FMO approach cannot be used to quantify reac-tivities (the hypothesized species in Scheme 2 b). Obtainingreactivity parameters here is similar to getting substitutionparameters from benzoic acid derivatives by Hammett,[17]

Scheme 2. A schematic description of scope of the application of FMOto Equation (5)–(8). a) FMO can be applied; electronic effects dominatereactivity. Equations (5)–(8) can be applied. b) FMO cannot be applied;both electronic and steric effects affect reactivity. Equations (5)–(8)should be modified.

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FULL PAPERZhi-Xiang Yu et al.

where the substituents can only be in the para or meta posi-tion of the benzoic acid.

Below, we will show that the present FMO approach cangreatly expand the data sets of the Mayr equation in manysystems if electronic effects are dominant in affecting reac-tivity.

2. Computational Support of the Relationship betweenLUMO and Electrophilicity[14]

Equation (6) should obey a linear relationship between Ei

and Li because the coefficients are constants, as supportedby the data shown in Table 1, independent of electrophile i.We were pleased to observe (Figure 1) that the LUMO en-

ergies[14] of benzhydrylium ions (a–v) have a very excellentliner relationship with the experimentally measured Mayrelectrophilicities, thus confirming the validity of Equa-tion (6).[15,16] Such linear relationship was also observed byLiu et al.[8i] Similar linear relationships were also found forMichael acceptors, and this discussion will be given in Sec-tion 5 below.[18] The good correlation here is quite under-standable considering that the species in Figure 1 differ inreactivity mainly in electronic effects (it is obvious that thesteric effects caused by R group in a–v can be ignored).

3. Computational Support of the Relationship BetweenHOMO and Nucleophilicity

Since Nj, mj, nj, and C2j are different for different nucleo-philes j, in principle, a general linear relationship cannot beobtained between Hj and Nj [Eq. (8)]. This situation sug-gests that we can have very random distribution of Hj andNj for compounds A–D (Table 1) and their derivatives,which also explains why Liu et al. could not obtain linearrelationships.[8i] Not only electronic effects but also other ef-fects influence the reactivities of compounds A–D. Conse-

quently, Equation (8) cannot be used, and no linear rela-tionship can be found.

However, a set of similar nucleophiles in which the reac-tivity differences are governed mainly by electronic effectscould have very similar values of mj, nj, C2j, and conse-quently a good relationship of Nj and Hj could be obtained(only electronic effects of R groups are dominant in affect-ing reactivity. This is the case (a) in Scheme 2). In otherwords, very different nucleophiles can be divided into sev-eral subsets, and in each subset, a good relationship be-tween nucleophilicities (Nj) and HOMO energies (Hj)could be obtained. We calculated the Hj values of the arenecompounds[6c,19] as an example to show a linear relationshipbetween HOMO energies and the experimentally measurednucleophilicities. We were excited to find that there isa linear correlation between Nj and Hj after classifying allaromatic compounds into either the five-membered hetero-aromatic subset (E1–E7) or the benzene derivative subset(B and F1–F4 ; Figure 2). We observed that the nucleophi-licities of five-membered heteroaromatics and benzene de-rivatives cover a range of more than seven orders of magni-tude in nucleophilicity. Furthermore, plotting Hj versus Nj

of indoles[20] also gave two separate linear correlations for2-methylindoles (G1–G4) and indoles (H1–H11; Fig-ure 2 c,d).

We must point out here that the HOMO is usually thereacting orbital considered in FMO theory. But in somespecial cases, the HOMO-1 or HOMO-2 could be the fron-tier molecular orbitals as the reacting orbitals. Therefore, inthis case, we have to use HOMO-1 or HOMO-2 orbitals inEquation (8). An example for this special case is found forpara-substituted pyridines (I1–I6, Figure 3), where somespecies have their HOMO-1 orbitals as the reacting orbitals(the lone pair orbitals of the nitrogen atoms).[21] Here wesee again that Equation (8) is kept in the present case.

4. Further Consideration of the FMO and MayrNucleophilicity: The Orbital Coefficients Are also

Important

In the above development, we did not consider the contri-bution of orbital coefficients, with the assumption that theorbital coefficients of the reaction centers in HOMO (orLUMO) are very close for a set of similar nucleophiles (orelectrophiles). This assumption proved to be true in mostcases (see the Supporting Information for more discussion).In addition, for most nucleophiles, the HOMO orbital coef-ficients at the reacting centers are much higher than thoseof the LUMOs. Similarly, for most electrophiles, theHOMO orbital coefficients at the reacting centers arelower than LUMO coefficients. Therefore, the orbitalshapes of HOMOs and LUMOs are not considered in ob-taining nucleophilicities and electrophilicities for most nu-cleophiles and electrophiles.

However, for some nucleophiles, the orbital coefficientsin their LUMOs are not small and negligible, and it is nec-essary to consider the orbital coefficients to get a better re-

Figure 1. Correlation between experimentally measured electrophilicity(Ei) and B3LYP/6-31G(d) computed LUMO energy (Li) of electrophilei (carbocations a–v) with Ei =�6.52 Li�43.67, R2 =0.9817.

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Quantifying Nucleophilicity and Electrophilicity

lationship between the Mayr equation and FMO theory.For example, we found that anionic nucleophiles unexpect-edly exhibited a loose correlation between Hj with Nj (seethe Supporting Information). In this case, we built a newmodified HOMO energy (H’j) by considering both the orbi-tal coefficients of FMOs at the reaction centers and theHOMO and LUMO energies of the electrophile [Eq. (9)]:

H 0j ¼ðcHÞ2Hj þ ðcLÞ2Lj

ðcHÞ2 þ ðcLÞ2ð9Þ

where cH and cL are the molecular orbital coefficients ofHOMOj and LUMOj, respectively, of the carbon atom inthe reaction center of the nucleophile. If cH @ cL, H’j�Hj. inthis case, there is no need to consider the LUMO, and thisis confirmed by the results from nucleophiles in Figure 2.For those nucleophiles with significant LUMO contribu-tions, the relative coefficients of the HOMOs and LUMOsat the reaction centers play an important role the in reactiv-ity (Figure 4).

A linear correction can be obtained between the modi-fied HOMO energies (H’j) and the nucleophilicities, asdemonstrated by the case of aryl acetonitrile anions (Fig-ure 4 a).[22a] The data in Figure 4 b give a regression coeffi-cient of 0.9977, while it is 0.88 (before the correction) inFigure 4 a. For trifluoromethylsulfonyl stabilized carbanions(K1–K5)[22b] and sulfur ylides (L1–L5),[22c] the linear rela-tionships between the modified HOMO energies and nucle-ophilicities are shown in Figure 4 c,d.

5. Relationship of Ei/Li for Sets of Michael Acceptors

Similar to the analysis of the relationship Nj/Hj, we ex-plored whether a linear Ei/Li relationship can be found forMichael acceptors. The electrophilicities of these Michaelacceptors were obtained using the Mayr equation, wherethe reaction rates were measured by the reactions of theseMichael acceptors toward carbocations (a–v) with knownnucleophilicities (Scheme 1). We found that when electro-philes are divided into different sets, for example, the x1–x10,[18a–c] y1–y4,[18d] and z1–z7[18e] sets, a linear relationshipbetween LUMO energies and electrophilicities can be ob-tained in each subset (Figure 5). This result is a further sup-port of Equation (6) and suggests that for electrophiles, wecan have different sets of electrophiles within which theelectrophiles have different reactivities due to electronicreasons. In each subset, the Mayr electrophilicities and theHOMOs correlate well.

6. Influences of Steric Effects on the Correlation BetweenHOMOs and Nucleophilicities

In Figures 1–5, the substituents affect the reactivity mainlythrough electronic effects, and Equations (5)–(8) hold in allthese cases. In this part, we will show that when both elec-tronic and steric effects are considered, linear relationshipbetween HOMO and nucleophilicity cannot be obtained.

Figure 2. Correlation between nucleophilicity (Nj) and the HOMOenergy (Hj) computed at the B3LYP/6-31G(d) level. a) The five-mem-bered heteroaromatic (E1–E7) subset; Nj =7.33 Hj +45.7, with R2 =

0.9989. b) The benzene derivative (B and F1–F4) subset; Nj =7.85 Hj +

45.9, with R2 =0.9883. c) The 2-methylindole derivatives (G1–G4)subset; Nj = 2.13 Hj +18.10, with R2 =0.9517. d) The indoles derivative(H1–H11) subset; Nj =3.53 Hj +24.50, with R2 =0.9876.

Figure 3. Correlation between Nj and Hj for para-substituted pyridines(I1–I6); Nj =5.24 Hj +49.35, with R2 =0.9751.

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For example, for M,[23a] a linear relationship is obtained ina wide range since the electronic effects are dominant in af-fecting their reactivities (Figure 6 a). However, the HOMOenergies in M2–M8 are close but different for steric reasons(and maybe other possible reasons such as distortions inthe transition states or solvation), and no satisfactory linearrelationship can be found. Also for N,[23b] O,[23c] and P,[23d]

we believe that steric effects introduced by the substituentsare very significant, and consequently, the N values andHOMO energies are not correlated well in these cases.(Figure 6 b–d)

Conclusions

In conclusion, we have proposed herein a new understand-ing of the Mayr equation based on FMO theory and theEyring equation and suggest a new approach by combining

FMO theory and the Mayr equation to quantify nucleophi-licity and electrophilicity. We have derived that the nucleo-philicity of a nucleophile is related to the HOMO energy ofthe nucleophile, while the electrophilicity is related to theLUMO energy of an electrophile. Calculation results sup-port these linear relationships between LUMO energiesand the Mayr electrophilicities (Es) and HOMO energiesand the Mayr nucleophilicities (Ns) for sets of electrophilesand nucleophiles, respectively. For each set of nucleophilesor electrophiles, the different reactivities are mainly con-trolled by the electronic effects of the substituents. If othereffects such as sterics play a role in affecting reactivity fora set of similar electrophiles or nucleophiles, the linear rela-tionships between HOMO levels and N values and LUMOlevels and E values cannot be secured. The theory hereinand Equations (5)–(8), which are based on the existing andwell-accepted principles of chemical reactions, are verypowerful tools to understand the Mayr equation and its cor-

Figure 4. a) Correlation between Nj and Hj of for aryl acetonitrile anions (J1–J6); Nj =6.49 Hj +27.44, with R2 =0.8813. b–d) Correlation between Nj

and H’j (the modified HOMO energy computed by the B3LYP/6-31G(d) method, see Equation (9)) for aryl acetonitrile anions (J1–J6), Nj =

11.81 H’j + 23.46, with R2 =0.9977 (b); for trifluoromethylsulfonyl stabilized carbanions (K1–K5), Nj =7.65 H’j +24.77, with R2 =0.9933 (c); and forsulfur ylides (L1–L5), Nj =4.28 H’j +34.91, with R2 =0.9816 (d).

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Quantifying Nucleophilicity and Electrophilicity

responding nucleophilicities and electrophilicities. The pres-ent discovery also suggests that we can predict the reactivi-ties of new species that share similar reaction patterns andsteric effects but differ by electronic effects (induced by the

different substituents) relative to the experimentally mea-sured species in the data sets from the Mayr group. TheMayr nucleophilicities and electrophilicities can be easilyobtained by computing these new species� frontier molecu-lar orbitals and then correlating these calculated orbital en-ergies with the experimentally measured Mayr parameters.In addition, implementation of the semi-quantitative FMOexpression of activation energy [Eq. (2)] can be envisionedin many areas of chemistry. Further studies to include moreeffects into Equation (2) to investigate a molecule�s reactiv-ity are ongoing.

Computational Method

All calculations were performed with the Gaussian 03 pro-gram.[15a] All gas-phase stationary points were optimizedusing the B3LYP[15b,c] functional with the 6-31G(d) basis setfor all atoms. Orbital energies were calculated by B3LYP/6-31G(d), B3LYP/6-31 +G(d), HF/6-31G(d), HF/6-31+ G(d),and BP86/6-31G(d) methods, based on the gas-phase geo-metries optimized at the B3LYP/6-31g(d) level. All the or-bital energies discussed in the text are B3LYP/6-31G(d)values.[16] Full Hessian matrixes in Gaussian 03 were calcu-lated to verify the nature of all stationary points as either

Figure 5. Correlation between experimentally measured electrophilicity(Ei) and B3LYP/6-31G(d) computed LUMO energy (Li) of Michael ac-ceptors i. a) x1–x10 subset, Ei =�7.69 Li�29.01, R2 =0.9573; b) y1–y4subset, Ei =�5.32 Li�28.49, R2 =0.9938; c) z1–z7 subset with Ei =

�6.87 Li�33.06, R2 =0.9850.

Figure 6. Correlation between nucleophilicity (Nj) and the HOMO energy (Hj) computed at the B3LYP/6-31G(d) level. a) Amide ions (M1–M10).b) Amines (N1–N13). c) Methylenecycloalkanes (O1–O7). d) imidazoles (P1–P7).

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FULL PAPERZhi-Xiang Yu et al.

minima or first-order saddle points. The first-order saddlepoints were further characterized by intrinsic reaction coor-dinate (IRC)[15d–f] calculations to confirm that the stationarypoints are correctly connected to the corresponding reac-tants and products.

Acknowledgements

We thank the Natural Science Foundation of China (20825205-NationalScience Fund for Distinguished Young Scholars, and 21232001) for fi-nancial support. We thank Prof. Xinhao Zhang of Peking UniversityShenzhen Graduate School for his insightful comments.

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[8] a) C. Schindele, K. N. Houk, H. Mayr, J. Am. Chem. Soc. 2002, 124,11208; b) P. P�rez, A. Toro-Labbe, A. Aizman, R. Contreras, J.Org. Chem. 2002, 67, 4747; c) P. Campodonico, J. G. Santos, J.Andres, R. Contreras, J. Phys. Org. Chem. 2004, 17, 273; d) L. R.Domingo, P. P�rez, R. Contreras, Tetrahedron 2004, 60, 6585; e) R.Contreras, J. Andres, L. R. Domingo, R. Castillo, P. P�rez, Tetrahe-dron 2005, 61, 417; f) B. Denegri, O. Kronja, J. Org. Chem. 2007,72, 8427; g) E. Chamorro, M. Duquenorena, P. P�rez, J. Mol. Struct.THEOCHEM 2009, 896, 73; h) E. Chamorro, M. Duque-NoreÇa, P.P�rez, J. Mol. Struct. THEOCHEM 2009, 901, 145; i) C. Wang, Y.Fu, Q. X. Guo, L. Liu, Chem. Eur. J. 2010, 16, 2586 –2598.

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[10] For discussions of distortion energy in transition state within thecontext of FMO theory, see: a) K. Kitaura, K. Morokuma, Int. J.Quantum Chem. 1976, 10, 325; b) S. Nagase, K. Morokuma, J. Am.Chem. Soc. 1978, 100, 1666; c) G. T. de Jong, F. M. Bickelhaupt,ChemPhysChem 2007, 8, 1170; d) D. H. Ess, K. N. Houk, J. Am.Chem. Soc. 2007, 129, 10646; e) A. P. C. Bento, F. M. Bickelhaupt,

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[11] Equation (2) can be regarded as a very rough liner free energy rela-tionship (LFER) between activation energy and FMO orbital ener-gies with an assumption that CA, which could include many knownand unknown factors affecting the activation energy, is a constant.So far, only the empirical valence band (EVB) method has beenwell developed to quantify LFER, see a) A. Warshel, ComputerModeling of Chemical Reactions in Enzymes and Solutions, Wiley-Interscience, Maiden, 1991; b) “Empirical valence bond”: A. War-shel, J. Florian in Encyclopedia of Computational Chemistry, Wiley,Hoboken, 2004 ; c) S. Braun-Sand, M. H. M. Olsson, A. Warshel,Adv. Phys. Org. Chem. 2005, 40, 201.

[12] In principle, another FMO quantitative analysis using activationfree energy DG 6¼

i=A could also be proposed as:DG 6¼

i=A ¼ mALi � nAHA þ C0

A (2’)This will give similar results as we have derived if we have CA’ asa constant.

[13] The CA is also dependent on the set of electrophiles i. These elec-trophiles are similar but are differ mainly through electronic effectsintroduced by their different R substituents, which are far awayfrom the reacting sites of the electrophiles (see also Scheme 2).

[14] The LUMO orbitals here refer to the reacting orbitals responsiblefor the elecrophilicity. In calculations with different basis sets, thesereaction orbitals could be changed to LUMO + 1, LUMO +2 orothers; in these cases, we have to use these corresponding reactingorbitals instead of the LUMO orbitals in Equation (6).

[15] All of the geometry optimizations and frequency calculations wereperformed with the B3LYP functional and the 6-31G(d) basis setimplemented in Gaussian 03; a) Gaussian 03 (Revision C.02), M. J.Frisch, et al., Gaussian, Inc.: Pittsburgh, PA, 2004 ; b) C. Lee, W.Yang, R. G. Parr, Phys. Rev. B 1988, 37, 785; c) A. D. Becke, J.Chem. Phys. 1993, 98, 5648; d) K. Fukui, J. Phys. Chem. 1970, 74,4161; e) C. Gonzalez, H. B. Schlegel, J. Chem. Phys. 1989, 90, 2154;f) C. Gonzalez, H. B. Schlegel, J. Phys. Chem. 1990, 94, 5523.

[16] All discussed orbital energies are referred to the B3LYP/6-31G(d)values. We found that using orbital energies of HF/6-31G(d) andBP86/6-31G(g) gave the same conclusion (see the Supporting Infor-mation).

[17] a) L. P. Hammett, Physical Organic Chemistry, 2nd ed. ; McGraw-Hill: New York, 1970 ; b) Y.-D. Wu, C.-L. Wong, K. W. K. Chan, G.-Z. Ji, X.-K. Jiang, J. Org. Chem. 1996, 61, 746.

[18] a) S. T. A. Berger, F. H. Seeliger, F. Hofbauer, H. Mayr, Org.Biomol. Chem. 2007, 5, 3020; b) F. Seeliger, S. T. A. Berger, G. Y.Remennikov, K. Polborn, H. Mayr, J. Org. Chem. 2007, 72, 9170;c) O. Kaumanns, H. Mayr, J. Org. Chem. 2008, 73, 2738; d) R.Lucius, R. Loos, H. Mayr, Angew. Chem. 2002, 114, 97; Angew.Chem. Int. Ed. 2002, 41, 91; e) O. Kaumanns, R. Lucius, H. Mayr,Chem. Eur. J. 2008, 14, 9675.

[19] M. F. Gotta, H. Mayr, J. Org. Chem. 1998, 63, 9769.[20] S. Lakhdar, M. Westermaier, F. Terrier, R. Goumont, T. Boubaker,

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2007, 13, 336.[22] a) O. Kaumanns, R. Appel, T. Lemek, F. Seeliger, H. Mayr, J. Org.

Chem. 2009, 74, 75; b) R. Appel, N. Hartmann, H. Mayr, J. Am.Chem. Soc. 2010, 132, 17894; c) S. T. A. Berger, A. R. Ofial, H.Mayr, J. Am. Chem. Soc. 2007, 129, 9753.

[23] a) M. Breugst, T. Tokuyasu, H. Mayr, J. Org. Chem. 2010, 75, 5250;b) F. Brotzel, Y. C. Chu, H. Mayr, J. Org. Chem. 2007, 72, 3679; c)M. Roth, C. Schade, H. Mayr, J. Org. Chem. 1994, 59, 169; d) M.Baidya, F. Brotzel, H. Mayr, Org. Biomol. Chem. 2010, 8, 1929.

Received: September 3, 2012Published online: November 22, 2012

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