Chemical Physics Letters 607 (2014) 75–80

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The singlet–triplet energy splitting of p-nucleophiles as a measureof their reaction rate with electrophilic partners

http://dx.doi.org/10.1016/j.cplett.2014.05.0400009-2614/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Fax: +380 472 37 21 42.E-mail address: bondchem@cdu.edu.ua (S.V. Bondarchuk).

Sergey V. Bondarchuk a,⇑, Boris F. Minaev a,b

a Department of Organic Chemistry, Bogdan Khmelnitsky Cherkasy National University, blvd. Shevchenko 81, 18031 Cherkasy, Ukraineb Department of Physics, Tomsk State University, pr. Lenina 36, 634050 Tomsk, Russian Federation

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 April 2014In final form 15 May 2014Available online 22 May 2014

The recently introduced nucleophilicity index (N) of unsaturated compounds, which are typical p-nucle-ophiles, correlates (R = 0.9229) with their singlet–triplet energy splitting (DS–T) between the ground sin-glet and the excited 3(pp⁄) state. The latter quantity is proposed for the activation barrier (Ea)determination of the nucleophile–electrophile addition reactions. A new approach for calculation ofthe Ea value is developed for reactions between alkenes and the p-methoxybenzhydryl cation. Thisscheme involves some simple DFT parameters, namely, the DS–T of the reactants and products. The exper-imentally available Ea values for the above reaction correlate well (R = 0.9944) with the calculated data.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Recently, we have proposed the triplet mechanism of theMeerwein arylation reactions [1], which are now intensively stud-ied under photochemical conditions [2–4]. The mechanismassumes the probability of admixture of the triplet states of unsat-urated compounds (UCs) in a vicinity of the transition state; theseare formed due to exchange interaction between UCs and the trip-let aryl cations [5–8]. For example, the triplet aryl cation (I) canreact with the triplet ethylene molecule (II) to form the singletdiradical cation (III) followed by transformation to the closed-shellcarbocation (IV) (Scheme S1; this is a simple way to illustratedeformation of the open-shell wave function) [1,6–9].

Taking into account formation of the cation III, the barrierheight DG– should be proportional to the singlet–triplet energysplitting (DS–T) of the UCs:

DG– / DS�T ð1Þ

To quantify the efficiency of the above reaction we earlier useda correlation between the DS–T values and the isolated yields of thearylation products [1]. The regression coefficient of the correlationobtained was rather low (R2 = 0.6709). This is explained by theindirect dependence of the latter quantities. Moreover, the finalyield value depends on a variety of factors which cannot beaccounted in the calculations. Based on the foregoing, the moreaccurate correlation is expected to exist between the DS–T energyand the reaction rate logarithm (log k). Since the aryl cation is a

typical electrophile species and the studied UCs are typicalp-nucleophiles the reaction rate logarithm can be expressed bythe Mayr–Patz equation [10,11], which was modified by Albiniet al. [12] for the aryl cations:

log k ¼ sðEþ eNexperÞ ð2Þ

where Nexper is the relative reactivity (nucleophilicity parameter), Eis the nucleophile-independent electrophilicity parameter, s (1–1.5for alkenes) and e (0.33 for the aryl cations) are the empiricalparameters [12].

If one use the same aryl cation as an electrophile, then E = const.Moreover, since p-nucleophiles are only used for the correlationand the s parameter varies in a rather tight range [12], one canassume that s ffi const. In Eq. (2) the nucleophilicity index (N)[13–17] can be used instead of the experimental Nexper parameter,since Nexper / Ncalcd. The index N is usually calculated in terms ofvertical approach (the Koopmans’ theorem) as the following:

Nvert ¼ EHOMOðNuÞ � EHOMOðTCEÞ ð3Þ

where the EHOMO(Nu) and EHOMO(TCE) are the highest occupied molec-ular orbital energies of a given nucleophile and tetracyanoethylene,TCE (as a reference), respectively. Recently, Yu et al. [17] made a sig-nificant advance to understand the nucleophilicity and electrophi-licity by combining transition state theory and frontier molecularorbital theory. It was found that one can derive the relative reactiv-ity of the interested species using HOMO and LUMO energies. Therelative nucleophilicity does correlate with the HOMO energy ofnucleophiles; [17] this also follows from Eq. (3). On the other hand,estimation of the N values by means of the HOMO energies (vertical

76 S.V. Bondarchuk, B.F. Minaev / Chemical Physics Letters 607 (2014) 75–80

approach) is only reasonable when using so-called ‘standard’ basisset, namely, 6-31G(d) for the calculations [13–17].

As it follows from our recent unpublished results moreextended basis sets, especially with added diffuse functions, signif-icantly affects the HOMO/LUMO energies. Thus, calculations of ion-ization energy of nucleophiles (HOMO energy) in terms of thevertical approach need some compromise between relatively lowaccuracy during geometry optimization and more or less satisfac-tory prediction of the desired value. We have modified Eq. (3) toavoid the vertical values and to use the adiabatic approach (adia-batic energy difference) in estimating the ionization energy (IE)of the studied compounds [6]. Thus, Eq. (3) can be rewritten asthe following:

Nad ¼ U�þTCE � U0TCE

� �� U�þNu � U0

Nu

� �

¼ E�þelecðTCEÞ þ ZPVE�þTCE

� �� E0

elecðTCEÞ þ ZPVE0TCE

� �h i

� E�þelecðNuÞ þ ZPVE�þNu

� �� E0

elecðNuÞ þ ZPVE0Nu

� �h ið4Þ

Herein, the U�þTCE, U0TCE, U�þNu and U0

Nu are the total energies of the oxi-dized radical cationic and the starting neutral state of TCE and of thegiven nucleophile at 0 K, respectively. These are the sums of the cor-responding energies for the stationary points on the Born-Oppen-heimer potential energy surfaces (Eelec) and the zero-pointvibration energies (ZPVE). Thus, for the series of p-nucleophileswhich react with the same aryl cation we have a simple relationlog k / Nad.

Taking into account Eq. (1) the rate constant can also beexpressed in logarithmic form as the following [18]:

ln kðTÞ / lnkBThco

� �� DS�T

RTð5Þ

Combining Eqs. (2) and (5) and taking into account thatNexper / Nad (omitting all the constants for simplicity) we obtainthe final correlation:

Nad /1

DS�Tð6Þ

To check this relation we have calculated the DS–T and Nad values fora series of 35 various UCs including alkenes, alkynes and alkadienes.The obtained results convincingly suggest that admixture of thetriplet UCs electronic shell should be expected in a vicinity of thetriplet aryl cation during the transition complex formation andthe activation barrier height does correlate with the DS–T value ofUCs. Actually, it does not mean that the triplet states of UCs arereally formed along the reaction path. The correlation reveals amultireference nature of the transition states of nucleophile–electrophile addition reactions with a significant contribution ofthe so-called ‘double–triplet’ singlet state configuration. In thisLetter we report results based on a new activation barrier calcula-tion methodology involving simple electronic descriptors. Thisallows one the fast and easy estimation of the electrophile–nucleophile reaction rate, and hence the reaction efficiency.

2. Computational details

Geometry optimization of all the studied unsaturated sub-strates was performed using the DFT [19]/B3LYP [20,21] methodwith the Pople’s split-valence double-f basis set (6-31 G) and addi-tion of both polarization (d,p) and diffuse (+) functions [22]. Inorder to evaluate the stationary points of the minima (number ofimaginary frequencies NIMAG = 0) the vibrational frequencyanalysis was carried out. We did not use polar media simulationsfor general molecular benchmark set, since the correlation has acommon character and does not relate to the systems reacting in

specific media. But in some particular cases needed, these simula-tions were specified using the polarizable continuum model (PCM)[23]. The optimization of all studied molecules was performedwithout imposing symmetry constrains. The ZPVE correctionswere always used in the total energy estimation. The time-dependent density functional theory (TDDFT) [24] calculationswere performed to obtain the excited states energy. Semi-empirical calculations were done with the recently developedPM7 method [25,26].

Since the DS–T and N are the relative quantities, i.e., Nad includesthe arithmetical differences between the calculated values, the sys-tematic errors of the method are almost quenched. The relativelysmall basis set used, namely, the 6-31+G(d,p) can be reasonablyimplemented to carry out these calculations [13–17]. The DFTstudies of the closed-shell species were performed using thespin-restricted Kohn–Sham formalism (RKS), while the open-shellspecies were treated in terms of the spin-unrestricted Kohn–Sham(UKS) method. All DFT geometry optimizations (the total numberof stationary states is 160) have been performed with the GAUSSIAN

09 suite of programs [27] and the semiempirical PM7 calculationswere done with the MOPAC2012 program package [28,29].

3. Results and discussion

3.1. Nature of the correlation

The singlet–triplet excitation of UCs, which is considered in thepresent study is related to the electron transition from the bondingto antibonding molecular orbital (MO) with respect to the multiplechemical bond forming the excited triplet state 3(pp⁄). In the caseof unsubstituted ethylene, acetylene and 1,3-butadiene these arethe ~A3B1u, ~A3Rþu and ~A3Bu vertical excited states, respectively. TheMOs involved in the triplet state formation of the above com-pounds are illustrated in Figure 1. On the other hand, the adiabaticstates corresponding to this type triplet excitation are known tohave the bent or twisted geometry. The ethylene, acetylene and1,3-butadiene molecules belong to the D2d, C2v and CS symmetrypoint group. The triplet states have two unpaired electrons andthe broken multiple bond. Thus, the molecule in this state is readyfor reaction with the triplet reactant to form a total singlet spinstate of the whole reacting system (see Scheme S1, for example).

In order to verify the foregoing statement we have calculatedthe adiabatic singlet–triplet energy splitting (DS–T) between theground singlet and the excited triplet state of the series of 35 var-ious unsaturated substrates. The results are collected in Table 1.The compounds have been selected among those previously usedfor the reaction with the aryl precursors [30,31]; the latter can gen-erate both the aryl radicals and cations depending on the reactionconditions [1,32,33]. As one can see in Table 1, the DS–T value,nucleophilicity index (Nad) and ionization energy (IEad) are in astrong linear dependence between each other; the correlation plotis illustrated in Figure 2. This is characterized by the Pearson coef-ficient equals to 0.9229, which is high enough to justify the corre-lation. For the most studied compounds the considered triplet stateoccurs due to an electron transition from the highest occupiedmolecular orbital (HOMO) to the lowest unoccupied molecularorbital (LUMO). Thus, a simple UKS procedure for the first tripletstate geometry optimization was employed in order to obtain thecorresponding DS–T energies. We also estimated the Yu’s approach[17] in order to check a correlation between the experimentallymeasured nucleophilicity and the HOMO energy. The resultsrevealed this correlation (see the Supplementary Material fordetails), but provided somewhat lower regression coefficient(0.8545) that is connected with the above discussion.

Figure 1. Molecular orbitals involved in the triplet excitation.

Table 1The adiabatic singlet–triplet energy splitting (DS–T) in kcal mol�1, nucleophilicityindex (Nad) in eV and the ionization energy IEad in eV for the series of 35 organicmolecules studied in this work.

Entry Name of the compound DS–T Nad IEad

1 trans,trans-1,3,5,7-Octatetraene 33.0 3.97 7.232 (1,2,2-Trifluorovinyl)benzene 41.5 2.92 8.283 2-Furylnitroethylene 43.5 2.57 8.624 1-Bromo-1-phenylethylene 46.8 3.01 8.195 Chloroprene 47.5 2.41 8.796 Trichloroethylene 48.2 1.95 9.257 1-Chloro-1-phenylethylene 48.4 2.93 8.268 1-Chloro-1,3-butadiene 49.7 2.72 8.489 Isoprene 50.4 2.77 8.43

10 1,3-Butadiene 50.4 2.54 8.6511 Isopropenyl acetylene 50.8 2.42 8.7712 Vinylacetylene 51.5 2.05 9.1513 2-Vinylpyridine 51.5 2.75 8.4514 trans-1,2-Dichloroethylene 52.1 1.81 9.3815 cis-1,2-Dichloroethylene 52.2 1.78 9.4216 Methyl methacrylate 53.5 1.99 9.2117 Methyl vinyl ketone 53.6 1.78 9.4118 1,1-Dichloroethylene 54.6 1.64 9.5619 1,4-Benzoquinone 54.9 2.23 8.9620 1,2-Dichloroacetylene 55.4 1.44 9.7621 Methyl acrylate 56.0 1.55 9.6422 Acrylic acid 56.0 1.22 9.9823 1,3-Dichloro-2-butene 56.4 2.17 9.0324 Chloroethylene 57.6 1.44 9.7525 1,1-Difluoroethylene 59.0 1.03 10.1726 Propylene 60.0 1.81 9.3927 Ethylene 60.1 0.98 10.2228 Chloroacetylene 66.7 0.86 10.3329 Acrolein 68.7 1.02 10.1830 Bromoacetylene 69.9 1.15 10.0531 Maleinimide 76.5 0.19 11.0032 Propiolic acid 77.7 0.77 10.4333 Propiolaldehyde 78.8 0.74 10.4534 Acetylene 82.1 0.04 11.1535 Ferrocene 115.8 �0.65 11.85

Figure 2. Correlation between the adiabatic nucleophilicity index of various UCs(Nad) in eV and the inverse of their singlet–triplet energy splitting DS–T inkcal mol�1; R is the regression coefficient; SD is a standard deviation; n is thenumber of points. The filled markers correspond to the values calculated by the firstexcited triplet state energy and the ground radical cation state level of the 1,4-benzoquinone (d), acrolein (j), maleinimide (N) and ferrocene (�) molecules.

Table 2The corresponding excited triplet and radical cation electronic states which are usedto calculate the DS–T and Nad quantities for several compounds.

Entry Triplet Radical cation

State Transition nature State Transition nature

19 4 3B3g HOMO-2 ? LUMO 2 2B2g SOMOa–1 ? LUMO29 2 3A0 HOMO-1 ? LUMO 1 2A00 SOMO ? LUMO31 3 3B2 HOMO-1 ? LUMO 1 2A2 SOMO–1 ? LUMO35 14 3B2 HOMO-4 ? LUMO+1 12 2B1 SOMO–3 ? LUMO

a SOMO is the singly occupied molecular orbital.

S.V. Bondarchuk, B.F. Minaev / Chemical Physics Letters 607 (2014) 75–80 77

Meanwhile, the compounds 19, 29, 31 and 35 indicate the3(pp⁄) state being higher in energy than the first triplet state. Thus,for these compounds the relative position of the 3(pp⁄) state wasdetermined by means of the TDDFT calculations. The results arepresented in Table 2. Along with the 3(pp⁄) state, the position ofthe corresponding radical cation (doublet) state, which occurs afterthe electron withdrawal from the p-bonding MO, is also obtainedby the TDDFT method (Table 2). These higher triplet and doubletstates structures were subsequently optimized and used for com-putation of the DS–T and Nad quantities. Though the TDDFTapproach is not relevant for the doublet excited states calculations,we have found that this method provides quite reasonable resultsfor the optimized excited triplet and doublet state energies(Figure 2). For comparison, we have illustrated the DS–T and Nad

values calculated using the first excited triplet state and theground doublet state structures in Figure 2 (the filled markers).

As one can see in Figure 2, only correct account of the higherexcited states during calculations of the DS–T and Nad values pro-vides a good correlation. It follows form Table 1 and Figure 2,that the UC molecules with relatively small DS–T value (ca. 30–50 kcal mol�1) represent the strong p-nucleophiles; since the Nad

value is relatively high (equal to ca. 4–2 eV).

3.2. Activation barrier height calculation using the DS–T energy gap

Herein, we develop a method for prediction of the activationenergy (Ea) for reactions of p-nucleophiles with electrophiles. Asa model reaction, the coupling between five different alkenes andthe p-methoxybenzhydryl cation was chosen because of availableexperimental data about the rate constant [34,35]. This methodis based on the previous suggestion about importance of accountof the reactants triplet excited states in bimolecular reaction bar-rier height formation [9,36]. Former quantum-chemical calcula-tions have indicated a possibility of the S–T crossings in the

Figure 3. Energy diagram illustrating how the DS–T value of UCs determines the activation barrier height for the p-methoxybenzhydryl cation reaction with olefins.Geometrically, the Ea value is the energy projection of the ad and bc intersection point with respect to the reference level a.

78 S.V. Bondarchuk, B.F. Minaev / Chemical Physics Letters 607 (2014) 75–80

region of the activated complex [36,37]. Schematically, the methodof the Ea value calculation is illustrated in Figure 3. There are threekey energy parameters affecting the final Ea energy. We have spec-ified these as the parameters h1, h2 and h3. According to Figure 3the h1 is the difference between the total energy levels of the trip-let-state (b) and the singlet-state reactants (a); h2 is the reactionheat and h3 is the difference between the reference level (a) andthe singlet-state level of the reaction product. Thus, the Ea energycan be easily calculated using the following equation:

Figure 4. Plot of the activation barrier height in kcal mol�1 calculated using Eq. (7)Ea(calcd) versus the ones calculated using Eq. (5) Ea(exper); the log k values forestimations of the Ea(exper) data are taken from Refs. [34,35].

EaðcalcdÞ ¼ h1h3

h1 � h2 þ h3; kcal mol�1 ð7Þ

We have applied the transition state theory expression for thereaction rate in its original form [18] (in Eq. (5) the DS–T is replacedby the DG– value) to estimate the Ea(exper) values for the experi-mental log k benchmark set [34,35]. Since the calculations of theparameters h1, h2 and h3 require significant computational costwe have decided to use recently introduced reparametrized semi-empirical PM7 method. This was recently developed as the mostrobust tool among the semiempirical methods, suitable for model-ing a wide range of species [38], partially due to inclusion of bothdispersion and hydrogen bond corrections [39]. This method wasalso found to be suitable for accurate prediction of the molecularoptical activities [40].

Surprisingly, the latter exhibits the best fit of the calculatedEa(calcd) to the Ea(exper) quantities (Table 3). Furthermore, wehave used PCM/B3LYP/6-31+G(d,p) method in CH2Cl2 medium tocalculate the Ea(calcd) quantities according to Eq. (7). These resultsprovides a good correlation with the Ea(exper) values (R = 0.9737),but somewhat higher Ea(calcd) values (ca. 30–32 kcal mol�1). Seethe Supplementary Materials for details on the B3LYP results. Incontrast, the PM7 method provides higher regression coefficientto be equal to 0.9944 (Figure 4) and much closer Ea(calcd) values(Table 3).

Table 3The energy levels in the geometrical scheme in Figure 3. (h1, h2 and h3) in kcal mol�1,and the corresponding Ea(calcd) and Ea(exper) values in kcal mol�1 calculated by thePM7 method.

Entry h1 h2 h3 Ea(calcd) Ea(exper) log k

1 59.9 5.6 16.0 13.6 14.7 �3.0272 59.6 3.7 16.4 13.5 14.4 �2.7333 59.6 �6.7 13.8 10.3 10.7 1.2654 59.1 �5.9 13.2 10.0 10.6 1.3675 57.1 �6.0 11.2 8.6 9.6 2.456

Originally, the geometrical scheme (Figure 3) includes the sin-glet diradical state of the reaction product [9,36] (like the adductIII in Scheme S1) to be present in the right side. On the other hand,the calculations of such species are rather complicated with aknown spin contamination problem caused by the use of a bro-ken-symmetry method [41–43]; to treat this problem properly,one should use the multireference methods (viz., CASSCF [44],CASPT2 [45], NEVPT2 [46], etc.). In this Letter, we have applied aprocedure developed in GAUSSIAN 09 for calculations of antiferro-magnetic coupling. The obtained triplet and singlet diradicaladducts of propylene with the p-methoxybenzhydryl cation are illus-trated in Figure 5. As expected, the alpha and beta spins in the singletdiradical 5b are localized on the separated molecular fragments. Onespin almost completely localizes on the propylene moiety, while theopposite spin is spread predominantly on the benzene rings and theoxygen atoms (Figure 5). This is in accord with the simple Scheme S1considering the aryl cations as electrophiles.

It is known that unrestricted Kohn–Sham DFT (UKS DFT) usuallypredicts diradical singlet ground state [47,48]. Herein, we havefound that the singlet diradical adduct (Figure 5b) is lower inenergy than the corresponding triplet adduct. The DS–T separationhas been found to be �3.9 kcal mol�1, respectively (B3LYP/6-31+G(d,p)). However, we should stress that these values are verysensitive to the basis set chosen and CAS size [49]. When we usethe 3-21G basis set, the ground state is also predicted to be the sin-glet but the DS–T energy separation is equal to �4.6 kcal mol�1

(Figure 5b). Because of the problem with the singlet diradical cal-culation technique and due to a relative proximity of the singlet

Figure 5. The main contribution of the Mulliken spin densities in the triplet (a) andsinglet (b) diradical adducts of propylene to the p-methoxybenzhydryl cationcalculated by the UB3LYP/6-31+G(d,p) and the UB3LYP/3-21G (values in parenthe-ses) methods.

S.V. Bondarchuk, B.F. Minaev / Chemical Physics Letters 607 (2014) 75–80 79

and triplet energy levels in diradical [50], we suggest to use thetriplet diradical energy instead the S1 in the Scheme in Figure 3.

Summing up, in the present work we have performed DFT cal-culations of 35 UCs that revealed a strong dependence(R = 0.9229) between two complementary molecular properties,namely, nucleophilicity index (N) and the singlet–triplet energysplitting (DS–T). These results suggest that the UC’s triplet 3(pp⁄)state energy can be proposed for measuring the reaction rate ofUCs with electrophiles (electrophile–nucleophile addition). Onthe basis of the above correlation a simple graphical scheme isdeveloped for prediction of activation energy (Ea) for such reac-tions. This scheme includes three important energy values, namely,the singlet–triplet energy separation of the reactants (h1), products(h3–h2) and the reaction heat (h2). (For the diradical product the S1

and T1 state energies are quasidegenerate).An awesome agreement between the available experimental

rate constant data and the calculated values is found for therecently introduced semiempirical PM7 method (R = 0.9944). ThePCM/B3LYP/6-31+G(d,p) results also exhibit a good (but somewhatlower) accuracy (R = 0.9737). Applicability of this procedure to thefast barrierless reactions as well as for the low-barrier reactions,like those involving the aryl cations [12], need further detailedstudy that was also concluded by Yu et al. [17] Certainly, one canapply procedure for a direct singlet–triplet reaction ‘seam’optimization in order to estimate the Ea value for the electro-phile–nucleophile interaction, but such routine calculations donot reveal the electronic nature of the transition state. In contrast,

the present results convincingly suggest that the ‘double–triplet’singlet state configuration significantly contributes the multirefer-ence transition state of the titled reaction; this configuration isformed upon electron excitation to the 3(pp⁄) triplet state. Thus,the energy of such triplet electron transition as well as the natureof molecular orbitals, which are involved in the nucleophilicitycomputation scheme in terms of the vertical approach [13–17], isimportant when considering the nucleophile–electrophile additionreaction mechanism.

Acknowledgments

This work was supported by the Ministry of Education and Sci-ence of Ukraine, Research Fund (Grant No. 0113U001694). Wethank Professor Hans Ågren (KTH, Stockholm) for the PDC super-computer use. The computations were performed on resourcesprovided by the Swedish National Infrastructure for Computing(SNIC) at the Parallel Computer Center (PDC) through the project‘Multiphysics Modeling of Molecular Materials’, SNIC 020/11-23.

Appendix A. Supplementary data

Supplementary data associated with this article can be found,in the online version, at http://dx.doi.org/10.1016/j.cplett.2014.05.040.

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