On the question of zwitterionic intermediates in 1,3-dipolar cycloadditions between hexafluoroacetone and sterically crowded diazocompounds

On the question of zwitterionic intermediates in 1,3-dipolar cycloadditions between hexafluoroacetone and sterically crowded diazocompounds

Accepted Manuscript Title: On the question of zwitterionic intermediates in 1,3-dipolar cycloadditions between hexafluoroacetone and sterically crowde...

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Accepted Manuscript Title: On the question of zwitterionic intermediates in 1,3-dipolar cycloadditions between hexafluoroacetone and sterically crowded diazocompounds Author: Radomir Jasi´nski PII: DOI: Reference:

S0022-1139(15)00133-5 http://dx.doi.org/doi:10.1016/j.jfluchem.2015.04.020 FLUOR 8565

To appear in:

FLUOR

Received date: Revised date: Accepted date:

27-2-2015 28-4-2015 29-4-2015

Please cite this article as: R. Jasi´nski, On the question of zwitterionic intermediates in 1,3-dipolar cycloadditions between hexafluoroacetone and sterically crowded diazocompounds, Journal of Fluorine Chemistry (2015), http://dx.doi.org/10.1016/j.jfluchem.2015.04.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the question of zwitterionic intermediates in 1,3-dipolar cycloadditions between hexafluoroacetone and sterically crowded diazocompounds

ip t

Radomir Jasiński*

Cracow University of Technology, Institute of Organic Chemistry and Technology,

Ac ce p

te

d

M

an

us

cr

Warszawska 24, 31-155 Cracow

*

email: [email protected]

Page 1 of 17

HIGHLIGHTS

Ac ce p

te

d

M

an

us

All attempts at optimization of zwitterions ended unsuccessfully

cr

DFT calculations do not provide confirmation for stepwise mechanism

ip t

Reactions of hexafluoroacetone and diazocompounds have polar nature

Page 2 of 17

On the question of zwitterionic intermediates in 1,3-dipolar cycloadditions between hexafluoroacetone and sterically crowded

ip t

diazocompounds

cr

Radomir Jasiński*

us

Cracow University of Technology, Institute of Organic Chemistry and Technology,

M

an

Warszawska 24, 31-155 Cracow

ABSTRACT

1,3-dipolar

DFT calculations – regardless of theory level - do not provide confirmation for the

cycloaddition,

hypothetical, stepwise mechanism of 1,3-dipolar cycloaddition of hexafluoroacetone

Fluorinated

to sterically crowded diazocompounds. These reactions take place according to a

dipolarophiles,

polar, but non-ionic, scheme. All attempts to locate paths leading to these from

Ac ce p

te

d

KEYWORDS

DFT study,

addents to hypothetical zwitterionic structures were unsuccessful.

Mechanism

1. Introduction

Recently it is more and more often said that the classical Huisgen [1] mechanism may

not be applied a priori to all reactions of 1,3-dipolar cycloaddiction. It has been shown that in reactions of nucleophilic 1,3-dipoles and -deficient dipolarophiles, a stepwise, zwitterionic mechanism may compete with the one-step mechanism. As shown in recent years, this

*

email: [email protected]

Page 3 of 17

mechanism takes place, e.g. in the case of 1,3-dipolar cycloaddition of azomethine ylides with dialkyl-2,3-dicyanobut-2-enedioates

[2],

di(tert-butyl)diazomethane

with

1,2-

bis(trifluoromethyl)ethene-1,2-dicarbonitrile [3] as well as various substituted nitrones with

ip t

fluorinated alkenes [4] and 1-EWG-substituted nitroethenes [5,6]. Thus, the mechanistic aspects of the 1,3-dipolar cycloaddition attract concern from many research groups.

cr

Some time ago, Shimizu and Barlett [6] worked on the reaction of 1,3-dipolar cycloaddition between hexafluoroacetone (1) and a series of sterically crowded

us

diarylodiazomethanes (2a-c), which take place regiospecificially and lead to corresponding 3,3-diaryl-5,5-bis(trifluoromethyl)-2,5-dihydro-1,3,4-oxadiazoles. They suggest that reactions

an

between diazocompounds and ketones can take place via formation of zwitterionic

Ar

C N

+

N

O C

CF3 Ph C O C+ N N CF3

Ar O

Ph N N

CF3 CF3

3a-c

Ac ce p

1

CF3

Ar

te

F3C

2a-c

d

Ph

M

intermediate (Scheme 1):

Ar = Ph (a), 4-Me-C6H4 (b), 4-Cl-C6H4 (c)

Scheme1. Hypethetically, zwitterionic mechanism of 1,3-dipolar cycloaddition between hexafluoroacetone and sterically crowded diarylodiazomethanes

Unfortunately, they do not present any confirmation of zwitterionic presence in the reaction paths. On the other hand, a stepwise mechanism seems to be very probable in the case of these cycloaddition reactions. This results, first and foremost, from two factors: (i) the CF3 groups thanks to their electron-withdrawing nature (Hammet constant σ=0.53) stimulate an increase in the -deficit of the C=O bond participating in the cycloaddition reaction (it must be noted, that the CF3 is an electron-withdrawing group able to increase the electrophilicity, but it does not change the regioselectivity), (ii) it is known [3,4,7-9] that CF3 groups stabilize zwiterionic intermediates occurring in the paths of cycloaddition reactions. In the analysed case, the

Page 4 of 17

zwitterionic mechanism should be facilitated additionally by the clearly nucleophilic nature of the 1,3-dipole and the non-uniform shielding of reaction centres of the addents. In particular, the presence of two aryl substituents at carbon atom of CNN moiety of diazocompounds is a

ip t

sufficient evidence to consider these compounds as sterically crowded, and, in consequence, to enforce zwitterionic reaction mechanism. This is confirmed in the case of 1,3-dipolar

cr

cycloaddition between diphenyldiazomethane and 1,1-dinitroethene which were described some years ago by Fridman and co-workers [10].

us

Taking the aforementioned issues into account, this work is aimed at explaining interpreting defining on the nature of transformations taking place along the path of

an

conversion of the title addents into adducts. Thanks to data obtained from DFT calculations it

M

was possible to better understand the mechanical aspects of the synthesis of perfluoralkyl-

te

2. Calculation methods

d

functionalized five-membered heterocycles with potential biological activity [11,12].

All calculations reported in this paper were performed on an SGI-Altix 3700 computer

Ac ce p

in the CYFRONET regional computational centre in Cracow. Hybrid functional B3LYP with the 6-31G(d) basis set included in the GAUSSIAN 09 package [13] was used. Recently published reports show, that the same theoretical level was used successfully for the mechanistic study of series one-step [14-19] and two-step [5,6,20] 1,3-dipolar cycloadditions as well as different reactions involving fluororganic compounds [21-23]. In addition, similar simulations using more advanced B3LYP/6-31+G(d), B3LYP/6-31G(d,p) as well as B3LYP/6-311G(d) theoretical levels were performed. Optimizations of the stable structures were performed with the Berny algorithm, whereas the transition states were calculated using the QST2 procedure followed by the TS method. Stationary points were characterised by frequency calculations. All reactants, and

Page 5 of 17

products had positive Hessian matrices. All transition states showed only one negative eigenvalue in their diagonalized Hessian matrices, and their associated eigenvectors were confirmed to correspond to the motion along the reaction coordinate under consideration. For

ip t

all reactions, IRC calculations were performed to connect previously computed transition structures with suitable minima. For the calculations of solvent effect on the reaction paths

cr

the polarizable continuum model (PCM) [24]. Values of the global electron density transfer

GEDT=-ΣqA

us

(GEDT) [25] were calculated according to the formula:

an

where qA is the net charge and the sum is taken over all the atoms of dipolarophile.

M

Global electronic properties of reactants were estimated according to the equations recommended earlier by Parr and Domingo [26-28]. In particular, the electronic chemical

d

potentials (and chemical hardness (η) were evaluated in terms of one-electron energies of

te

FMO (HOMO andLU ) using the following equations: LUMO 

ηLUHOMO

Ac ce p

Next, the values of  and η were then used for the calculation of global electrophilicity (according to the formula:

 η

Subsequently, global nucleophilicity (N) [29] of diazocompounds can be expressed in terms of equation:

diazocompoundHOMO (tetracyanoethene) The local electrophilicity (k) condensed to atom k was calculated by projecting the index  onto any reaction centre k in the molecule by using Parr functions P+k [30]: k = P+k·

Page 6 of 17

The local nucleophilicity (k) condensed to atom k was calculated using global nucleophilicity N and Parr functions P-k [30] according to the formula:

ip t

k = P-k·

cr

3. Results and discussion

us

First and foremost, it was decided to analyze the nature of interaction between reagent molecules in the elementary act of cycloaddition. The recent, intensively developed

an

[5,16,17,20,26,27,31,32] DFT reactivity indices theory was used towards this goal. It turned out that hexafluoroacetone 1 is characterised by high global electrophilicity, in excess of

M

2.5eV. The global nucleophilicity index for this molecule is only 0.48eV. This compound should be included in the group of strong electrophiles within the reactivity scale proposed by

d

Domingo [26]. On the other hand, global electrophilicities of diazocompounds 2a-c are in the

te

range of 1.47-1.70eV, whilst global nucleophilicities are in excess of 3.7eV. The nucleophilic nature of the diazocompound becomes simultaneously more profound with the increase of the

Ac ce p

electrodonating properties of the substituent. Table 1. Global and local electronic properties for hexafluoroacetone 1 and diazocompounds 2a-c.

Global properties 

Local properties



η

[eV]

[eV]

[eV] [eV]

1

-5.71

5.87

2.78

0.48

2a

-3.36

3.70

1.52

3.91

0.23

0.41

0.91

1.62

2b

-3.28

3.67

1.47

4.00

0.21

0.41

0.86

1.63

2c

-3.54

3.67

1.70

3.75

0.22

0.40

0.81

1.50

N

P-C

P-N

NC

NN

[eV] [eV]

P+O 0.38

P+C 0.50

O

C

[eV]

[eV]

1.07

1.39

Page 7 of 17

Thus, it can be assumed that the cycloaddition process will be controlled in the analysed cases by GEDT from the diazocompound to hexafluoroacetone as the electrons acceptor. A similar conclusion can be reached from comparison of the values of chemical

ip t

electronic potentials (see Table 1). It should be noted that in cases of all reagent pairs, the difference in global electrophilicities exceeds 1eV. This should lead to inclusion of these

cr

reactions in the group of clearly polar processes [33].

Having indices of global reactivity, distributions of local electrophilicity for

us

hexafluoroacetone 1 and those of local nucleophilicity for diazocompounds 2a-c were determined. The analysis of local reactivity indices recently allowed

interpretation of

an

regioselectivity for a series of various cycloaddition reactions [16,17,26,27].

M

It turned out that the most electrophilic center of 1 is the carbon atom of the >C=O group. On the other hand, the strongest nucleophilic reaction centre in the molecules of

d

diazocompounds 2a-c is always located at the terminal nitrogen atom of the >C=N=N part.

te

Thus, the nature of local electrophile-nucleophile interactions should facilitate creation of oxadiazoles 3a-c, which – as experimental work has shown [6] – are created during the

Ac ce p

studied reactions.

The analysis of global electrophilicity indices suggests that the tested reactions have a

polar nature. However, it does not allow confirmation, in any manner, of the presence of zwitterionic intermediates along the path of substrate conversion into products. Thus, the next step included performance of DFT simulation of reaction paths. These simulations took into account (using the PCM model [24]) the presence of n-pentane, which has been used as a reaction medium in experimental conditions [6].

Page 8 of 17

ip t cr

Fig. 1. Key structures of 1,3-dipolar cycloaddition reaction of hexafluoroacetone 1 and

us

diphenydiazomethane 2a in n-pentane according to B3LYP/6-31G(d) (PCM) calculations

an

As the data of B3LYP/6-31G(d) calculations show, in the medium of n-pentane, reagent interactions lead in the first step of the reaction to a pre-reaction complex (PC) (Fig.

M

1) formation, in a barrier-free manner. It is accompanied by lowering the enthalpy of the reacting system by 2.9 kcal/mol. It must be noted that the energetic valley of the PC is only of

d

an enthalpic nature (Table 2). A radical entropy change of the reacting system results the

te

process of PC formation to have G>0. At room temperature, this excludes the possibility of a

Ac ce p

PC structure existing as a stable intermediate. Within the PC, no new bonds  are yet formed (Table 3). However, already at this stage the addent molecules become arranged in a manner determining process regioselectivity. PC is not a CT-type complex (Charge-Transfer Complex). This is indicated by a string GEDT. The existence of similar orientation prereaction complexes is suggested by DFT calculations related to 1,3-dipolar cycloadditions taking place in the case of gem-dinitroethene [5] and nitroacetylene [34]. The existence of such structures along the reaction path was also shown experimentally in the case of 1,3dipolar cycloaddition of ozone to etene [35].

Page 9 of 17

Table 2. Kinetic and thermodynamic parameters for 1,3-dipolar cycloaddition reaction of hexafluoroacetone 1 and diphenydiazomethane 2a according to DFT (PCM) calculations (H, G are in kcal/mol; S are in cal/molK).

Transition

H

G

1+2a→LM

-2.9

7.7

1+2a→TS

5.7

1+2a→3a

-17.4

-1.7

-52.7

1+2b→LM

-6.3

6.1

-41.4

()

(1.8371)

-48.8

5.2

19.6

-48.2

1+2b→3b

-17.6

-1.9

-52.4

1+2c→LM

-2.5

7.6

-34.1

1+2c→ TS

6.3

20.7

-48.5

-17.0

-1.5

-52.0

1+2a→LM

-2.4

8.5

-36.4

1+2a→ TS

4.9

19.7

-49.7

1+2a→3a

-16.6

-1.2

-51.8

1+2a→LM

-2.9

7.7

-35.4

1+2a→ TS

5.6

20.3

-49.1

1+2a→3a

-7.3

8.4

-52.7

1+2a→LM

-0.8

11.2

-40.5

1+2a→ TS

7.8

22.5

-49.4

1+2a→3a

-12.7

3.1

-53.0

1+2a→LM

-1.9

8.3

-34.3

d

n-Pentane

20.2

1+2b→ TS (1.8371) B3LYP/6-31G(d)

-35.6

M

n-Pentane

an

(1.8371)

us

n-Pentane

S

ip t

level

Solvent

cr

Theory

te

1+2c→3c

Ac ce p

Nitromethane (36.562)

n-Pentane

B3LYP/6-31G(d,p)

(1.8371)

n-Pentane B3LYP/6-31+G(d) (1.8371) B3LYP/6-311G(d)

n-Pentane

Page 10 of 17

(1.8371)

1+2a→ TS

8.2

22.9

-49.3

1+2a→3a

-11.8

3.4

-51.2

ip t

Further conversion of the reacting system along the reaction coordinate leads to the transition state (TS). As the data of B3LYP/6-31G(d) calculations show, this is related to an

cr

enthalpy increase of just a few kcal/mol. This observation correlates well with experimental data. As Shimizu and Barlett showed [6], the analyzed reaction can be realized in practice

us

under very mild conditions. Both new bonds required for the formation of the oxadiazole ring

an

are formed within the TS. The degree of their progression is variable, though. In particular, the C5-N1 bond reaches the length of 1.895 Å (l=0.735) within the TS. The C3-O4 shows

M

slower formation (r=2.181Å, l=0.488). The TS is thus a largely asynchronous structure. However it is not asynchronous enough to make a zwitterion formation possible. TS shows a

d

clearly polar nature (which was suggested by the earlier analysis of global nucleophile-

te

electrophile interactions). A proof of this nature is provided by a strong electron transfer towards the dipolarophile structure. This effect is illustrated quantitatively by the value of the

Ac ce p

GEDT (0.31e). IRC calculations relate the TS structure to the valley of the PC complex (on one side) and of the 3a product (on the other side). All attempts at locating the hypothetical, zwitterionic intermediates along with reaction paths have been unsuccessful. In addition, no stable zwitterionic structures with "extended" conformations were obtained (as the DFT calculations data suggest, such structures may be created during polar cycloaddition – e.g. nitroacetylene with of allenyl type 1,3-dipoles [34] and methyl dinitrocinnamate with alkylvinyl ethers [36]). Thus, the energy changes of the reacting system during the cycloaddition reaction may be illustrated as depicted in Fig. 2. Subsequently, regioisomeric reaction channel leading to 4,5-dihydro-1,2,3-oxadiazole 4a has been also analyzed. It was found however, that under analyzed conditions this reaction

Page 11 of 17

path should be considered as forbidden from kinetic point of view (H>16kcal/mol, G>33kcal/mol). Table 3. Key structures for 1,3-dipolar cycloaddition reaction of hexafluoroacetone 1 and

ip t

diphenydiazomethane 2a according to B3LYP/6-31G(d) (PCM) calculations (H, G are in kcal/mol;

C3-O4

Solvent

1+2b

2.181 0.488

3a

1.443

LM

3.177

TS

Nitromethane (36.562)

*)

l XY

1+2a

[e]

frequence [cm -1]

0.00

1.895 0.735 0.25

0.31

1.498

0.20

2.589

0.00

2.189 0.484

1.889 0.739 0.26

0.32

3b

1.444

1.498

0.21

LM

3.273

2.790

0.00

Ac ce p 1+2c

an

TS

te

(1.8371)

2.703

M

n-Pentane

3.320

l *)

GEDT

d

1+2a

LM

r [Å]

l

us

l *)

ture r [Å]

()

Imaginary

C5-N1

StrucReaction

cr

S are in cal/molK).

TS

2.176 0.490

1.900 0.731 0.24

0.31

3c

1.442

1.497

0.20

LM

3.245

2.652

0.00

TS

2.222 0.461

1.857 0.760 0.30

0.35

3a

1.444

1.498

0.20

-276.325

-269.156

-279.700

-243.340

rXTS Y  rXP Y  1 rXP Y

where rTSX-Y is the distance between the reaction centers X and Y in the transition structure and rPX-Y is the same distance in the corresponding product.

Page 12 of 17

ip t cr us an

Fig. 2. Enthalpy profiles for 1,3-dipolar cycloaddition reaction of hexafluoroacetone 1 and

M

diphenydiazomethane 2a in n-pentane according to DFT (PCM) calaulations

A very similar picture of the analysed reaction is provided by calculations in more

d

theoretically advanced function bases 6-31G(d,p), 6-31+G(d) and 6-311G(d). In particular,

te

the shape of the energy profiles of the reaction is virtually identical from the qualitative point

Ac ce p

of view. Only the quantitative description of individual critical points shows slight changes. However, geometry parameters of the critical structures remain virtually identical to those obtained from B3LYP/6-31G(d) calculations. It was then decided to check what influence may be exerted by substituents in the

phenyl ring of the 1,3-dipole on the course of cycloaddition. These studies employed diazocompounds 2b,c, which have also been tested under experimental conditions as components of 1,3-dipolar cycloaddition with hexafluoroacetone 1. Of turned out that introduction of electrodonating methyl group stimulates lowering the activation barrier by 0.5 kcal/mol, whilst introduction of a chlorine atom results in the increased activation barrier. To a degree, the nature of the substituent also influences the synchronicity of new bonds formation within the TS. In particular, the presence of an electrodonating methyl group

Page 13 of 17

facilitates asynchronicity of the TS. However, this effect is relatively weak (see l values in Table 3) and does not make a change of reaction mechanism possible. Finally,

simulations

of

a

reaction

between

hexafluoroacetone

1

with

ip t

diphenyldiazomethane 2a in the medium of nitromethane as a model, strongly polar medium (=36.562) were performed. It can occur [5] that the increase of medium polarity stimulates

cr

differentiation of advancement of the new -bonds formation that the single-step mechanism

us

of 1,3-dipolar cycloaddition gives way to the two-step mechanism. B3LYP/6-31G(d) calculations showed that asynchronicity of the reaction centres getting closer is increased in

M

enforce a zwitterionic, stepwise mechanism.

an

strongly polar nitromethane (see l values in Table 3). However, it is not increased enough to

4. Conclusion

d

Analysis of the nucleophile-electrophile interactions suggests a polar nature of 1,3-

te

dipolar cycloaddition of hexafluoroacetone to sterically crowded diazomethanes. Simulations of reaction paths, regardless of theory levels, show, however, that this proccess - even if

Ac ce p

taking place via polar structures - takes place according to the one-step mechanism. The asynchronicity of the process on formation of the two single bonds is mainly controlled by the favorable nucleophilic/electrophilic interaction taking place at the TSs of the reaction.

Acknowledgements

The regional computer center "Cyfronet" in Cracow (Grant No. MNiSW/Zeus_lokalnie/PK/009/2013) is thanked for the allocation of computing time.

Page 14 of 17

References [1] R. Huisgen R., in: 1,3-Dipolar Cycloaddition Chemistry, (Ed. A. Padwa), Wiley Interscience, New York (1984).

ip t

[2] A.F. Khlebnikov, A.S. Koneva, A.A. Virtseva, D.S. Yufit, G. Mlostoń, H. Heimgartner, Helv. Chim. Acta 97 (2014) 453-470.

cr

[3] R. Huisgen, P. Pöchlauer, G. Młostoń, K. Polsborn, Helv. Chim. Acta 90 (2007) 983-998.

[5] R. Jasiński, Tetrahedron 69 (2013) 927-932.

an

[6] R. Jasiński, Tetrahedron Lett. 56 (2015) 532-535.

us

[4] H. Wójtowicz-Rajchel, H. Koroniak, J. Fluor. Chem. 135 (2012) 225–230.

M

[6] N. Shimizu, P.D. Barlett, J. Am. Chem. Soc. 100 (1978) 4260-4267. [7] R. Huisgen, H. Giera, K. Polborn, Tetrahedron 61 (2005) 6143–6153.

te

998.

d

[8] R. Huisgen, P. Pöchlauer, G. Mlostoń, K. Polborn, Helvetica Chim. Acta 90 (2007) 983-

Ac ce p

[9] V.Yu. Korotaev, A.Yu. Barkov, P.A. Slepukhin, M.I. Kodess, V.Ya. Sosnovskikh, Tetrahedron Letters 52 (2011) 5764–5768.

[10] A.L Fridman, F.A Gabitov, W.D Surkov, Zh. Org. Khim. 8 (1972) 2457-2462. [11] C.R. Burkholder, W.R. Dolbier, M Medebielle, J. Fluor. Chem. 109 (2001) 39-48. [12] C. Alonso, E.M. de Marigorta, G. Rubiales, F. Palacios, Chem. Rev. 115 (2015) 1847– 1935.

[13] M.J. Frisch , G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery, T.Jr. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, Y.

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Nakajima, O. Honda, O. Kitao, H. Nakai, M. Klene, X Li, J.E. Knox, H.P. Hratchian, J.B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador,

ip t

J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, M.C., Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G.

cr

Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D. J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A.

us

Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C.

an

Gonzalez,. J.A. Pople, Gaussian 09 rev A.1, Gaussian, Inc., Wallingford CT (2009). [14] R. Jasiński, Coll. Czech. Chem. Commun. 74 (2009) 1341-1349.

M

[15] R. Jasiński, J. Heterocyclic Chem. 52 (2015) 185-192.

[16] A. Szczepanek, E. Jasińska, A. Kącka, R. Jasiński, Current Chem. Lett. 4 (2015) 33-44.

te

d

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