Ghenia Bentabed-Ababsa

Aicha Derdour

Thierry Roisnel

José A Saez

Patricia Pérez

Edouardo Chamorro

Luis R Domingo
email: domingo@utopia.uv.es

Florence Mongin
email: florence.mongin@univ-rennes1.fr

Jose A Sáez

Eduardo Chamorro

Andres Bello

A

Introduction

Cycloaddition reactions are fundamental synthetic processes, with both synthetic and mechanistic interest in organic chemistry. Among them, 1,3-dipolar cycloadditions, the general concept of which was introduced by Huisgen and co-workers in the 1960s, 1 are versatile tools for building fivemembered heterocycles. 2 The 1,3-dioxolane 3 and oxazolidine 4 moieties represent important skeletons present in molecules endowed with biological activities. Derivatives can be synthesized by reaction of carbonyl ylides, generated by thermal electrocyclic ring opening of epoxides, with -bonds of aldehydes 5 and imines. 6 Current understanding of the underlying principles in reactions such as 1,3-dipolar cycloaddition has grown from a fruitful interplay between theory and experiment. 2 Through a recent study of the [3+2] cycloaddition reaction between carbonyl ylides and ketones in order to synthesize spirocyclic dioxolane indolinones, 7 we have embarked on theoretical calculations using DFT methods to depict the mechanism of these reactions using carbonyl ylides generated from epoxides. These reactions are domino processes that comprise two consecutive reactions (see Scheme 1). 7 The first one is the thermal ring opening of the epoxide 1a to yield the carbonyl ylide intermediate CYa, whereas the second reaction is a concerted [3+2] cycloaddition of CYa with the ketone function of N-methylisatin to yield the final [3+2] spirocycloadducts SCAa. The activation energy associated with the thermal ring opening of the epoxide 1a, 26.9 kcal/mol, proved to be slightly higher than that associated with the thermal opening of the cyclopropane ring on the 2,2-dimethoxy-3,3-dicyanospiro [cyclopropane-1,9'-[9H]fluorene recently reported by Warkentin, 24.4 kcal/mol. 8 However, the formation of the corresponding carbonyl ylide CYa was lesser endothermic, 11.7 kcal/mol, that the corresponding zwitterion intermediate, 22.8 kcal/mol. Interestingly, the zwitterion proposed by Warkentin was captured by benzaldehyde to yield a formally [3+2] cycloadduct, which showed the same regioselectivity than that found on the reaction of CYa with isatin.

In spite of the cycloaddition has a lower activation energy than that for the opening of the epoxide, the large endothermic character of the formation of the intermediate CYa together with the bimolecular nature of the cycloaddition, make this reaction to be the rate-limiting step of the overall domino process. The energetic results indicated that the [3+2] cycloaddition reaction present a poor stereoselectivity and a large regio-and chemoselectivity, in agreement with the experimental results.

The most favorable regioisomeric channels were associated with the nucleophilic attack of the isatin carbonyl oxygen atom to the phenyl substituted carbon atom of the carbonyl ylide CYa. The larger electrophilicity of the carbonyl ylide CYa,  = 4.29 eV, with respect to N-methylisatin,  = 2.66 eV, allowed us to explain the nucleophilic attack of the carbonyl oxygen of the isatin to the phenyl substituted carbon atom of CYa. However, a charge transfer (CT) analysis at the transition state structures (TSs) showed a slight net CT to the ketone framework. If reactions between carbonyl ylides, generated by thermal electrocyclic ring opening of epoxides, with -bonds of aldehydes 5a,b and imines 6 have previously been described with similar regiospecificity and stereoselectivity, the identification of the cis and trans cycloadducts products has not been studied

unequivocally. Here we describe similar syntheses, 9 but most of all the structural analysis of the cycloadducts as well as their formation mechanism.

The discrepancy between the electrophile/nucleophile interaction and the CT observed at the TSs of the cycloaddition of the carbonyl ylide CYa with N-methylisatin has encouraged us to perform subsequent studies on the mechanism of these [3+2] cycloaddition reactions.

Results and Discussion

Synthetic Aspects

Reactions were first carried out between 2,2-dicyano-3-(4-substituted)phenyloxiranes 1a-c 10 and benzaldehydes 2 (1 molar equivalent) in order to get 2,5-diphenyl-1,3-dioxolane-4,4-dicarbonitriles (Table 1). The conversion to the derivatives 3-5 using 3,4,5-trimethoxybenzaldehyde (2a) was monitored by NMR, and showed that the reactions carried out in refluxing toluene were completed after 35-68 h, depending on the R 1 group on the epoxide 1. The cis products 3a-5a were isolated from the crude mixture by recrystallization from petrol/Et 2 O in yields ranging from 40 to 52%, and identified by NMR. NOESY, HMBC and HMQC sequences performed on (CD 3 ) 2 CO solutions which allowed for the assignments of the 1 H and 13 C NMR signals. The NOESY experiment showed the relationship between H2 (singlets at 6.59, 6.57 and 6.53 ppm for 3a, 4a and 5a, respectively) and H2'-H6' (at 7.80, 7.74 and 7.68 ppm), and between H2 and H5 (singlets at 6.04, 6.00 and 5.99 ppm) (see Table 1, a). Cis 5a was then identified unequivocally by X-ray structure analysis. The trans compounds 3b-5b were identified using 1 H NMR spectra of enriched fractions. The diastereoisomeric ratios were determined from 1 H NMR spectra of the crude mixtures. The cis products predominate over the trans with about 3 to 1 (a:b) ratios.

A rising number of articles have advocated the use of microwave technology in organic synthesis.

Long reaction times often required for cycloaddition reactions could generally be reduced using this technique. 11 Thus, several experiments were performed using microwave irradiation at various powers and irradiation times. [START_REF]Using a monomode reactor (Prolabo Synthewave ® 402) with accurate control of power and temperature (infrared detection)[END_REF] The best conditions were obtained without solvent (power: 60 W), with significant reduction of reaction times (45 min instead of 35 h for 3 (R 1 = H), 40 min instead of 68 h for 4 (R 1 = Cl), and 35 min instead of 45 h for 5 (R 1 = OMe)) and similar a:b ratios (68/32 instead of 77/23

for 3 (R 1 = H), 71/29 instead of 72/28 for 4 (R 1 = Cl), and 70/30 instead of 68/32 for 5 (R 1 = OMe)) in comparison to reaction in toluene at reflux 13 (Table 1, Entries 1-3).

Replacing 3,4,5-trimethoxybenzaldehyde (2a) by piperonal (2b) slightly disfavored the formation of the cis compounds 6a and 7a over the trans 6b and 7b (average a:b ratios of 62/38 for 6 against 72/28 for 3 (R 1 = H), and 60/40 for 7 against 71/29 for 4 (R 1 = Cl)) whereas the same ratio (69/31) was obtained for 8 and 5 (R 1 = OMe) (Entries 4-6). The trans 7b and 8b could only be identified using 1 H NMR spectra of enriched fractions, but crystals of 6b suitable for X-ray structure analysis were obtained.

O CN CN R 1 1 O O CN CN R 1 + a b H O O CN CN R 1 H 2 5 4 1' 2' 3' 4' 5' 6' 2 5 4 2 O H 1' 2' 3' 4' 5' 6' R 2 R 3 R 4 R 2 R 3 R 4 R 2 R 3 R 4 1" 2" 3" 4" 5" 6" 1" 2" 3" 4" 5"
6" H H The cis products 6a-8a were isolated from the crude mixture by recrystallization from petrol/Et 2 O in moderate to medium yields. Their structures were elucidated as for compounds 3a-5a using NMR experiments, and confirmed by X-ray analysis.

In order to reach differently 5-substituted 2-(4-substituted phenyl)-1,3-dioxolane-4,4-dicarbonitriles, reactions were performed with 1-naphthaldehyde ( 9), indole-3-carboxaldehyde (10), furan-2carboxaldehyde (11) and thiophene-2-carboxaldehyde (12) (1 molar equivalent) (Table 2). The reactions were completed in refluxing toluene after reaction times of 35-84 h, depending on both the R group on the epoxide 1 and the nature of the aldehyde. The reactions were also carried out under microwave irradiation, allowing reaction times to be reduced from 54-84 h to 30-45 min.

The cis products 13a-15a proved to be favored over the trans for the reactions carried out between 2,2-dicyano-3-(4-substituted)phenyloxiranes 1a-c 10 and 1-naphthaldehyde (9). They were isolated from the crude mixtures by column chromatography over silica gel in yields ranging from 27 to 55%, and identified by NMR. NOESY, HMBC and HMQC sequences performed on the racemic 14a allowed the assignments of the main 1 H and 13 C signals. The proximity between H2 and H5, in accordance with a cis stereochemistry, was shown by conducting the NOESY experiment in CDCl 3 , after assignment of the singlets at 6.38 and 6.48 ppm 14 to H2 (correlation between H2 and H2'-H6') and H5 (correlation between H5 and H2"), respectively (see Table 2, a). After identification of the trans compounds 13b-15b from 1 H NMR spectra of the crude mixtures, the diastereoisomeric ratios were calculated using the integration. The a:b ratio is about 63/37 when R = H, against 71/29 when R = Cl and OMe, and is slightly favored using classical heating conditions (Entries 1-3).

The reactions carried out under classical heating conditions between 2,2-dicyano-3-(4substituted)phenyloxiranes 1a-c 10 and indole-3-carboxaldehyde (10) resulted in different diastereoisomeric ratios than using 1-naphthaldehyde (9). Indeed, 50/50 (16, R = H), 43/57 (17, R = Cl) and 84/16 (18, R = OMe) were obtained as cis/trans ratios using the heterocyclic aldehyde against 65/35 (13, R = H), 75/25 (14, R = Cl) and 76/24 (15, R = OMe) for the non-heterocyclic one. Recourse to microwave irradiation favors the formation of 16a-17a over 16b-17b, but disfavors the formation of 18a over 18b (Entries 4-6). The cis products were isolated from the crude mixture by column chromatography over silica gel in 30 to 46% yields, and were identified by NMR as before, by identifying a correlation relation between H2 and H5 (see Table 2, a). The structure of 17a was confirmed by X-ray analysis of crystals obtained by slowly evaporating an acetone solution. The products 16b-17b were identified using 1 H and 13 C NMR spectra of enriched fractions, and 18b from the 1 H NMR spectra of the crude mixture. Reactions were finally carried out between 2,2-dicyano-3-(4-substituted)phenyloxiranes 1a-d 10 and imines 25 15 (1 molar equivalent) in order to get substituted 2,4-diphenyloxazolidine-5,5-dicarbonitriles (Table 3). The conversion to the derivatives 26-28 using N-(phenylmethylene)methanamine (25a), 30-32 using N-(1,3-benzodioxol-5-ylmethylene)propylamine (25b), 34-36 using N-(1,3-benzodioxol-5ylmethylene)butylamine (25c), and 38-40 using N-(1,3-benzodioxol-5-ylmethylene)benzylamine (25d)

were monitored by NMR, and showed that the reactions carried out in refluxing toluene were finished in 5-40 h with imines 25, against 35-72 h with benzaldehydes 2. The conversion to derivatives 29, 33, 37 and 41 proved possible by reaction of imines 25 with 2,2-dicyano-3-(4-nitro)phenyloxirane (1d), an epoxide unable to react with benzaldehydes 2. Also in contrast to what has been observed using aldehydes, the 1 H NMR spectra of the crude mixtures showed cis diastereoisomers were always mainly formed with ratios > 90/10. The main compounds 26a-41a were isolated from the crude reaction mixtures by chromatography over silica gel or/and recrystallization from Et 2 O. Yields were satisfactory for R 1 = H, Cl, OMe (48-80%), and lower for R 1 = NO 2 (18-40%). Even if reaction times were shorter using imines, the reactions were carried out under microwave irradiation without significant changes (32-85% for R 1 = H, Cl, OMe, and 18-33% for R 1 = NO 2 ). As before, recourse to NMR HMBC, HMQC and NOESY sequences allowed the assignments of 1 H and 13 C signals, as well as the detection of a correlation between H2 and H4 (see Table 3, a), in accordance with a cis stereochemistry. Cis 29a, 31a-33a and 39a were then identified unequivocally by X-ray structure analysis.

Theoretical Study of the [3+2] Cycloaddition Reactions of the Carbonyl Ylide CYa with Aldehydes and Imines

a) Analysis based on the global and local reactivity indices at the ground state of the reagents.

Recent studies devoted to Diels-Alder 16 and [3+2] cycloaddition [START_REF] Pérez | [END_REF] reactions have shown that the analysis of the global indices defined within the context of conceptual DFT 18 is a powerful tool to understand the behavior of polar cycloadditions. In Table 4, we report the static global properties, namely, electronic chemical potential , chemical hardness , global electrophilicity , and nucleophilicity N, of the carbonyl ylides CYa-c, the aromatic aldehydes 2a-c and the imines 25a and 25e.

The electronic chemical potentials, , of the carbonyl and imine derivatives, which range from -0.1434 to -0.1199 au, are higher than those for the carbonyl ylides, where  values range from -0.1576 to -0.1746 au, indicating that along these [3+2] cycloadditions the net CT will take place from the carbonyl and imine derivatives to the carbonyl ylides.

The carbonyl ylide CYa has a high electrophilicity value, 19  = 4.29 eV. Electron-releasing (ER)

substitution by a -OMe group decreases the electrophilicity of CYc,  = 3.80 eV, while electronwithdrawing (EW) substitution by a Cl atom increases the electrophilicity of CYb,  = 4.67 eV. Note that these carbonyl ylides present the larger electrophilicity of this series, indicating that along these

[3+2] cycloaddition reactions they will act as strong electrophiles. Unsubstituted benzaldehyde (2c) and the corresponding methyl imine 25a also have large electrophilicity values,  = 1.79 and 1.32 eV, respectively, being classified as strong electrophiles within the electrophilicity scale. 16,[START_REF] Pérez | [END_REF] Substitution on the aromatic ring by ER -OR groups decreases the electrophilicity of the carbonyl and imine derivatives.

entry compound (R)    N 1 CYb (R 1 = Cl) 1 3 2 CN O CN R 1 -0.1746 0.0889 4.67 3.16 2 CYa (R 1 = H) -0.1693 0.0908 4.29 3.28 3 CYc (R 1 = OMe) -0.1576 0.0890 3.80 3.62 4 2c (R 2 = R 3 = R 4 = H) O H R 2 R 3 R 4 4 5 -0.1590 0.1923 1.79 2.18 5 2a (R 2 = R 3 = R 4 = OMe) -0.1434 0.1719 1.63 2.88 6 2b (R 2 R 3 = OCH 2 O, R 4 = H) -0.1392 0.1656 1.59 3.08 7 25a (R 5 = R 6 = H) NMe H R 5 R 6 4 5 -0.1379 0.1964 1.32 2.70 8 25e (R 5 R 6 = OCH 2 O) -0.1199 0.1705 1.15 3.54
The carbonyl ylides have very large nucleophilicity values, 20 N between 3.16 and 3.62 eV, respectively, and are also classified as strong nucleophiles. Note that the nucleophilicity is also sensitive to the substitution on the aromatic ring, the methoxy derivative CYa being the most nucleophilic species of the series given in Table 4. Interestingly, as can be concluded of the  and N values, these carbonyl ylides could act as strong electrophiles and nucleophiles in polar cycloaddition reactions. Benzaldehyde (2c) and the imine 25a have also a large nucleophilicity, which increases with the ER substitution on the phenyl ring. Thus, the imine 25e is the best nucleophile of the PhCH=X(R) reagent subseries.

An analysis of the global indices indicates that both reagents involved in these [4+3] cycloadditions can act as good electrophiles and nucleophiles in polar cycloadditions. It is expected that the more favorable polar interaction will take place between the strongest electrophilic reagent and the strongest nucleophilic one. Previously, we have proposed that the better interaction will take place between the pairs of reagents presenting the largest , 16 that is, between reagents located at the opposite sides of the electrophilicity scale. In Diels-Alder reactions, we have shown that a reagent located above other reagent in the electrophilicity scale forces the last to behave as a nucleophile. 21 Therefore, it is expected that in both reactions, CYa will act as the electrophile whereas 2c and 25a will act as nucleophiles.

Recent studies devoted to cycloaddition reactions with a polar character have shown that the analysis of the local electrophilicity index, 22  k , at the electrophilic reagent and the nucleophilic Fukui function, 23  k f at the nucleophilic one allows to explain the observed regioselectivity. Very recently, we have proposed the local nucleophilicity index N k , 24 which is able to measure the local nucleophilic activation between molecules. In this way, the k and N k at the carbonyl ylide CYa, at the aldehyde 2c

and at the imine 25a will be used to predict the best electrophile/nucleophile interaction in these polar cycloadditions, and therefore to explain the regioselectivity experimentally observed.

The carbonyl ylide CYa has the largest electrophilic activation at the phenyl substituted C1 carbon atom,  k = 1.30 eV, and the largest nucleophilic activation at the dicyano-substituted C3 carbon atom, N k = 1.13 eV (see Table 5). Therefore, C1 and C3 will be the most electrophilic and nucleophilic centers, respectively, of these carbonyl ylides. This picture is in agreement with a heterolytic C1-C3 bond breaking of the epoxide 1a, in which the phenyl substituted C1 position becomes the carbocationic center while the dicyano-substituted C3 position turns into the carbanionic center.

The aldehyde 2c has the largest electrophilic activation at the carbonyl C4 carbon atom,  k = 0.47 eV, and the largest nucleophilic activation at the carbonyl O5 oxygen atom, N k = 1.52 eV, whereas the imine 25a has both the largest electrophilic and nucleophilic activation sites at the imine N5 nitrogen atom,  k = 0.30 eV and N k = 0.50 eV (see Table 5). Then, while the carbonyl C4 carbon is the most electrophilic center of 2c, the N5 nitrogen is the most electrophilic center of the imine 25a. In addition, the nucleophilic activation of the carbonyl O5 oxygen of 2c is three times higher than that at the imine N5 nitrogen of 25a. The results obtained for the imine 25a, which locate the most nucleophilic and electrophilic centers over the same atom, are a consequence of the presence of the conjugated aromatic ring at the carbon atom, which modifies the electrophilic/nucleophilic behavior of the N5=C4 double bond. For the [3+2] cycloaddition reaction between CYa and the aldehyde 2c, the regioisomeric channels associated to the formation of the C1-O5 and C3-C4 bonds correspond to the approach that makes possible the interaction between the most electrophilic(A)/nucleophilic(B) centers of these reagents, respectively. If we consider the sum of the  k + N k values at the two feasible interactions that take place along the CYa + 2c reaction, 2.82 eV along the C1/O5 two-center interaction ( kC1 + N kO5 ) and 1.60 eV (N kC3 +  kC4 ) along the C3/C4 two-center interaction, we can see that the former is clearly favored.

This analysis allows to explain the fact that at the more favorable regioisomeric TSo1c and TSo1t, the C1-O5 bond-formation is more advanced than the C3-C4 one (see later). The activation energies of the most favorable reaction channels of these cycloadditions have very low values; 2.6 kcal/mol for TSo1c and 2.8 kcal/mol for TSn1t (see Table 6). The cycloadditions present a very low stereoselectivity in the case of aldehydes. For the reaction of CYa with the imine 25a, the computed trans stereoselectivity is opposite to that found experimentally. Further single point calculations at the MP3/6-31G*//B3LYP/6-31G* yielded a trans stereoselectivity similar to that obtained at DFT level. All these [3+2] cycloaddition reactions present a total regioselectivity, being the more unfavorable regioisomeric TSs between 10 and 12 kcal/mol higher in energy. In addition, all cycloadditions are strongly exothermic: between -30.0 to -36.0 kcal/mol. These energy results are similar to those recently obtained for the [3+2] cycloaddition reaction of the carbonyl ylide CYa with N-methylisatin (see Scheme 1). -934.355789 -36.4 -934.363842 -33.5 CAn1t -934.348124 -31.6 -934.356115 -28.6 CAn2c -934.350452 -33.1 -934.358006 -29.8 CAn2t -934.355171 -36.0 -934.362370 -32.5 As all these [3+2] cycloaddition reactions have a polar character and solvent can stabilize some species, solvent effects of toluene were considered through single point energy calculations over the gas-phase optimized geometries using the PCM method. Solvent effects stabilize all species between 2 and 7 kcal/mol (see Table 6), the reagents being more stabilized that the TSs. In consequence, the activation barrier for the cycloadditions increases in 2.0 and 1.0 kcal/mol. In addition, solvent effects do not change the gas-phase low stereo-and the large regioselectivity found. Therefore, solvent effects appear to have a low influence over the [3+2] cycloaddition reactions, remaining the trans selectivity for the reaction with the imine 25a. Further thermodynamic calculations in toluene showed that TSn1t remains 2.6 kcal/mol in free energy below TSn1c (see Table 7). One of the reviewers proposed that under the rather severe reaction conditions the imine E/Z interconversion should take place populating the less stable (Z) imine stereoisomer that is presumably a more reactive dipolarophile. In order to probe this suggestion, the TSs associated with the cis and trans approach modes of the imine 25a in the (Z) configuration to the carbonyl ylide CYa, TSn1c-Z and TSn1t-Z, were optimized. However, TSn1c-Z and TSn1t-Z were located on the potential energy surface 1.0 and 1.5 kcal/mol above TSn1t (the total energies and geometries of these TSs are given is supplementary material). For the reaction of the carbonyl ylide CYa with isatin, we found that thermodynamic calculations at the reaction conditions does not modify the regio-and chemoselectivity of the reaction. In order to probe if these calculations have some incidence on the cis/trans stereochemistry of the reactions, the free energies of the TSs involved in these cycloadditions were calculated at 110 ºC in toluene. The results are summarized in Table 7. A comparison of the free energies of the TSs involved in the cycloaddition of CYa with benzaldehyde (2c) and the imine 25a

b) Mechanistic study of the [3+2] cycloaddition reactions of the carbonyl ylide

with the (E) configuration indicates that the inclusion of the thermal corrections and entropy to the free energies does not modify the stereochemistry found with the gas-phase electronic energies. However, when the thermodynamic calculations were performed at the TSs of the imine 25a with the (Z) configuration an interesting result was found. Now, TSn1c-Z is located -1.5 kcal/mol below TSn1t in free energy, in agreement with the trans selectivity found experimentally. An analysis of the enthalpies and entropies of the TSs indicates that the large entropy associated with TSn1c-Z together with the high reaction temperature, 110 ºC, are responsible for the trans selectivity.

The The electronic structure of the TSs involved in the more favorable regioisomeric channels of these

[3+2] cycloaddition reactions was analyzed using the Wiberg bond order 25 indicates that this cycloaddition has a two-stage mechanism; 26 that is, while at the first stage of the reaction only the C1-N5 bond is being formed, the second stage is associated to the C3-C4 bond formation. We want to remark the HPn1t structure located at the half of IRC where the C1-N5 bond formation is very advanced, 1.697 Å, whereas the C3-C4 bond formation is very delayed, 2.778 Å. At HPn1t, the BO values of the C1-N5 and C3-C4 forming-bonds are 0.74 and 0.22, respectively. On the other hand, the IRC from the asynchronous TSo1t to CAo1t shows the concerted nature of this cycloaddition. Along the reaction, the C1-O5 bond-formation is slightly more advanced than the C3-C4 one.

The natural population analysis (NPA) allows the evaluation of the CT and its direction at these

[3+2] cycloaddition reactions. The B3LYP/6-31G* natural atomic charges at the TSs associated to the most favorable regioselective channels were shared between the fragments of the carbonyl ylide CYa and the PhCH=X(R) derivatives 2c and 25a. The net charge at the carbonyl ylide fragment at these TSs is predicted to be: +0.03e at TSo1c, +0.05e at TSo1t, and -0.13e at TSn1c and TSn1t. Along the IRC from TSn1t to CAn1t the CT increases until it reaches -0.20e at HPn1t; after this point, the CT decreases due to a back donation process from the carbonyl ylide to the imine. Some interesting conclusions can be obtained from these results: i) firstly, these values indicate that there is a change in the direction of the flux of the net CT at the TSs involving carbonyl or imine derivatives. Note that the CT obtained at the reaction of unsubstituted benzaldehyde (2c) is similar to that found in the [3+2] cycloaddition reaction of CYa with N-methylisatin; 7 ii) while at the reaction involving the imine 25a

the CT increases along the first stage of the reaction until HPn1t, the unexpected low CT observed in the reaction of benzaldehyde (2c) remains along the reaction.

Considering that the technical details and nomenclature of the ELF topological analysis are widely available, 27,28 we will concentrate our attention directly on its application 29,30 to the characterization of electron delocalization and the bonding pattern associated to TSo1t and TSn1t structures. Our aim is to further elucidate the electronic nature of charge rearrangement of these TSs associated to polar cycloaddition processes. The topological analysis of ELF for both TSs reveals the same structure of attractors (Figure 3). Focusing on the valence region, disynaptic basins associated to the bonding regions C4-X5, C1-O2 and O2-C3 appear. In addition, monosynaptic basins associated to the X5, and O2 atom in each TS emerge. A monosynaptic basin V(C3) associated to the atom C3 of CYa is also found in the reaction center of both TSs. This polarized monosynaptic basin V(C3) can be associated to the carbanionic center of the zwitterionic carbonyl ylide CYa. The basin populations associated to the V(C1,O2), V(C3,O2), and V(C4,X5) regions are 1.84e, 1.40e, and 1.98e for TSo1t and 1.81e, 1.46e, and 2.80e for TSn1t, respectively. The monosynaptic basins V(O2), V(X5), and V(C3) integrate 4.27e, 5.44e, and 1.05e in the case of TSo1t and 4.15e, 2.70e, and 0.94e in the case of TSn1t, respectively.

These populations indicate a highly polarized electronic rearrangement. A complete analysis of the delocalization (not included here) reveals a greater fluctuation of electron populations between the valence basins in TSo1t than those observed in TSn1t. The electron density accumulated on the C3 center is, as noted above, higher at TSo1t than at TSn1t. This result can be rationalized by a large polarization of the former due to a back donation process (see later). The ELF topological analysis of these two TSs does not reveal disynaptic basins that can be associated to the two new forming bonds (e.

g., C3-C4 and C1-X5). By considering a simple summation of electron and nuclei charges in the two separated reagent regions in both cases, 20 charge separations of 0.08e and 0.12e for TSo1t and TSn1t can be measured, respectively. The population analysis based on the topological regions shows that the ylide fragment is slightly positively charged in the case of TSo1t (+0.08e) and negatively charged in the case of TSn1t (-0.12e). These results are in complete agreement with those obtained through the NBO analysis made before. The polar nature and relative polarization of these TSs is also evident from the observed low bifurcation ( *  ) values occurring between the two fragments in each case. As it has been previously emphasized, 20 the higher ELF bifurcation between two regions is, the higher the electron delocalization between both regions is expected. In the present case, such bifurcations are associated to the C1-X5 and C3-C4 bonding domains, 4). For TSo1t, the bifurcation values are 0.375 and 0.358 for C1-X5 and C3-C4, respectively; while for TSn1t, the corresponding values are 0.460 and 0.100. Close bifurcation values for TSo1t can be traced out to a more synchronous electron charge rearrangement from O5 to C1 and from C3 to C4 (e. g., back donation as above described). This effect is absent in the attack of N5 to C1 in TSn1t. As a result, a slightly higher accumulation of charge results in the monosynaptic region associated to the carbon atom C3 in TSo1t. TSn1t becomes more polar and more asynchronous with the formation of the two new bonds. The ELF for HPn1t structure indicates that a disynaptic attractor can be associated to the formation of the C1-N5 bond, localizing 1.25e (see Figure 5). The monosynaptic valence attractor associated to C3 atom remains monosynaptic in nature and it integrates to 0.99e in HPn1t. Therefore, becomes clear that, on this pathway, the bond-formation process is more asynchronous than that for the TSo1t. This picture supports the non-concerted nature of the two-stage mechanism aforementioned: while appear a new disynaptic basin V(C1,N5) indicating the formation of the C1-N5 bond, the monosynaptic basin V(C3) remains at TSn1t. 30c The basin populations associated to the V(C1,O2), V(C3,O2), and V(C4,N5) regions are 1.46e, 1.36e and 2.48e, respectively. The monosynaptic attractors V(O2), V(N5) and V(C3) have populations of 4.61e, 2.00e and 0.99e. As compared to TSn1t, the HPn1t structure is in fact more polarized. The population analysis based on the topological regions indicates that the negative charge at the ylide fragment increases to 0.27e. This result is in complete agreement with the NBO analysis made at HPn1t and discussed above. This fact further supports the idea that along the reaction path associated with the TSn1t, the CT process is modulated by the initial attack of the N5 center to the C1 atom, opening the channel to the ring closure due to the attack of C3 atom on the ylide to the C4 imine center.

* ( 3, 4) V C C  and * ( 1, 5) V C X  (See Figure
The present theoretical study allows one to explain the mechanism of the [3+2] cycloaddition reactions of the carbonyl ylides as CYa as well as the unexpected reverse CT found at the reactions with aldehydes and ketones. The analysis of the reactivity indices indicates that both reagents involved in these [3+2] cycloadditions (carbonyl ylides and aldehydes or imines) have electrophilic/nucleophilic behaviors. However, the large electrophilic character of the carbonyl ylides makes us to think that these intermediates will act as electrophiles in polar processes. This analysis is in clear agreement with the NBO and ELF analysis of the electronic structure of TSn1c. At this nonconcerted TS, which is associated with the nucleophilic attack of the imine N5 nitrogen to the phenyl substituted C1 carbon of the ylide, the CT fluxes clearly from the imine 25a to CYa (see Scheme 3). In conclusion, we have shown that first 2,5-diaryl-1,3-dioxolane-4,4-dicarbonitriles and then 2,4diphenyloxazolidine-5,5-dicarbonitriles can be prepared by regioselective cycloadditions between carbonyl ylides generated from epoxides. In contrast to the use of aldehydes, the reactions performed with imines proceed diastereoselectively.

The mechanism of the [3+2] cycloaddition reactions of the carbonyl ylides with aldehydes and imines has been theoretically studied using DFT methods. The analysis of the reactivity indices indicates that both reagents involved in these [3+2] cycloadditions (carbonyl ylides and aldehydes or imines) have electrophilic/nucleophilic behaviors. However, the large electrophilic character of the carbonyl ylides makes them act as strong electrophiles in these polar processes. This behavior is supported by the NBO and ELF analysis of the electronic structure of TSs involved in these reactions.

The cycloaddition consists of the nucleophilic attack of the aldehyde oxygen or imine nitrogen atom to the carbonyl ylide. For the reaction with aldehydes, a back donation effect that balances the net CT towards the carbonyl carbon atom allows one to explain the unexpected reverse CT found in the [3+2] cycloaddition reactions involving these strong electrophilic carbonyl ylides and carbonyl compounds acting as dienophiles.

Experimental Section

General Procedure 1 for the reaction between epoxides and aldehydes using classical heating:

A mixture of epoxide (2.0 mmol) and aldehyde (2.0 mmol) in dry toluene (30 mL) was heated at reflux under N 2 . The mixture was then evaporated to dryness and purified as specified in the product description.

General Procedure 2 for the reaction between epoxides and aldehydes using microwave irradiation: A mixture of epoxide (2.0 mmol) and aldehyde (2.0 mmol) was heated in the microwave oven (power, temperature and time are given in the product description). The residue was purified as specified in the product description.

General Procedure 3 for the reaction between epoxides and imines using classical heating: A mixture of epoxide (2.0 mmol) and imine (2.0 mmol) in dry toluene (30 mL) was heated at reflux under N 2 . The mixture was then evaporated to dryness. The residue was dissolved in a minimum of Et 2 O.

Upon addition of petrol, the precipitate formed was collected by filtration before recrystallization from Et 2 O.

General Procedure 4 for the reaction between epoxides and imines using microwave irradiation: A mixture of epoxide (2.0 mmol) and imine (2.0 mmol) was heated in the microwave oven (power, temperature and time are given in the product description). The residue was crystallized in Et 2 O/petrol 20:80 and purified as specified in the product description.

Diastereoisomers of 5-(1-naphthyl)-2-phenyl-1,3-dioxolane-4,4-dicarbonitrile (13a,b).

The general procedure 1 (reflux of 80 h), using 3-phenyloxirane-2,2-dicarbonitrile (1a, 0.34 g) and 1naphthaldehyde (9, 0.31 g), gave a 65/35 mixture from which the preponderant diastereoisomer 13a was isolated by column chromatography over silica gel (eluent: petrol/AcOEt 90:10) in 27% yield as a white powder: mp 142°C; 1 H NMR ((CD 3 ) 2 CO)  6.70 (s, 1H), 7.07 (s, 1H),

1H, J = 8.4); 13 C NMR ((CD 3 ) 2 CO)  70. 4, 83.4, 108.1, 112.8, 114.0, 123.1, 126.3, 126.4, 127.5, 127.6, 128.3, 128.5 (2C) The general procedure 1 (reflux of 72 h), using 3-(4-chlorophenyl)oxirane-2,2-dicarbonitrile (1b, 0.41 g) and 1-naphthaldehyde (9, 0.31 g), gave a 75/25 mixture from which the preponderant diastereoisomer 14a was isolated by column chromatography over silica gel (eluent: petrol/AcOEt 90:10) in 32% yield as a beige powder: mp 178°C; 1 H NMR ((CD 3 ) 2 CO)  6.74 (s, 1H), 7.09 (s, 1H), 7.7 (m, 5H), 7.84 (d, 2H, J = 8.7), 8.10 (d, 2H, J = 7.5), 8.14 (d, 1H, J = 8.4), 8.37 (d, 1H, J = 8.4); 13 C NMR ((CD 3 ) 2 CO)  70. 4, 84.3, 107.2, 112.6, 113.9, 123.1, 126.3, 126.3, 126.4, 127.5, 128.3, 130.1 (2C), 130.3 (2C), 130.3, 131.1, 131.9, 133.0, 134.8, 137.5; HRMS, m/z: 360.0669 13 C NMR ((CD 3 ) 2 CO)  55. 8, 70.1, 84.1, 108.3, 112.9, 114.2, 115.2 (2C), 123.2, 125.9, 126.3, 126.4, 127.4, 127.8, 128.2, 130.2 (2C), 130.2, 131.2, 131.8, 134.8, 162.4; HRMS, m/z: 356.1189 (br s, 1H); 13 C NMR ((CD 3 ) 2 CO)  70. 7, 85.0, 105.3, 108.3, 113.0, 113.7, 114.0, 119.9, 121.1, 123.4, 126.4, 126.4, 128.4 (2C), 129.8 (2C), 131.8, 134.7, 137.7; HRMS, m/z: 315.1035 1, 82.6, 106.1, 107.9, 112.9, 114.0, 114.0, 120.0, 121.0, 123.4, 126.6, 127.4, 127.5 (2C), 129.7 (2C), 131.1, 136.3, 137.7 The general procedure 1 (reflux of 80 h), using 3-(4-chlorophenyl)oxirane-2,2-dicarbonitrile (1b, 0.41 g) and indole-3-carboxaldehyde (10, 0.29 7, 85.0, 105.1, 107.3, 113.0, 113.6, 113.8, 119.9, 121.1, 123.4, 126.3, 126.4, 130.0 (2C), 130.1 (2C), 133.6, 137.3, 137.7; HRMS, m/z: 349.0637 1, 82.6, 106.0, 107.0, 112.8, 112.9, 113.9, 120.0, 121.0, 123.4, 126.5, 126.6, 129.3 (2C), 129.8 (2C), 135.3, 136.6, 137.6 13 C NMR ((CD 3 ) 2 CO)  55. 8, 70.6, 84.8, 105.4, 108.4, 113.0, 113.9, 114.1, 115.1 (2C), 120.0, 121.0, 123.4, 126.3, 126.3, 126.5, 130.0 (2C), 137. Et 2 O/heptane 30:70) in 36% yield as a white powder: mp 72°C; 1 H NMR ((CD 3 ) 2 CO)  6.17 (s, 1H), 6.53 (s, 1H), 6.64 (dd, 1H, J = 3.4 and 1.8), 6.99 (d, 1H, J = 3.4), 7.6 (m, 3H), 7.7 (m, 2H), 7.84 (d, 1H, J = 1.7); 13 C NMR ((CD 3 ) 2 CO)  68. 7, 82.4, 108.7, 112.1, 112.8, 113.2, 113.4, 128.4 (2C) 1, 81.3, 109.6, 111.9, 112.0, 114.3, 114.4, 128.1 (2C), 129. The general procedure 1 (reflux of 40 h), using 3-(4-chlorophenyl)oxirane-2,2-dicarbonitrile (1b, 0.41 g) and 2-furaldehyde (11, 0.39 g, 0.33 mL, 4.0 mmol instead of 2.0 mmol), gave a 67/33 mixture from which the minor diastereoisomer 20b was isolated by column chromatography over silica gel (eluent:

Et 2 O/heptane 30:70) in 21% yield as white needles: mp 100°C; 1 H NMR ((CD 3 ) 2 CO)  6.48 (s, 1H), 6.83 (s, 1H), 6.61 (dd, 1H, J = 3.4 and 1.9), 6.95 (d, 1H, J = 3.4), 7.57 (d, 2H, J = 8.4), 7.68 (d, 2H, J = 8.4), 7.80 (d, 1H, J = 1.8); 13 C NMR ((CD 3 ) 2 CO)  68. 9, 81.3, 108.7, 111.9, 111.9, 114.3, 114.4 111.9, 112.5, 112.9, 113.4, 129.8 (2C), 129.9 (2C), 132.6, 137.4, 144.0, 146.3.

Diastereoisomers of 5-(furyl)-2-(4-methoxyphenyl)-1,3-dioxolane-4,4-dicarbonitrile (21a,b).

The general procedure 1 (reflux of 35 h), using 3-(4-methoxyphenyl)oxirane-2,2-dicarbonitrile (1c, 0.40 g) and 2-furaldehyde (11, 0.39 g, 0.33 mL, 4.0 mmol instead of 2.0 mmol), gave a 65/35 mixture from which the minor diastereoisomer 21b was isolated by column chromatography over silica gel (eluent:

Et 2 O/heptane 30:70) in 15% yield as a beige glitter: mp 85°C; 1 H NMR ((CD 3 ) 2 CO)  3.85 (s, 3H),

The crystals were obtained by slowly evaporating acetone solutions.

Crystal data for 5a (colorless prisms): C 21 H 20 N 2 O 6 , M r = 396.39, monoclinic, space group P 2 1 , a = 7.8658( 7 Crystallographic data were deposited in CSD under CCDC registration number 688229.

analysis of both geometric and electronic properties in cycloaddition reactions. The stationary points were characterized by frequency calculations in order to verify that TSs had one and only one imaginary frequency. The intrinsic reaction coordinate (IRC) [START_REF] Fukui | [END_REF] path was traced in order to check the energy profiles connecting each TS to the two associated minima of the proposed mechanism using the second order González-Schlegel integration method. 35 The electronic structures of stationary points were analyzed by the NBO method 36 and the topological analysis of the ELF, ()  r . 27 The ELF study was performed with the TopMod program 37 using the corresponding monodeterminantal wave functions of transition state structures.

Solvent effects have been considered by B3LYP/6-31G* single-point calculations on the gas phase structures using a self-consistent reaction field (SCRF) 38 based on the polarizable continuum model (PCM) of Tomasi's group. 39 Since these reactions are carried out in toluene, we have selected its dielectric constant at 298.0 K,  = 2.38.

The global electrophilicity index, 19 , which measures the energy stabilization when the system acquires an additional electronic charge N from the environment, is given by the following simple expression, where k f  and k f  are the Fukui functions for a nucleophilic and electrophilic attacks, respectively. 23

  SCHEME 1. Thermal Ring Opening of the Epoxide 1a, and Cycloaddition Reaction between CYa and N-methylisatin.

  CYa with benzaldehyde (2c) and the imine 25a.For each one of these [3+2] cycloaddition reactions, four reactive channels have been studied. They are related to the two regioisomeric approaching modes of CYa to the aldehyde, x = o, or to the imine, x = n, named as 1 and 2, and the two stereoisomeric approaching modes related to the cis or trans rearrangement of the two phenyl substituents on the final [3+2] cycloadducts, named as c and t. Note that the carbonyl ylide CYa can adopt the (E) or (Z) configuration by the restricted rotation of the C1-O2 bond. While (E)-CYa adopts a planar rearrangement, 7 (Z)-CYa is twisted as a consequence of the hindrance between the phenyl and one cyano group. This hindrance makes (Z)-CYa 8.6 kcal/mol higher in energy than (E)-CYa. In addition, the barrier height associated with the C1-O2 bond rotation is very large, 27.1 kcal/mol. Consequently, only the (E) configuration of the carbonyl ylide CYa was considered in the present study. An analysis of the stationary points associated with the reaction channels of these cycloadditions indicates that they have a one-step mechanism. Therefore, eight TSs and eight cycloadducts were located and characterized (see Scheme 2).

  SCHEME 2. Reaction Channels Studied for the Cycloaddition Reaction between CYa and 2c or 25a

FIGURE

  FIGURE 1. Transition Structures Involved in the Reaction of the Carbonyl Ylide CYa with Benzaldehyde (2c)

  (BO), the natural charges obtained by a NBO (Natural Bond Order) analysis, and the topological analysis of the electron localization function (ELF). The C1-X5 (X = O or N) BO values at the TSs are 0.32 at TSo1c, 0.32 at TSo1t, 0.40 at TSn1c and 0.34 at TSn1t, while the C3-C4 BO values are 0.20 at TSo1c, 0.24 at TSo1t, 0.14 at TSn1c and 0.01 at TSn1t. At these TSs, the C1-X5 bond formation is more advanced than the C3-C4 one, in clear agreement with the analysis carried out with the local electrophilicity and nucleophilicity indices. The [3+2] cycloaddition reactions involving the imine 25a are more advanced and more asynchronous than those involving the aldehyde 2c.

FIGURE 2 .

 2 FIGURE 2. Transition Structures Involved in the Reaction of the Carbonyl Ylide CYa with the Imine 25a

FIGURE

  FIGURE 3. Spatial Localization of the Maxima (e. g., Attractors) of the Electron Localization Function (ELF) for TSo1t and TSn1t (only Attractors at the Reaction Center are Labeled)

FIGURE 5 .

 5 FIGURE 5. (a) Spatial Localization of the Maxima (e. g., Attractors) of the Electron Localization Function (ELF) for HPn1t Point Along the Reaction Path Associated to TSn1t. Only Attractors at the Reaction Center are Labeled. (b) ELF Isosurface ( = 0.728) Pictures for HPn1t

  ), b = 8.6374(8), c = 15.3824(16) Å,  = 99.533(5) °, V = 1030.65(17) Å 3 , Z = 2,  calcd = 1.277 g.cm -3 ,  = 0.095 mm -1 . A final refinement on F 2 with 2497 unique intensities and 262 parameters converged at R(F 2 ) = 0.0812 (R(F) = 0.0359) for 2359 observed reflections with I > 2(I).Crystallographic data were deposited in CSD under CCDC registration number 688228.Crystal data for 6a (colorless prisms): C 18 H 12 N 2 O 4 , M r = 320.30, monoclinic, space group P 2 1 /a, a = 13.4303(7), b = 7.8903(4), c = 13.9246(6) Å,  = 91.425(3) °, V = 1475.12(12) Å 3 , Z = 4,  calcd = 1.442 g.cm -3 ,  = 0.104 mm -1 . A final refinement on F 2 with 3377 unique intensities and 217 parameters converged at R(F 2 ) = 0.0885 (R(F) = 0.0391) for 2781 observed reflections with I > 2(I). Crystallographic data were deposited in CSD under CCDC registration number 688219. Crystal data for 6b (colorless prisms): C 18 H 12 N 2 O 4 , M r = 320.30, orthorhombic, space group P b c a, a = 8.2939(5), b = 12.8750(6), c = 27.3006(14) Å, V = 2915.3(3) Å 3 , Z = 8,  calcd = 1.46 g.cm -3 ,  = 0.105 mm -1 . A final refinement on F 2 with 3346 unique intensities and 217 parameters converged at R(F 2 ) = 0.1126 (R(F) = 0.0471) for 3129 observed reflections with I > 2(I). Crystallographic data were deposited in CSD under CCDC registration number 688216. Crystal data for 7a (colorless prisms): C 18 H 11 ClN 2 O 4 , M r = 354.74, monoclinic, space group P 2 1 /a, a = 13.7192(7), b = 7.4109(4), c = 15.3928(7) Å,  = 93.948(2) °, V = 1561.30(14) Å 3 , Z = 4,  calcd = 1.509 g.cm -3 ,  = 0.272 mm -1 . A final refinement on F 2 with 3547 unique intensities and 226 parameters converged at R(F 2 ) = 0.0894 (R(F) = 0.0353) for 2987 observed reflections with I > 2(I). Crystallographic data were deposited in CSD under CCDC registration number 688218. Crystal data for 8a (colorless prisms): C 19 H 14 N 2 O 5 , M r = 350.32, monoclinic, space group P 2 1 /a, a = 13.9650(6), b = 7.3292(3), c = 15.6911(6) Å,  = 95.712(2) °, V = 1598.05(11) Å 3 , Z = 4,  calcd = 1.456 g.cm -3 ,  = 0.107 mm -1 . A final refinement on F 2 with 3654 unique intensities and 236 parameters converged at R(F 2 ) = 0.0912 (R(F) = 0.0358) for 3085 observed reflections with I > 2(I). Crystallographic data were deposited in CSD under CCDC registration number 688223. Crystal data for 17a (colorless prisms): C 19 H 12 ClN 3 O 2 , M r = 349.77, monoclinic, space group C 2/c, a = 8.7943(18), b = 18.422(4), c = 19.814(4) Å,  = 92.353(9) °, V = 3207.3(12) Å 3 , Z = 8,  calcd = 1.449 g.cm -3 ,  = 0.256 mm -1 . A final refinement on F 2 with 3662 unique intensities and 229 parameters converged at R(F 2 ) = 0.1389 (R(F) = 0.0525) for 3113 observed reflections with I > 2(I). Crystallographic data were deposited in CSD under CCDC registration number 688214. Crystal data for 20b (colorless prisms): C 15 H 9 ClN 2 O 3 , M r = 300.69, monoclinic, space group P 2 1 /a, a = 5.8491(3), b = 27.7737(12), c = 8.3610(4) Å,  = 95.084(2) °, V = 1352.91(11) Å 3 , Z = 4,  calcd = 1.476 g.cm -3 ,  = 0.294 mm -1 . A final refinement on F 2 with 3097 unique intensities and 190 parameters converged at R(F 2 ) = 0.0955 (R(F) = 0.0375) for 2776 observed reflections with I > 2(I). Crystallographic data were deposited in CSD under CCDC registration number 688222. Crystal data for 24a (colorless prisms): C 16 H 12 N 2 O 3 S, M r = 312.34, triclinic, space group P -1, a = 8.7692(5), b = 10.0582(6), c = 10.4679(6) Å,  = 115.933(2),  = 92.721(3),  = 112.118(2) °, V = 743.87(7) Å 3 , Z = 2,  calcd = 1.394 g.cm -3 ,  = 0.231 mm -1 . A final refinement on F 2 with 3371 unique intensities and 200 parameters converged at R(F 2 ) = 0.081 (R(F) = 0.0315) for 3117 observed reflections with I > 2(I). Crystallographic data were deposited in CSD under CCDC registration number 688221. Crystal data for 29a (colorless prisms): C 18 H 14 N 4 O 3 , M r = 334.33, monoclinic, space group P 2 1 /n, a = 7.2945(6), b = 12.4389(12), c = 18.2616(18) Å,  = 96.102(5) °, V = 1647.6(3) Å 3 , Z = 4,  calcd = 1.348 g.cm -3 ,  = 0.095 mm -1 . A final refinement on F 2 with 3766 unique intensities and 226 parameters converged at R(F 2 ) = 0.1076 (R(F) = 0.0431) for 3118 observed reflections with I > 2(I).

  of the electronic chemical potential  and the chemical hardness .These quantities may be approached in terms of the energies of the frontier molecular orbital HOMO and LUMO, H Recently, we have introduced an empirical (relative) nucleophilicity index, N based on the HOMO energies obtained within the Kohn-Sham scheme, and defined as

TABLE 1 . Reactions between Epoxides 1 and Benzaldehydes 2

 1

TABLE 2 . Reactions between Epoxides 1 and Aldehydes 9-12

 2 Determined from the 1 H NMR spectra of the crude mixture. b Reactions performed without solvent.

					H	Ar
		R	1	O CN CN		O 9-12	R	4' 3'	5' 2'	6' 1' H	a O O H 2 5 4	CN CN Ar	+	R	4' 3'	5' 2'	6' 1' H	b H O O 2 5 4	Ar CN CN
	entry R (1)	aldehyde	a:b		conditions					a:b ratio a isolated product(s), yield(s)
	1	H (1a)			13a:13b toluene, reflux, 80 h		65/35			13a, 27%
							MW, b 90 W, 120°C, 40 min	62/38			13a, 35%
	2	Cl (1b)			14a:14b toluene, reflux, 72 h		75/25			14a, 32%
			H				MW, b 90 W, 120°C, 40 min	63/37			14a, 42%
	3	OMe (1c)	O		15a:15b toluene, reflux, 57 h		76/24			15a, 48%
				9
							MW, b 90 W, 80°C, 30 min	71/29			15a, 55%
	4	H (1a)			16a:16b toluene, reflux, 84 h		50/50			16a, 40%
							MW, b 90 W, 120°C, 45 min	60/40			16a, 42%
	5	Cl (1b)	H	NH	17a:17b toluene, reflux, 80 h		43/57			17a, 37%
							MW, b 90 W, 120°C, 45 min	76/24			17a, 39%
			O
	6	OMe (1c)			18a:18b toluene, reflux, 54 h		84/16			18a, 46%
				10			MW, b 120 W, 80°C, 40 min	66/34			18a, 30%
	7	H (1a)			19a:19b toluene, reflux, 50 h		66/34			19a, 36%
			H
	8	Cl (1b)		O	20a:20b toluene, reflux, 40 h		67/33			20b, 21%
			O
	9	OMe (1c)		11	21a:21b toluene, reflux, 35 h		65/35			21b, 15%
	10	H (1a)			22a:22b toluene, reflux, 48 h		73/27			22a, 40%; 22b, 20%
			H
	11	Cl (1b)		S	23a:23b toluene, reflux, 41 h		74/26			23a, 45%; 23b, 25%
			O
	12	OMe (1c)		12	24a:24b toluene, reflux, 36 h		62/38			24a, 50%; 24b, 20%

a

, 20b, 21b, 22a,b, 23a,b and 24a,b were

	isolated from the crude mixtures by column chromatography over silica gel and/or recrystallization
	from petrol/Et 2 O, and all the structures were first determined on the basis of 1 H and 13 C NMR chemical
	shifts. This was consistent with a complete NMR study including HMBC, HMQC and NOESY
	experiments performed on 19a. In addition, attempts to get crystals suitable for X-ray analysis from an
	acetone solution were successful for trans 20b and cis 24a.

TABLE 3 . Reactions between Epoxides 1 and Imines 25

 3

								R 5	R 6			2"	R 5 3" 4"	R 6
		R 1	1	O	CN CN	H	NR	25	R 1 4' 3'	5' 2'	6' 1' H R	N O a H 2 4 5	1"	CN CN 6"	5"
	entry	R 1 (1)	imine			conditions						isolated product, yield
	1	H (1a)					toluene, reflux, 24 h					26a, 52%
							MW, b 120 W, 125°C, 55 min				26a, 35%
	2	Cl (1b)					toluene, reflux, 20 h					27a, 60%
			H				MW, b 120 W, 125°C, 50 min				27a, 48%
	3	OMe (1c)	NMe			toluene, reflux, 5 h					28a, 78%
					25a		MW, b 60 W, 80°C, 25 min					28a, 85%
	4	NO 2 (1d)					toluene, reflux, 35 h					29a, 40%
							MW, b 150 W, 125°C, 60 min				29a, 33%
	5	H (1a)					toluene, reflux, 27 h					30a, 48%
					O		MW, b 120 W, 125°C, 55 min				30a, 32%
	6	Cl (1b)			O		toluene, reflux, 25 h					31a, 59%
							MW, b 120 W, 125°C, 45 min				31a, 40%
			H
	7	OMe (1c)					toluene, reflux, 9 h					32a, 60%
			NPr			MW, b 60 W, 80°C, 30 min					32a, 80%
					25b
	8	NO 2 (1d)					toluene, reflux, 45 h					33a, 35%
							MW, b 150 W, 125°C, 65 min				33a, 30%
	9	H (1a)					toluene, reflux, 32 h					34a, 56%
					O		MW, b 120 W, 125°C, 60 min				34a, 40%
	10	Cl (1b)			O		toluene, reflux, 32 h					35a, 59%
							MW, b 120 W, 125°C, 60 min				35a, 49%
			H
	11	OMe (1c)					toluene, reflux, 9 h					36a, 60%
			NBu			MW, b 60 W, 80°C, 30 min					36a, 75%
					25c
	12	NO 2 (1d)					toluene, reflux, 65 h					37a, 18%
							MW, b 150 W, 125°C, 75 min				37a, 22%
	13	H (1a)					toluene, reflux, 40 h					38a, 60%
					O		MW, b 120 W, 125°C, 70 min				38a, 42%
	14	Cl (1b)			O		toluene, reflux, 37 h					39a, 50%
							MW, b 120 W, 125°C, 60 min				39a, 37%
			H
	15	OMe (1c)					toluene, reflux, 18 h					40a, 80%
			NBn			MW, b 60 W, 80°C, 35 min					40a, 70%
					25d

a Determined from the 1 H NMR spectra of the crude mixture. b Reactions performed without solvent.

TABLE 4 . Electronic Chemical Potential (, in au), Chemical Hardness (, in au), Global Electrophilicity (, in eV) and Global Nucleophilicity (N, in eV) values of the Carbonyl Ylides CYa-c and the Aromatic Aldehydes 2a-c and Imines 25a and 25e (by decreasing values of

 4

.

TABLE 5 . Local Electrophilicity,  k , and Nucleophilicity, N k , Values (in eV) of the Carbonyl Ylide CYa, Benzaldehyde (2c), and the Imine 25a

 5

			 k		N k
	CYa	C1	C3	C1	C3
		1.30	0.45 0.46	1.13
	2c	C4	O5	C4	O5
		0.47	0.35 0.10	1.52
	25a	C4	N5	C4	N5
		0.25	0.30 0.11	0.50

  7

TABLE 6 . Total (E, in au) and Relative (E, in kcal/mol) Energies, in Gas-Phase and in Toluene, of the Stationary Points Involved in the [3+2] Cycloaddition Reactions of the Carbonyl Ylide CYa with Benzaldehyde (2c) and the Imine 25a

 6

		E	E	E toluene	E toluene
	CYa	-569.289033		-569.298872
	2c	-345.573442		-345.577940
	TSo1c -914.858355 2.6	-914.869424 4.6
	TSo1t -914.858333 2.6	-914.869037 4.9
	TSo2c -914.841138 13.4	-914.850215 16.7
	TSo2t -914.841635 13.1	-914.850095 16.8
	CAo1c -914.913839 -32.2 -914.922338 -28.6
	CAo1t -914.910561 -30.2 -914.918724 -26.3
	CAo2c -914.904879 -26.6 -914.912830 -22.6
	CAo2t -914.909770 -29.7 -914.917550 -25.6
	25a	-365.008701		-365.011670
	TSn1c -934.290181 4.7	-934.300925 6.0
	TSn1t -934.293307 2.8	-934.304485 3.8
	TSn2c -934.274360 14.7	-934.282835 17.4
	TSn2t -934.274440 14.6	-934.282450 17.6
	CAn1c

Table 7 . Total and Relative (Relative to TSo1t and TSn1t) Enthalpies (H, in au, and H, in kcal/mol), Entropies (S and S, in eu), and Free Energies (G, in au, and G, in kcal/mol) at 110 ºC in Toluene of the TSs Involved in the [3+2] Cycloaddition Reactions of the Carbonyl Ylide CYa with Benzaldehyde (2c) and the Imine 25a.

 7

		H	H	S	S	G	G
	TSo1c	-914.616511	-0.1	165.36	1.5	-914.717702	-0.7
	TSo1t	-914.616273	0.0	163.83	0.0	-914.716527	0.0
	TSn1c	-934.007402	2.3	180.64	-0.8	-934.117946	2.6
	TSn1t	-934.011091	0.0	181.41	0.0	-934.122104	0.0
	TSn1c-Z	-934.011051	0.0	185.50	4.1	-934.124567	-1.5
	TSn1t-Z	-934.011018	0.0	176.33	-5.1	-934.118923	2.0

Acknowledgment. We are grateful to the Spanish Government (project CTQ2006-14297/BQU), the Algerian MESRS for PROFAS financial support to G. B.-A. and the Fondecyt Projects Nos. 1060961 (P. P.) and 1070378 (E. C.). E. C. and P. P. also thank the Universidad Andrés Bello (UNAB) for support through project DI 21-06/R and 45-08/R, respectively. L. R. D. also thanks the Fondecyt Grant No. 7080026 (Cooperación Internacional) for financial support and the Universidad Andrés Bello for its hospitality. We thank Jean Pierre Bazureau for generous microwave access. We thank Sourisak Sinbandhit for his contribution to this study.

donation back donation donation

TSo1t TSn1t

-0.05e Nucleophiles Electrophile

Crystal data for 31a (colorless plates): C 21 H 18 ClN 3 O 3 , M r = 395.83, monoclinic, space group P 2 1 , a = 7.1324 (9), b = 7.4170 (9), c = 18.265(2) Å,  = 101.203(6) °, V = 947.8(2) Å 3 , Z = 2,  calcd = 1.387 g.cm -3 ,  = 0.229 mm -1 . A final refinement on F 2 with 3807 unique intensities and 255 parameters converged at R(F 2 ) = 0.1125 (R(F) = 0.049) for 3239 observed reflections with I > 2(I).

Crystallographic data were deposited in CSD under CCDC registration number 688215. Crystallographic data were deposited in CSD under CCDC registration number 688226.

Computational methods:

All calculations were carried out with the Gaussian 03 suite of programs. 31 DFT calculations were carried out using the B3LYP 32 exchange-correlation functionals, together with the standard 6-31G* basis set. [START_REF] Hehre | Ab initio Molecular Orbital Theory[END_REF] This level of theory has shown to be suitable to provide good enough performance in the Supporting Information Available: General methods, starting materials, compound characterizations, crystallography, ORTEP diagrams, copies of the 1 H and 13 C NMR spectra for compounds 13a-24a, geometries of the TSs with the (Z)-imine 25a, B3LYP/6-31G* computed total energies, unique imaginary frequencies, cartesian coordinates of the TSs and cycloadducts, and CIF files of 5a, 6a, 6b, 7a, 8a, 17a, 20b, 24a, 29a, 31a, 32a, 33a and 39a. This material is available free of charge via the Internet at http://pubs.acs.org.

References and Footnotes

(1) Huisgen, R.; Grashey, R.; Sauer, J. in The chemistry of Alkenes, Interscience: New York, 1964.

(2) (a) 1,3-Dipolar Cycloaddition Chemistry (Ed.: Padwa, A.); Wiley: New York, 1984; Vols. 1,2. (b) Gothelf, K. V.; Jorgenson, K. A. Chem. Rev. 1998, 98, 863-909. (c) B. Bioorg. Med. Chem. 2004, 12, 6237-6247; (b) Liang, Y.; Narayanasamy, J.; Schinazi, R. F.; Chu, C. K. Bioorg. Med. Chem. 2006, 14, 2178-2189; (c) Wender, P. A.; Verma, V. A. Org. Lett. 2006, 8,


Aude Bolopion
email: bolopion@isir.upmc.fr

Barthelemy Cagneau
email: barthelemy.cagneau@uvsq.fr

Stephane Redon
email: stephane.redon@inria.fr

Stéphane Régnier
email: regnier@isir.upmc.fr

Barth Élemy Cagneau

St

Variable gain haptic coupling for molecular simulation

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

The design of new drugs, the synthesis of new alloys or the prototyping of bio-nanorobots are made possible thanks to molecular simulators [START_REF] Hamdi | Prototyping bio-nanorobots using molecular dynamics simulation and virtual reality[END_REF]. Since intuitiveness is one of the key point of such tools, haptic devices are widely used. They provide natural manipulation modes of the molecule, and molecular forces are easily interpretable by operators when displayed haptically [START_REF] Brooks | Project GROPE -haptic displays for scientific visualization[END_REF], [START_REF] Persson | Designing and evaluating a haptic system for biomolecular education[END_REF]. The performance of such systems is measured considering the stability, the ease and intuitiveness of manipulation, and the realism of the haptic feedback.

Stability conditions for haptic feedback of molecular interactions are very strict. This is due in particular to long computation times and the scale difference between the macro and the nanoworlds. Approaches currently used to ensure stability either decreases the accuracy of the simulation, the perception of interaction forces or the ease of manipulation. The most common methods are the reduction of the computational load and the limitation of the scaling factors.

To reduce the computational load, molecules are rigidified [START_REF] Nagata | Concept and prototype of protein-ligand docking simulator with force feedback technology[END_REF]. However, the accuracy of computed forces is deteriorated since the reconfiguration of the molecular system is not considered. A simplified model of the force field such as a spring can also be used to repel atoms, but it decreases the realism of the force transmitted to operators [START_REF] Lee | Smoothing haptic interaction using molecular force calculations[END_REF].

To select both force and displacement scaling factors, there is an important trade-off between stability, perception of interaction forces and ease of manipulation. If force scaling is low, attractive forces cannot be felt by operators. If it is high, the haptic force must be truncated, which leads to a loss of information [START_REF] Wollacott | Haptic applications for molecular structure manipulation[END_REF].The displacement scaling factor might create instability if large displacements are allowed. If they are limited, ease of manipulation is decreased. To deal with this issue, several techniques are proposed, from the concept of clutching (freezing the displacement of the virtual object while enabling the user to modify the position of the haptic handle), to the Bubble technique (position control for fine positioning, and rate control for large displacements) [START_REF] Dominjon | The "Bubble" technique: interacting with large virtual environments using haptic devices with limited workspace[END_REF]. As an alternative the concept of Active Haptic Workspace (AHW) is considered [START_REF] Subasi | A new haptic interaction and visualization approach for rigid molecular docking in virtual environments[END_REF]. As for the Bubble technique, position control is used inside the AHW, and easy translations and rotations of objects are ensured outside. These techniques present good results, but are based on user actions, who choose to stay or not within the position control range. They do not take into account the environment (position with respect to objects on the scene).

As explained above, a constant scaling factor does not enable to transmit both sufficiently amplified attractive forces and non truncated repulsive ones. This parameter should thus be adapted depending on molecular interaction forces. A large displacement scaling factor induces time consuming experiments and deteriorates the ease of manipulation. A small one results in an unstable system. One solution is to change the value of this scaling factor at runtime. Variable gain haptic coupling seems to be a promising solution to overcome these problems. It is investigated in this paper. Some works dealing with variable gains have been done to ensure stability [START_REF] Mobasser | Adaptive teleoperation control using online estimate of operator's arm damping[END_REF], [START_REF] Zhu | Stability guaranteed teleoperation: an adaptive motion/force control approach[END_REF], [START_REF] Vander Poorten | Robust variable-scale bilateral control for micro teleoperation[END_REF], or to improve user perception for biomedical applications [START_REF] De Gersem | Reliable and enhanced stiffness perception in soft-tissue telemanipulation[END_REF]. A history dependent force scaling has also been proposed [START_REF] Persson | Improved feature detection over large force ranges using history dependent transfer functions[END_REF]. However, to our knowledge, none of them concern the improvement of both force rendering and ease of manipulation for nanoscale applications.

This paper is based on previous works on molecular simulation and haptic coupling. The dynamics simulator has been presented in [START_REF] Rossi | Adaptive torsion-angle quasi-statics: a general simulation method with applications to protein structure analysis and design[END_REF]. Only a few examples of haptic manipulation on that simulator are presented in this paper. More can be found in [START_REF] Bolopion | Comparing position and force control for interactive molecular simulators with haptic feedback[END_REF], which details the possible applications of such systems. It also analyzes and compares several constant-gain coupling schemes for molecular simulations in terms of transparency and stability. This paper is organized as follows. In Section 2, the specificity of haptic feedback for molecular simulation is highlighted. The influence of the gains of the haptic coupling is determined. Based on these conclusions, a variable gain system is proposed in Section 3, and evaluated by user tests in Section 4. The variable gain system is tested on a HIV molecule. Results and limits of the system are presented in Section 5. Section 6 concludes the paper.

MOLECULAR INTERACTIONS AND HAPTIC COUPLING

Using molecular simulators, users interact with virtual objects of hundreds or thousands of degrees of freedom. To improve the intuitiveness of such simulators, haptic feedback is a promising solution. However, interaction forces are specific since at these scales attractive forces cannot be neglected and are of utmost importance for the determination of molecular conformational states. In particular, both attractive and repulsive forces must be considered to find the equilibrium state of the system.

To highlight the specificity of molecular interactions, a simple experiment is performed. In the next sections, it will be used to evaluate the proposed variable gain coupling. This experiment is composed of two water molecules. One is fixed. The displacement of the second one is constrained along one direction, only the interaction forces aligned with this direction are haptically sent to users. The manipulated water molecule is moved towards the second one, and then retracted. The molecular interaction force between the two molecules is plotted with respect to the time in Figure 1 1 . It must be noted on this plot that attractive forces are limited whereas repulsive ones may become arbitrarily large. This point is of utmost importance since these two phenomena should be rendered to users through haptic feedback.

Haptic coupling for molecular simulators

The simplest structure to connect the haptic device to the simulation software is to use the position of the haptic handle x h as the desired position of the molecule x i . The user feels the force F h through the haptic interface, corresponding to the molecular interaction force F m (Figure 2). When connecting a haptic device to a molecular simulator delays due to long computation times of complex interactions must be taken into account. Transfer functions are thus represented using the discrete time variable z. Since the variables coming from the simulator are delayed by one sampling period (period of the simulator), an explicit one step delay z -1 is introduced between the simulation and the coupling. The discrete time transfer function of the haptic interface H(z) is composed of M h and B h , respectively the master's inertia and viscosity [START_REF] Diolaiti | Stability of haptic rendering: Discretization, quantization, time delay, and coulomb effects[END_REF]. Several parameters need to be considered in that coupling:

F m F h x i F op H(z) A f Delay A -1 d z -1 + - x h
• T s : simulator sampling time. It corresponds to the maximum time needed for the software to perform one step of the simulation for a given experiment (moving the molecule according to the haptic device's handle position and computing interaction forces). It is the time that the user must wait to feel the updated force. The worst case is considered here by taking the largest delay. For a given simulation, it mainly depends on the number of activated degrees of freedom. Limiting the number of activated degrees of freedom with partial rigidification of the molecules decreases the sampling time,

• A f : force scaling factor. It amplifies the molecular interaction forces to render it to users,

• A d : displacement scaling factor. It maps the position of the haptic handle and the position of the molecule. These force and displacement scaling coefficients are necessary due to the scale difference between characteristic parameters of the haptic device and the molecular simulatior.

To determine stability conditions, a simplified model of the environment is considered (an equivalent spring constant k e ). This is obviously not enough to model complex interactions between molecules, but first indications of stability can be derived. Considering a first order taylor approximation near zero of the exponential function, a necessary stability condition is derived [START_REF] Bolopion | Analysis of stability and transparency for nanoscale force feedback in bilateral coupling[END_REF]:

A f A d T 2 s k e < M h (1)
The delays due to the computation time are thus limited to ensure stability. The relation also highlights the trade off between force amplification and ease of manipulation. To highly amplify forces in order to haptically transmit attractive phenomena, A f must be high. However, to ensure stability A d must also be increased. This means that the displacements performed by users will be scaled down, and the corresponding displacements of the molecule will be small. Ease of manipulation is thus deteriorated. Adding damping could improve stability. However, since transparency is an important requirement to render to users small variations of molecular interactions, this solution is discarded.

Material

The molecular simulator used in the following experiments, SAM-SON (System for Adaptative Modeling and Simulation Of Nano-Objects), is based on a quasi-statics method to simulate the motion of the molecular system (Figure 3). The force field used is derived from a well known molecular mechanics force field, CHARMM [START_REF] Brooks | CHARMM: A program for macromolecular energy, minimization, and dynamics calculations[END_REF], which models interactions through van der Waals, electrostatic and dihedral contributions. Thanks to the equation of motion solved, the value of the force applied to each atom is directly known. More information about the simulator can be found in [START_REF] Rossi | Adaptive torsion-angle quasi-statics: a general simulation method with applications to protein structure analysis and design[END_REF].

The first solution to ensure stability is to reduce the computational cost of the simulation. A solution is to rigidify part of the molecule. To limit the subsequent loss of accuracy, SAMSON uses an adaptive algorithm which relies on a divide-and-conquer representation of a molecular system. In order to tune the complexity of the simulation, the number of active degrees of freedom at each time step is chosen by users, and the adaptive simulation algorithm automatically determines the set of active nodes. This enables the adaptive algorithm to focus on the degrees of freedom with the largest accelerations, and avoid spending time on parts of the molecular system with little internal motion. In Section 5 this feature will be used to simulate a molecular system composed of hundreds of degrees of freedom in less than 100ms.

Note that any other simulator which gives access to the required variables (position, force) may have been used. The only specificity is the adaptive feature of SAMSON which enables to decrease the computational cost of the simulations. The haptic interface is a Phantom Omni haptic device from Sensable 2 . This master arm is a 6 degrees of freedom (dof ) movement and a 3 dof force feedback device. The maximum force that it can provide is 3.3N. To avoid damages, forces higher than 3N are truncated. This is a key point in this work since the variations of forces higher than this maximum value are not rendered to users. The force feedback accuracy is thus deteriorated. To interact with large virtual environments, the user can freeze the displacement of the virtual object and modify the position of the haptic handle as explained in [START_REF] Dominjon | The "Bubble" technique: interacting with large virtual environments using haptic devices with limited workspace[END_REF]. To connect the haptic device to the simulator, the OpenHaptics development toolkit is used. The computer which runs the simulation and controls the haptic device has a 3GHz Intel processor, and uses Windows operating system.

VARIABLE GAIN HAPTIC COUPLING

To ensure stability, adequate haptic feedback and ease of manipulation, variable gain haptic coupling is detailed in next sections. The variable gains will be noted using the tilde symbol.

Variable force scaling factor à f

As seen in Figure 1, attractive forces are limited (amplitude less than 0.35nN). They should be highly amplified to be transmitted to users. On the contrary, repulsive forces can become arbitrarily large. However, forces sent to the haptic device must be limited to avoid damages. To feel attractive forces while avoiding the truncation of repulsive ones the force scaling factor must be non constant. One solution is to compute the haptic force from the molecular interaction force as follows:

F h = α 1 atan(δ 1 F m ) + α 2 atan(δ 2 (F m -F 0 ))+ α 2 atan(δ 2 (F m + F 0 )) (2)
where

α 1 = F h1 π/2 , α 2 = F h2 -F h1 π , δ 1 = A fmax α 1 , δ 2 = A f min 2α 2 . F h1 (resp. F h2
) is the value of the haptic force to which the amplified low magnitude interaction force (resp. high magnitude interaction force) tends. F 0 is the limit between what is considered to be low and high magnitude interaction forces. The resulting force profile is depicted in Figure 4:

F 0 -F 0 A f max F m Molecular force F m (N) Haptic force F h (N)
• for F m → 0: F h = A f max F m .
For low interaction forces, the amplification is linear, and the force scaling factor is

à f = A f max • for F m → F 0 : F h = F h1 + A f min F m .
For interaction forces around F 0 , the amplification is linear.

• for F m → +∞ (resp. F m → -∞): F h → F h2 (resp. F h → -F h2 ).
For high magnitude interaction forces, the haptic force tends to a maximum value set as the maximum force of the haptic interface. Even if the variations of high amplitude forces are smoothed, they are not truncated.

This function enables to highly amplify small amplitude forces without truncating high amplitude ones, and presents the advantage of having two linear amplification range of forces. However, other functions could have been used to compute the variable force scaling. The aim of this work is not to compare exhaustively all the possible functions, but to analyze the use of variable gains 3 .

To test this function, the water molecule experiment described in Section 2 is performed. The numerical values of the parameters F h1 , F h2 , F 0 , A f min and A f max are the same as in Figure 4. The haptic force F h , as well as the molecular interaction force F m , are plotted in Figure 5. It can be seen that attractive forces are amplified so that an untrained user can distinguish them easily, whereas no truncation of repulsive forces is necessary.

Variable displacement scaling factor Ãd

To avoid the stability and ease of manipulation trade off, a variable displacement scaling factor is used. It is defined so that the ease of manipulation is increased while interaction forces between molecules are low (typically when the molecules are far from each other). Compared to existing solutions that combine position control inside a given range, and other control modes such as rate mode control outside, the solution investigated in this work takes into account the environmental conditions. The choice of the adequate mode is done automatically depending on molecular interactions, and does not depend on user actions. The proposed variable displacement scaling factor is:

Ãd = A d min + β atan(( F m -F o f f )γ) -β atan(-F o f f γ) (3)
3 This remark is also true for the variable displacement scaling function. where β = A dmax -A d min π/2-atan(-γF o f f ) . A d min (resp. A d max ) is the lower (resp. higher) value of the displacement factor. It should be noted that:

• for F m → 0: Ãd = A d min . For low molecular interaction forces (typically when the molecules are far from each other), large displacements are enabled to improve ease of manipulation.

• for F m → +∞ or F m → -∞: Ãd = A d max . For high molecular interaction forces (typically when the molecules are close to each other), displacements are limited to preserve stability (see Equation ( 1)).

• γ determines the slope of the variation of the scaling factor while changing from low to high (or high to low) values. The higher is γ, the faster is the transition.

• F o f f is the magnitude of the interaction force for which the transition between low and high scaling is made.

The water molecule experiment is performed using the variable Ãd to test the proposed function (see Figure 6). The system remains stable and no oscillation appears on the force sent to the user (blue solid line). Ease of manipulation is ensured since large displacements are possible when interactions between molecules are small. To avoid deterioration of the user perception, the displacement scaling has been chosen so that is constant in the neighborhood of the second water molecule, and the transition between low and high scaling is made at the very beginning of molecular interactions.

USER TESTS

To evaluate this variable gain coupling, user tests have been performed on the water molecule experiment. Tests have been conducted on five voluntary people, aged from 25 to 29. They were asked to determine the equilibrium position of the two water molecules, using the haptic device. Five different modes were proposed:

1. variable force scaling, variable displacement scaling, 2. variable force scaling, constant displacement scaling, 3. constant force scaling, variable displacement scaling, 4. constant force scaling, constant displacement scaling, 5. no force feedback, constant displacement scaling. Each group of five modes were proposed three times, and the overall experiment lasted around 20 minutes for each participant. In each group, modes were proposed randomly. People were asked to determine as precisely as possible the molecular equilibrium position, and press a key when they were done. Both the distance to the actual equilibrium position, and the time they needed to position the molecule were recorded. In addition to these quantitative data, users were asked to give qualitative comments at the end of the test.

All the participants had knowledge about interaction forces at the nanoscale. They all had previously manipulated haptic devices, but they discovered the task and none of them had previously tested variable feedback. They were given as much time as they needed on a preliminary experiment (involving other molecules) to get familiar with the simulator and the Phantom device.

(b) Average time needed to determine the equilibrium position.

Figure 7: Average distance to the equilibrium position, and average time to complete the experiment. The standard deviation is represented with a scale 1/2, its value is given into parenthesis.

Results of these tests are plotted in Figure 7. Figure 7(a) represents the average distance to the actual equilibrium position. The average time needed to determine the equilibrium position is represented in Figure 7(b). For the sake of clarity, the standard deviation is represented with a scaling 1/2, and its value is given inside parenthesis. In Figure 7(a), results for haptic mode 5 are not displayed since the average distance to the equilibrium position was much higher with no force feedback than with any other mode. Values of the average distance and the associated standard deviation are specified in the caption.

As seen in Figure 7(a), variable force feedback enables a more precise determination of the molecule's equilibrium position. This is illustrated in Figure 8 where the repartition of the positions determined by one of the user for the different modes are depicted. On the contrary, variable displacement scaling factors do not seem to decrease the completion time. This might be because this mode is not intuitive for users who are not fully confident with haptic devices, since movements they perform are not scaled linearly to visual feedback. In addition to these quantitative evaluations, people were asked a few questions. The first one was about which mode they considered the most appropriated to find the equilibrium position. All the participants mentioned modes including variable force scaling (either mode 1 or mode 2) as the most appropriated. The second question was which mode they preferred to manipulate the molecule (move it around, and set it at a given position). Most of the users (3 out of 5) preferred to get linear displacement scaling, as they think it provides better intuitiveness. This is consistent with quantitative results. However, two people pointed out that variable displacement scaling was much more adapted when large displacements were to be made. This disparity may be related to the degree of confidence users have while manipulating haptic interfaces.

FULLY ADAPTIVE EXPERIMENT 5.1 The HIV example

The variable gain coupling is then tested by a trained user on complex molecules, the HIV protease and its inhibitors (Protein Data Bank code: 2AZ8). This experiment was not selected for user tests since participants were not familiar with molecular simulations. To better highlight the effects of variable gain coupling, an experiment that did not require knowledge about molecular interactions was selected. In addition, the water molecule experiment enables to determine precise metrics to measure the completion of the task and compare the coupling modes.

Experiment on the HIV molecule consists in extracting one of the inhibitor from the whole system, and then reinserting it. To limit the simulation time to a few tens of milliseconds, the adaptive algorithm detailed in Section 2.2 is used to activate only 40 degrees of freedom out of 848. This leads to a computation time T s = 83ms.

The haptic force, and the molecular interaction force are plotted in Figure 9(a). The system remains stable during the whole experiment, and small forces are highly amplified. The position of the molecule during the experiment and the value of the displacement scaling factor with respect to the time are represented in Figure 9(b). While not in contact with the rest of the molecule, the displacement of the inhibitor is easier for the user since the displacement scaling factor is smaller.

For that experiment, the system remains stable since the sampling period is low, as adaptive simulation is used. The accuracy of the computed forces is ensured by the adaptive feature of the simulator. With the variable force amplification the user can feel all molecular interactions. Ease of manipulation is preserved for trained users by variable displacement scaling.

Discussion

This work highlights the potential applications of variable gain systems for micro and nano teleoperations. Even if it is only the first step towards generalized use of such couplings, some conclusions can be made:

• To render small amplitude phenomena to users with no truncation of higher magnitude interaction forces, a variable force amplification is proposed. The function used presents two linear amplification range (for forces of small magnitude, and forces around a given value F 0 ). The haptic force does not enable to compare the whole range of forces since amplification is non linear outside the two specified ranges. However, new molecular interactions can be transmitted to operators using this variable scaling. This feature has been widely appreciated by users, and proves to lead to better results than constant force amplification for a task involving the determination of the equilibrium position between two molecules.

• The function used to compute the variable displacement scaling factor enables to get two different values for low (resp. high) molecular forces. Numerical parameters are chosen so that the transition between these two values is clearly marked, and happens for very low molecular interaction forces. The user perception is not deteriorated by these changes since the scaling factor is almost constant apart from this transition. However, only experienced users benefit from this variable displacement scaling. Existing solutions based on rate mode control must be considered, in particular for non-expert users.

Involving users in the decision of changing the way of controlling the molecule may be more appropriated for people not confident with haptic manipulation.

The main advantage of this variable coupling is that it relies on a simple and widely used haptic coupling. It can thus be adapted to any haptic feedback teleoperation system dealing with micro or nanoscale objects (real or virtual). It can in particular benefit micromanipulation through atomic force microscope since the same issues of scale difference between forces arise [START_REF] Onal | A scaled bilateral control system for experimental one-dimensional teleoperated nanomanipulation[END_REF]. Attractive forces would give information about imminent contact, and would avoid damages of fragile objects and tools.

CONCLUSION

Classical non-variable haptic feedback systems for molecular simulators may have limited applications due to the difficulty to render high-dynamic-range molecular interactions. In particular, attractive forces cannot be sufficiently amplified with a constant force factor if no truncation of repulsive forces is required. Ease of manipulation cannot be ensured with constant displacement scaling factor without leading to instability. A solution based on variable gain haptic coupling has been proposed, and evaluated by user tests. Adaptive force scaling factor is appreciated by all participants, and gives good results for a task based on the determination of the equilibrium position of two molecules. Variable displacement scaling factor only benefits users confident with haptic manipulation, while others describe it as counterintuitive. This is supported by quantitative results since the completion time is not significantly reduced. Existing solutions based on rate mode control should be considered, since they are based on user decision to modify the control mode. This may be more appropriated for non trained users. Extraction and reinsertion of an HIV inhibitor have been conducted, and the system remained stable.

This work presents the first step of variable gain coupling for haptic feedback at micro and nanoscale. To use it for the manipulation of both real and virtual objects, two main issues need to be addressed in future works. The first one is the determination of the most appropriated functions to compute the variable gains. User perception should be taken into account. Then, a detailed stability analysis will be performed. The non linearity of variable gains must be taken into account in this study. Based on the defined functions, and these stability results, it will be possible to propose a fully adaptive gain coupling. For a given task, users will define the phenomena they need to feel as well as the desired accuracy of the simulation. The gains will be adjusted on-line to maintain a stable system with appropriated scaling of forces and displacements.

Figure 1 :

 1 Figure 1: Molecular interaction between two water molecules.

Figure 2 :

 2 Figure 2: Haptic coupling control scheme

Figure 3 :

 3 Figure 3: The HIV molecule and its inhibitors, 848 degrees of freedom. A haptic device is used to control the inhibitors and to transmit molecular interactions.

Figure 4 :

 4 Figure 4: Profile of the force amplification using the non linear function (Equation (2)). F h1 = 1N, F h2 = 3N, F 0 = 1.5 • 10 -9 N, A f min = 0.64 • 10 9 , A fmax = 5.1 • 10 9 .

Figure 5 :

 5 Figure 5: Water molecule interactions using variable force amplification, T s = 14ms. The parameters used are: F h1 = 1N, F h2 = 3N, F 0 = 1.5 • 10 -9 N, A f min = 0.64 • 10 9 , A fmax = 5.1 • 10 9 . Displacement scaling factor: see Section 3.2.

Figure 6 :

 6 Figure 6: Water molecule interactions using variable displacement amplification Ãd , T s = 14ms. The parameters used are: γ = 10 • 10 11 , F o f f = 20 • 10 -12 N, A d min = 0.005 • 10 9 , A dmax = 0.25 • 10 9 . Force scaling factor: see Section 3.1.

  Average distance to the actual equilibrium position. Mode 5 -average distance: 1.0018 Å, standard deviation: 1,026 Å.

Figure 8 :

 8 Figure 8: Repartition of the determined equilibrium positions of one of the user represented on the reference force/position interaction plot. The equilibrium position (null interaction force) is represented by dashed lines. Modes 1 and 2 (variable force scaling): red; modes 3 and 4 (constant force scaling): blue: mode 5 (no force feedback): green.

  Parameters for the force scaling function: F h1 = 1N, F h2 = 3N, F 0 = 1.5 • 10 -9 N, A f min = 0.25 • 10 9 , A fmax = 5.1 • 10 9 . Parameters for the displacement scaling function: γ = 10 • 10 11 , F o f f = 20 • 10 -12 N, A d min = 0.005 • 10 9 , A dmax = 0.25 • 10 9 .

Figure 9 :

 9 Figure 9: Extraction and reinsertion of an HIV inhibitor: 40 degrees of freedom out of 848 are activated. (a): Haptic and molecular interaction forces. (b): Position of the HIV inhibitor and scaling factor profile.

A plot of the force against the relative position of the two molecules might be more appropriated. However, since the time taken to perform the experiments is representative of ease of manipulation, all plots in this paper (except if otherwise mentioned) will be represented as a function of time.

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ACKNOWLEDGEMENTS

This work was supported by the French National Agency of Research, through the NANOROL project.