A bonding evolution theory study of the mechanism of [3+2] cycloaddition reactions of nitrones with electron-de ﬁ cient ethylenes †

The mechanism of zw-type [3+2] cycloaddition (32CA) reactions of nitrones with electron-de ﬁ cient (ED) ethylenes has been studied using density functional theory (DFT) methods at the MPWB1K/6-31G(d) level of theory. An exploration of the potential energy surfaces associated with the four competitive reactive channels of the 32CA reaction between C -phenyl- N -methyl nitrone and acrolein indicates that the cycloaddition reaction takes place through a one-step mechanism. This cycloaddition reaction presents a moderate meta regioselectivity and a complete endo stereoselectivity, which is diminished in dichloromethane. Analysis of the DFT reactivity indices of the reagents allows an explanation of the participation of nucleophilic nitrones in zw-type 32CA reactions towards ED ethylenes. A bonding evolution theory (BET) study of the two endo regioisomeric reactive channels allows establishing the molecular mechanism of these relevant 32CA reactions. Both regioisomeric channels topologically take place along eight di ﬀ erentiated phases. While the formation of the C – C single bond follows Domingo's recently proposed model, the formation of the O – C single bond takes place at the short distance of 1.6 ˚ A through the donation of some electron density of the oxygen lone pairs of the nitrone to the b -conjugated carbon atom of acrolein. BET supports the non-concerted nature of these zw-type 32CA reactions and makes it possible to reject the pericyclic mechanism proposed for them.


Introduction
The [3+2] cycloaddition (32CA) reaction of nitrones with alkenes is one of the most interesting methods for the synthesis of isoxazolidines, 1 which are important precursors for the synthesis of a large number of biologically active molecules with a diversity of applications, mainly as antiviral agents. 2 The use of non-symmetric alkenes opens up the possibility of the formation of two isomeric isoxazolidines (see Scheme 1).
Depending on the electronic nature of the three-atomcomponents (TACs) involved in the 32CA reactions, these cycloadditions have recently been classied into pseudodiradical-type (pr-type) reactions involving TACs with a high pseudodiradical character, e.g. carbonyl ylides, which take place easily through an earlier TS with non-polar character, and zwitterionic-type (zw-type) reactions involving TACs with a high zwitterionic character, e.g. nitrones, characterised by favourable nucleophilic/electrophilic interactions, taking place through polar transition state structures (TSs). 3 A recent study of a series of the most common TACs used in organic synthesis, 4 including nitrile ylide 1, nitrile imine 2, nitrile oxide 3, diazoalkane 4, azide 5, nitrone 6 and methyl nitronate 7, has shown that they present a high nucleophilic character, participating in zw-type reactions towards electron-decient (ED) ethylenes (see Scheme 2). 5 Thus, the simplest nitrone 6 has a nucleophilicity index of N ¼ 2.95 eV, 5 being on the borderline of strong nucleophiles. 6 Scheme 1 32CA reactions of nitrones.
Computed activation energies indicated that these nonsubstituted TACs react quickly towards the ED 1,1-dicyanoethylene (DCE) 8. Interestingly, the most favourable regioisomeric TS associated with the 32CA of nitrone 6 with ED ethylene 8, TS1-m, was found to be 1.0 kcal mol À1 below that of the separated reagents (see Scheme 3). This 32CA reaction takes place with a high global electron density transfer (GEDT), 7 0.26e, in clear agreement with a zw-type 32CA reaction, 3 compared with the non-polar 32CA reaction between nitrone 6 and ethylene 11 presenting an activation energy of 13.1 kcal mol À1 and a GEDT of 0.01 eV. 5 Most 32CA reactions of nitrones take place via a one-step mechanism in which the two new single bonds are formed in a more or less asynchronous manner. In addition, due to the non-symmetry of the ethylene component, two regioisomeric channels are feasible, each one of them being initialised by the formation of the O-C or the C-C single bond. As the formation pattern of the O-C and C-C single bonds along the two regioisomeric channels is different, it is expected that these channels have a dissimilar molecular mechanism.
There is growing interest in explaining chemical reactivity arising from the redistribution of the electron density along the corresponding reaction pathway. The advantage of this selection is based on the fact the electron density is a local function dened within the exact many body theory, and it is also an experimentally accessible scalar eld, allowing a sound description of bonding changes to characterise a reaction mechanism 8 To perform these analyses quantitatively, the bonding evolution theory (BET), consisting of the joint-use of the electron localisation function (ELF) of Becke and Edgecombe 9 and Thom's catastrophe theory 10 (CT), was proposed 11 as a new tool for analysing the electronic changes in chemical processes. BET has been applied to different elementary reactions, allowing the molecular mechanism to be established. 12 Very recently, Merino et al. studied the mechanism of the 32CA reaction between nitrone 12 and lithium ynolate 13 using DFT methods (see Scheme 4). 13 An ELF topological analysis of selected points of the IRC associated with this 32CA reaction showed that aer the formation of a starting molecular complex MC2-o without energy barrier, in which the lithium atom is coordinated to both nitrone 12 and ynolate 13, the 32CA reaction begins with the initial formation of the C-C single bond via a two-stage one-step mechanism, 14 the activation energy being 14.7 kcal mol À1 . Along this non-concerted process, the O-C single bond formation takes place at the end of the reaction path, when the rst C-C single bond has been completely formed. 13 Herein, a DFT study of the zw-type 32CA reaction of nitrone 15 with acrolein 16 yielding isoxazolidines 17 and/or 18, as a model of 32CA reactions of nitrones with ED ethylenes, is carried out in order to characterise the molecular mechanism of this signicant class of 32CA reactions (see Scheme 5). A BET study of the two competitive endo regioisomeric reaction channels of the 32CA reaction between nitrone 15 and acrolein 16 is performed, characterising the molecular mechanism of these CA reactions.

Computational methods
Several works have shown that the B3LYP functional 15 is relatively accurate for kinetic data, although the reaction exothermicities are underestimated. 16 Truhlar's group has proposed some functionals, such as the MPWB1K, 17 which improve thermodynamic calculations. Therefore, in this study the MPWB1K functional was selected together with the standard 6-31G(d) basis set. 18 Optimisations were performed using the Berny analytical gradient optimisation method. 19 The stationary points were characterised by frequency computations in order to verify that TSs have one and only one imaginary frequency. The IRC paths 20 were traced in order to check the energy proles connecting each TS to the two associated minima of the proposed mechanism using the second order González-Schlegel integration method. 21 Solvent effects of dichloromethane (DCM) in the single-point energy calculations were taken into account using the polarisable continuum model (PCM) Scheme 2 TACs used in organic synthesis. 5 Scheme 3 32CA reaction between the simplest nitrone 6 and DCE 8. B3LYP/6-31G(d) relative energies are given in parentheses. 5 Scheme 4 Reaction between nitrone 12 and ynolate 13. B3LYP/6-31G(d) relative energies are given in parentheses. 13 developed by Tomasi's group 22 in the framework of the selfconsistent reaction eld (SCRF). 23 The electronic structures of the stationary points were analysed by the natural bond orbital (NBO) method 24 and by the ELF topological analysis, h(r). 25 The ELF study was performed with the TopMod program 26 using the corresponding monodeterminantal wavefunctions of the selected structures of the IRC. All computations were carried out with the Gaussian 09 suite of programs. 27 Conceptual DFT provides different indices to rationalise and understand chemical structure and reactivity. 28 The global electrophilicity index, 29 u, is given by the following expression, u ¼ (m 2 /2h), in terms of the electronic chemical potential m and the chemical hardness h. Both quantities may be approached in terms of the one-electron energies of the frontier molecular orbitals HOMO and LUMO, 3 H and 3 L , as m ¼ (3 H + 3 L )/2 and h ¼ (3 L À 3 H ), respectively. 30 The global nucleophilicity index, 31 N, based on the HOMO energies obtained within the Kohn-Sham scheme, 32 is dened as N ¼ E HOMO (Nu) À E HOMO (TCE), where tetracyanoethylene (TCE) is the reference because it presents the lowest HOMO energy in a long series of molecules already investigated in the context of polar organic reactions. This choice allows handling conveniently a nucleophilicity scale of positive values. The electrophilic P k + and nucleophilic P k À Parr functions, 33 which allow for the characterisation of the electrophilic and nucleophilic centers of a molecule, were obtained through the analysis of the Mulliken atomic spin density (ASD) of the radical anion of acrolein 16 and the radical cation of nitrone 15, respectively, by single-point energy calculations from the optimised neutral geometries.

Results and discussion
The present theoretical study has been divided into three parts: (i) rst, an analysis of the DFT reactivity indices at the ground state of the reagents involved in the 32CA reaction of nitrone 15 with acrolein 16 is carried out; (ii) second, the potential energy surface (PES) associated with this 32CA reaction is explored and characterised; (iii) and nally, a BET study of the endo regioisomeric channels associated with the 32CA reaction between nitrone 15 and acrolein 16 is performed in order to characterise its molecular mechanism.
(i) Analysis of the global and local DFT reactivity indices at the ground state of the reagents involved in the 32CA reaction of nitrone 15 with acrolein 16 Studies devoted to DA and 32CA reactions have shown that the analysis of the reactivity indices dened within conceptual DFT 28,34 is a powerful tool to understand the reactivity in polar cycloadditions. The global indices, namely, electronic chemical potential m, chemical hardness h, global electrophilicity u and global nucleophilicity N, for the reagents involved in this 32CA reaction are given in Table 1. It can be observed that the electronic chemical potential of nitrone 15, m ¼ À3.41 eV, is higher than that of acrolein 16, m ¼ À4.58 eV, indicating that along a polar reaction the GEDT will take place from the nitrone framework towards the acrolein one.
The simplest nitrone 6 presents a nucleophilicity N index of 2.95 eV and an electrophilicity u index of 0.73 eV, being clas-sied on the borderline of strong nucleophiles, 6 and as a moderate electrophile. 35 Substitution of the N-hydrogen atom by a methyl group and the incorporation of a phenyl group at the C carbon of nitrone noticeably increases the nucleophilicity N index of nitrone 15 to N ¼ 3.77 eV, and slightly increases its electrophilicity index to u ¼ 0.96 eV. In consequence, nitrone 15 is considered a strong nucleophile participating in polar reactions.
Polar cycloaddition reactions require the participation of good electrophiles and good nucleophiles. Ethylene 11 is one of the poorest electrophilic, u ¼ 0.57 eV, and nucleophilic, N ¼ 1.83 eV, species involved in cycloaddition reactions, being classied as a marginal electrophile and a marginal nucleophile. Therefore, it cannot participate in polar reactions. 5 Inclusion of a carbonyl group in the ethylene framework notably increases the electrophilicity u index of acrolein 16, u ¼ 1.34 eV, being now classied as a strong electrophile, and slightly decreases its nucleophilicity index to N ¼ 1.76 eV, remaining classied as a marginal nucleophile.
Along a polar reaction involving the participation of nonsymmetric reagents, the most favourable reactive channel is that involving the initial two-center interaction between the most electrophilic center of the electrophile and the most nucleophilic center of the nucleophile. 36 Recently, we have proposed the electrophilic P k + and nucleophilic P k À Parr functions, derived from the changes of the electron density reached via the GEDT process 7 from the nucleophile to the electrophile, as powerful tools in the study of the local reactivity in polar processes. 33 Analysis of these functions accounts for the most favourable single bond formation between the most electrophilic and nucleophilic centers of the reagents. 36 Hence, in order to characterise the most nucleophilic and the most electrophilic centers of the species involved in this 32CA reaction, the nucleophilic P k À Parr functions of nitrone 15 and the electrophilic P k + Parr functions of acrolein 16 were analysed. Fig. 1 shows the 3D representation of the ASD of the radical cation 15 +$ and the nucleophilic P k À Parr functions of nitrone 15, and the 3D representation of the ASD of the radical anion 16 À$ and the electrophilic P k + Parr functions of acrolein 16. Analysis of the nucleophilic P k À Parr functions at nitrone 15 indicates that the O1 oxygen is the most nucleophilic center of this species presenting the maximum value, P k À ¼ 0.61 (see Scheme 5 for atom numbering). At this oxygen, the value of the local nucleophilicity N k index is N O1 ¼ 2.28 eV. 33 Note that the local nucleophilicity N k index at the C3 carbon in nitrone 15 presents a very low value, N C3 ¼ 0.31 eV. On the other hand, the electrophilic P k + Parr functions of acrolein 16 indicates that the terminal C4 carbon is ca. twice as electrophilically activated, P k + ¼ 0.55, as the carbonyl C6 carbon, P k + ¼ 0.27. The corresponding local electrophilicity u k values are u C4 ¼ 0.73 eV and u C3 ¼ 0.36 eV. 33 Therefore, the most favourable electrophilenucleophile interaction along the nucleophilic attack of nitrone 15 on acrolein 16 will take place between the most nucleophilic center of nitrone 15, the O1 oxygen, and the most electrophilic center of acrolein 16, the C4 carbon.
(ii) Study of the PES of the 32CA reaction of nitrone 15 with acrolein 16 Due to the non-symmetry of the reagents, four reactive channels are feasible along the nucleophilic attack of nitrone 15 on acrolein 16. The two stereoisomeric reactive channels, the endo and the exo ones, are related to the two stereoisomeric approach modes of the carbonyl C]O group of acrolein 16 relative to the sp 2 hybridised N2 nitrogen of nitrone 15. Along the endo channel, the carbonyl group is placed over the nitrogen atom of the nitrone framework. The two regioisomeric reaction pathways are related to the two regioisomeric approach modes of the O1 oxygen or the C3 carbon of nitrone 15 toward the b-conjugated C4 carbon of acrolein; depending on the relative position of the carbonyl group of acrolein 16 with respect to the oxygen O1 atom of nitrone 15 in the isoxazolidines, the two regioisomeric possibilities were named meta and ortho. An exploration of the PES for the zw-type 32CA reaction between nitrone 15 and acrolein 16 indicates that this reaction takes place through a one-step mechanism. Consequently, the reagents, one TS and one cycloadduct were located and characterised along each one of the four reactive channels (see Scheme 5). Relative electronic energies of the stationary points involved in this 32CA reaction, in gas phase and in DCM, are displayed in Table 2, while the total electronic energies are given in Table S1 in the ESI. † The gas phase activation energies associated with the 32CA reaction between nitrone 15 and acrolein 16 are 7.9 (TS3-mn), 11.9 (TS3-mx), 10.3 (TS3-on) and 13.5 (TS3-ox) kcal mol À1 , the reaction being exothermic by between 28 and 30 kcal mol À1 . Formation of CA 17-n is kinetically favoured, while that of 18-n is thermodynamically favoured. These energy results indicate that this 32CA reaction is kinetically and moderately regioselective, as the most favourable meta TS3-mn is 2.4 kcal mol À1 below the ortho TS3-on, and displays a complete endo stereoselectivity as the most favourable endo TS3-mn is 4.0 kcal mol À1 below the exo TS3-mx. The strong exothermic character of the reaction makes the formation of the four CAs irreversible; therefore, the major product of the reaction, isoxazolidine 17-n, is formed by kinetic control.
Activation and reaction energies of the endo stereoisomeric channel increase by between 1 and 3 kcal mol À1 (see Table 2) when solvent effects of DCM are considered. This behaviour is a consequence of a better solvation of the reagents than the TSs and CAs. 37 In DCM, while the moderate regioselectivity found in gas phase remains unchanged, the complete endo stereoselectivity found in gas phase diminishes; the most favourable endo TS3-mn is now 2.1 kcal mol À1 below the exo TS3-mx.
The geometries of the TSs involved in the zw-type 32CA reaction between nitrone 15 and acrolein 16 are shown in Fig. 2. At the meta TSs, the lengths of the O1-C4 and C3-C5 forming bonds are 1.784 and 2.454Å at TS3-mn and 1.822 and 2.496Å at TS3-mx, respectively, while at the ortho TSs the lengths of the O1-C5 and C3-C4 forming bonds are 2.224 and 2.015Å at TS3on and 2.183 and 2.066Å at TS3-ox. These lengths indicate that the TSs involved in the most favourable meta channel are more asynchronous than those involved in the ortho one. The IRCs from the ortho TSs toward [3+2] cycloadducts 18-n and 18-x show that the two ortho stereoisomeric channels are associated to a two-stage one-step mechanism 14 initialised by the nucleophilic attack of the O1 oxygen of nitrone 15 on the C4 carbon of acrolein 16.
The electronic nature of the TSs associated with the one-step 32CA reaction of nitrone 15 with acrolein 16 was studied by analysing the bond order (BO) 38 and computing the GEDT at the corresponding TSs. In gas phase, the BO values of the O1-C4 and C3-C5 forming bonds at the meta TSs associated with the formation of [3+2] CAs 17-n and 17-x are 0.50 and 0.21 at TS3-mn and 0.47 and 0.21 at TS3-mx, respectively, while the BO values of the O1-C5 and C3-C4 forming bonds at the ortho TSs associated with the formation of CAs 18-n and 18-x are 0.25 and 0.44 at TS3-on and 0.27 and 0.41 at TS3-ox. These values suggest asynchronous bond formation processes, in which the bond formation at the C4 carbon of acrolein 16 is more advanced than that at the C5 carbon atom.
In order to calculate the GEDT at the TSs, the natural atomic charges at the TSs for the meta/ortho regioisomeric channels and the endo/exo stereoisomeric ones, obtained through a natural population analysis (NPA), were shared between the nitrone and the acrolein frameworks. In gas phase, the GEDT that uxes from the nitrone moiety toward the acrolein one is 0.16e at TS3-mn, 0.14e at TS3-mx, 0.10e at TS3-on and 0.07e at TS3-ox. The most favourable meta/endo channel via TS3-mn is the most polar one, in clear agreement with a zw-type mechanism. 3 These low values denote a low polar character of these zw-type 32CA reactions, although suggest a slight acceleration of the 32CA reaction due to the strong nucleophilic and electrophilic character of nitrone 15 and acrolein 16, respectively. Note that the GEDT at the TS associated with the non-polar 32CA reaction between the simplest nitrone 6 and ethylene 11 is 0.01e. 5 When comparing the 32CA reaction between nitrone 15 and acrolein 16 with that between the simplest nitrone 6 and DCE 8, the larger acceleration found in the latter 32CA reaction is in agreement with the high GEDT computed at the corresponding TS, 0.26e. In spite of the higher nucleophilic character of C-phenyl nitrone 15 than that of the simplest nitrone 6, the higher electrophilic character of DCE 8 than that of acrolein 16 is responsible for the higher polar character of the 32CA reaction involving ED ethylenes.
Finally, the energy results and geometries obtained by using the MPWB1K functional were compared with those obtained by using the B3LYP one. For this purpose, the stationary points associated with the 32CA reaction between nitrone 15 and acrolein 16 were fully optimised at the B3LYP/6-31G(d) level. Gas phase total and relative energies are given in Table S2, † while the TS geometries are shown in Fig. S1 in the ESI. † A comparison of the gas phase relative energies indicates that while the B3LYP activation energies are found to be between 3.4  and 4.4 kcal mol À1 higher than the MPW1K ones, the reaction energies are underestimated by 17 kcal mol À1 . On the other hand, both functionals yield a similar endo stereoselectivity, while the B3LYP functional reduces the moderate meta regioselectivity. A comparison of the B3LYP and MPWB1K lengths of the O-C and C-C forming bonds indicates that both functionals give similar values, thus giving similar bonding changes at the TSs.
(iii) Characterisation of the molecular mechanisms of zw-type 32CA reactions of nitrones with ED ethylenes Several theoretical studies have shown that the ELF topological analysis of the changes of electron density along a reaction path can be used as a valuable tool to understand the bonding changes along the reaction path, and consequently to establish the molecular mechanisms. 12 Aer an analysis of the electron density, ELF provides basins of attractors, which are the domains in which the probability of nding an electron pair is maximal. 39 The spatial points in which the gradient of ELF has a maximum value are designated as attractors. The basins are classied as core basins and valence basins. The latter are characterised by the synaptic order, i.e. the number of atomic valence shells in which they participate. Thus, there are monosynaptic, disynaptic, trisynaptic basins and so on. 40 Monosynaptic basins, labelled V(A), correspond to the lone pairs or non-bonding regions, while disynaptic basins, labelled V(A,B), connect the core of two nuclei A and B and, thus, correspond to a bonding region between A and B. This description recovers the Lewis bonding model, providing a very suggestive graphical representation of the molecular system. ELF topological analysis of the structures involved in an elementary step allows characterising mainly three types of valence basins: (i) protonated basins, V(A,H), (ii) monosynaptic basins, V(A), associated with non-bonding regions, and (iii) disynaptic basins, V(A,B), associated with bonding regions. A set of valence basins characterises the ELF topology of a particular molecular structure. Analysis of changes in number or type of valence basins along the IRC allows the characterisation of the IRC points, Pi, dening the different phases that topologically characterise a molecular mechanism.
Recently, Domingo has shown that the formation of the C-C single bond in organic reactions involving sp 2 hybridised carbons begins in the short C-C distance range of 1.9-2.0Å by merging two monosynaptic basins, V(Cx) and V(Cy), into a new disynaptic basin, V(Cx,Cy), associated with the formation of the new Cx-Cy single bond. 7 The Cx and Cy carbons characterised by the presence of the monosynaptic basins V(Cx) and V(Cy) are called pseudoradical centers. 7 The 32CA reaction between nitrone 15 and acrolein 16 can take place along four competitive channels; two pairs of endo/exo stereoisomeric channels and two pairs of meta/ortho regioisomeric channels. Although the energy differences between the TSs associated with the endo/exo stereoisomeric channels are higher than those associated with the meta/ortho regioisomeric ones, analysis of the geometric and electronic properties of the four TSs indicates that the endo/exo stereoisomeric TSs present similar behaviours, suggesting analogous molecular mechanisms. Consequently, only the regioisomeric reaction paths associated with the most favourable endo regioisomeric channels were selected to perform the BET studies of the 32CA reaction of nitrone 15 with acrolein 16 and, thus, to establish the molecular mechanism of 32CA reactions of nitrones with ED ethylenes.
Finally, an ELF topological analysis of the regioisomeric TSs involved in the 32CA reaction between the simplest nitrone 6 and DCE 8, and the TS involved in the 32CA reaction between dimethylnitrone 12 and ynolate 13, recently studied by Merino et al., 13 was also performed in order to characterise the most signicant bonding changes of these TSs. Table 3 Valence basin populations N calculated from the ELF of the IRC points, P1-P7, defining the eight phases characterising the molecular mechanism associated with the meta/endo regioisomeric channel of the 32CA reaction between nitrone 15 and acrolein 16. The stationary points MC3-mn and 17-n are also included. Distances are given inÅ, while the GEDTs obtained by NBO analysis are given in e (a) BET study of the meta/endo regioisomeric channel associated with the 32CA reaction between nitrone 15 and acrolein 16. The BET study of the meta/endo regioisomeric path associated with the nucleophilic attack of nitrone 15 on the ethylene moiety of acrolein 16 indicates that this reaction is topologically characterised by eight differentiated phases. The population of the most signicant valence basins of the selected points of the IRC is displayed in Table 3. A schematic picture of the attractor positions of the ELF for the relevant points along the IRC is given in Fig. 3, while the basin-population changes along the reaction path are graphically represented in Fig. 4.
Phase III, 1.89Å $ d(O1-C4) > 1.82Å and 2.53Å $ d(C3-C5) > 2.49Å, begins at P2. The most signicant occurrence in this phase is the creation of a new V(C5) monosynaptic basin at P2, integrating 0.08e (see P2 in Fig. 4). The electron density of the  new V(C5) monosynaptic basin is mainly reached through the depopulation of the V(C4,C5) disynaptic basin, which continues its depopulation until reaching a population of 3.02e. This loss of electron density is mainly gathered at the V(C5,C6) disynaptic basin, which has increased its integration, 2.47e. At P2 the maximum GEDT along the 32CA reaction is observed, 0.16e, indicating that this zw-type 32CA has a low polar nature.
At phase V, 1.67Å $ d(O1-C4) > 1.64Å and 2.36Å $ d(C3-C5) > 2.33Å, the most notable change is the creation of a new V(C3) monosynaptic basin at P4, integrating 0.18e, whereas the V(C5) monosynaptic basin created in phase III has reached a population of 0.45e, as a consequence of the strong depopulation experienced by the V(C4,C5) disynaptic basin, 0.53e. Note that the two V(C3) and V(C5) monosynaptic basins are required for the formation of the C3-C5 single bond in phase VII. Meanwhile, the V(N2) monosynaptic basin has increased its population to 1.45e as the consequence of the depopulation of both the V(O1,N2) and V(N2,C3) disynaptic basins. GEDT has slightly decreased to 0.13e.
Phase VI, 1.64Å $ d(O1-C4) > 1.58Å and 2.33Å $ d(C3-C5) > 2.26Å, begins at P5. The most relevant change along this short phase is the remarkable creation of a V 00 (O1) monosynaptic basin integrating 0.71e as a result of the decrease of the electron density of the V 0 (O1) monosynaptic basin from 3.34e to 2.64e. GEDT scarcely varies.
Phase VII, 1.58Å $ d(O1-C4) > 1.49Å and 2.26Å $ d(C3-C5) > 2.02Å, begins at P6. The rst most relevant change along the reaction path takes place at this point. While the V 00 (O1) monosynaptic basin created in the previous phase has disappeared, a new V(O1,C4) disynaptic basin is created with an initial population of 0.88e (see the V(O1,C4) disynaptic basin in P6 in Fig. 3, and the change from V 00 (O1), in green in P5, to V(O1,C4), in blue in P6, in Fig. 4). This notable topological change indicates that the formation of the new O1-C4 single bond has already begun at a length of ca. 1.58Å. GEDT decreases to 0.08e.
Finally, phase VIII, 1.49Å $ d(O1-C4) $ 1.42Å and 2.02Å $ d(C3-C5) $ 1.56Å, begins at P7 and ends at isoxazolidine 17-n, d(O1-C4) ¼ 1.419Å and d(C3-C5) ¼ 1.555Å. The second most relevant change along the reaction path takes place at P7. The two V(C3) and V(C5) monosynaptic basins present in the previous phase are merged into a new V(C3,C4) disynaptic basin, which presents a population of 1.24e (see the V(C3,C5) disynaptic basin in P7 in Fig. 3, and the merger of V(C3) and V(C5), in green in P6, into the new V(C3,C5), in blue in P7, in Fig. 4). This notable topological change indicates that the formation of the new C3-C5 single bond has already begun at a length of ca. 2.02Å, by a C-to-C coupling of the two C3 and C5 pseudoradical centers. With the start of the C3-C5 bond formation, GEDT decreases until reaching a slightly negative value, À0.06e, as the consequence of a back donation process. From P7 to 17-n only changes in basin populations are observed. Some topological features in cycloadduct 17-n are noteworthy: the O1-C4 and O1-N2 single bond regions are very polarised toward the oxygen atom, the population of the V(O1,C4) and V(O1,N2) disynaptic basins and that of the V(O1) and V 0 (O1) monosynaptic basins integrating 1.29e and 0.95e, and 2.54e and 2.52e, respectively. The nitrogen N2 lone pair and Table 4 Valence basin populations N calculated from the ELF of the IRC points, P1-P7, defining the eight phases characterising the molecular mechanism associated with the ortho/endo regioisomeric channel of the 32CA reaction of nitrone 15 with acrolein 16. The stationary points MC3-on and 18-n are also included. Distances are given inÅ, while the GEDTs obtained by NBO analysis are given in e (b) BET study of the ortho/endo regioisomeric channel associated with the 32CA reaction between nitrone 15 and acrolein 16. The BET study of the ortho/endo regioisomeric reaction path associated with the nucleophilic attack of nitrone 15 on acrolein 16 indicates that this reaction can also be topologically characterised by eight differentiated phases. The populations of the most signicant valence basins of the selected points of the IRC are compiled in Table 4. A schematic picture of the attractor positions of the ELF for relevant points along the IRC is displayed in Fig. 5, while the basin-population changes along the reaction path are graphically represented in Fig. 6.
Phase    6 Graphical representation of the basin-population changes along the ortho/endo regioisomeric channel of the 32CA reaction between nitrone 15 and acrolein 16. Point-dotted curves in grey represent the sum of disynaptic basins describing a double bond region or monosynaptic basins describing lone pairs. and 1.35e. In this phase, a new V(N2) monosynaptic basin appears with a population of 0.96e. This monosynaptic basin is associated with the nitrogen N2 lone pair present at isoxazolidine 18-n. It is also observed that the two V(C4,C5) and V 0 (C4,C5) disynaptic basins belonging to the C4-C5 double bond region of the acrolein framework have merged into one V(C4,C5) disynaptic basin integrating 3.19e.
The short Phase III, 2.02Å $ d(C3-C4) > 1.91Å and 2.22Å $ d(O1-C5) > 2.17Å, begins at P2, d(C3-C4) ¼ 2.015Å and d(O1-C5) ¼ 2.224Å. Structure P2 corresponds to the TS of the reaction, TS3-on. The most signicant change occurring at this phase is the creation of a new V(C4) monosynaptic basin at the acrolein moiety, integrating 0.28e. Note that the C4 carbon of acrolein is the most electrophilic center of this molecule. On the other hand, the populations of the V(C3) and V(N2) monosynaptic basins increase to 0.53e and 1.21e.
Phase IV, 1.91Å $ d(C3-C4) > 1.64Å and 2.17Å $ d(O1-C5) > 1.91Å, begins at P3. A second signicant event along the reaction path occurs in this phase. While the two V(C3) and V(C4) monosynaptic basins disappear, a new V(C3,C4) disynaptic basin, integrating 1.08e, is created (see the V(C3,C4) disynaptic basin in P3 in Fig. 5, and the merger of V(C3) and V(C4), in green in TS3-on, into the new V(C3,C4), in blue in P3, in Fig. 6). This notable topological change indicates that the formation of the new C3-C4 single bond has already begun at a length of ca. 1.91Å, by a C-to-C coupling of the two C3 and C4 pseudoradical centers. On the other hand, the population of the V(N2) monosynaptic basin increases to 1.42e along the reaction progress, whereas the population of V(C4,C5) decreases to 2.64e.
Phase VIII, 1.57Å $ d(C3-C4) $ 1.52Å and 1.63Å $ d(O1-C5) $ 1.40Å, begins at P7 and ends at 18-n. Another relevant change along the reaction path takes place at P8. Whereas the population of the V(O1) monosynaptic basins decreases to 2.73e, a new V(O1,C5) disynaptic basin is created with an initial integration of 0.79e (see the V(O1,C5) disynaptic basin in P7 in Fig. 5, and the creation of V(O1,C5) in blue in P7 in Fig. 6). This notable topological change indicates that the formation of the new O1-C5 single bond has already begun at a length of ca. 1.63Å. Along this last phase, the population of the created V(O1,C5) disynaptic basin is increased until reaching 1.32e in 18-n, while the populations of the V(O1) and V 0 (O1) monosynaptic basins decrease to 2.61e and 2.48e, respectively. These values indicate that the created O1-C5 single bond is strongly polarised toward the O1 oxygen atom. In 18-n, the C3-C4 single bond has reached a population of 1.85e.
Some appealing conclusions can be drawn from the BET study of the two endo regioisomeric channels: (i) the BET study of the two endo regioisomeric reaction channels allows distinguishing eight differentiated phases associated with the creation or disappearance of valence basins; (ii) formation of the C-C single bond follows the recently proposed pattern: 7 (a) depopulation of the double bond regions; (b) formation of two non-bonding V(C) and V(C 0 ) monosynaptic basins; and (c) formation of a new V(C,C 0 ) disynaptic basin through the merger of the electron density of the aforementioned monosynaptic basins. The formation of the C-C single bond begins at 2.02Å along the meta channel and at 2.17Å along the ortho one; (iii) a different behaviour is found for the formation of the O-C single bond. Along the most favourable meta channel, the formation of the O1-C4 single bond begins with the creation of a new V(O1,C4) disynaptic basin at the very short distance of 1.58Å. Interestingly, the electron population of this new disynaptic basin mainly proceeds from the electron density of one V(O1) monosynaptic basin associated with the lone pairs of the nitrone O1 oxygen atom. Likewise, along the ortho regioisomeric channel, the formation of the O1-C5 single bond begins with the creation of a new V(O1,C5) disynaptic basin at the very short distance of 1.63Å. The electron density of this new V(O1,C5) disynaptic basin proceeds mainly from the electron density of one V(O1) monosynaptic basin associated with the lone pairs of the nitrone O1 oxygen atom. This pattern for the formation of these O-C single bonds is similar to that recently found for the formation of the N-C single bond along keteneimine Staudinger reactions, in which the electron density of the new N-C single bond comes from the nitrogen lone pair; 41 (iv) interestingly, while the formation of the C-C single bond begins at ca. 2.0Å, the formation of the O-C single bond begins at the very short distance of 1.6Å. These behaviours indicate that in most 32CA reactions the asynchronicity of the C-C and O-C single bond formation cannot be measured by using geometrical parameters. Thus, while the analysis of the bond lengths of the two O-C and C-C forming bonds suggests that TS3-mn is more asynchronous than TS3-on, analysis of the basin-population changes in Fig. 4 and 6 indicates that the O-C and C-C bond formation is more synchronous along the most favourable meta regioisomeric channel. Similarly, the BO values of the O1-C4 and C3-C5 forming bonds at TS3-mn and those of the O1-C5 and C3-C4 forming bonds at TS3-on suggest an asynchronous bond formation process, in which the bond formation at the C4 carbon of acrolein 16 is more advanced than that at the C5 carbon atom. However, the ELF topological analysis at TS3-mn indicates that the formation of both single bonds is almost synchronous. Moreover, the BO value of the O1-C4 forming bond at TS3-mn, 0.50, and that of the C3-C4 forming bond at TS3-on, 0.44, show that they have high values, but the ELF topological analysis at the corresponding TSs indicates that the formation of these single bonds has not begun. Consequently, while the BO values in the range from 1.0 to 3.0 can be used as a measure of the bonding electron density between two neighbouring atoms, BO values in the range from 0.0 to 1.0 cannot account for any information about the bonding evolution along the reaction coordinate.
(v) along the two endo regioisomeric channels, the formation of the O1-C4 or C3-C4 single bonds begins by a two-center interaction involving the most electrophilic center of acrolein 16, the C4 carbon. Note that at the most favourable meta/endo channel, the O1-C4 two-center interaction also involves the most nucleophilic center of nitrone 15, the O1 oxygen, a behaviour anticipated by the analysis of the electrophilic and nucleophilic Parr functions; (vi) along the ortho/endo regioisomeric channel, formation of the second O1-C5 single bond begins at the seventh phase, in which the rst C3-C4 single bond is practically formed. This behaviour indicates that the ortho reaction takes place through a two-stage one-step mechanism; 14 along the most favourable meta/endo regioisomeric channel, formation of the O1-C4 and C3-C5 single bonds is more synchronous.
(c) ELF topological analysis of the regioisomeric TSs involved in the 32CA reaction between the simplest nitrone 6 and DCE 8, and the TS involved in the 32CA reaction between dimethylnitrone 12 and ynolate 13. Finally, an ELF topological analysis of the regioisomeric TSs involved in the 32CA reaction between the simplest nitrone 6 and DCE 8, and the TS involved in the 32CA reaction between dimethylnitrone 12 and ynolate 13, recently studied by Merino et al., 13 was performed in order to characterise the most signicant bonding changes of the TSs of these 32CA reactions. The population of the most signicant valence basins of the corresponding stationary points is displayed in Table 5, while the attractor positions of the valence basins of the TSs aforementioned are presented in Fig. 7.
Similarly, the ELF picture of the ortho TS, TS1-o, d(C3-C4) ¼ 2.074Å and d(O1-C5) ¼ 2.312Å, shows the presence of two V(O1,N2) and V(N2,C3) disynaptic basins, with populations of 1.37e and 3.41e, belonging to the O1-N2 and N2-C3 bonding regions of the nitrone moiety, as well as one V(C4,C5) disynaptic basin, integrating 3.06e, associated with the C4-C5 bonding region of DCE 8. The most relevant feature of this TS is the presence of two V(C3) and V(C4) monosynaptic basins integrating 0.48e and 0.22e, which are demanded for the subsequent formation of the C3-C4 single bond.
Otherwise, the ELF picture of the TS involved in the 32CA reaction between dimethylnitrone 12 and ynolate 13, TS2-o, d(C3-C4) ¼ 2.123Å and d(O1-C5) ¼ 3.231Å, shows that the O1-N2 and N2-C3 bonding regions remain characterised by two V(O1,N2) and V(N2,C3) disynaptic basins whose populations are 1.12e and 2.88e, while the C4-C5 triple bond region of ynolate 13 is characterised by the presence of two V(C4,C5) and V 0 (C4,C5) disynaptic basins, which integrate a total of 4.86e. The most relevant feature of this TS is the presence of one V(C4) monosynaptic basin, integrating 0.89e, and one V(N2) monosynaptic basin, with a population of 1.44e, which is related to the N2 nitrogen lone pair present in CA 14. While the electron population of the V(C4) monosynaptic basin at this TS comes from the depopulation of the C4-C5 triple bond region of ynolate 13, the V(N2) monosynaptic basin comes from the depopulation of the N2-C3 double bond region of nitrone 12.
Some signicant conclusions can be drawn from the ELF topological analysis of the TSs involved in the 32CA reactions of nitrones 6, 12 and 15 towards ED ethylenes 8 and 16 and electron-rich ethylene 13: (i) ELF pictures of the most favourable meta TS1-m and TS3mn show the presence of a V(C5) monosynaptic basin with a similar population. No V(C4) monosynaptic basin is observed at both TSs; (ii) ELF pictures of ortho TS1-o and TS3-on show the presence of two V(C3) and V(C4) monosynaptic basins, with similar populations. Note that these monosynaptic basins are involved in the subsequent formation of the C3-C4 single bond.
(iii) unlike TS3-mn and TS3-on, no V(N2) monosynaptic basin is observed at TS1-m nor at TS1-o. This behaviour, together with the larger population of the V(N2,C3) disynaptic basin at the latter TS1-m and TS1-o, indicates that the changes in the N2-C3 bonding region are more delayed at the simplest nitrone 6; (iv) in spite of these topological differences, the two regioisomeric TSs involved in the 32CA reaction of the simplest nitrone 6 with DCE 8 show a great similarity along the formation of the O-C and C-C single bonds with that observed in the 32CA reaction of nitrone 15 with acrolein 16. This nding supports a similar description for the formation of the O-C and C-C single bonds along the two regioisomeric channels of 32CA reactions of nitrones with ED ethylenes; (v) a comparison of the ELF features of ortho TS2-o with those of TS1-o and TS3-on shows signicant dissimilarities. Thus, TS2-o shows the presence of a V(C4) monosynaptic basin with high electron population, 0.89e. This behaviour could be due to a notable difference between these 32CA reactions. Analysis of the electronic chemical potential of dimethyl nitrone 12 and ynolate 13, À2.93 and À1.79 eV (see Table 1), indicates that along a polar reaction the GEDT will ux from ynolate 13 toward nitrone 12 (see the change of the sign of the GEDT at TS2-o in Table 5). Consequently, in this zw-type 32CA reaction nitrone 12 will be the electrophile, while ynolate 13 will be the nucleophile. Interestingly, in spite of the high GEDT found at TS2-o, 0.30e, the activation energy associated with this polar process is very high, 14.7 kcal mol À1 , as a consequence of the marginal electrophilic character of nitrone 12, 0.56 eV (see Table 1).
In 1965, Woodward and Hoffmann established the pericyclic reactions 42 as "reactions in which all rst order changes in bonding relationship take place in concert on a close curve". 43 For 32CA reactions, the pericyclic model assumes that the six electrons involved in the cyclisation, four of the TAC and two of the alkene, move in a concerted fashion around the ve atoms. BET analysis of the zw-type 32CA reaction between nitrone 15 with acrolein 16 permits to establish that bonding changes are not neither concerted, but sequential along the nucleophilic attack of the nitrone 15 to the ED ethylene 16, nor on a close curve, but the electron density at the beginning of the reaction uxes from the nucleophilic nitrone to the ED ethylene. In addition, any bonding disynaptic basins in the C-C and O-C forming bond regions exist at TS1-m, TS1-o, TS2o, TS3-mn and TS3-on, such as it would be expected at the TSs associated with a pericyclic reaction. Consequently, as in Diels-Alder reactions, the BET study of these zw-type 32CA reactions allows rejecting the pericyclic mechanism proposed for them. 44

Conclusions
The mechanisms of zw-type 32CA reactions of nitrones with ED ethylenes have been studied using DFT methods at the MPWB1K/ 6-31G(d) level, and the obtained results are supported by the combination of the analysis of the values of reactivity indices at the ground state of the reagents, derived from conceptual DFT, the exploration and characterisation of the PESs associated with the cycloaddition reaction of nitrone 15 with acrolein 16, and the BET study of the two endo regioisomeric channels.
Nitrones such as 15 present a high nucleophilicity N index, being classied as strong nucleophiles. Consequently, it is expected that they participate in zw-type 32CA reactions towards ED ethylenes such as acrolein 16 or DCE 8. In addition, analysis of the nucleophilic P k À Parr functions allows characterising the O1 oxygen atom as the most nucleophilic center of nitrone 15. An exploration of the PESs associated with the four competitive reactive channels of the 32CA reaction of nitrone 15 with acrolein 16 indicates that the cycloaddition reaction takes place through a one-step mechanism. This cycloaddition reaction presents moderate meta regioselectivity and complete endo stereoselectivity. In DCM, the most favourable meta/endo reactive channel presents an activation energy of 9.0 kcal mol À1 , while the formation of the corresponding cycloadduct is exothermic by 29.1 kcal mol À1 .
A BET study of the endo regioisomeric channels allows establishing the molecular mechanism of these relevant 32CA reactions involving nitrones and ED ethylenes, and provides new insight into the bonding changes taking place along the two regioisomeric reaction channels. While the formation of the C-C single bond begins at ca. 2.0Å, the formation of the O-C bond begins at the very short distance of 1.6Å. Unlike the formation of the C-C bond, which follows Domingo's recently proposed model, 7 the formation of the O-C bond takes place at the short distance of 1.6Å through the donation of some electron density of the nitrone oxygen lone pairs to the acceptor carbon atom of the ED ethylene. This model for the formation of heteroatom-carbon single bonds is similar to that found in ketene-imine Staudinger reactions. 41 Fig. 7 ELF attractor positions for the regioisomeric TSs involved in the 32CA reaction between the simplest nitrone 6 and DCE 8, and the TS involved in the 32CA reaction between dimethylnitrone 12 and ynolate 13.