The Diels–Alder (DA) reaction is arguably one of the most powerful reactions in the arsenal of the synthetic organic chemist [1,2]. By varying the nature of the diene and dienophile, many different types of carbocyclic structures can be built. Since the discovery of the DA reaction in the 1920s by Otto Diels and Kurt Alder [3] a tremendous amount of experimental and theoretical work has been devoted to the study of the mechanism and the selectivity of these cycloaddition reactions.

In DA reactions, two new C-C single bonds are formed from two C=C double bonds of both a diene and an ethylene, a behaviuor that makes these reactions thermodynamically favourable. However, the bonding changes in these cycloadditions are more complex, as along with the formation of the two new C-C single bonds, one new C=C double bond at the cycloadduct is also formed, whereas three C=C double bonds are broken in the reagents.

The first transition state structure (TS) for the DA reaction between butadiene 1 and ethylene 2 (Scheme 1) was proposed by Wassermann in 1935 [4]. It was suggested that the lengths of the two forming bonds in the symmetric TS were 2.0 Å, a distance close to the 2.2 Å currently obtained (Figure 1) [5]. Today, this synchronous TS is presented as a prototype for DA reactions in all textbooks.

Scheme 1: DA reaction between butadiene 1 and ethylene 2.

Figure 1: B3LYP/6-31G* transition structure TS1 associated with the DA reaction between butadiene 1 and ethylene 2. The distances are given in Angstroms.

During the period from 1965 to 1969, R. B. Woodward and R. Hoffmann developed the concept of pericyclic reactions [6]. They defined pericyclic reactions as “reactions in which all first order changing in bonding relationship take place in concert on a close curve” [7]. They uncovered the principles of orbital symmetry conservation [8], according to which allowed reactions could be concerted and forbidden ones could not. The importance of the symmetry of frontier orbitals in pericyclic reactions was discovered by Fukui [9], who made the fundamental assumption that a majority of chemical reactions should take place at the position and the direction of maximum overlapping of the HOMO and the LUMO frontier orbital of the reacting species. The orbital symmetry together with the frontier molecular orbital (FMO) theory provided theoretical backing to the notion that pericyclic reactions occur by means of a concerted mechanism (Scheme 2).

Scheme 2: Pericyclic concerted mechanism for the DA reaction between butadiene 1 and ethylene 2.

The pericyclic model supposes that the six electrons involved in the cycloaddition, four of butadiene 1 and two of ethylene 2, move in a concerted fashion around the six carbon atoms (see the arrows in Scheme 2). The model developed from this picture was applied to other organic reactions such as electrocyclic and sigmatropic reactions. As a consequence of the proposed concerted electron movement around six atoms, an aromatic character was attributed to these TSs, and the nucleus-independent chemical shifts (NICS) were used to characterize pericyclic reactions [10].

The irrelevance of the attributed aromatic character of TSs and the synchronicity in the bond formation process is supported by the fact that there are very few synchronous DA reactions [11]. Any simple substitution in ethylene breaks the synchronicity of the C-C single bond formation. Note that in the DA between cyclopentadiene (Cp) 4 and styrene 5, although slightly more unfavourable than the DA reaction between Cp 4 and ethylene 2, the synchronicity in the C-C bond formations is broken (see TS3 in Figure 2).

Figure 2: TSs associated with the DA reaction between Cp 4 and ethylene 2, TS2, and between Cp 4 and styrene 5, TS3.

For the simplest DA reaction between butadiene 1 and ethylene 2, which is presented in all text books [12-14] as the prototype of this cycloaddition type but is not carried out experimentally in laboratories due to its unfavourable activation energy, 27.5 kcal mol^-1 [15], two different mechanisms have been proposed (Scheme 3) [5,7]:

i) a one-step mechanism through a synchronous C-C single bond formation process; and

ii) a two-step or stepwise mechanism via a diradical intermediate. Analysis of the activation energies involved in these mechanisms using the Transition State Theory [16-18] (TST) at different quantum chemical levels has estimated that the activation energy associated with the formation of the diradical intermediate is 2.3–7.7 kcal mol^-1 higher than that for the one-step mechanism, being in good agreement with the experimental estimates of 2.7 kcal mol^-1 [5]. These energy differences rule out the stepwise mechanism for the DA reaction between 1 and 2.

Scheme 3: One-step and stepwise mechanisms for the DA reaction between butadiene 1 and ethylene 2.

Adequate substitution in both the diene and the dienophile, which increases the electrophilic or/and the nucleophilic behaviours of one or both reagents, favours DA reactions usually via a polar mechanism [19], characterized by a global charge transfer along the cycloaddition. These DA reactions, named polar Diels-Alder (P-DA) reactions, take place also through a one-step mechanism. Only in the extreme case in which the substitution stabilizes zwitterionic intermediates, the reactions become stepwise [20,21]. However, the short life of these intermediates makes them experimentally undetectable, and consequently, this behaviour is irrelevant from a synthetic and kinetic points of view.

In the study of mechanisms of DA reactions using the TST, computational chemists claim that most of DA reactions “have a concerted mechanism instead of a stepwise one”. However, the word “concerted”, derived from the description of the “concerted” TS shown in Scheme 2 within the pericyclic model, is unrelated to the word “stepwise” used for the diradical mechanism shown in Scheme 3; while the word “concerted” refers to the bonding changes taking place in an elementary step, the word “stepwise” refers to those reactions taking place in more than one elementary step, and consequently, the opposite of “stepwise” is “one-step”, unless we assume that one-step reactions had concerted bonding changes [5,12]. This observation is crucial because most P-DA reactions [19], which take place via high asynchronous TSs, have a two-stage one-step mechanism in which the two C-C single bonds are formed in two distinct stages of the reaction, being non-concerted [22]. Note that the unique difference between a two-stage one-step mechanism and a stepwise mechanism in most P-DA reactions is the stabilization of a feasible zwitterionic intermediate located after formation of the first C-C single bond. Sometimes, this stabilization is attained by the use of polar solvents; in these cases, the inclusion of solvent effects on the geometrical optimization can change the two-stage one-step mechanism found in gas-phase to a stepwise mechanism in solution [23].

Along the nucleophilic approach to the electrophile only one C-C single bond is being formed, and when it is practically formed, the formation of the second C-C single bond begins through a ring-closure process in the second stage of the reaction. Consequently, P-DA reactions involving asymmetric electrophiles are non-concerted processes [19]. However, the significant question whether the prototype DA reaction between butadiene 1 and ethylene 2 is a concerted process remains unsolved.

In the 1990s, Savin and Silvi [24-27] developed some theoretical tools based on the electron localisation function (ELF) established by Becke and Edgecombe [28] to analyse the chemical bonding in molecules. The ELF bonding analysis has become a powerful tool for the organic chemist because it allows the establishment of a relationship between the quantum chemical electronic structure of a molecule with its Lewis structure, [29] thus describing the electronic structure in terms of lone pairs and located bonds within the valence bond theory (VBT) [30-32]. This scenario is very appealing for organic chemists since it permits an easy interpretation of molecular properties.

The characterization of electron pair rearrangements for describing the changes in the bonding scheme along the reaction pathway can be considered the most desirable way to analyse a reaction mechanism [33]. However, this description can only be obtained through a quantitative assessment of electronic pairing rather than intuitive or qualitative descriptions. Moreover, a robust mathematical treatment for the structural electronic changes is required when there is a change in the number or type of electron pairs. To fulfill these requirements, the bonding evolution theory (BET), consisting of the joint use of ELF and the Rene Thom`s catastrophe theory (CT) [34-36] was proposed by Krokidis et al. [37] as a new tool for the contemporary understanding of electronic rearrangements in chemical processes and applied to different elementary reactions [33].

In this way, in 2003 [38], the bonding changes along the onestep pathway of the prototype DA reaction between butadiene 1 and ethylene 2, and further in 2010 [39] in the DA reaction between Cp 4 and ethylene 2 and with styrene 5 were studied using BET. These studies allowed redrawing some significant conclusions about the bonding changes in these non-polar Diels-Alder (N-DA) reactions [19]. The most relevant conclusions can be summed up in the following four points:

i) ELF bonding analysis along the intrinsic reaction coordinate (IRC) associated with the one-step pathway of the N-DA reaction between butadiene 1 and ethylene 2 establishes that the IRC can be shared in seven phases (Figure 3) [38]. Each one of these phases is characterized by a bonding change in the Lewis structure with respect to the previous phase.

Figure 3: Reaction path calculated by means of the IRC method for the DA reaction between butadiene 1 and ethylene 2 [38]. A bonding between atoms in all phases is demonstrated by the standard Lewis representation; however, in the case of phases IV and V, ellipses reflect the non-bonding electron density concentrated in the C atoms. The four colours represent the four main groups in which the bonding changes can categorize.

ii) The formation of the C-C single bond in these N-DA reactions takes place through the C-to-C coupling between the pseudoradical centers generate through the breaking of the C=C double bonds (Figure 4) [39]. It is interesting to remark that this behaviour is also observed in P-DA reactions [11,40-42].

Figure 4: Most relevant ELF attractors in the structures of phase V, (a), and phase VI, (b), involved in the synchronous C-C single bond formation in the DA reaction between butadiene 1 and ethylene 2. The C-C distances are given in Armstrong.

iii) The seven phases, which are also observed in other cycloaddition reactions, can be categorized in four groups (Figure 3):

a) in the first one, A (in red colour), the three C=C double bonds present in butadiene 1 and ethylene 2 break, phases I to III;

b) in the second group, B (in green colour), the formation of two pseudodiradical structures [43] takes place by gathering electron density at the end carbons of the two unsaturated reagents, phases IV and V (see the four monosynaptic basins V(C), integrating ca 0.5e each, one in Figure 4a). The electron density demanded for the creation of the pseudoradical centers comes from the depopulation of the C=C double bonds of the butadiene and ethylene moieties;

c) in the third group, C (in blue colour), which is constituted only by the most relevant phase VI, the formation of the two new C-C single bonds takes place through the C-to-C coupling between the pseudoradical centers generated in the previous phase V (see the two disynaptic basins V (Cx,Cy), integrating ca 1.0e each, one, in Figure 4b) [39];

d) the fourth group, D (in violet colour), while the formation of the two C-C single bond is completed, the formation of the new C=C double bonds takes place at the end of the IRC. Consequently, BET analysis for the prototype DA reaction between butadiene 1 and ethylene 2 clearly shows that the bonding changes along the one-step pathway are non-concerted [39].

iv) Finally, these bonding changes take place symmetrically in the plane P1 that divides butadiene 1 and ethylene 2 into two equivalent fragments, but they take place in the opposite direction with respect plane P2, and, consequently, not in a cyclic fashion as was proposed in the definition of pericyclic reactions (Figure 5) [6].

Figure 5: Electron-density rearrangement for the prototype DA reaction between butadiene 1 and ethylene 2 by a) concerted cyclic bond formation, as proposed in pericyclic reactions; and b) symmetric, P1, and in the opposite direction, P2, bonding changes resulting from BET.

Moreover, ELF bonding studies carried out in polar reactions have shown that the C-C bond formation takes place also via a C-to-C coupling between two pseudoradical centers formed at the most electrophilic and nucleophilic sites of the reagents [11,40-42]. Interestingly, in all cases, non-polar and polar processes, the C-C single bond formation begins in a short region of the IRC located between 1.9 and 2.0 Å.

The synchronous TSs associated with the N-DA reactions between butadiene 1 and ethylene 2 and Cp 4 and ethylene 2, TS1 and TS2, are characterized by a C-C distance of ca 2.2 Å (Figures 2 and 3). At this point of the reaction, ELF indicates that the formation of the two new C-C single bonds has not begun, whereas the three C=C double bonds present in the reagents have already broken. Note that the C-C single bond formation in the N-DA reaction between butadiene 1 and ethylene 2 begins at C-C distance of 2.04 Å (Figure 4) The ELF electronic structures of TS1 and TS2 are given in Figure 6. This figure shows that the formation of the four pseudoradical centers responsible for the formation of the new C-C single bonds in phase VI, characterized by the presence of four monosynaptic basins V(C), has not begun, while the three C=C double bond, characterized in the reagents by the presence of two disynaptic basins V(C,C’) and V’(C,C’), have already broken.

Figure 6: Most relevant ELF attractors at the TSs of the DA reactions between
i) Butadiene 1 and ethylene 2, TS1,
ii) Cp 4 and ethylene 2, TS2, and
iii) Cp 2 and styrene 5, TS3.

These ELF analyses do not only allow to ascertain that the bonding changes in these synchronous processes are non-concerted, but also rule out the pericyclic mechanism since there is non bonding region in the space between the two reactant molecules in these synchronous TSs. Note that although the Natural Bond Orbital [44] (NBO) analysis yields a bond order [45] of ca 0.36 in the C-C forming bond region, ELF indicates that these data are not indicative of a bonding region.

On the other hand, the reaction model for the C-C bond formation in these N-DA reactions given by BET accounts for the asynchronicity found in the reaction between Cp 4 and styrene 5 (Figures 2 and 6) [39]. Analysis of the distance between the two carbons involved in the formation of the C-C single bond at the asynchronous TS3 indicates that it is more advanced; note that the shortest C-C distance at this TS is 2.02 Å. As a consequence, at this point of the IRC, the two monosynaptic basins V(C) and V(C’) have already been formed (Figure 6). Interestingly, as can be observed in this figure any monosynaptic basin appears at the substituted olefinic carbon of styrene 5 and at the symmetric carbon of Cp 4. The presence of the aromatic ring in ethylene produces a delocalisation of the electron density of the pseudoradical center formed at the substituted carbon of styrene 5, favouring the asynchronous bond formation of the two C-C single bonds, taking place in two different stages of the reaction. Note that in the Cp 4 framework, the corresponding pseudoradical is delocalised through the formation of allylic framework.

Analysis of the TSs associated with these DA reactions allows for the definition of two significant planes already mentioned in Figure 5, to describe the bonding changes in cycloaddition reactions (Figure 7); plane P1 divides the two reagents in two equivalent fragments that allows for the characterization of the synchronicity in the formation of the two C-C single bonds, while plane P2 divides the space in which the two new C-C single bonds will be formed along the cycloaddition. DA reactions involving symmetrically substituted reagents in plane P1, take place via synchronous TSs with identical d1 and d2 distances [38,39]. Any asymmetric substitution on butadiene or ethylene, such as styrene 5, breaks the symmetry upon formation of the two C-C single bond causing d1 ≠ d2, and thus yielding asynchronous TSs (Figure 2) [11,40-42].

Figure 7: Relevant P1 and P2 planes at the synchronous TS1 associated with the DA reaction between butadiene 1 and ethylene 2. While P1 defines the synchronicity, P2 defines the bonding region in the C-C single bond formation.

On the other hand, ELF bonding analysis of diverse organic reactions suggests that the formation of the new C-C single bonds takes place in the short distance comprised between 2.0 and 1.9 Å, topologically characterized by the creation of one disynaptic basin V(C,C’) associated with a boding region [11,38-42]. As previously commented, most synchronous TSs, including that for the N-DA reaction between butadiene 1 and ethylene 2, are characterized by a distance d > 2.1 Å (see TS1 and TS2 in Figures 1 and 2). Consequently, while the C=C double bonds presents in the diene and ethylene are already broken at these TSs, the formation of the two C-C single bond has not yet begun, a behavior that unequivocally establishes that these synchronous C-C single bond formation processes are non-concerted.

Another important feature derived from the ELF bonding analysis is that in these synchronous processes, the changes in electron density take place symmetrically with respect to plane P1, and not through a circle movement as proposed in the pericyclic model. Figure 5 shows that at the begin of the cycloaddition the electron density must flux from the region of the C=C double bonds of butadiene 1 and ethylene 2 towards the terminal carbons of the two reagents to gather the electron density required for the formation of the two pseudoradical centers, before the formation of the C-C single bond [38,39].

Why is the DA reaction between butadiene 1 and ethylene 2 nonconcerted? In the net-changes in bonding in the DA reaction between butadiene 1 and ethylene 2, two C-C single bonds are formed from two C=C double bonds. Within the Huckel bonding model [46], the two new σ bonds are formed at the end carbons of two π systems. BET indicates that the C−C bond formation takes place by coupling of two pseudoradical centers formed at the end carbons of the two π systems. Considering that the electron density of these pseudoradical centers comes from the depopulation of the electron density of the π bonds, the high energy required for the breaking of the three C=C double bonds is responsible for the high activation energy associated with these N-DA reactions [39]. In addition, as the formation of these pseudoradical centers demands the previous breaking of the π bonds, these processes can not be concerted with the C-C σ bond formation. Note that ELF analysis of the structures involved in the IRC of these and related organic reactions [11,40-42] indicates that the two disynaptic basins associated with the C=C double bonds do not coexist with the monosynaptic basins associated with the formation of pseudoradical centers.

This finding is supported by the very low activation energy found in the non-polar [3+2] cycloaddition (32CA) reactions of the simplest azomethine ylide 6 with ethylene 2 [47] and the 32CA reaction between carbonyl ylide 8 and tetramethylethylene 9 [48]: 1.2 and 4.7 kcal mol^-1, respectively (Scheme 4). The ELF bonding analyses for these 32CA reactions have shown that the pseudodiradical character of the threeatom- components (TACs) azomethine ylide 6 and carbonyl ylide 8 is responsible for the high reactivity of these species even with the nonactivated ethylene 2 and tetremethylethylene 8 [47,48] (see the Lewis structures of TACs 6 and 8 in Scheme 4). The earlier character of the corresponding synchronous TSs, d1 = d2 = 2.54 Å (6) and 2.66 Å (8), are in complete agreement with the very low activation energies found in these 32CA reactions [49] since TACs 6 and 8 do not require any drastic bonding changes to reach the corresponding pseudoradical centers, a behaviour that differs from the high energy required in the N-DA reaction between butadiene 1 and ethylene 2.

Scheme 4: Non-polar 32CA reactions of azomethine ylide 6 and carbonyl ylide 8.

In conclusion, the present study based on BET analyses of some N-DA reactions, including the prototype DA reaction between butadiene 1 and ethylene 2, highlights that the bonding changes in these cycloadditions, which involve the breaking and formation of several C-C double and single bonds, are non-concerted as a consequence of the electronic changes demanded for the formation of the new C-C single bonds.

Only in the few cases in which the two reagents are symmetric with respect to plane P1, the formation of the two C-C single bonds is synchronous. However, any asymmetric substitution in one of the two reagents breaks the synchronicity. These two significant behaviours, non-concerted bond breaking and bond formation together with a symmetric electron density reorganisation along plane P1 for the prototype DA reaction between butadiene 1 and ethylene 2 and related cycloaddition reactions, which are contrary to the electronic changes proposed by Woodward and Hoffmann [6] in pericyclic reactions, in which “all first order changing in bonding relationship take place in concert on a close curve”, refute the pericyclic mechanism proposed in all text books for this relevant organic reaction.

DFT calculations were carried out using the B3LYP [50,51] exchange-correlation functionals, together with the standar 6-31G* basis set [52]. The IRC [53] paths were 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 [54,55]. The electronic structures of IRC points along these one-step mechanisms were analyzed by the topological analysis of the ELF, η(r) [24-27]. The ELF study was performed with the TopMod program [56] using the corresponding monodeterminantal wavefunctions of the selected structures of the IRC. All calculations were carried out with the Gaussian 09 suite of programs [57].

Bonding changes along the Diels-Alder reaction between butadiene 1 and ethylene 2 and related N-DA reactions have been analysed using the BET. Changes in electron density instead of molecular orbitals are used to rationalise the reaction mechanism. The ELF analysis along these reactions indicates that C-C bond formation takes place by the C-to-C coupling of two pseudoradical centers formed along the reaction.

The present study permits the establishment of two significant findings:

i. the breaking of the C=C double bonds in butadiene 1 and ethylene 2 and the formation of the C-C single bonds in cycloadduct are non-concerted due to the changed in electron density required for the formation of the pseudoradical centers, and

ii. the symmetric changes in electron density along these cycloadditions do not have a cyclic movement.

These behaviours, which are opposite to the “concerted and close curve bonding changes” proposed by Woodward and Hoffmann [6] for the pericyclic reactions, allow refuting this mechanism for Diels-Alder reactions.

This work was supported by research funds provided by the University of Valencia (project UV-INV-AE13-139082).