In our first step, provided that we have established the difficulties of directly finding the value of ΔEd (C), we decided to explore the feasibility of calculating the value of ΔEi by other methods and, with ΔEd (R1) and ΔEd (R2) obtained by the same methodology previously described , then it should be possible to calculate indirectly ΔEd (C) from the sum of those three terms and the difference with respect to the total energy, ΔE. The most critical point of this approach is to find a way to accurately calculate ΔEi. In this regard, let us assume that this interaction energy can be decomposed into four different terms, ΔEi = ΔEi (R1–R2) + ΔEi (R1-C) + ΔEi (R2-C) + ΔEdisp, where the first three terms are associated to the pair-wise interactions and the last one to non-local, dispersion reorganizations. In the event of a chemical reaction between R1 and R2, where C is an inert spectator (in the sense that it is not chemically modified in any way), the first term should be clearly predominant and ΔEi ≈ ΔEi (R1–R2).
To corroborate this hypothesis, we took a closer look to our particular systems. In this case, the condition of C being an inert connector weakly interacting with the two reaction centers is clearly fulfilled, as it is as simple as an alkyl chain. In fact, the influence of C over the total interaction energy should be remarkably close to that of a methane molecule (see Fig. 4). Therefore, we decided to explore the intermolecular Diels–Alder reaction between 1-methoxy-1,3-pentadiene and 1-nitro-1-propylene in the presence of methane, given that, in this situation, all distortion and pair-wise interaction energies (see SI for the procedure for the calculation of these latter terms) can be calculated very easily as all reagents are separate entities. We found that the diene–dienophile interaction energy is not only the clear predominant term but recovers up to 99.2% of the total interaction energy (see SI for further information about energy contributions), thus allowing us to infer that ΔEi ≈ ΔEi (R1–R2) is an accurate approximation. Performing this study using the same DFT functional but with the augmented aug-cc-pVDZ basis set (using M06-2X/cc-pVDZ geometries), which should describe better long-range interactions, yielded a similar result (100.0%). Substitution of NO2 by CF3 led to a mild overestimation of the total interaction energy (105.0%), due to a larger, positive value of ΔEi (R1-C) (see SI). Before moving on, it is worth mentioning that this approximation has one noticeable flaw. If there exists a strong interaction between C and R1/R2 (or both), then the corresponding pair-wise terms will be no longer negligible, and the approximation fails. Indeed, substituting the methane molecule by 1,3-propanediol, and placing it in a position where it can form hydrogen bonds with both OMe and NO2 groups (see SI), led to the recovery of 83.3% of the total interaction energy by just considering ΔEi (R1-R2), which slightly increases to 84.7% with the aug-cc-pVDZ basis set.
With a method to obtain all ΔEd contributions and ΔEi, we calculated the reaction profiles for the Diels–Alder reaction that yields the structures shown in Fig. 3. Entries 1–8 vary in the strength of the EWG, entry 9 extends the π system of the dienophile to study the effect of the EWG at long distances, entries 10–11 increase the size of the connector to yield larger cycles, and entry 12 is a transannular reaction, a special type of intramolecular reactions in which the reaction centers are present at opposite places of a macrocycle, and the process leads to the formation of polycyclic scaffolds in a single step [37,38,39]. Entries 9–12 contain an aldehyde group as EWG as the best compromise between relative strength and size (in terms of number of atoms and electrons), offering a consistent electron-withdrawing effect with little computational effort. Thus, their results should be compared with entry 5 only. For the calculation of all energies, we took as reference the acyclic conformer which is directly connected to the TS of the Diels–Alder reaction, without regard of it being the most stable conformer or not. For all entries, we found that the reaction follows a concerted but asynchronous mechanism: the δ1-β2 bond (see Fig. 3) is formed in first place, quickly followed by the formation of the α1-α2 bond and the cyclohexene motif. This is in perfect agreement with the Lewis model, in which the OMe group can increase the electron density at the δ1 position of the diene while the EWG decreases it at the β2 position of the dienophile.
Table 1 shows the activation (ΔE≠) and reaction energies (ΔEreac) for entries 1–12, and the values of ΔEd for each component of the molecule, the total ΔEd, and ΔEi, calculated at the TS points taking as reference the acyclic structures (reagents). For entries 1–8, the value of σ− of the EWG is also provided. This value is a variation of the Hammett constant, more accurate to describe the effect of the functional group over the reactivity when it is conjugated with a negative charge/partial charge during the reaction pathway . For the calculation of ΔEd (R1), ΔEd (R2) and ΔEi, the connector was replaced by methyl groups (see Fig. 5 and SI), which are nearly identical to the connector (a –CH2– group has been replaced by a –CH3); thus, any possible X–X′ interaction that might arise for the calculation of ΔEi (see Fig. 2C) was present in the original molecule. This holds for the main difference with respect to Houk and coworkers’ approach . In their procedure to calculate ΔEd (C), they substituted a nitrone, one of the reaction centers, by a primary amine. While uncapable of evolving via 1,3-dipolar cycloaddition, it gives rise to a spurious amine-alkene interaction that contaminates the results of both ΔEd (C) and ΔEi. In our case, replacing the alkyl chain that forms the connector by methyl groups does not add new interactions and therefore can lead to more accurate results.
In all cases, the largest contributor to the total ΔEd is R1, and ΔEd (C) takes much smaller values. This is not surprising, given that R1 is larger than R2 and contains more bonds to be reorganized during the reaction, and that the connector only experiences a small compression and/or light conformational torsions. If we take a closer look at the values of ΔEd (C), its negative value for entry 9 can be explained with the IRC analysis, which is presented later on. Compared to entry 5 (EWG = CHO, formation of a 5-membered ring), entry 11 (formation of a 7-membered ring) shows a larger value of ΔEd (C), which can be related to a larger connector; on contrary, entry 10 (formation of a 6-membered ring) shows a lower value of ΔEd (C) despite having a larger connector than entry 5. This can be explained by the greater stability of the arising cyclohexane with respect to both cyclopentanes and cycloheptanes. Finally, ΔEd (C) for entry 12 (transannular) is 2.4 times greater than that of entry 5, in good agreement with the fact that it contains the same connector twice and therefore a value of ΔEd (C) twice of that with only one connector could be expected. However, given that the “total” ΔEd (C) is calculated, there is no systematic way with this procedure to obtain the individual distortion energies of each connector.
For entry 5 (EWG = CHO), we compared the results for ΔEd (C) and ΔEi obtained with our approach and confronted them to the results using Houk’s methodology . To do so, we calculated ΔEd (C) directly by substituting R1 and R2 for other functional groups. As inferred from these results, shown in Table 2, this procedure is highly reliant on the chosen replacements for the reaction centers, with ΔEd (C) values that oscillate from 1.05 to 31.40 kcal/mol. Entry 5b represents the closest example to the Houk and coworkers’ approach, R2 has been simplified by removal of the formyl moiety and R1 was replaced by a vinyl group, electronically similar to the original structure but unable to undergo Diels–Alder cycloaddition. The strong π-π interaction overestimates ΔEd (C), thus yielding a much larger value for ΔEi. Moreover, such large values of ΔEd (C) are not consistent with the mild deformations experienced by the connector from the initial structure to the TS. While entry 1b (X = X′ = H) produces a similar value of ΔEd (C) that the one obtained with our methodology, we cannot ensure that this is not a fortuitous coincidence and it will be observed for other reactions and/or connectors.
Entries 6b and 7b show the results obtained using Bickelhaupt’s approach . Given that this procedure involves a homolytic bond breaking in the connector to separate the reaction centers, the value of ΔEd (C) is not calculated explicitly but it is included in ΔEd (R1) and ΔEd (R2). Therefore, one may expect that these values, obtained with Bickelhaupt’s approach, should be slightly higher than the same values obtained with our approach, for which ΔEd (C) is considered separately. However, homolytic cleavage of Ca bond (see Fig. 3) leads to lower values of ΔEd (R1) and ΔEi, while ΔEd (R2) gives a nearly identical result. We propose that this arises from the conjugation of the generated CH2 radical with the diene moiety (revealed by analysis of the spin density, see SI), that perturbs the electronic structure of the π system compared to the original, non-disconnected molecule. On the other hand, Cb bond cleavage, for which the conjugation of the CH2 radical with the dienophile is less efficient and does not provoke a substantial perturbation of the electron density, leads to the expected trend for the values of the distortion energies. Moreover, all energy terms of entries 5 and 7b are in particularly good agreement. This suggests that, as long as an adequate bond is selected for its homolytic cleavage, both Bickelhaupt’s approach and ours yield identical results. While Bickelhaupt’s approach is less computationally demanding (requires a lower amount of calculations), our approach allows to quantitatively assign the effect of the connector over the activation barrier. It also provides additional evidence to confirm that the approximation we used to calculate ΔEi is reliable.
By taking entries 1–8, we can analyze the effect of the strength of the EWG on the activation, distortion and interaction energies. As can be seen from Fig. 6, there is a linear correlation (R2 = 0.91) between σ− and ΔE≠. Stronger EWGs modify the electron density of the dienophile in a more pronounced way, making it more susceptible to undergo normal electron-demand Diels–Alder reactions, decreasing ΔE≠. This relation between the strength of a functional group and its influence in the reaction rates is precisely the usefulness of Hammett constants, and it has been observed in previous studies of this kind [9, 22]. There also exists a linear correlation (R2 = 0.87) between σ− and ΔEi. Following the same reasoning, a stronger EWG should provide the dienophile a larger positive partial charge that, combined with the negative partial charge of the diene (provided by the OMe group), leads to greater values of ΔEi due to simple electrostatic interactions. On the other hand, ΔEd is less affected by the strength of the EWG (R2 = 0.56). As the main geometry deformations (pyramidalization of the α1, δ1, α2 and β2 positions, elongation of the α1-β1, γ1-δ1 and α2-β2 bonds, and shortening of the β1-γ1 bond) are a constant in all variations of the Diels–Alder reaction, then one should expect a weak dependence of these distortions with respect to the substitution, as we observed.
We conducted an in-depth study by analyzing the evolution of all these energy parameters along the reaction pathway. For this purpose, we calculated the IRC for all reactions and projected them onto the variation of the central CCC angle of the connector (see Fig. 3), which provides a description of how the connector must be distorted to bring the two reaction centers closer in space. Only for entries 10 and 11 was it necessary to project the IRC onto a different parameter, the C–C distance between the two methylene groups directly attached to R1 and R2 (see also Fig. 3). While there are many possible CCC angles in those structures, this parameter also provides a description of the connector compression during the course of the reaction. Figure 7 shows the results for entry 5 (EWG = CHO), all the remaining profiles can be found on the SI (IRC analysis using Houk’s and Bickelhaupt’s approaches for entry 5 can also be found on the SI). The early stage of the reaction is characterized by a positive ΔEi, even larger than the total ΔEd (Fig. 7, inset on left). This can be related with the increasing steric repulsions between the two reaction centers. On the other hand, the required ΔEd (C) is nearly energetically free (or even favorable), as it takes very low values (Fig. 7, right). We call this first stage (0° to − 5° for entry 5) the oncoming phase of the reaction, no relevant distortions take place in the reaction centers and steric repulsions predominate. After this stage, ΔEi quickly starts to decrease to take negative values and becomes the driving force. While the distortion energies increase at a higher rate than before, it is not enough to overcome ΔEi. This is also true for ΔEd (C), which also experiences a steady increase. We refer to this second stage as the conversion phase of the reaction (− 5° to − 9.5° for entry 5), where all the chemical transformations (bond breakings and formations) take place, and therefore the TS is located here. Finally, the reaction enters a stage in which the growths of ΔEi and all ΔEd are slowed down, and the reaction energy gets just a mild stabilization. This is the relaxation phase of the reaction, in which conformational rearrangements are carried out to reduce steric repulsions (− 9.5° to − 11° for entry 5). These rearrangements are less energetically demanding that any process of bond formation/breaking, which explains the energy growth reduction.
Interestingly, ΔEd (C) does not grow slowly in the relaxation phase, but in fact it starts to decrease. While this was quite subtle for some systems (like the one presented here, Fig. 7 right), it was more evident in others like entry 2 (EWG = CF3, see SI). To provide further insights into the behavior of ΔEd (C) at the conversion and relaxation phases, we performed a 100 fs BOMD calculation, so the different times at each event takes place can be assigned. Due to computational cost, this analysis was carried out exclusively for entry 5 (EWG = CHO). Free evolution (no additional excitation energy) from the TS structure toward the Diels–Alder adduct led to the formation of the first C–C bond at 18 fs , while the second C–C bond is formed at 37 fs, demonstrating the asynchronous mechanism. At this stage, the HCCH dihedral angle in the alkyl chain that forms the connector has been reduced with respect to the TS (see Fig. 8). In fact, it reaches its minimum value at 72 fs, which explains why ΔEd (C) keeps increasing after surpassing the TS, as shown in Fig. 7. Above 72 fs, the dihedral angle starts to increase and reaches its maximum value, 48.6°, at 93 fs, higher than that of the TS (39.9°). Adopting a more staggered conformation softens steric repulsions, leading to the reduction of ΔEd (C) previously characterized. Therefore, the BOMD simulation is in good agreement with the reaction profile extracted from the IRC. While we did not perform BOMD calculations for the rest of the systems, we corroborated that their relaxation stage, where ΔEd (C) starts to decrease, is associated with an increase of the HCCH dihedral angle to get a nearly staggered conformation.