Understanding the 1,3‐Dipolar Cycloadditions of Allenes

Abstract

We have quantum chemically studied the reactivity, site‐, and regioselectivity of the 1,3‐dipolar cycloaddition between methyl azide and various allenes, including the archetypal allene propadiene, heteroallenes, and cyclic allenes, by using density functional theory (DFT). The 1,3‐dipolar cycloaddition reactivity of linear (hetero)allenes decreases as the number of heteroatoms in the allene increases, and formation of the 1,5‐adduct is, in all cases, favored over the 1,4‐adduct. Both effects find their origin in the strength of the primary orbital interactions. The cycloaddition reactivity of cyclic allenes was also investigated, and the increased predistortion of allenes, that results upon cyclization, leads to systematically lower activation barriers not due to the expected variations in the strain energy, but instead from the differences in the interaction energy. The geometric predistortion of cyclic allenes enhances the reactivity compared to linear allenes through a unique mechanism that involves a smaller HOMO–LUMO gap, which manifests as more stabilizing orbital interactions.

All about allenes! Quantum chemical activation strain analyses reveal that the cycloaddition reactivity of linear allenes decreases as the number of heteroatoms is increased. Moreover, cyclic allenes experience a significant rate enhancement due to the stronger orbital interactions and not to the reduced activation strain, as previously reported.

Introduction

Allenes are a class of unsaturated hydrocarbon that contain two cumulated double bonds and have received significant attention in the past decade due to their privileged role in the synthesis of natural products through cycloaddition reactions.1 The simplest allene, propadiene (CCC), for instance, reacts with both cyclopentadiene and 1,3‐dipoles to form either a substituted norbornene2 or a heterocycle,3 respectively (Scheme 1 a and b), both of which are common motifs in natural products. Intramolecular Diels–Alder reactions4 as well as 1,3‐dipolar cycloadditions5 (Scheme 1 c and d) of allenes provide strategies for the construction of complex polycyclic molecules.6 In addition, the cycloaddition reactivity of allenes can be broadened to heteroallenes, such as ketenimine (CCN),7 ketene (CCO),8 carbodiimide (NCN),9 isocyanate (NCO),10 and even to carbon dioxide (OCO).11

Scheme 1.

Inter‐ and intramolecular Diels–Alder reactions and 1,3‐dipolar cycloadditions of propadiene.

In contrast, strained allenes, that is, cyclic allenes, have received less attention in the field likely due to their lower kinetic stabilities.12 Nevertheless, experimental studies have shown that strained allenes can be formed in situ and trapped instantaneously by either dienes or 1,3‐dipoles.13 For example, Houk and co‐workers studied the formation and subsequent trapping of 1,2‐cyclohexadiene in a Diels–Alder reaction (Scheme 2 a).13d Lofstrand et al. synthesized and subsequently trapped 1,2‐cyclohexadiene, through a 1,3‐dipolar cycloaddition under mild conditions (Scheme 2 b).13e Houk and Garg recently carried out a systematic study on the synthesis of azacyclic allenes as well as their reactivity towards cycloadditions (Scheme 2 c).13g These examples clearly illustrate that cyclic allenes can serve as prominent building blocks in the construction of polycyclic compounds and may also engage in rapid reactions in analogy with strained alkenes and alkynes.14

Scheme 2.

Diels–Alder reactions and 1,3‐dipolar cycloadditions of strained allenes.

A number of theoretical studies have shed light on the cycloaddition reactivity of allenes. A concerted asynchronous mechanism has been proposed to be a more energetically favorable reaction pathway for cycloadditions of allenes.13g, 15 Gandolfi and co‐workers proposed that the differences in the extent of structural deformations determine the trends in reaction barrier heights of the cycloadditions of allenes.15a, 15b On the contrary, an activation strain analysis on transition structures, by Garg and co‐workers, concluded just the opposite, namely, that the strength of the interaction plays a large role in determining the regioselectivity of the Diels–Alder reactions of azacyclic allenes.13g To the best of our knowledge, a thorough investigation into the reactivity, site‐, and regioselectivity of cycloadditions of allenes has not yet been reported.

We have performed a systematic computational study of the concerted 1,3‐dipolar cycloaddition reactions of allenes, including the linear (hetero)allenes propadiene (CCC; L3), ketenimine (CCN), ketene (CCO), carbodiimide (NCN), isocyanate (NCO), and carbon dioxide (OCO) and a series of cyclic allenes 1,2‐cyclooctadiene (C8), 1,2‐cycloheptadiene (C7), and 1,2‐cyclohexadiene (C6). These cyclic allenes have all been synthesized16 and might be relevant reactive dienophiles/dipolarophiles in bioorthogonal chemistries in the future.17 As azides are common reactants in 1,3‐dipolar cycloadditions,18 as well as strain‐promoted azide–alkyne cycloadditions (SPAACs),14a‐14c methyl azide (Az) was chosen as the model 1,3‐dipole in this study. The activation strain model (ASM)19 in combination with quantitative Kohn–Sham molecular orbital (KS‐MO) theory and the matching energy decomposition analysis (EDA)20 were employed to provide insight into the factor controlling the reactivity in these cycloaddition reactions. This approach has proven valuable for understanding of the reactivity of related pericyclic reactions and continues our current research line into the reactivity of cyclic dienophiles and dipolarophiles.21

Computational Methods

All calculations were carried out in ADF2017,22 using the BP8623 functional with the TZ2P basis set.24 The exchange‐correlation (XC) functional has been proven to be accurate in calculating the relative trends in activation and reaction energies for cycloadditions.25 Additionally, single‐point energies were computed at BP86‐D3(BJ)/TZ2P,26 M06‐2X/TZ2P,27 and COSMO(toluene)BP86/TZ2P28 on the fully optimized BP86/TZ2P geometries in order to assess the effect of a meta‐hybrid functional, dispersion‐corrections, and solvation on the computed reactivity trends. Frequency calculations were performed in order to characterize the nature of the stationary points. Local minima present only real frequencies, whereas transition state structures have one imaginary frequency. The potential energy surface (PES) was calculated using the intrinsic reaction coordinate (IRC) method, which follows the imaginary eigenvector of the transition structure towards the reactant and product. The resulting PES was analyzed with the aid of the PyFrag 2019 program.29 All chemical structures were illustrated using CYLview.30

Quantitative analyses of the activation barriers associated with the studied reactions are obtained by means of the activation strain model (ASM) of reactivity.19 Herein, the PES, ΔE(ζ), is decomposed into the strain energy, ΔE _strain(ζ), and the interaction energy, ΔE _int(ζ) [Eq. (1)]. All energy terms are projected onto the reaction coordinate ζ, the average distance of newly forming bonds, which undergoes a well‐defined change during the course of the reactions and has been proven to provide reliable results for cycloaddition reactions.21a, 25b, 25c, 31

$${\displaystyle \Delta E\left(\zeta \right)=\Delta E{}_{\mathrm{strain}}\left(\zeta \right)+\Delta E{}_{\mathrm{int}}\left(\zeta \right)}$$ (1)

The ΔE _strain(ζ) is associated with the rigidity as well as the structural deformation of the reactants from their equilibrium geometry to the geometry acquired along the reaction coordinate. The total ΔE _strain(ζ) can be further divided into the strain energy associated with deforming each respective reactant [Eq. (2)].

$${\displaystyle \Delta E{}_{\mathrm{strain}}\left(\zeta \right)=\Delta E{}_{\mathrm{strain},\mathrm{reactant},\mathrm{A}}\left(\zeta \right)+\Delta E{}_{\mathrm{strain},\mathrm{reactant},\mathrm{B}}\left(\zeta \right)}$$ (2)

The ΔE _int(ζ) is related to the electronic structure of the reactants and their spatial orientation and takes the mutual interaction between the deformed reactants into account. In order to obtain a deeper insight into the physical mechanism behind the interaction energy, we employ the canonical energy decomposition analysis (EDA).20 This analysis method decomposes the interaction energy between the two deformed reactants, within the framework of Kohn–Sham DFT, into three physically meaningful terms [Eq. (3)].

$${\displaystyle \Delta E{}_{\mathrm{int}}\left(\zeta \right)=\Delta V{}_{\mathrm{elstat}}\left(\zeta \right)+\Delta E{}_{\mathrm{Pauli}}\left(\zeta \right)+\Delta E{}_{\mathrm{oi}}\left(\zeta \right)}$$ (3)

The electrostatic interaction, ΔV _elstat(ζ), corresponds to the classical electrostatic interaction between the unperturbed charge distributions of the deformed reactants. The Pauli repulsion, ΔE _Pauli(ζ), comprises the repulsion between closed‐shell occupied orbitals and is, therefore, destabilizing. The orbital interaction, ΔE _oi(ζ), accounts for the stabilizing orbital interactions such as electron‐pair bonding, charge transfer (interaction between the occupied orbitals of fragment A with the unoccupied orbitals of fragment B, and vice versa), and polarization (e.g., occupied‐unoccupied orbital mixing on fragment A due to the presence of fragment B and vice versa). A detailed step‐by‐step protocol on how to perform the activation strain and energy decomposition analysis can be found in ref. 19a.

The magnitude of the orbital interaction of a 1,3‐dipolar cycloaddition mainly comes from two distinct orbital interaction mechanisms, namely, the normal electron demand (NED) interaction, occurring between occupied orbitals of the dipole and unoccupied orbitals of the dipolarophile (the allene in this study), and the inverse electron demand (IED) interaction, originating from the interaction between the unoccupied orbitals of the dipole with occupied orbitals of the dipolarophile. The stabilization of a specific orbital interaction mechanism is proportional to the orbital overlap squared divided by their respective orbital energy gap, that is, S ²/Δϵ.32 Thus, with the help of this relation, we can quantify the importance of the individual orbital interaction mechanisms.

The atomic charge distribution was analyzed by using the Voronoi deformation density (VDD) method.33 The VDD method partitions the space into so‐called Voronoi cells, which are non‐overlapping regions of space that are closer to nucleus A than to any other nucleus. The charge distribution is determined by taking a fictitious promolecule as reference point, in which the electron density is simply the superposition of the atomic densities. The change in density in the Voronoi cell when going from this promolecule to the final molecular density of the interacting system is associated with the VDD atomic charge Q. The VDD atomic charge Q _A of atom A is calculated according to Equation (4):

$${\displaystyle Q\genfrac{}{}{0pt}{}{\mathrm{VDD}}{\mathrm{A}}=-\underset{\mathrm{V}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{f}\mathrm{A}}{\int }[\rho \left(r\right)-{\rho }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{e}}(r\left)\right]\mathrm{d}r}$$ (4)

So, instead of computing the amount of charge contained in an atomic volume, we compute the flow of charge from one atom to the other upon formation of the molecule. The physical interpretation is therefore straightforward. A positive atomic charge Q _A corresponds to the loss of electrons, whereas a negative atomic charge Q _A is associated with the gain of electrons in the Voronoi cell of atom A.

Results and Discussion

1,3‐Dipolar cycloadditions of linear allenes

As a starting point, we studied the 1,3‐dipolar cycloaddition reaction between methyl azide (Az) and the following linear (hetero)allenes: propadiene (CCC), ketenimine (CCN), ketene (CCO), carbodiimide (NCN), isocyanic acid (NCO), and carbon dioxide (OCO). For each 1,3‐dipolar cycloaddition, two regiospecific cycloadducts can be formed, the 1,5‐adduct with the methyl group adjacent to the second double bond and the 1,4‐adduct with them on opposite sides (Scheme 3). Additionally, the asymmetric heteroallenes CCN, CCO, and NCO are able to form two site‐specific adducts, i.e., coordinating with either of the two double bonds.

Scheme 3.

The 1,3‐dipolar cycloaddition between Az and a linear allene.

Table 1 lists the activation energies, ΔE ^≠, and reaction energies, ΔE _rxn, for the studied 1,3‐dipolar cycloadditions between Az and linear (hetero)allenes. Three clear trends can be observed. In the first place, the cycloadditions towards the 1,5‐adducts are kinetically and thermodynamically favored over the formation of the 1,4‐adducts. Secondly, for the asymmetric heteroallenes, Az preferentially attacks at the more electropositive of the two terminal atoms. The only exception, however, is CCO, which has a slightly lower ΔE ^≠ for the attack at the CO (19.2 kcal mol⁻¹) than the CC (20.0 kcal mol⁻¹). Thirdly, the cycloaddition reactivity decreases when heteroatoms are introduced in the linear allene, from CCC to CCN and CCO, as well as from CCO to NCO to OCO and from CCN to NCN. The computed trends in reactivity at BP86/TZ2P agree well with those calculated at BP86‐D3(BJ)/TZ2P//BP86/TZ2P and M06‐2X/TZ2P//BP86/TZ2P. We note that, when using M06‐2X/TZ2P//BP86/TZ2P, the two site‐selective cycloadditions of NCO, forming the 1,5‐adduct, have nearly identical reaction barriers. Furthermore, the observed trends in reactivity, site‐, and regioselectivity also hold when solvent effects in toluene are included at COSMO(toluene)BP86/TZ2P//BP86/TZ2P (Table S1 in the Supporting Information).

Table 1.

Electronic reaction barriers ΔE ^≠ and reaction energies ΔE _rxn [kcal mol⁻¹] for the 1,3‐dipolar cycloadditions between Az and linear allenes leading to 1,4‐ and 1,5‐adducts computed at various levels of theory.^[d]

Allene	Site	1,5‐Adduct						1,4‐Adduct
		ΔE ≠[a]	ΔE ≠[b]	ΔE ≠[c]	ΔE rxn [a]	ΔE rxn [b]	ΔE rxn [c]	ΔE ≠[a]	ΔE ≠[b]	ΔE ≠[c]	ΔE rxn [a]	ΔE rxn [b]	ΔE rxn [c]
OCO		32.1	28.8	35.8	20.6	17.2	14.8	53.8	50.6	62.6	37.8	34.8	35.7
NCN		28.0	23.7	30.9	−15.3	−19.8	−25.2	35.8	31.6	43.5	1.6	−2.6	−6.6
NCO	CO	27.9	24.1	28.8	14.6	10.7	8.3	45.0	41.4	54.0	32.4	29.0	27.0
	NC	26.4	22.7	28.9	−14.9	−18.8	−24.0	42.8	39.0	50.8	6.6	2.9	−0.1
CCO	CO	19.2	15.0	21.7	3.5	−0.8	−2.6	27.1	23.4	40.2	16.5	12.9	11.5
	CC	20.0	15.7	26.2	−32.2	−36.4	−42.1	29.1	24.8	37.2	−24.5	−28.6	−32.2
CCN	CN	22.6	18.0	27.1	−20.3	−25.1	−29.1	26.4	22.2	34.3	−6.1	−10.4	−13.7
	CC	20.0	15.7	27.0	−30.8	−35.6	−41.3	23.5	19.0	30.8	−28.1	−32.6	−36.7
CCC		19.0	14.1	24.4	−34.0	−39.1	−43.7	19.5	14.8	25.2	−31.5	−36.2	−39.7

[a] Computed at BP86/TZ2P. [b] Computed at BP86‐D3(BJ)/TZ2P//BP86/TZ2P. [c] Computed at M06‐2X/TZ2P//BP86/TZ2P. [d] See Table S1 for computed enthalpies and Gibbs free energies.

1,5‐ versus 1,4‐regioselectivity

Next, we turn to the activation strain model (ASM)19 of reactivity to gain a quantitative insight into the physical factors governing the 1,5‐ versus 1,4‐regioselectivity in the 1,3‐dipolar cycloadditions presented herein. In Figure 1, we focus on the ASM diagram for the 1,5‐ vs. 1,4‐regioselectivity of OCO for which the difference in ΔE ^≠ is the largest (Table 1). The ASM diagrams of the other linear allenes possess the same characteristics, only less pronounced (Figures S1–S7). The lower ΔE ^≠ for the formation of the 1,5‐adduct originates mainly from a more stabilizing ΔE _int term, whereas the ΔE _strain is nearly identical (Figure 1 a). The canonical energy decomposition analysis (EDA)20 reveals that both the more stabilizing ΔV _elstat and ΔE _oi are the causes of the more favorable ΔE _int term for the 1,5‐adduct formation compared to the 1,4‐adduct (Figure 1 b).

Figure 1.

a) Activation strain and b) energy decomposition analysis of the 1,3‐dipolar cycloaddition between Az and OCO, projected onto the average newly forming C/O⋅⋅⋅N bond, computed at BP86/TZ2P.

The more stabilizing ΔE _oi for the 1,3‐dipolar cycloaddition yielding the 1,5‐adduct can be entirely described to the more effective orbital overlap of the normal electron demand (NED) interaction occurring between the HOMO−1_Az and LUMO_OCO. Only the lower‐lying HOMO−1_Az participates in the NED interaction, because its lobes are oriented towards the LUMO of OCO, while the lobes of HOMO_Az are orthogonal to the LUMO_OCO (Scheme S1). As shown in Scheme 4, the HOMO−1_Az has the largest lobe on the nitrogen next to the methyl group, due to a methyl‐induced mix of the π‐atomic orbitals of the N₃ fragment of Az (Scheme S1e). The LUMO_OCO has a larger lobe on the carbon atom than on the terminal oxygens, due to the more diffuse nature of the 2p atomic orbital of carbon compared to oxygen, which, in turn, leads to a better HOMO−1_Az–LUMO_OCO orbital overlap when forming the 1,5‐adduct compared to the 1,4‐adduct. The computed overlaps of the HOMO−1_Az–LUMO_OCO NED interaction for formation of both adducts on a consistent geometry with an average newly forming C/O⋅⋅⋅N bond length of 1.86 Å amounts S _1,5=0.30 and S _1,4=0.16. The larger orbital overlap for the formation of the 1,5‐adduct is responsible for the more stabilizing ΔE _oi compared to the 1,4‐adduct counterpart (Figure 1 b). The NED orbital energy gaps, on the other hand, are identical for the formation of the 1,5‐ and 1,4‐adduct, because the orbital interactions take place between the same molecular orbitals. In addition, the cycloaddition resulting in the 1,5‐adduct also has a stronger electrostatic attraction between the more negatively charged nitrogen and the positively charged carbon atom (Scheme 4) and, therefore, a significantly more stabilizing ΔV _elstat term (Figure 1 b).

Scheme 4.

Schematic diagrams of the orbital interaction between the HOMO−1 of Az and the LUMO of OCO for the 1,4‐ and 1,5‐adducts. VDD charges (red, in electrons) of key atoms in isolated fragment computed at BP86/TZ2P.

Site‐selectivity of asymmetric heteroallenes

After having established that the 1,3‐dipolar cycloaddition between Az and (hetero)allene preferentially form the 1,5‐adduct, we have analyzed the site‐selectivity of the asymmetric heteroallenes CCN, NCO, and CCO. First, we discuss the site‐selectivity of CCN by applying the ASM analysis. From Figure 2 a we can clearly see that the attack of Az at the more electropositive CC bond is favored exclusively due to a more stabilizing ΔE _int term compared to the attack at the CN bond. The more stabilizing ΔE _int for attack at CC compared to CN compensates for the destabilizing ΔE _strain for this pathway. Our EDA indicates that the more stabilizing ΔE _int term for the attack at CC over CN originates mainly from a more favorable ΔE _oi supported by a moderately stronger ΔV _elstat (Figure 2 b).

Figure 2.

a) Activation strain and b) energy decomposition analysis of site‐specific 1,3‐dipolar cycloadditions between Az and CCN, projected onto the average newly forming C/N⋅⋅⋅N bond, computed at BP86/TZ2P. The vertical dotted line indicates the point along the reaction coordinate at which the average length of newly forming C/N⋅⋅⋅N bond is 2.19 Å.

The more stabilizing ΔE _oi term for the Az attack at CC can exclusively be ascribed to its significantly more favorable inverse electron demand (IED) interaction term (Figure 3 a), which overcomes its less stabilizing NED interaction (Figure 3 b). The IED energy gap for the attack at CC is considerably smaller compared to the attack at CN, 1.7 and 5.0 eV, respectively, while the orbital overlap is also larger for the attack at CC. This manifest in an orbital stabilization term, that is,10³×S ²/Δϵ, of 17.0 and 4.0 for the attack at CC and CN, respectively. In contrast, the NED interaction is slightly weaker for the attack at CC than for CN, due to a larger NED energy gap and a poorer orbital overlap. This, however, can easily be overcome by the much stronger IED interaction, which leads to a more stabilizing ΔE _oi and thus a lower reaction barrier for the attack at the CC bond (Figure 2 b).

Figure 3.

FMO diagrams with calculated orbital energy gaps and overlaps of a) the IED (LUMO_Az–π‐MO_CCN) interaction and b) the NED (HOMO−1_Az–π*‐MO_CCN) interaction for site‐specific 1,3‐dipolar cycloadditions between Az and CCN at consistent geometries with the average newly forming C/N⋅⋅⋅N bond of 2.19 Å computed at BP86/TZ2P.

In the case of the asymmetric linear heteroallene NCO, the underlying mechanism behind the preference for the attack at NC over CO is identical to the above discussed CCN (Figure S8 a). For heteroallene CCO, the ΔE _strain for the attack at CC is more destabilizing than for CO, because the terminal carbon atom needs to deform from a trigonal planar to a tetrahedral geometry, which overcomes the more favorable ΔE _int, leading to nearly identical reaction barriers (Figure S8 b). But, the cycloaddition at CO is reversible and goes with a positive reaction energy (Table S1 for Gibbs free reaction energies), therefore, the reaction at CC will be preferred thermodynamically. The finding that allenes prefer to undergo cycloadditions at the more electropositive terminal atom is in line with several experimental reports.7, 8, 10

Influence of heteroatoms on the reactivity

In this section, we discuss the effect of heteroatoms on the reactivity of linear allenes towards the 1,3‐dipolar cycloaddition with Az yielding the 1,5‐adduct, by systematically modifying the nature and number of heteroatoms. Upon going from CCC to CCN and CCO, while attacking at the kinetically preferred CC site, the ΔE ^≠ increases from 19.0 to 20.0 kcal mol⁻¹ solely due to a more destabilizing ΔE _strain (Figure 4 a). Even though CCC requires a larger extent of bending over the course of the reaction compared to CCN and CCO (CCC: 24°; CCN: 18°; CCO: 22°), the difference in ΔE _strain can be ascribed to the more rigid heteroallene CCX (X=N, O) backbone, which can be reflected by the calculated bending vibrational frequencies of allenes (CCC: 361 cm⁻¹; CCN: 471 cm⁻¹; CCO: 503 cm⁻¹) as well as the analysis of the strain energy upon artificially bending of the heteroallene (Figure S9a). The increased rigidity from CCC to CCN to CCO is due to the increased bond strength between carbon and the heteroatom along the same series.34 The ΔE _int term, on the other hand, shows a trend which is opposite to the strain energy, namely, CCO goes with the most stabilizing interaction energy followed by CCN and CCC. This trend in interaction energy is exclusively determined by the orbital interactions (see Figure S9b for EDA diagrams), which, in turn, can be traced to a less stable LUMO_allene going from CCO to CCN and CCC (−2.4, −1.5, and −0.9 eV, respectively), and, therefore, a larger HOMO−1_Az–LUMO_allene energy gap. This trend in ΔE _int term, however, is overruled by the larger differences in ΔE _strain.

Figure 4.

a) Activation strain analysis for the 1,3‐dipolar cycloadditions between Az and CCC, CCN, and CCO attacking at the CC site and b) equilibrium geometries of the allene and respective consistent geometries with the average newly forming C/N⋅⋅⋅N bond of 2.20 Å with internal bending angles [°] computed at BP86/TZ2P.

For the series CCO, NCO, and OCO, Az attacks at different sites, namely, CC, NC, and OC, respectively, but the neighboring heteroatom is always oxygen. Along this series, the ΔE ^≠ systematically increases (Table 1), due to a less stabilizing ΔE _oi (Figure S10). Similar to the analysis of the site‐selectivity of the asymmetric heteroallenes (Figure 3), going from CC to the hetero double bonds NC and OC causes a remarkably destabilized IED interaction (LUMO_Az–HOMO_allene), due to the increased IED orbital energy gap supported by less efficient orbital overlap (Figure S11). This exact rationale also holds for the comparison of CCN and NCN (Figure S12).

Summarizing, we have analyzed and compared the 1,3‐dipolar cycloaddition reactivity of linear (hetero)allenes with Az, which is all cases prefers to form the 1,5‐adduct. The archetypal allene, CCC, is the most reactive. By introducing a heteroatom, the heteroallene becomes less reactive due to the increased rigidity of the CCX (X=N, O) backbone. Additionally, a second heteroatom diminishes the stabilizing ΔE _oi, making them even less reactive towards Az.

1,3‐Dipolar cycloaddition of cyclic allenes

At last, we also analyzed and compared the 1,3‐dipolar cycloaddition reactions between methyl azide (Az) and a series of cyclic allenes, namely, 1,2‐cyclooctadiene (C8), 1,2‐cycloheptadiene (C7), and 1,2‐cyclohexadiene (C6) as well as propadiene (L3), the most reactive linear allene (vide supra). These cyclic allenes have all been synthesized and featured in cycloaddition reactions.13

Figure 5 shows the transition state structures of the 1,3‐dipolar cycloadditions of Az with the linear allene (Az‐L3) and the cyclic allenes (Az‐C8–Az‐C6). The transition structures are concerted asynchronous and become earlier, with regard to the average forming bond distances, as the ring size of the cyclic allene decreases. The cycloaddition of the linear L3 is predicted to proceed with the highest reaction barrier (ΔE ^≠=19.0 kcal mol⁻¹) and has the least favorable reaction energy (ΔE _rxn=−34.0 kcal mol⁻¹). The reaction barrier height decreases along the series L3>C8>C7>C6, and the cycloaddition reaction becomes more exergonic when going from L3 to C6, which is in line with the Hammond–Leffler postulate35 (Figure S13). The computed trends at BP86/TZ2P agree well with those calculated at BP86‐D3(BJ)/TZ2P//BP86/TZ2P and M06‐2X/TZ2P//BP86/TZ2P, as well as when solvent effects are included at COSMO(toluene)BP86/TZ2P//BP86/TZ2P (Table S2).

Figure 5.

Transition structures with forming bond lengths [Å], computed reaction barriers (ΔE ^≠ [kcal mol⁻¹], blue) with relative reaction rate constants (k _rel, black), and reaction energies (ΔE _rxn [kcal mol⁻¹], red) for 1,3‐dipolar cycloadditions of Az with L3 and C8–C6 computed at BP86/TZ2P.

In order to understand the intrinsic differences in reactivity between linear and cyclic allenes in the 1,3‐dipolar cycloaddition with Az, we performed an ASM analysis. Figure 6 a graphically represents how the ΔE _strain and ΔE _int components evolve along the reaction coordinate for 1,3‐dipolar cycloadditions of Az with L3 and C8–C6. Surprisingly, the origin of the increased reactivity as the ring size of allene decreases can be entirely attributed to the differences in ΔE _int, which becomes more stabilizing from L3 to C6 (Figure 6 a). The total ΔE _strain for all studied allenes are nearly identical (Figure 6 a). As expected upon decreasing the size of the ring, the cyclic allene becomes more predistorted towards the cycloaddition reaction with Az, which leads to a smaller contribution of the cyclic allene to the total ΔE _strain, consistent with the earlier literature.13f The contribution of the 1,3‐dipole Az to the total ΔE _strain, however, is more destabilizing for C6 than for L3 (Figure S14), due to the fact that the more reactive allenes (vide infra) deform Az to a larger degree (Figure S15).

Figure 6.

a) Activation strain and b) energy decomposition analysis of 1,3‐dipolar cycloadditions of Az with L3 and C8–C6, projected onto the average newly forming C⋅⋅⋅N bond, computed at BP86/TZ2P. The vertical dotted line indicates the point along the reaction coordinate where the average newly forming C⋅⋅⋅N bond is 2.48 Å.

The origin of the differences in ΔE _int was uncovered by means of the EDA method, and the results are shown in Figure 6 b. It is apparent that the ΔE _oi is the major contributor to the trend in ΔE _int, guided by a smaller contribution of ΔV _elstat. The ΔE _Pauli shows a reverse trend, and, therefore, is not responsible for the trend in ΔE _int. To further probe the key orbital interactions, that cause this difference in ΔE _oi, involved in the 1,3‐dipolar cycloadditions of Az with L3 and C8–C6, we analyzed the FMOs participating in these interactions on consistent geometries with an average newly forming C⋅⋅⋅N bond of 2.48 Å (Figure 7).

Figure 7.

FMO diagrams with calculated key orbital energy gaps and overlaps of a) the NED (HOMO−1_Az–LUMO_allene) interaction and b) the IED (LUMO_Az–HOMO_allene) interaction for 1,3‐dipolar cycloadditions between Az with L3 and C8–C6 at consistent geometries with the average newly forming C⋅⋅⋅N bond of 2.48 Å computed at BP86/TZ2P.

The FMOs participating in the NED and IED reveal that the more stabilizing orbital interactions when going from L3 to C8 to C6 are exclusively determined by a reduction in orbital energy gap (Figure 7). The NED interaction between Az and L3 and C8–C6 occurs between the HOMO−1_Az and LUMO_allene (Figure 7 a). The least reactive allene L3 has the largest and least favorable NED orbital energy gap (Δϵ=6.4 eV). As the ring size decreases from L3 to C8 to C6, the NED orbital energy gap continuously decreases from 6.4 to 4.8 eV. The orbital overlap in the NED interaction are identical for all reactions (S=0.15). The IED interaction takes place between the LUMO_Az and HOMO_allene (Figure 7 b). Again, L3 has the largest and, therefore, least favorable IED orbital energy gap (Δϵ=3.7 eV). The IED gap also systematically decreases from 3.7 eV for L3 to 2.7 eV for C6. The increasingly stabilizing ΔE _oi term (Figure 6 b), as the ring size of allene decreases, therefore, is a direct result of the diminishing energy gap for both the NED and IED interaction, resulted from the continuously stabilizing LUMO and destabilizing HOMO of allene (Figures 7 and S16).

In order to quantify the effect of allene predistortion on the HOMO and LUMO, we chose to bent our model system L3. Figure 8 a shows the optimized undistorted structure (top) and the distorted, bent, structures of L3 (middle and bottom). Bending of the allene backbone causes a loss in orthogonality of the two adjacent π systems, because it is accompanied with a twist in the structure, reducing the dihedral angle from 90°, for the linear allene, to 64.1°, for the 130° bent allene. This observation not only holds for L3, but also for the cyclic allenes C6–C8 (Figure S17) and is in line with earlier reported literature.13g As the backbone of L3 becomes distorted, the LUMO is stabilized while the HOMO is destabilized (Figure 8 b).

Figure 8.

a) Front and right‐side views of the pristine and constrained optimized structures of L3. b) FMO energies associated with the internal angle computed at BP86/TZ2P.

Detailed Kohn–Sham molecular orbital (KS‐MO) analysis of the formation of the HOMO and LUMO of the undistorted (linear) and distorted (bent to 130°) H₂C=C=CH₂ (L3), in terms of an H₂C=C^.. and a ^..CH₂ fragment, is shown in Figure 9. For the archetypal L3 (Figure 9 a), one LUMO (the bold LUMOs in Figure 9) is solely formed by the π* orbital of H₂C=C^.., whereas the other degenerate LUMO, which is orthogonal to the former, is a result of the antibonding combination of the p orbitals of two individual fragments. Furthermore, the HOMO (the bold HOMOs in Figure 9) originates from the antibonding combination between the π orbital of H₂C=C^.. and the C−H σ orbital of ^..CH₂, meanwhile the other degenerate HOMO is the bonding combination of the p orbitals of both fragments. When L3 is bent to 130° (Figure 9 b), the π* orbital of H₂C=C^.. has an in‐phase overlap with the σ* orbital of ^..CH₂ which leads to a stabilization of the LUMO. In addition, due to the prior mentioned twisting effect, the fragment p orbitals mix into the LUMO which results in the additional stabilization. The HOMO, on the other hand, is stabilized due to the decreased anti‐bonding π–σ overlap owing to the bending and twisting of the backbone, but, at the same time, obtains a slightly stronger destabilization from the mixing of the fragmental p orbitals. This destabilization effect overcomes the stabilizing counterpart, resulting in the overall destabilization of the HOMO.

Figure 9.

Diagrams for the Kohn–Sham MO analyses of a) the archetypal and b) 130° bent L3, where the fragments are H₂C=C^.. and ^..CH₂ computed at BP86/TZ2P.

These analyses were further verified by investigating both the pure bending (no twisting) or twisting (no bending) of L3. Solely bending L3 and maintaining orthogonality of the structure stabilizes the LUMO due to the enhanced π*–σ* overlap and also stabilizes the HOMO because of the decreased π–σ overlap (Figure S18). On the other hand, solely twisting L3 and maintaining a linear backbone induces a stabilization of the LUMO, because of an in‐phase mixing between the π* and p orbitals of the fragments, and a significantly destabilization of the HOMO, due to the mixing between the π and p orbitals (Figure S19).

Conclusions

1,3‐Dipolar cycloadditions of linear allenes and heteroallenes with methyl azide (Az) favor the formation of the 1,5‐adduct over the 1,4‐adduct. In addition, bond formation to the asymmetric heteroallene is preferred at the more electropositive terminal atom. This process becomes less reactive as the number of heteroatoms in the allene increases. Cyclic allenes experience a significant rate enhancement compared to their linear allene counterparts. These findings emerge from our quantum chemical study based on density functional theory calculations.

Our activation strain analyses furthermore identified that the site‐selective preference for the 1,5‐adduct compared to the 1,4‐adduct is exclusively determined by a more favorable orbital overlap and thus more stabilizing orbital interactions between the reactants. Furthermore, in the case of the asymmetric heteroallenes, the preference for attacking at the more electropositive atoms is caused by a significantly stronger inverse electron demand (IED) orbital interaction. This is due to the fact that double bonds involving more electropositive atoms have lower‐lying acceptor orbitals, leading to smaller IED energy gaps and, thus, more stabilizing orbital interactions with Az. The archetypal allene, propadiene (CCC) was found to be the most reactive linear allene. Introducing a heteroatom to CCC makes the allene less reactive, due to a more destabilizing ΔE _strain, originating from a more rigid backbone, as well as less stabilizing orbital interactions.

The enhanced reactivity of cyclic allenes with respect to linear ones originates from an enhancement of donor–acceptor orbital interactions, which become more stabilizing as the ring size of the cyclic allene decreases, and not from a previously reported reduced activation strain. Our activation strain analyses reveal that, in smaller rings, the allene moiety is more bent; this goes with a smaller HOMO–LUMO gap in the π‐electron system and, hence, with the aforementioned stabilization of the transition state by stronger donor–acceptor orbital interactions.

Conflict of interest

The authors declare no conflict of interest.

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Supplementary

S. Yu, P. Vermeeren, K. van Dommelen, F. M. Bickelhaupt, T. A. Hamlin, Chem. Eur. J. 2020, 26, 11529.