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Published in final edited form as: Nature. 2025 Feb 12;640(8059):683–690. doi: 10.1038/s41586-025-08745-1

σ-Bond insertion reactions of two strained diradicaloids

Arismel Tena Meza 1, Christina A Rivera 1,2, Huiling Shao 1,2, Andrew V Kelleghan 1, K N Houk 1,✉, Neil K Garg 1,✉
PMCID: PMC12457010  NIHMSID: NIHMS2111289  PMID: 39938569

Abstract

The development of new synthetic methodologies is instrumental for enabling the discovery of new medicines. The methods that provide efficient access to structural alternatives for aromatic compounds (that is, saturated arene bioisosteres) have become highly coveted1–4. The incorporation of these bioisosteres typically leads to favourable drug-like properties and represents an emerging field of research. Here we report a new synthetic method that furnishes a coveted motif, the bicyclo[2.1.1]hexane scaffold5,6, using mild reaction conditions and an operationally simple protocol. The methodology proceeds through the uncommon coupling of two strained fragments: transiently generated cyclic allenes and bicyclo[1.1.0]butanes, which possess considerable strain energies of about 30 kcal mol−1 (ref. 7) and about 60 kcal mol−1 (ref. 6), respectively. The reaction is thought to proceed by a σ-bond insertion through a diradical pathway. However, rather than requiring an external stimulus to generate radical species, reactivity is thought to arise as a result of innate diradical character present in each reactant. This diradicaloid character8, an underused parameter in reaction design, arises from the severe geometric distortions of each reactant. Our studies provide a means to access functionalized bicyclo[2.1.1]hexanes of value for drug discovery, underscore how geometric distortion of reactants can be used to enable uncommon modes of reactivity and should encourage the further exploration and strategic use of diradicaloids in chemical synthesis.

The presence of aromatic rings (1; Fig. 1a) has become dominant in medicinally relevant compounds over the past several decades. This is largely attributed to the prevalence of metal-catalysed cross-coupling reactions in contemporary synthetic chemistry, in which functionalized aromatics can be easily prepared by the union of two fragments (for example, 2 and 3) (refs. 9,10). However, the products from these couplings largely possess unsaturated frameworks, which often have undesirable drug-like properties1. As a result, there has been a recent movement to discover methods that allow for the efficient assembly of rigid, three-dimensional structural motifs that may serve as arene bioisosteres with favourable drug-like properties1–4.

Fig. 1 |. Background and overview.

Fig. 1 |

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a, Aromatics 1 commonly accessed by conventional metal-catalysed cross-coupling reactions. b, BCHs as bioisosteres of aromatic compounds and present study involving the coupling of two highly strained intermediates, cyclic allenes 9 and BCBs 10, to give BCHs 12 by σ-bond insertion. Both 9 and 10 possess diradical character and are considered diradicaloids, which allow for the reaction to take place without an external stimulus, such as light, high temperature, radical initiator, or one-electron oxidant or reductant. [M], transition metal; Y, halide or pseudohalide; Me, methyl; Oi-Bu, isobutyl group; X, O, CH2, N–Boc (tert-butyloxycarbonyl), or N–Ts, 4-methylbenzenesulfonyl; and R1, alkyl or ester group.

A specific rigid structural motif that has become highly coveted in medicinal chemistry is the bicyclo[2.1.1]hexane (BCH) (4; Fig. 1b) that can mimic para-, meta-, or ortho-substituted aromatics (for example, 5–7, respectively) depending on specific substitution patterns. A representative compound bearing the [2.1.1] BCH motif with 1,4-disubstitution is the anti-inflammatory compound 8 (ref. 11). The most common approach to synthesizing [2.1.1] BCHs involves reactions of bicyclo[1.1.0]butanes (BCBs)5,6 with unstrained olefins, with many elegant advances being reported recently, a sampling of which we cite here12–22. Thermal approaches have been shown to require high temperatures, with limited examples being available23. Recent efforts involve the use of Lewis acid mediation (with appropriate coordinating functional groups)14,15,20. Other breakthroughs involve the intentional generation of radical intermediates12,22, including reactions that use photochemical activation, as shown in refs. 13,16–18,21,24.

To complement these pioneering studies, we questioned whether the chemistry of transient strained intermediates could provide an entryway to access highly functionalized [2.1.1] BCH scaffolds. Our anticipated reaction would involve the uncommon coupling of two types of highly strained molecules, namely, in situ generated strained cyclic allenes (about 30 kcal mol−1 strain energy) 9 (ref. 7) and BCBs (about 60 kcal mol−1 strain energy) 10 (ref. 6) (Fig. 1b). The two would be merged in a bimolecular coupling reaction to afford [2.1.1] BCH products 12 through a C–C σ-bond insertion. As discussed later, the geometric distortion (Fig. 2a) leads to some innate diradical-like character8 in each reactant, ultimately allowing for this direct insertion reaction to take place without an external stimulus (see transition structure 11). The strained structures are diradicaloids, a concept that we discuss later.

Fig. 2 |. Geometric distortions and reaction parameters.

Fig. 2 |

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a, Geometric distortions of strained cyclic allene 14 (for example, bending, twisting and pyramidalization); geometry-optimized structures calculated using DFT (ωB97X-D/def2SVP). b, Structure and properties of BCBs calculated using DFT (ωB97X-D/def2SVP). c, Survey of conditions for the reaction between the two strained compounds: cyclic allene 14 and BCB 18. aConditions: allene precursor 17 (1 equiv.), 18 (3 equiv.), Bu4NF (2 equiv.) or CsF (5 equiv.), additive (if applicable, 1 equiv.), 16 h and sealed vessel. bYields were determined by 1H NMR analysis using 1,3,5-trimethoxybenzene as an external standard. Et, ethyl; OTf, trifluoromethanesulfonate; Me, methyl; Bu4NF, tetrabutylammonium fluoride; THF, tetrahydrofuran; DME, 1,2-dimethoxyethane; Bu, butyl; and M, molarity.

Strained cyclic allenes (9; Fig. 1b), known since the 1960s (refs. 25,26), and other related transient species7,27–29 have become valuable synthetic intermediates as they (1) can be harnessed to achieve challenging fragment couplings through the formation of multiple bonds; (2) are typically used under operationally simple and mild fluoride-based reaction conditions; and (3) have been used in the synthesis of structurally complex carbocycles and heterocycles27,30–35, DNA-encoded library synthesis36 and natural product synthesis37. Although most known cyclic allene trappings involve engagement of the allene with reactants that contain π-bonds or lone pairs of electrons (that is, cycloadditions with olefins, 1,3-dipoles or dienes), reactions in which cyclic allenes engage with C–C σ-bonds are unknown. Moreover, direct σ-bond insertion reactions of BCBs and strained alkenes, such as those present in cyclic allenes, have not been reported.

Several issues render our predicted transformation challenging, but also provide an opportunity for reaction discovery and mechanistic understanding. Bringing together two highly strained intermediates that could otherwise be prone to dimerization or decomposition is kinetically challenging. For example, strained cyclic allenes are well-known to readily undergo dimerization7, whereas BCBs have limited shelf-life5,6. Cyclic allenes are typically formed in low concentration in the presence of a stable reactant to enable productive reactions25. No examples of strained cyclic allenes undergoing controlled reaction with other highly strained compounds are available. Furthermore, previous attempts to trap BCBs with transient strained intermediates such as benzyne and strained cyclopropenes predominantly lead to Alder-ene reactions rather than σ-bond insertion38–40.

We now describe the σ-bond insertion reaction between strained cyclic allenes and BCBs. We show that reactions proceed efficiently and regioselectively at ambient temperatures to deliver highly substituted BCH products. Each strained reactant is a diradicaloid8 and we suggest that this property, which is underused in chemical synthesis, is important for enabling the complexity-generating reaction that we have discovered.

Geometric distortion of reactants

At the outset of our studies, we considered the ground state structure of each reactant. Figure 2a provides an analysis of cyclic allene 14 in comparison to a simple linear allene 13. The constraint of the linear allene unit into the six-membered ring leads to a strain energy of about 30 kcal mol−1 (ref. 29). The geometry-optimized structure of 14 was obtained using density functional theory (DFT) (ωB97X-D/def2SVP) to examine three different types of geometric distortion that are present. Cyclic allenes are most commonly associated with bending; more specifically, whereas the C=C=C bond angle in linear allene 13 is 180°, the corresponding angle in cyclic allene 14 is compressed to 133°. The other forms of geometric distortion that are present, but not commonly discussed, are twisting and pyramidalization associated with the individual alkenes41,42. In a typical alkene (that is, unstrained), cis substituents on opposite termini of the alkene are coplanar, with a dihedral angle of 0°. In cyclic allene 14, each alkene is substantially twisted; for example, the C4′ = C3′ twist angle (that is, the C5′–C4′ = C3′ = C2′ dihedral angle) is 21° as shown. Pyramidalization is a third type of distortion; the geometry typically associated with an alkene terminus is trigonal planar, with the individual angles of about 120° between substituents on each terminus. However, in the case of cyclic allene 14, these angles are distorted to 109° and 113° at C4′ and C2′, respectively. C4′ and C2′ exhibit pyramidalization, as the CHs bend out of plane to increase the π overlap. The geometries deviate from trigonal planarity by 5.5° (C4′) and 4.0° (C2′), also known as the pyramidalization angles (Φp) (ref. 43). Having these three types of geometric distortion present in a compound is rare and ultimately renders strained cyclic allenes highly reactive.

The strain and geometric distortion present in BCB 16 are shown in Fig. 2b (refs. 5,6). Restricting a saturated framework into the small, rigid ring system of the BCB leads to a high strain energy of about 64 kcal mol−1. Although cyclic allenes have a lower strain energy compared with BCBs, they are transient species, whereas BCBs are typically isolable. As such, strain energy should be used as a guide, but not necessarily an indicator of stability44. Nonetheless, the primary geometric distortion seen in the BCB is angle bending at the carbons of the highly strained central bond, as shown by the difference in angles seen in linear alkane 15 and BCB 16. Unstrained alkanes are tetrahedral, with angles about 109.5°, but BCB 16 has six C–C–C angles of about 60°, typical of cyclopropanes. Complementarily, the external angles are enlarged, as shown by the angles of 130° and 129° for H–C1–C4 and H–C1–C3, respectively. The central C1–C3 bond is 1.5 Å in length and possesses 96% p-character6. The strain energy is about 10 kcal mol−1 greater than that of two cyclopropanes.

Experimental validation

Although intermolecular coupling reactions between two highly strained intermediates are not common, we were intrigued by the possibility of achieving a productive transformation between cyclic allenes and BCBs. As noted earlier, cyclic allenes have been used in intermolecular reactions that invoke π-systems or atoms bearing lone pairs of electrons, but not directly with σ-bonds or other strained compounds. Figure 2c shows key parameters and results observed for the reaction using cyclic allene precursor 17 and BCB 18 to give BCH 19. As shown in entries 1 and 2, the reaction proceeds in modest yield using either CsF or Bu4NF as the fluoride source using acetonitrile as solvent at 23 °C. However, a notable amount of allene dimerization product is also detected under these and other conditions (up to about 30%). Given that the poor solubility of CsF generally leads to slower generation of strained intermediates and consequently more controlled reactions, we largely focused our attention on CsF-mediated reactions. We observed that the use of higher temperatures leads to increased yields and several solvents could be used for the transformation (entries 3–6). At 80 °C, the highest yield of 93% is obtained when 1,2-dimethoxyethane (DME) is used as solvent (entry 6). Given that some BCBs are thermally unstable, we also evaluated the use of tetrabutylammonium salt additives at ambient temperature, as these salts can be used to improve the solubility of CsF in organic solvents45. The use of Bu4NBr at 23 °C was most successful (entry 8). Finally, we found that concentration has a notable effect; simply doubling the concentration leads to the formation of 19 in 98% yield (entry 9). Our optimal reaction conditions for BCH formation are fairly mild and involve treatment of 17 and 18 with CsF at 23 °C, with Bu4NBr as additive, in DME as solvent. Of note, the BCH product shown, 19, is formed as the main constitutional isomer (>20:1 regioselectivity (r.r.)); the origins of selectivity parameters are discussed later.

Scope of the σ-bond insertion reaction

As shown in Fig. 3, both mono- and disubstituted BCBs 10 can be used in the σ-bond insertion reaction using cyclic allene 14. Reactions are synthetically useful and uniformly proceed with high regioselectivities (>20:1 r.r.) to form polysubstituted BCH products 20. With regard to mono-substituted BCBs, aryl ketones are suitable substrates, giving rise to BCH products in 79–94% yield (entries 1–3). The methodology also shows promise for the use of extended aromatic groups and heteroaryl substituents (entries 4 and 5, respectively). However, the fragment coupling is not restricted to aryl ketones. Alkyl ketone 31 and Weinreb amide 33 can be used, giving rise to BCHs 32 and 34 (entries 6 and 7, respectively). An aryl-sulfone-substituted BCB (35) was also tested, ultimately leading to sulfone 36 in 82% yield (entry 8). The use of disubstituted BCBs in this methodology provides an effective means to access tetrasubstituted BCHs. For example, by using BCB 37, which contains a t-butyl ester and phenyl group as substituents, BCH 38 is accessible in 88% yield (entry 9). Both the ester group and the aryl group can be varied, as shown by the results shown in entries 10–12. The use of BCB 42 is notable, as it allows for the introduction of the medicinally privileged pyridine ring present in BCH 43 (entry 12). Finally, use of morpholine amide-containing disubstituted BCB 44 furnishes BCH 45 in 96% yield (entry 13).

Fig. 3 |. Scope of trapping reactions with carbocyclic allene 14 and BCBs 10.

Fig. 3 |

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aConditions: 17 (1 equiv.), trapping agent (2.7–4.0 equiv.), CsF (5 equiv.), Bu4NBr (1 equiv.), DME (0.2 M), 23 °C, 16 h and sealed vessel. Observed regioselectivities are reported as a ratio of isomers (r.r.). bConditions: CsF (9 equiv.). cConditions: 33 (6 equiv.) and CsF (9 equiv.). dConditions: Bu4NBr omitted and CsF (9 equiv.). eConditions: Bu4NBr omitted and 80 °C. Et, ethyl; OTf, trifluoromethanesulfonate; DME, 1,2-dimethoxyethane; Bu, butyl; Ph, phenyl; and Me, methyl.

The cyclic allene fragment used in the σ-bond insertion can also be varied. As shown in Fig. 4, substituted carbocyclic allenes and heterocyclic allenes serve as building blocks to access highly functionalized BCHs. Reactions proceed with high regioselectivity with regard to the disubstituted BCB fragment (>20:1 r.r.) and, in most cases, with regard to the strained cyclic allene as well. Interception of carbocyclic allenes 48 or 51 gives BCHs 49 and 52 (entries 1 and 2, respectively). In both cases, the transformation favours the reaction of the alkene positioned distal to the substituent in the cyclic allene fragment, presumably because of steric factors. With regard to heterocyclic allenes, oxygen- and nitrogen-containing cyclic allenes undergo the σ-bond insertion with BCBs (entries 3–7). For example, products 55 and 58 (entries 3 and 4) highlight the use of oxacyclic allenes in the methodology, whereas cycloadducts 61 and 64 (entries 5 and 6) showcase the use of azacyclic allenes. We also examined the generation and trapping of unsymmetrical azacyclic allene 66, the isomer of 60 that has the N-substituent directly attached to the allenic carbons. The reaction delivers a 1:1 ratio of isomeric products 67 and 69 (entry 7), which we surmise arises from diradical intermediate 68 undergoing indiscriminate cyclization on either terminus of the allylic radical shown (that is, proximal or distal to the N-substituent of the cyclic allene fragment, giving 67 or 69, respectively).

Fig. 4 |. Reactions of substituted or heterocyclic allenes with disubstituted BCB 18.

Fig. 4 |

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aConditions: allene precursor (1 equiv.), 18 (3–4 equiv.), CsF (5 equiv.), Bu4NBr (1 equiv.), DME (0.2 M), 23 °C, 16 h and sealed vessel. Observed regioselectivities are reported as a ratio of isomers (r.r.). Ar = p-Br-Phenyl; DME, 1,2-dimethoxyethane; Me, methyl; Bu, butyl; R, alkyl or ester group; Et, ethyl; OTf, trifluoromethanesulfonate; Ts, 4-methylbenzenesulfonyl; and Boc, tert-butyloxycarbonyl.

The results shown in Figs. 3 and 4 demonstrate the productive fragment coupling between two types of strained compounds: transiently generated cyclic allenes and strained BCBs. It is notable that the reactions proceed at ambient temperatures under mild conditions, do not require an external stimulus (for example, high temperatures, Lewis acid, light, radical initiators or one-electron oxidants or reductants) and take place without notable decomposition or dimerization. Generally, either an electron-withdrawing group or an aromatic substituent on the BCB ring juncture is required for the reaction to take place efficiently. We believe that these substituents are essential for enabling C–C bond rupture (of the BCB) and stabilizing the developing radical intermediate through resonance. By contrast, alkyl substituents on the BCB were not tolerated (see the Supplementary Information, part 1, section E) and the origins of these empirical findings are subject to further investigation. Nonetheless, we also highlight that the reactions discussed here occur regioselectively to deliver polysubstituted BCH products that are well poised for further elaboration, thus offering a valuable new synthetic tool.

Computational study and diradical character

The σ-bond insertion reaction of the transient cyclic allenes and BCBs proceeds readily and efficiently under relatively mild conditions. We conducted DFT and complete active space self-consistent field (CASSCF) calculations to show the reaction mechanism, interrogate the origins of observed regioselectivities and understand why the reactions occur without an external stimulus.

Using DFT, we computed the reaction coordinate profiles for the reactions of mono- and disubstituted BCBs with 1,2-cyclohexadiene (14) (Extended Data Figs. 1 and 2). The favourable pathway is discussed here, whereas pathways deemed less favourable are discussed in the Supplemental Information, part 2, sections H and I. The computed reaction coordinate profile of mono-substituted benzoyl BCB 21 (Extended Data Fig. 1) predicts that the σ-bond insertion of 1,2-cyclohexadiene 14 and 21 favours formation of the observed BCH product 22 with a relatively low free-energy barrier of 22.6 kcal mol−1. The cyclic allene reacts at the central allenic carbon (C3′), perhaps similar to a nucleophilic allyl radical46, cleaving the C1–C3 BCB σ-bond to form diradical intermediate 71. Diradical 71 undergoes a conformational change to bring the two radical centres into proximity and, through what is expected to be a barrierless or low-barrier process (TS2a), form the BCH product 22, which is the only experimentally observed constitutional isomer (Fig. 3, entry 1). The competing process involving C3′– C1 bond formation by TS1b to form diradical 70 is kinetically disfavoured by 3.6 kcal mol−1 (22.6 kcal mol−1 for TS1a compared with 26.2 kcal mol−1 for TS1b). Because both ground state reactants 14 and 21 are highly strained, formation of the diradical intermediate 71 is exergonic by 31.0 kcal mol−1. The DFT-optimized transition states for the first step of the transformation, TS1a and TS1b, are shown in Extended Data Fig. 1. The regioselectivity is controlled by the stability of the developing radicals at these transition states, favoured when stabilized by the C1 benzoyl group. We find notable diradical character at these transition states, as discussed later. We also note that attempts to perform the σ-bond insertion reaction in the presence of TEMPO or galvinoxyl shut down the desired reaction pathway, typically leading to complex mixtures. In previous studies, both cyclic allenes47 and BCBs16–18 have independently been shown to react with radical species, which provides some experimental support for the likelihood of the proposed diradical pathway.

The corresponding reaction coordinate profile for the reactions of disubstituted BCB 18 with 1,2-cyclohexadiene (14) is shown in Extended Data Fig. 2. The favoured reaction pathway involves C3′–C3 bond formation by TS3a to form diradical intermediate 73 with a low free-energy barrier of 22.5 kcal mol−1. Subsequent radical–radical recombination forms the experimentally observed BCH product 39. The competing pathway, TS3b, is kinetically disfavoured by 2.6 kcal mol−1, because the C1 bromophenyl substituent in TS3a better stabilizes the developing radical compared with the C3 carbomethoxy substituent in TS3b.

Our DFT calculations predict a low free-energy barrier for the σ-bond insertion transition state of cyclic allene 14 and BCBs. The remarkable facility of these reactions, requiring little or mild heating, but no light or catalyst, bears closer examination. The concept of a diradicaloid, namely, a stable molecule with diradical character, is common in theoretical and materials chemistry48–51, but not yet commonplace in the synthetic community. The concepts of diradical character and diradicaloids were recently reviewed in ref. 8. We propose that the innate diradicaloid character of both cyclic allene 14 and BCB 21 is the key to the low free-energy barrier of this reaction.

For all species along the lowest energy reaction pathway of the σ-bond insertion reaction of 14 and 21, we calculated the diradical characters (y0) from the occupation number of the lowest unoccupied natural orbital (nLUNO) using CASSCF calculations52,53 (Fig. 5a) (see Supplementary Information, part 2, section E). Both reactants, cyclic allene 14 and BCB 21, show notable diradical character in the ground state (14% and 6%, respectively). The diradicaloid nature of each is thought to be important, as substituting either reactant with a species having less diradical character does not lead to a productive fragment coupling under our reaction conditions (e.g. substituting the cyclic allene with an unactivated alkene or a linear allene, or substituting the BCB fragment with a cyclopropane or housane). We posit that the diradicaloid nature of the reactants contributes to the subsequent low transition state barrier in which the allene disrupts the strained σ-bond. The consequently very early transition state TS1a has 16% diradical character. The subsequent diradical 71 is calculated to have 100% diradical character.

Fig. 5 |. Analysis of diradical character and reactivity.

Fig. 5 |

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a, Computed diradical character of all species along the lowest energy reaction pathway of 14 and 21. Percentage diradical character is based on nLUNO, which is calculated at the CASPT2/CASSCF(6,6)/cc-pVDZ level of theory. b, DIAS analysis for the first step of the transformation (TS1a). c, Diradical character along the same reaction coordinate and depiction of TS1a-HONO and TS1a-LUNO. d, Calculations associated with stepwise geometric distortions (pyramidalization, twisting and bending) of linear allene 74. Calculations performed at the ωB97X-D/def2-TZVPP level of theory. Ph, phenyl; BCB, bicyclo[1.1.0]butanes; Φp, pyramidalization angle; N/A, not applicable; LUNO, lowest unoccupied natural orbital; HONO, highest occupied natural orbital; and TS, transition state.

To further analyse the nature of TS1a, we conducted distortion–interaction activation strain (DIAS)54 analysis for the first step of the reaction that leads to fragment coupling (Fig. 5b) using CASSCF(6,6)/CASPT/cc-pVDZ single-point energy calculations. Owing to the highly strained nature of both reactants, we found that minimal distortion is needed to form the early transition state TS1a (Eallene-dist = 0.3 kcal mol−1 and EBCB-dist = 2.3 kcal mol−1); a detailed analysis of the geometric distortions seen in the reactants, TS1a, and intermediate 71, are provided in the Supplementary Information, part 2, section G. Consequently, although the interaction energy is relatively small (Einteraction = −1.3 kcal mol−1) for this early transition state, the low distortion energies contribute to an overall low kinetic barrier (ΔE‡ = 1.3 kcal mol−1). The marked diradicaloid character of both reactants decreases the bonding energy of the π (cyclic allene) or σ (BCB) bonds, as compared with non-distorted analogues53.

Moreover, we studied the change of diradical character of reacting fragments 14 and 21, both separately and together, along the reaction coordinate (Fig. 5c). Consistent with the low distortion energy observed in DIAS analysis, changes of the diradical characters of the cyclic allene and BCB fragments are small before the transition state TS1a, but increase rapidly along the reaction coordinate after the transition state. Strain in the substrates induces diradicaloid behaviour, including low distortion energies due to low bonding energy. Once the notable diradicaloid character in the transition state is achieved (16% in TS1a-LUNO alone or 28% if considering three lowest unoccupied natural orbitals in TS1a; see Supplementary Information, part 2, section E), there is no barrier to subsequent motions along the energy surface to eliminate strain energy and achieve a stabilized diradical structure. We also show TS1a-HONO and TS1a-LUNO. The shapes of the computed highest occupied natural orbital (HONO) and lowest unoccupied natural orbital (LUNO) further revealed the nature of TS1a. Because TS1a is a very early transition state with minimal distortion on both the BCB and cyclic allene fragments, the HONO of TS1a is similar in shape to the HONOs of diradicaloid ground states of the BCB and cyclic allene. The computed LUNO of TS1a demonstrates the growing σ-bonding character between the centre allenic carbon and the unsubstituted BCB carbon.

The origins of diradical character in each reactant warrant discussion. The diradical character of BCBs (6% for 21) is because of the strained nature of the C–C bond common to the two fused cyclopropanes, which decreases the σ orbital bonding and σ* antibonding (see Fig. 2b for geometry-optimized structure). To better understand the origin of high diradical character of the cyclic allene, we systematically constrained linear allene 74 to the cyclic allene geometry by sequentially adding pyramidalization, bending and twisting distortions (Fig. 5d). Linear allene 74 has some delocalized diradical character (y0 = 0.08, 8% diradical character) in the ground state. Adding pyramidalization at C2 and C4 gives 74-P, which is only slightly higher in energy compared with 74 (ΔErel = 1.5 kcal mol−1). Notably, this geometric distortion has minimal impact on the diradical character of the allene. Next, bending was added to give 74-PB. In this structure, the C2–C3–C4 bond angle is constrained to 133° instead of 180°. The relative energy of 74-PB compared with 74 is 16.4 kcal mol−1 higher and the diradical character increases to y0 = 0.12 (12% diradical character), localized on the central carbon of the allene, C3. Finally, twisting of the C3=C4 alkene55 by 24.7° (that is, constraining the C2–C3 = C4–C7 dihedral angle) gives 74-PBT, which has the cyclic allene geometry. The energy increases by an additional 5.9 kcal mol−1 and, overall, 74-PBT is 22.3 kcal mol−1 higher in energy compared with linear allene 74. The diradical character of 74-PBT is calculated to be y0 = 0.12 (12% diradical character). Overall, bending is the predominant geometric distortion that leads to the localized diradicaloid nature of cyclic allenes. This understanding of geometric distortion and diradical character is expected to enable the future design of diradicaloid reactants. Moreover, an examination of known reactions in which diradicaloids may be invoked, such as reactions between benzynes and BCBs38–40, will be pursued in due course.

This connection between diradicaloid nature and distortion of reactants leads to a low barrier to reach the transition state geometry with greater diradical character compared with the radical character present in either reactant. The initial C–C bond formation that takes place resembles to some extent the collapse of two radicals to form a σ-bond and is the key to understanding the high reactivity of strained cyclic allenes with BCBs. Transition states formed from reactants with very low or zero diradicaloid character, for example, undistorted allenes or acetylenes (see the Supplemental Information, part 2, section L), are expected to have much greater activation barriers to undergo the main distortions necessary to produce the diradicaloid transition states.

Conclusion

We have developed a fragment coupling reaction of two types of strained compounds: strained cyclic allenes (about 30 kcal mol−1 of strain energy) and BCBs (about 60 kcal mol−1 of strain energy). These reactions between two strained intermediates are uncommon, but this new transformation proceeds in the absence of an external stimulus and efficiently at room temperature, because of the severe geometric distortions present in each reactant. The specific impact of geometric distortions is two-fold: (1) distortion leads to a small percentage of diradical character in each reactant, rendering them diradicaloids that can engage in unconventional reactivity; and (2) distortions of reactant geometries allow for low kinetic barriers of the desired σ-bond insertion reactions, as reactants are pre-distorted towards transition state geometries. Our studies not only provide a means to access functionalized bicyclo[2.1.1]hexanes that are of value for drug discovery as arene bioisosteres but also underscore how geometric distortion of reactants can be used to enable uncommon modes of reactivity. Moreover, we hope these efforts prompt the further exploration and strategic use of diradicaloids in chemical synthesis.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-025-08745-1.

Extended Data

Extended Data Fig. 1 |. Computational study using BCB 21.

Extended Data Fig. 1 |

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Reaction pathway and calculated transition states TS1a and TS1b for the σ-bond insertion reaction of strained cyclic allene 14 and monosubstituted BCB 21. ΔG and ΔH are calculated at ωB97X-D/def2TZVPP/SMD(DME)//ωB97X-D/def2SVP level of theory. Ph, phenyl; TS, transition state.

Extended Data Fig. 2 |. Computational study using BCB 18.

Extended Data Fig. 2 |

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Reaction pathway and calculated transition states TS3a and TS3b for the σ-bond insertion reaction of strained cyclic allene 14 and disubstituted BCB 18. ΔG and ΔH are calculated at ωB97X-D/def2TZVPP/SMD(DME)//ωB97X-D/def2SVP level of theory. Me, methyl; Ph, phenyl; TS, transition state.

Supplementary Material

supp info

The online version contains supplementary material available at https://doi.org/10.1038/s41586-025-08745-1.

Acknowledgements

We thank the NIH-NIGMS (R35 GM139593 to N.K.G. and F31 GM149161 to A.T.M.), the NSF (CHE–2153972 to K.N.H. and DGE-2034835 to A.V.K.), the UCLA Cota Robles Fellowship program (C.A.R.), the Foote family (A.V.K. and A.T.M.) and the Trueblood family (N.K.G.). These studies were supported by shared instrumentation grants from the NSF (CHE-1048804), the NIH NCRR (S10RR025631) and the NIH ORIP (S10OD028644). Calculations were performed on the Hoffman2 cluster and the UCLA Institute of Digital Research and Education (IDRE) at UCLA and the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation (OCI-1053575). We thank Z. G. Walters (UCLA) and A. Wong (UCLA) for computational assistance. We are indebted to our recently departed colleague and dear friend, F. Stoddart (1942–2024), for inspiring us, by his example, to explore the unknown with curiosity and passion, and to support and cherish the next generation of brilliant scientific minds.

Footnotes

Competing interests The authors declare no competing interests.

Data availability

Experimental procedures, characterization data, computational methods and computational data are provided in the Supplementary Information.

References

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

supp info
NIHMS2111289-supplement-supp_info.pdf (13MB, pdf)

Data Availability Statement

Experimental procedures, characterization data, computational methods and computational data are provided in the Supplementary Information.

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