Ring Expansion of Alkylidenecarbenes Derived from Lactams, Lactones, and Thiolactones into Strained Heterocyclic Alkynes: A Theoretical Study

Strained cycloalkynes are of considerable interest to theoreticians and experimentalists, and possess much synthetic value as well. Herein, a series of cyclic alkylidenecarbenes—formally obtained by replacing the carbonyl oxygen of four-, five-, and six-membered lactams, lactones, and thiolactones with a divalent carbon—were modeled at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** levels of theory. The singlet carbenes were found to be more stable than the triplets. The strained heterocyclic alkynes formed by ring expansion of these singlet carbenes were also modeled. Interestingly, the C≡C bonds in the five-membered heterocycles, obtained from the rearrangement of β-lactam- and β-lactone-derived alkylidenecarbenes, displayed lengths intermediate between formal double and triple bonds. Furthermore, 2-(1-azacyclobutylidene)carbene was found to be nearly isoenergetic with its ring-expanded isomer, and 1-oxacyclopent-2-yne was notably higher in energy than its precursor carbene. In all other cases, the cycloalkynes were lower in energy than the corresponding carbenes. The transition states for ring-expansion were always lower for the 1,2-carbon shifts than for 1,2-nitrogen or oxygen shifts, but higher than for the 1,2-sulfur shifts. These predictions should be verifiable using carbenes bearing appropriate isotopic labels. Computed vibrational spectra for the carbenes, and their ring-expanded isomers, are presented and could be of value to matrix isolation experiments.


Alkylidenecarbenes Derived from β-Lactam, β-Lactone, and β-Thiolactone
The singlet-triplet energy gap (∆E S-T ) in the alkylidenecarbenes derived from β-lactam, β-lactone, and β-thiolactone were computed at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** levels of theory. In all calculations, the singlets were found to be significantly more stable than the triplets. This gap is consistent with the known preference of alkylidenecarbenes to adopt the singlet state, in which the lone pair can be nested in an approximately sp-hybridized orbital of the divalent carbon [6]. The PES for ring expansion of the singlet carbenes into the corresponding 1-X-cyclopent-2-ynes, which could occur by a 1,2-shift of either the heteroatom X or carbon (Scheme 4), were also modeled. Results are discussed below.

Alkylidenecarbenes Derived from β-Lactam, β-Lactone, and β-Thiolactone
The singlet-triplet energy gap (ES-T) in the alkylidenecarbenes derived from β-lactam, β-lactone, and β-thiolactone were computed at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** levels of theory. In all calculations, the singlets were found to be significantly more stable than the triplets. This gap is consistent with the known preference of alkylidenecarbenes to adopt the singlet state, in which the lone pair can be nested in an approximately sp-hybridized orbital of the divalent carbon [6]. The PES for ring expansion of the singlet carbenes into the corresponding 1-X-cyclopent-2-ynes, which could occur by a 1,2-shift of either the heteroatom X or carbon (Scheme 4), were also modeled. Results are discussed below. Scheme 4. Two pathways for the ring expansion of singlet alkylidenecarbenes derived from β-lactam, β-lactone, and β-thiolactone.
As seen in Figure 1, singlet 12 is virtually isoenergetic with 13 and lies 18.6 kcal/mol below the triplet. Another distinctive feature of 13 is that its triple bond is significantly elongated with a calculated length of 1.28 Å. This value is intermediate between the length of a C=C bond in cyclopentene (~1.32 Å) [51] and our calculated value for the CC bond (1.22 Å) in cyclopentyne [15]. Thus, 13 appears to display a partial triple bond, rather than a formal one, in order to alleviate ring strain. Furthermore, the conversion of singlet 12 into 13 appears to favor a 1,2-carbon shift, which has a significantly lower barrier than for a 1,2-nitrogen shift. This preference could stem from at least two factors. One is the "bystander effect" [52] of nitrogen that accelerates the 1,2-migration of carbon. In other words, as the carbon begins to shift, a decrease in electron density at the migration origin in the transition state, as evident from a Natural Population Analysis (NPA) [53,54] (see Supplementary Materials), could be stabilized by electron donation from the nitrogen. A second possible explanation is that the resonance contributions in 12 imparts a partial double bond character to the N-C (sp 2 ) bond ( Figure 2) which would thus become even more resistant to cleavage. Importantly, such conjugation can be maintained during the 1,2-carbon shift. The PES for ring expansion of singlet 12 into 1-aza-cyclopent-2-yne (13), which could occur by a 1,2-shift of either the nitrogen or carbon (Scheme 4), is depicted in Figure 1 using CCSD(T)/cc-pVTZ//CCSD/6-311+G** energies and structures. A similar diagram, based on CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations, is provided in the Supplementary Materials ( Figure S1).
As seen in Figure 1, singlet 12 is virtually isoenergetic with 13 and lies 18.6 kcal/mol below the triplet. Another distinctive feature of 13 is that its triple bond is significantly elongated with a calculated length of 1.28 Å. This value is intermediate between the length of a C=C bond in cyclopentene (~1.32 Å) [51] and our calculated value for the C≡C bond (1.22 Å) in cyclopentyne [15]. Thus, 13 appears to display a partial triple bond, rather than a formal one, in order to alleviate ring strain. Furthermore, the conversion of singlet 12 into 13 appears to favor a 1,2-carbon shift, which has a significantly lower barrier than for a 1,2-nitrogen shift. This preference could stem from at least two factors. One is the "bystander effect" [52] of nitrogen that accelerates the 1,2-migration of carbon. In other words, as the carbon begins to shift, a decrease in electron density at the migration origin in the transition state, as evident from a Natural Population Analysis (NPA) [53,54] (see Supplementary Materials), could be stabilized by electron donation from the nitrogen. A second possible explanation is that the resonance contributions in 12 imparts a partial double bond character to the N-C (sp 2 ) bond ( Figure 2) which would thus become even more resistant to cleavage. Importantly, such conjugation can be maintained during the 1,2-carbon shift.  Vibrational spectra of 12 and 13 computed at CCSD/6-311+G** are shown in Figure 3. The C=C stretch in singlet 12 appears as a weak absorbance at 1711 cm −1 . The two prominent bands at 763 cm −1 and 665 cm −1 are associated with the wagging motion of the NH bond coupled to ring oscillations. Triplet 12, on the other hand, shows a much stronger C=C stretch at 1566 cm −1 , and another strong absorbance at 447 cm −1 corresponding to the out of-plane wagging motions of the N-H bond and the methylene groups. The wagging of these two moieties in the plane of the ring is observed at 1175 cm −1 . The spectrum of 13 shows a strong absorption at 1656 cm −1 associated with the stretching of the NCC moiety. The wagging vibrations of the NH and methylene groups, in a direction perpendicular to the plane of the ring also show a strong absorbance at 716 cm −1 .   Vibrational spectra of 12 and 13 computed at CCSD/6-311+G** are shown in Figure 3. The C=C stretch in singlet 12 appears as a weak absorbance at 1711 cm −1 . The two prominent bands at 763 cm −1 and 665 cm −1 are associated with the wagging motion of the NH bond coupled to ring oscillations. Triplet 12, on the other hand, shows a much stronger C=C stretch at 1566 cm −1 , and another strong absorbance at 447 cm −1 corresponding to the out of-plane wagging motions of the N-H bond and the methylene groups. The wagging of these two moieties in the plane of the ring is observed at 1175 cm −1 . The spectrum of 13 shows a strong absorption at 1656 cm −1 associated with the stretching of the NCC moiety. The wagging vibrations of the NH and methylene groups, in a direction perpendicular to the plane of the ring also show a strong absorbance at 716 cm −1 . Vibrational spectra of 12 and 13 computed at CCSD/6-311+G** are shown in Figure 3. The C=C stretch in singlet 12 appears as a weak absorbance at 1711 cm −1 . The two prominent bands at 763 cm −1 and 665 cm −1 are associated with the wagging motion of the NH bond coupled to ring oscillations. Triplet 12, on the other hand, shows a much stronger C=C stretch at 1566 cm −1 , and another strong absorbance at 447 cm −1 corresponding to the out of-plane wagging motions of the N-H bond and the methylene groups. The wagging of these two moieties in the plane of the ring is observed at 1175 cm −1 . The spectrum of 13 shows a strong absorption at 1656 cm −1 associated with the stretching of the NC≡C moiety. The wagging vibrations of the NH and methylene groups, in a direction perpendicular to the plane of the ring also show a strong absorbance at 716 cm −1 .  Figure 2S). The cycloalkyne 15, however, was found to be not a minimum at the DFT level of theory, instead reverting without barrier to 14. On the other hand, singlet and triplet 14, as well as 15, were all found to be minima on the CCSD/6-311+G** PES. Their structures and CCSD(T)/cc-pVTZ//CCSD/6-311+G** energies, along with the transition states for converting singlet 14 into 15 are depicted in Figure 4. Notably, 15 was found to be 5.6 kcal/mol higher in energy than singlet 14, and features a partial triple bond (~1.26 Å) much like 12. Similar to observations with the 12 to 13 rearrangement, the 1,2-carbon shift in singlet 14 is  Figure S2). The cycloalkyne 15, however, was found to be not a minimum at the DFT level of theory, instead reverting without barrier to 14. On the other hand, singlet and triplet 14, as well as 15, were all found to be minima on the CCSD/6-311+G** PES. Their structures and CCSD(T)/cc-pVTZ//CCSD/6-311+G** energies, along with the transition states for converting singlet 14 into 15 are depicted in Figure 4. Notably, 15 was found to be 5.6 kcal/mol higher in energy than singlet 14, and features a partial triple bond (~1.26 Å) much like 12. Similar to observations with the 12 to 13 rearrangement, the 1,2-carbon shift in singlet 14 is considerably more facile than the 1,2-oxygen shift. Thus, the reactivity of 14 appears to mirror that of 12, for the same reasons as those discussed above.
Molecules 2019, 24, 593 6 of 26 considerably more facile than the 1,2-oxygen shift. Thus, the reactivity of 14 appears to mirror that of 12, for the same reasons as those discussed above. Vibrational spectra of 14 and 15 computed at CCSD/6-311+G** are shown in Figure 5. In the spectrum of singlet 14, an especially strong absorbance was found at 1102 cm −1 , which corresponds to the in-plane, side-to-side motion of the endocyclic sp 2 -hybridized carbon. A similar motion was also observed for triplet 14, with a strong absorbance at 1138 cm -1 . The spectrum of triplet 14 also displays another moderately strong absorbance at 970 cm −1 corresponding to the in-plane oscillation of the oxygen. Conspicuously, only very weak absorbances were found for C=C stretch in both singlet and triplet 14. The spectrum of 15 shows strong bands at 1803 cm −1 and 453 cm −1 associated with vibrations of the OCC moiety, and a moderate absorbance at 1153 cm −1 corresponding to the in-plane stretching of the OC(sp)bond. Vibrational spectra of 14 and 15 computed at CCSD/6-311+G** are shown in Figure 5. In the spectrum of singlet 14, an especially strong absorbance was found at 1102 cm −1 , which corresponds to the in-plane, side-to-side motion of the endocyclic sp 2 -hybridized carbon. A similar motion was also observed for triplet 14, with a strong absorbance at 1138 cm −1 . The spectrum of triplet 14 also displays another moderately strong absorbance at 970 cm −1 corresponding to the in-plane oscillation of the oxygen. Conspicuously, only very weak absorbances were found for C=C stretch in both singlet and triplet 14. The spectrum of 15 shows strong bands at 1803 cm −1 and 453 cm −1 associated with vibrations of the OC≡C moiety, and a moderate absorbance at 1153 cm −1 corresponding to the in-plane stretching of the OC(sp)bond.

Ring Expansion of --Thiocyclobutylidene)carbene (16) into 1-Thiocyclopent-2-yne (17)
Given the larger size of sulfur, and the attendant incorporation of longer bonds in the ring, one might expect that the thio carbene 16 and 1-thiocyclopent-2-yne (17) might behave somewhat differently than the aza and oxa analogs discussed above. These expectations are indeed borne out by CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations ( Figure 6). As with 12 and 14, singlet 16 is more stable than the triplet (ES-T = −28.7 kcal/mol). The PES for the conversion of singlet 16 into 17, however, does show some important differences compared to the profiles seen in Figures 1 and 4. For one, the carbon-carbon triple bond 16 is shorter (~1.24 Å) than in either 12 or 14. Furthermore, 16 Given the larger size of sulfur, and the attendant incorporation of longer bonds in the ring, one might expect that the thio carbene 16 and 1-thiocyclopent-2-yne (17) might behave somewhat differently than the aza and oxa analogs discussed above. These expectations are indeed borne out by CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations ( Figure 6). As with 12 and 14, singlet 16 is more stable than the triplet (∆E S-T = −28.7 kcal/mol). The PES for the conversion of singlet 16 into 17, however, does show some important differences compared to the profiles seen in Figures 1 and 4. For one, the carbon-carbon triple bond 16 is shorter (~1.24 Å) than in either 12 or 14. Furthermore, 16 is lower in energy than singlet 15 by −7.2 kcal/mol. This could be attributed to lesser ring strain in 16 compared to 12 and 14, a benefit accorded by the incorporation of longer carbon-sulfur bonds in the ring. A second important difference is that the ring expansion of 16 to 17 prefers to occur by a 1,2-sulfur shift, which is almost barrier-free, compared to a 1,2-carbon shift that needs an activation energy of almost 10 kcal/mol. This behavior may be attributed to the greater nucleophilicity and size of sulfur. The 3p orbital on sulfur is less effective for resonance stabilization due to poor overlap with the 2p orbital on the adjacent carbon. On the other hand, a lone pair on the larger sulfur atom can interact with the empty orbital on the carbenic center to initiate bonding at a longer distance than possible with either nitrogen or oxygen. A similar profile, obtained at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** level of theory, is provided in the Supplementary Materials ( Figure S3). is lower in energy than singlet 15 by −7.2 kcal/mol. This could be attributed to lesser ring strain in 16 compared to 12 and 14, a benefit accorded by the incorporation of longer carbon-sulfur bonds in the ring. A second important difference is that the ring expansion of 16 to 17 prefers to occur by a 1,2sulfur shift, which is almost barrier-free, compared to a 1,2-carbon shift that needs an activation energy of almost 10 kcal/mol. This behavior may be attributed to the greater nucleophilicity and size of sulfur. The 3p orbital on sulfur is less effective for resonance stabilization due to poor overlap with the 2p orbital on the adjacent carbon. On the other hand, a lone pair on the larger sulfur atom can interact with the empty orbital on the carbenic center to initiate bonding at a longer distance than possible with either nitrogen or oxygen. A similar profile, obtained at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** level of theory, is provided in the Supplementary Materials ( Figure 3S). Vibrational spectra of 16 and 17 computed at CCSD/6-311+G** are shown in Figure 7. The IR spectrum of singlet 16 is relatively clean and shows a strong absorbance at 1686 cm −1 for the C=C stretch. The spectrum of triplet 16 shows a moderately strong absorbance at 1265 cm −1 for the wagging motions of the two methylene groups in the ring. The stretching vibrations of the bond between O and the endocyclic sp 2 carbon have an absorbance at 691 cm −1 , and another absorbance at 997 cm −1 is seen for the stretching vibrations of the bond between the CH2 group and the double bond. The alkynyl group in 17 shows two distinct bands in the IR spectrum, a weak absorbance at 1970 cm −1 for Vibrational spectra of 16 and 17 computed at CCSD/6-311+G** are shown in Figure 7. The IR spectrum of singlet 16 is relatively clean and shows a strong absorbance at 1686 cm −1 for the C=C stretch. The spectrum of triplet 16 shows a moderately strong absorbance at 1265 cm −1 for the wagging motions of the two methylene groups in the ring. The stretching vibrations of the bond between O and the endocyclic sp 2 carbon have an absorbance at 691 cm −1 , and another absorbance at 997 cm −1 is seen for the stretching vibrations of the bond between the CH 2 group and the double bond. The alkynyl group in 17 shows two distinct bands in the IR spectrum, a weak absorbance at 1970 cm −1 for its stretching vibrations and a strong absorbance at 353 cm −1 for its wagging motions in the plane of the ring. Absorbances due to C-H stretches are seen 3080 cm −1 , and the stretching motion of the bond between O and the alkynyl group absorbs at 784 cm −1 .

Alkylidenecarbenes Derived from γ-Lactam, γ-Lactone, and γ-Thiolactone
The ∆E S-T gaps of the the alkylidenecarbenes in this series also favor the singlets at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** levels of theory. Structures and energies of these carbenes, and the PES for conversion of the singlet carbenes into corresponding 1-X-cyclohex-2-ynes, by a 1,2-shift of X or carbon (Scheme 5), are described below. As depicted in Figure 8, CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 18 is about 18.7 kcal/mol lower in energy than the triplet. Furthermore, in sharp contrast to what has been described above with 12 and 13, the cyclic alkyne 19 is lower in energy than singlet 18 by 10.0 kcal/mol. This result is understandable given that 19, which features a six-membered ring, is much less strained than the five-membered ring in 13. The conversion of singlet 18 into 19 preferentially occurs by a 1,2-carbon shift, which needs to overcome a barrier of 8.3 kcal/mol. A 1,2-nitrogen shift, on the other hand, has a much higher activation energy of 20.5 kcal/mol. A similar profile, obtained by CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations, is provided in the ( Figure 4S). CCSD/6-311+G** calculated vibrational spectra of singlet and triplet 18, as well as 19, are shown in Figure 9. The spectrum of singlet 18 shows strong absorbances at 1167 cm -1 , for the in-plane motion of the endocyclic sp 2 carbon, and 531 cm -1 for the out of plane N-H bend. The C=C stretch in singlet As depicted in Figure 8, CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 18 is about 18.7 kcal/mol lower in energy than the triplet. Furthermore, in sharp contrast to what has been described above with 12 and 13, the cyclic alkyne 19 is lower in energy than singlet 18 by 10.0 kcal/mol. This result is understandable given that 19, which features a six-membered ring, is much less strained than the five-membered ring in 13. The conversion of singlet 18 into 19 preferentially occurs by a 1,2-carbon shift, which needs to overcome a barrier of 8.3 kcal/mol. A 1,2-nitrogen shift, on the other hand, has a much higher activation energy of 20.5 kcal/mol. A similar profile, obtained by CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations, is provided in the ( Figure S4). As depicted in Figure 8, CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 18 is about 18.7 kcal/mol lower in energy than the triplet. Furthermore, in sharp contrast to what has been described above with 12 and 13, the cyclic alkyne 19 is lower in energy than singlet 18 by 10.0 kcal/mol. This result is understandable given that 19, which features a six-membered ring, is much less strained than the five-membered ring in 13. The conversion of singlet 18 into 19 preferentially occurs by a 1,2-carbon shift, which needs to overcome a barrier of 8.3 kcal/mol. A 1,2-nitrogen shift, on the other hand, has a much higher activation energy of 20.5 kcal/mol. A similar profile, obtained by CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations, is provided in the ( Figure 4S). CCSD/6-311+G** calculated vibrational spectra of singlet and triplet 18, as well as 19, are shown in Figure 9. The spectrum of singlet 18 shows strong absorbances at 1167 cm -1 , for the in-plane motion of the endocyclic sp 2 carbon, and 531 cm -1 for the out of plane N-H bend. The C=C stretch in singlet 18 appears at a frequency of 1706 cm -1 . Triplet 18 shows a band at 1568 cm -1 , for the vibrations of the C=C-N moiety, and another strong absorbance at 566 cm -1 for the out-of-plane wagging of the N-H  Singlet 20 was found to be 24.9 kcal/mol below the triplet, and 6.3 kcal/mol above cycloalkyne 21, according to CCSD(T)/cc-pVTZ//CCSD/6-311+G** (Figure 10). These calculations also reveal that

Ring Expansion of 2-(1-Oxacyclopentylidene)carbene (20) into 1-Oxacyclohex-2-yne (21)
Singlet 20 was found to be 24.9 kcal/mol below the triplet, and 6.3 kcal/mol above cycloalkyne 21, according to CCSD(T)/cc-pVTZ//CCSD/6-311+G** ( Figure 10). These calculations also reveal that the barrier for converting singlet 20 into 21 by a 1,2-carbon shift is 10.1 kcal/mol, whereas the transition state for accomplishing the ring expansion by a 1,2-oxygen shift is much higher at 28.8 kcal/mol. These results are also summarized in Figure 10. Structures and energies from the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations on this system are provided in the Supplementary Materials ( Figure S5).  The computed vibrational spectra (CCSD/6-311+G**) of 20 (singlet and triplet) and 21 are displayed in Figure 11. The strongest absorbances in the spectrum of singlet and triplet 20 appear at 1157 cm -1 and 1236 cm -1 respectively and correspond to the stretching vibration of the bond between oxygen and the endocyclic sp 2 carbon. The spectrum of 21 shows a strong absorbance at 2132 cm −1 characteristic of the alkynyl group. Stretching vibrations of the bond between oxygen and the alkyne moiety absorb at 1178 cm −1 , and the in-plane motions of the OCC group absorb at 407 cm −1 . The computed vibrational spectra (CCSD/6-311+G**) of 20 (singlet and triplet) and 21 are displayed in Figure 11. The strongest absorbances in the spectrum of singlet and triplet 20 appear at 1157 cm −1 and 1236 cm −1 respectively and correspond to the stretching vibration of the bond between oxygen and the endocyclic sp 2 carbon. The spectrum of 21 shows a strong absorbance at 2132 cm −1 characteristic of the alkynyl group. Stretching vibrations of the bond between oxygen and the alkyne moiety absorb at 1178 cm −1 , and the in-plane motions of the OC≡C group absorb at 407 cm −1 .
with CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations and are provided in the Supplementary Materials ( Figure 6S). Vibrational spectra of 22 and 23 computed at CCSD/6-311+G** are shown in Figure 13. Singlet 22 shows a particularly strong absorbance at 1688 cm −1 for the C=C stretch. Triplet 22 shows a moderate absorbance at 1389 cm −1 due to the C=C stretch, whereas the stretching vibration of the bond between S and the sp 2 carbon shows somewhat stronger absorbances at 1082 cm −1 and 681 cm −1 . The CC stretching frequency for 23 appears at 2134 cm −1 but is rather weak compared to the other members of the series, 19 and 21. This may be attributable to the diminished change in dipole moment, which is associated with that vibrational mode due to the lower electronegativity of sulfur compared to oxygen and nitrogen. The spectrum of 23 also shows moderately strong absorbances at 527 cm −1 for the coupled motions of the carbons in the ring, and 373 cm −1 for the in-plane vibrations of the S-CC group. Vibrational spectra of 22 and 23 computed at CCSD/6-311+G** are shown in Figure 13. Singlet 22 shows a particularly strong absorbance at 1688 cm −1 for the C=C stretch. Triplet 22 shows a moderate absorbance at 1389 cm −1 due to the C=C stretch, whereas the stretching vibration of the bond between S and the sp 2 carbon shows somewhat stronger absorbances at 1082 cm −1 and 681 cm −1 . The C≡C stretching frequency for 23 appears at 2134 cm −1 but is rather weak compared to the other members of the series, 19 and 21. This may be attributable to the diminished change in dipole moment, which is associated with that vibrational mode due to the lower electronegativity of sulfur compared to oxygen and nitrogen. The spectrum of 23 also shows moderately strong absorbances at 527 cm −1 for the coupled motions of the carbons in the ring, and 373 cm −1 for the in-plane vibrations of the S-C≡C group.

Alkylidenecarbenes Derived from δ-Lactam, δ-Lactone, and δ-Thiolactone
Members in this group are considerably less strained than their counterparts in the βand γseries. CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations reveal that the ES-T gaps of the alkylidenecarbenes in the δ-series are much larger than seen for analogously substituted species in the previous two series, and favor the singlet. The cycloalkynes in this series are also substantially more stable than their corresponding alkylidencarbene isomers. Structures and energies of these carbenes as well the PES for conversion of singlet carbenes into the corresponding 1-X-cyclohept-2-ynes, by a 1,2-shift of X or carbon (Scheme 6), are described below.

Alkylidenecarbenes Derived from δ-Lactam, δ-Lactone, and δ-Thiolactone
Members in this group are considerably less strained than their counterparts in the βand γ-series. CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations reveal that the ∆E S-T gaps of the alkylidenecarbenes in the δ-series are much larger than seen for analogously substituted species in the previous two series, and favor the singlet. The cycloalkynes in this series are also substantially more stable than their corresponding alkylidencarbene isomers. Structures and energies of these carbenes as well the PES for conversion of singlet carbenes into the corresponding 1-X-cyclohept-2-ynes, by a 1,2-shift of X or carbon (Scheme 6), are described below. Scheme 6. Ring expansion of singlet alkylidenecarbenes, derived from δ-lactam, δ-lactone, and δthiolactone, by a 1,2 shift of either X or carbon.
2.3.1. Ring Expansion of --Azacyclohexylidene)carbene (24) into 1-Azacyclohept-2-yne (25) As seen in Figure 14, CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 24 lies 24.6 kcal/mol below the triplet and 19.7 kcal/mol above the azacycloheptyne 25. Structurally, the ring in singlet 24 adopts the familiar chair conformation but in the triplet, the portion of the ring in the vicinity of the double bond is essentially planar. Another interesting feature in singlet 24 is that the double bond appears to sharply bend toward the nitrogen with the N-C=C angle compressing to 78.5° and the C-C=C angle increasing to 154.5°. This could be rationalized as a stabilizing interaction between the lone pair on nitrogen and the empty p orbital on the carbenic carbon. Somewhat counterintuitively, despite this structural distortion, the formation of 25 by a 1,2-nitrogen shift still has a rather large barrier of 24.4 kcal/mol compared to a much smaller activation energy of 10.8 kcal/mol required for the 1,2-carbon shift. Results obtained with CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations are provided in the Supplementary Materials ( Figure 7S). The vibrational spectra of 24 and 25 computed at CCSD/6-311+G** are shown in Figure 15. Singlet 24 has a strong band at 786 cm -1 caused by the out-of-plane wagging motion of the N-H bond. Scheme 6. Ring expansion of singlet alkylidenecarbenes, derived from δ-lactam, δ-lactone, and δ-thiolactone, by a 1,2 shift of either X or carbon.

Ring Expansion of 2-(1-Azacyclohexylidene)carbene (24) into 1-Azacyclohept-2-yne (25)
As seen in Figure 14, CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 24 lies 24.6 kcal/mol below the triplet and 19.7 kcal/mol above the azacycloheptyne 25. Structurally, the ring in singlet 24 adopts the familiar chair conformation but in the triplet, the portion of the ring in the vicinity of the double bond is essentially planar. Another interesting feature in singlet 24 is that the double bond appears to sharply bend toward the nitrogen with the N-C=C angle compressing to 78.5 • and the C-C=C angle increasing to 154.5 • . This could be rationalized as a stabilizing interaction between the lone pair on nitrogen and the empty p orbital on the carbenic carbon. Somewhat counterintuitively, despite this structural distortion, the formation of 25 by a 1,2-nitrogen shift still has a rather large barrier of 24.4 kcal/mol compared to a much smaller activation energy of 10.8 kcal/mol required for the 1,2-carbon shift. Results obtained with CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations are provided in the Supplementary Materials ( Figure S7). As seen in Figure 14, CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 24 lies 24.6 kcal/mol below the triplet and 19.7 kcal/mol above the azacycloheptyne 25. Structurally, the ring in singlet 24 adopts the familiar chair conformation but in the triplet, the portion of the ring in the vicinity of the double bond is essentially planar. Another interesting feature in singlet 24 is that the double bond appears to sharply bend toward the nitrogen with the N-C=C angle compressing to 78.5° and the C-C=C angle increasing to 154.5°. This could be rationalized as a stabilizing interaction between the lone pair on nitrogen and the empty p orbital on the carbenic carbon. Somewhat counterintuitively, despite this structural distortion, the formation of 25 by a 1,2-nitrogen shift still has a rather large barrier of 24.4 kcal/mol compared to a much smaller activation energy of 10.8 kcal/mol required for the 1,2-carbon shift. Results obtained with CCSD(T)/cc-pVTZ//B3LYP/6-311+G** calculations are provided in the Supplementary Materials ( Figure 7S). The vibrational spectra of 24 and 25 computed at CCSD/6-311+G** are shown in Figure 15. Singlet 24 has a strong band at 786 cm -1 caused by the out-of-plane wagging motion of the N-H bond. The vibrational spectra of 24 and 25 computed at CCSD/6-311+G** are shown in Figure 15. Singlet 24 has a strong band at 786 cm −1 caused by the out-of-plane wagging motion of the N-H bond. The spectrum of triplet 24 shows a strong band at 1589 cm −1 that corresponds to the in-plane vibrations of the C=C-N moiety, and another band at 1409 cm −1 that is associated with the coupled motions of atoms in the ring. The characteristic C≡C stretching frequency in 25 appears as a moderately strong band at 2264 cm −1 . Cycloalkyne 25 also shows a strong absorbance at 767 cm −1 and a weaker band at 750 cm −1 , both of which are associated with the out-of-plane bending motions of the N-H bond.
The spectrum of triplet 24 shows a strong band at 1589 cm −1 that corresponds to the in-plane vibrations of the C=C-N moiety, and another band at 1409 cm −1 that is associated with the coupled motions of atoms in the ring. The characteristic CC stretching frequency in 25 appears as a moderately strong band at 2264 cm −1 . Cycloalkyne 25 also shows a strong absorbance at 767 cm −1 and a weaker band at 750 cm −1 , both of which are associated with the out-of-plane bending motions of the N-H bond.  (27) CCSD(T)/cc-pVTZ//CCSD/6-311+G** structures and energies of 26 (singlet and triplet), and the PES for the rearrangement of singlet 26 into 27, are represented in Figure 16. These calculations show that singlet 26 is lower in energy than the triplet by 29.9 kcal/mol. Similar to what was observed with 24 above, the six-membered ring in singlet 26 adopts a chair conformation whereas the triplet features a somewhat flattened portion of the ring near the double bond. Unlike singlet 24, however, there is no inclination for the double bond in singlet 26 to 'lean' toward the oxygen. The cyclohexyne 27 is 22 kcal/mol below singlet 26. The transition state for the 1,2-carbon shift to rearrange from singlet 25 into 26 lies 8.1 kcal/mol above the carbene, whereas the barrier to form 27 by a 1,2-oxygen shift is much higher at 26.1 kcal/mol. Structures and energies for this system calculated at CCSD(T)/cc-pVTZ//B3LYP/6-311+G** are provided in the Supplementary Materials ( Figure S8).    CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 28 is 46.1 kcal/mol lower in energy than the triplet, and this ES-T gap is the largest observed for all carbene species in this study ( Figure 18). Singlet 28 also shows a chair conformation in the six-membered ring while the triplet has a significantly planarized ring around the double bond. The structure of singlet 28 closely resembles that of the aza analog 24 in that the carbenic center is tilted toward the sulfur. This leads to an unusually small S-C=C bond angle of 83.1° whereas the C-C=C angle widens to 150.8°. As also shown in Figure 18, cycloheptyne 29 lies 17.2 kcal/mol below singlet 28. Consistent with the behavior of the CCSD(T)/cc-pVTZ//CCSD/6-311+G** calculations show that singlet 28 is 46.1 kcal/mol lower in energy than the triplet, and this ∆E S-T gap is the largest observed for all carbene species in this study ( Figure 18). Singlet 28 also shows a chair conformation in the six-membered ring while the triplet has a significantly planarized ring around the double bond. The structure of singlet 28 closely resembles that of the aza analog 24 in that the carbenic center is tilted toward the sulfur. This leads to an unusually small S-C=C bond angle of 83.1 • whereas the C-C=C angle widens to 150.8 • . As also shown in Figure 18, cycloheptyne 29 lies 17.2 kcal/mol below singlet 28. Consistent with the behavior of the other two sulfur-containing singlet carbenes discussed above (16 and 22), the 1,2-sulfur shift in singlet 28 to form 29 has a lower barrier of 9.02 kcal/mol relative to a 1,2-carbon shift, which needs to surmount a barrier of 21.8 kcal/mol. Results of calculations on this system at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** level of theory are reported in the Supplementary Materials ( Figure S9). other two sulfur-containing singlet carbenes discussed above (16 and 22), the 1,2-sulfur shift in singlet 28 to form 29 has a lower barrier of 9.02 kcal/mol relative to a 1,2-carbon shift, which needs to surmount a barrier of 21.8 kcal/mol. Results of calculations on this system at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** level of theory are reported in the Supplementary Materials ( Figure 9S). Vibrational spectra of 28 and 29, computed at CCSD/6-311+G**, are shown in Figure 19. The spectrum of singlet 28 shows a prominent band for the C=C stretch at 1798 cm −1 and another weak absorbance at 630 cm −1 for the stretching of the bond between sulfur and the endocyclic sp 2 carbon. Triplet 28 shows bands at 1330 cm −1 and 1284 cm −1 that correspond to the C=C stretch coupled to motions of the ring carbons. The CC stretch in 29 shows a weak band at 2222 cm −1 , and the absorbance for the stretching vibrations of the bond connecting sulfur and the alkynyl group appears at 692 cm −1 . Vibrational spectra of 28 and 29, computed at CCSD/6-311+G**, are shown in Figure 19. The spectrum of singlet 28 shows a prominent band for the C=C stretch at 1798 cm −1 and another weak absorbance at 630 cm −1 for the stretching of the bond between sulfur and the endocyclic sp 2 carbon. Triplet 28 shows bands at 1330 cm −1 and 1284 cm −1 that correspond to the C=C stretch coupled to motions of the ring carbons. The C≡C stretch in 29 shows a weak band at 2222 cm −1 , and the absorbance for the stretching vibrations of the bond connecting sulfur and the alkynyl group appears at 692 cm −1 .

Computational Methods
All calculations were carried out with Gaussian 09 [55] and Gaussian 16 [56], using methods and basis sets described in the text. Graphics of structures used in the potential energy diagrams were generated with GaussView6 [57]. The vibrational spectra were produced using Avogadro [58]. Restricted methods were used for singlet species and unrestricted for triplet. The spin expectation values, <S 2 >, were found to be 2.000 for all the triplet carbenes. The stationary points obtained from geometry optimizations were verified as minima (zero imaginary frequency) or transition states (one imaginary frequency) by subsequent frequency calculations. Intrinsic reaction coordinate

Computational Methods
All calculations were carried out with Gaussian 09 [55] and Gaussian 16 [56], using methods and basis sets described in the text. Graphics of structures used in the potential energy diagrams were generated with GaussView6 [57]. The vibrational spectra were produced using Avogadro [58]. Restricted methods were used for singlet species and unrestricted for triplet. The spin expectation values, <S 2 >, were found to be 2.000 for all the triplet carbenes. The stationary points obtained from geometry optimizations were verified as minima (zero imaginary frequency) or transition states (one imaginary frequency) by subsequent frequency calculations. Intrinsic reaction coordinate calculations were also performed on all transition states found at the B3LYP/6-311+G** level of theory to ensure that they connected the correct minima.
The CCSD/6-311+G** and CCSD/cc-pVTZ//CCSD/6-311+G** T1 diagnostic [59] was computed for all structures, and the values were generally less than 0.02, as recommended. However, for a few cases, specifically in the beta series, the T1 value was in the range of 0.02-0.05: not alarmingly high, but slightly above the ideal range. In order to confirm that the computed energies were still reliable, we carried out additional single-point calculations for the beta series. We chose for this purpose the Brueckner doubles method, with quadruples and triples: BD(TQ) [60][61][62]. It has the advantage of representing a slightly different approach to configuration interaction than coupled cluster, and especially with quadruple as well as triple excitations included, should provide even more reliable energies [63]. Since BD(TQ)/cc-pVTZ would be quite demanding computationally, we instead performed BD(TQ)/cc-pVDZ and BD(T)/cc-pVTZ calculations, and estimated the BD(TQ)/cc-pVTZ energy as BD(TQ)/cc-pVDZ + BD(T)/cc-pVTZ -BD(T)/cc-pVDZ. The energy differences computed using this procedure differed from those obtained using CCSD/cc-pVTZ//CCSD/6-311+G** by −0.1 to +0.7 kcal/mol, i.e., they were generally 0.3-0.5 kcal/mol higher, but never more than 0.7 higher (or more than 0.1 kcal/mol lower). The similarity of results using a significantly different correlation procedure lends confidence to the energies reported above, despite the somewhat higher than ideal T1 values for a limited number of cases. A spreadsheet summarizing the computational results is provided in the Supplementary Materials.

Conclusions
A series of cyclic alkylidenecarbenes, formally obtained by replacing the carbonyl oxygen of four-, five-, and six-membered lactams, lactones, and thiolactones with a divalent carbon, were modeled at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** levels of theory. In all cases, the singlet carbenes were found to be considerably more stable than the triplets. The ∆E S-T gap increased with increasing ring size for each type of heteroatom substituent. Structures and energies of the cycloalkynes formed by ring expansion of the singlet carbenes were modeled using both levels of theory described above, although 1-oxacyclopent-2-yne (15) was found to be a minimum by CCSD/6-311+G** calculations but not at the B3LYP/6-311+G* level. The structures of 1-azacyclopent-2-yne (13) and 15 displayed elongated lengths for the alkynl bonds. Furthermore, 2-(1-azacyclobutylidene)carbene (12) was found to be nearly isoenergetic with its ring-expanded isomer 13, and 15 was notably higher in energy than 2-(1-oxacyclobutylidene)carbene (14). In all other cases, the cycloalkynes were lower in energy than the corresponding carbenes.
As ring expansion of the title alkylidenenecarbenes in this study could occur by a 1,2-shift of either the heteroatom or carbon, both pathways were modeled. For the nitrogen-and oxygen-substituted systems, the barrier for 1,2-carbon shifts were always lower in energy than those for the corresponding 1,2-nitrogen or oxygen shifts. In the case of sulfur-substituted carbenes, however, sulfur migration was significantly more facile than the carbon shift. These predictions, summarized in Figure 20, offer a platform for experimental verification using carbenes bearing appropriate isotopic labels. calculations were also performed on all transition states found at the B3LYP/6-311+G** level of theory to ensure that they connected the correct minima. The CCSD/6-311+G** and CCSD/cc-pVTZ//CCSD/6-311+G** T1 diagnostic [59] was computed for all structures, and the values were generally less than 0.02, as recommended. However, for a few cases, specifically in the beta series, the T1 value was in the range of 0.02-0.05: not alarmingly high, but slightly above the ideal range. In order to confirm that the computed energies were still reliable, we carried out additional single-point calculations for the beta series. We chose for this purpose the Brueckner doubles method, with quadruples and triples: BD(TQ) [60][61][62]. It has the advantage of representing a slightly different approach to configuration interaction than coupled cluster, and especially with quadruple as well as triple excitations included, should provide even more reliable energies [63]. Since BD(TQ)/cc-pVTZ would be quite demanding computationally, we instead performed BD(TQ)/cc-pVDZ and BD(T)/cc-pVTZ calculations, and estimated the BD(TQ)/cc-pVTZ energy as BD(TQ)/cc-pVDZ + BD(T)/cc-pVTZ -BD(T)/cc-pVDZ. The energy differences computed using this procedure differed from those obtained using CCSD/cc-pVTZ//CCSD/6-311+G** by -0.1 to +0.7 kcal/mol, i.e., they were generally 0.3-0.5 kcal/mol higher, but never more than 0.7 higher (or more than 0.1 kcal/mol lower). The similarity of results using a significantly different correlation procedure lends confidence to the energies reported above, despite the somewhat higher than ideal T1 values for a limited number of cases. A spreadsheet summarizing the computational results is provided in the Supplementary Materials.

Conclusions
A series of cyclic alkylidenecarbenes, formally obtained by replacing the carbonyl oxygen of four-, five-, and six-membered lactams, lactones, and thiolactones with a divalent carbon, were modeled at the CCSD(T)/cc-pVTZ//B3LYP/6-311+G** and CCSD(T)/cc-pVTZ//CCSD/6-311+G** levels of theory. In all cases, the singlet carbenes were found to be considerably more stable than the triplets. The ES-T gap increased with increasing ring size for each type of heteroatom substituent. Structures and energies of the cycloalkynes formed by ring expansion of the singlet carbenes were modeled using both levels of theory described above, although 1-oxacyclopent-2-yne (15) was found to be a minimum by CCSD/6-311+G** calculations but not at the B3LYP/6-311+G* level. The structures of 1azacyclopent-2-yne (13) and 15 displayed elongated lengths for the alkynl bonds. Furthermore, -azacyclobutylidene)carbene (12) was found to be nearly isoenergetic with its ring-expanded isomer 13, and 15 was notably higher in energy than --oxacyclobutylidene)carbene (14). In all other cases, the cycloalkynes were lower in energy than the corresponding carbenes.
As ring expansion of the title alkylidenenecarbenes in this study could occur by a 1,2-shift of either the heteroatom or carbon, both pathways were modeled. For the nitrogen-and oxygensubstituted systems, the barrier for 1,2-carbon shifts were always lower in energy than those for the corresponding 1,2-nitrogen or oxygen shifts. In the case of sulfur-substituted carbenes, however, sulfur migration was significantly more facile than the carbon shift. These predictions, summarized in Figure 20, offer a platform for experimental verification using carbenes bearing appropriate isotopic labels. This work bears a striking parallel to the experimental observations of Robson and Shechter who investigated the migratory aptitudes in carbenes of the type 30 ( Figure 21) generated from the corresponding diazo compounds [64]. They noted that neither nitrogen nor oxygen substituents migrated to the carbenic center in 30 but the sulfur group did undergo a 1,2-shift. This work bears a striking parallel to the experimental observations of Robson and Shechter who investigated the migratory aptitudes in carbenes of the type 30 ( Figure 21) generated from the corresponding diazo compounds [64]. They noted that neither nitrogen nor oxygen substituents migrated to the carbenic center in 30 but the sulfur group did undergo a 1,2-shift. As alluded to above, one explanation for the "nonmigration" of nitrogen and oxygen may be due to their ability to stabilize the transition state for 1,2-carbon shift. Results of NPA calculations [53,54] (see Supplementary Materials) reveal that in all of the title carbenes in this study (with the sole exception of the sulfur-containing 28), there is a depletion of electron density at the migration origin and an accumulation of negative charge at the migration terminus, during the carbon shift. Perhaps the nitrogen and oxygen, both of which are effective  donors, stabilize the developing positive charge at the adjacent carbon (migration origin). Another explanation is that the nitrogenand oxygen-containing alkylidene carbenes reported in this study enjoy resonance stabilization as exemplified in Figure 2. Such resonance effects strengthen the X-Csp2 bond making it harder to cleave. Furthermore, the lone pair on oxygen and nitrogen can remain in conjugation with the adjacent carbon-carbon  bond even during the 1,2-carbon shift. Sulfur, on the other hand, is less effective at  conjugation with carbon due to significant differences in the sizes and energies of the interacting p orbitals in the two atoms (3p vs 2p). Furthermore, given the larger size and nucleophilicity of sulfur, it can interact with the empty orbital on the carbenic center to initiate bonding en route to a 1,2-shift.
Vibrational spectra were calculated for all carbenes (singlets and triplets), and their ringexpanded isomers. These spectra are of potential value in matrix isolation experiments aimed at generating these species. They allow comparison of experimentally determined vibrational frequencies with those that are computed to facilitate identification.  [64] to probe migratory aptitudes of substituents at the β-position.
As alluded to above, one explanation for the "nonmigration" of nitrogen and oxygen may be due to their ability to stabilize the transition state for 1,2-carbon shift. Results of NPA calculations [53,54] (see Supplementary Materials) reveal that in all of the title carbenes in this study (with the sole exception of the sulfur-containing 28), there is a depletion of electron density at the migration origin and an accumulation of negative charge at the migration terminus, during the carbon shift. Perhaps the nitrogen and oxygen, both of which are effective π donors, stabilize the developing positive charge at the adjacent carbon (migration origin). Another explanation is that the nitrogen-and oxygen-containing alkylidene carbenes reported in this study enjoy resonance stabilization as exemplified in Figure 2. Such resonance effects strengthen the X-C sp2 bond making it harder to cleave. Furthermore, the lone pair on oxygen and nitrogen can remain in conjugation with the adjacent carbon-carbon π bond even during the 1,2-carbon shift. Sulfur, on the other hand, is less effective at π conjugation with carbon due to significant differences in the sizes and energies of the interacting p orbitals in the two atoms (3p vs. 2p). Furthermore, given the larger size and nucleophilicity of sulfur, it can interact with the empty orbital on the carbenic center to initiate bonding en route to a 1,2-shift.
Vibrational spectra were calculated for all carbenes (singlets and triplets), and their ring-expanded isomers. These spectra are of potential value in matrix isolation experiments aimed at generating these species. They allow comparison of experimentally determined vibrational frequencies with those that are computed to facilitate identification.