Coarctate and Möbius: The Helical Orbitals of Allene and Other Cumulenes

Abstract

As brought to the attention of the community by Hendon et al. and noted by previous workers, the π orbitals of the equilibrium geometry odd-carbon (even number of double bonds = n) [n]cumulenes may be written in either rectilinear or helical form. We trace the origins and detailed composition of the helical orbitals of cumulenes, which emerge in the simplest Hückel model and are not much modified in advanced computations. For the α,ω-disubstituted even [n]cumulenes, the helical representation is obligatory as the symmetry is reduced from D2d to C2. A relationship is apparent between these helical orbitals of the even [n]cumulenes, seen as a Herges coarctate system, and the corresponding Möbius cyclic polyene orbitals. The twist of the orbitals varies in interesting ways along the helix, and so does the contribution of the component atomic orbitals. Though the electronic structures of even [n]cumulenes and Möbius cyclopolyenes are closely related, they differ for higher n in intriguing ways; these are linked to the constrained rotation of the basis orbitals along the helical twist itinerary. Relations are constructed between the level patterns of the π-systems of even [n]cumulenes and ideas of Hückel and Möbius aromaticity.

Does the Electronic Structure of Möbius Annulenes Follow Heilbronner's Ideas?

“…The close correspondence between Heilbronner's ideas and the SCGVB (8,9) description of the monocyclic cyclononatetraenyl cation is confirmed by the overlap integrals between the SC orbitals from Figure 3(a). As can be seen from Table 2, the overlap integrals between consecutive SC orbitals around the ring, y m y mþ1 , are all positive with the exception of y 1 h jy 9 i, Figure 3.…”

“…The optimized geometry was verified as a local minimum through diagonalization of the MP2/aug-cc-pVDZ analytic nuclear Hessian. Fullyvariational SCGVB (8,9) and CASSCF (8,9) calculations were carried out for the 1 1 A electronic ground state at the MP2/augcc-pVDZ optimized geometry, using the aug-cc-pVDZ and augcc-pVTZ basis sets. The differences from the B3LYP/6-311 + G(d,p) geometry reported by Mauksch et al [20] are relatively minor: As shown in Figure 3, MP2/aug-cc-pVDZ yields slightly longer carbon-carbon bond lengths.…”

“…The lowest harmonic frequency, corresponding to a normal mode of B symmetry, was obtained as 179.6 cm À1 . For the SCGVB (8,9) calculations we used the CASVB algorithms [22][23][24][25] implemented in MOLPRO. Fullyvariational SCGVB (8,9) and CASSCF (8,9) calculations were carried out for the 1 1 A electronic ground state at the MP2/augcc-pVDZ optimized geometry, using the aug-cc-pVDZ and augcc-pVTZ basis sets.…”

“…The differences from the B3LYP/6-311 + G(d,p) geometry reported by Mauksch et al [20] are relatively minor: As shown in Figure 3, MP2/aug-cc-pVDZ yields slightly longer carbon-carbon bond lengths. [26,27] The CASSCF (8,9) calculations were performed both with GAUS-SIAN16 [21] and with MOLPRO, [26,27] which produced identical results; additionally, the nine active space orbitals from the CASSCF (8,9) calculations were localized using the Foster-Boys procedure [28] implemented in GAUSSIAN16. For the SCGVB (8,9) calculations we used the CASVB algorithms [22][23][24][25] implemented in MOLPRO.…”

“…The shapes of the nine non-orthogonal SC orbitals from the SCGVB (8,9) wavefunction are superimposed in Figure 3(a). Despite the twisted structure of C 9 H þ 9 , all of these orbitals are very similar in appearance.…”