How small is the smallest known carbon ring containing only double bonds?

How small is the smallest known carbon ring containing only double bonds?

The stress from the double bonds makes small rings impossible to make something like cyclohexahexaene so I want to know how large the ring would need to be so the stress is small enough such that the ring can be made.

TL;DR: Isolated derivatives of cyclo-$$\ce{C18}$$ most likely have the smallest isolated cumulenic 18-membered ring, though upon direct complexation of transition metal in/outside the ring the strain as well as its size can be reduced further.

In the past two decades a research group of Prof. Dr. François Diederich performed numerous studies on cyclo[n]carbons and monocyclic $$\mathrm{sp}$$-hybridized C-rings. He also published several reviews, e.g. see chapters "Cyclic and Linear Acetylenic Molecular Scaffolding" in [1, p. 43], and more recent "Oligoacetylenes" in [2, p. 443] devoted to versatile chemistry of cumulenes.

As @R.M. referred in the comment, early theoretical work [3] predicted special Hückel-aromatic cyclo-$$\ce{C18}$$ compound stabilized via two orthogonal $$(4n + 2)$$ $$\pi$$-electron systems, but later computational studies predict somewhat controversial electronic structures (from [2]):

Self-consistent field (SCF) calculations with a 3-21G or larger basis set predicted that the cyclic acetylenic $$D_{\mathrm{9h}}$$ structure 1a with alternating bond lengths represents the ground-state geometry[4].

But optimizations at the Møller-Plesset second-order perturbation theory (MP2) level including valence electron correlations as well as density functional theory calculations favor the cumulenic $$D_{\mathrm{18h}}$$ structure 1b as the most stable planar monocyclic geometry [5].

More from [1, p. 46]:

Accordingly, cyclo-$$\ce{C10}$$ is predicted to prefer a cumulenic $$D_{\mathrm{5h}}$$ structure, cyclo-$$\ce{C14}$$ is borderline, and the energies of its cumulenic $$D_{\mathrm{7h}}$$ and polyynic $$C_{\mathrm{7h}}$$ structures are nearly degenerate. It is clear that this fascinating theoretical controversy over the past years can only be definitively solved with the synthesis and characterization of cyclo-$$\ce{C18}$$.

cyclo-$$\ce{C18}$$ appeared to be a very elusive compound to isolate (on macroscoping scale typical products are anthracene and polymers), therefore only its derivatives were synthesized and characterized. Synthetic methods include the following paths: Retro-Diels-Alder; 3-Cyclobutene-l,2-dione route; transition metal complexation [2]:

Generally, cyclic cumulenes have two types of strains: distortion from linearity imposed by $$\mathrm{sp}$$-hybridization and distortion from orthogonality of adjacent double bonds (cyclo[2n]carbons). [6]

Attempts to liberate the cyclocarbons strain in cyclic cumulenes is similarly relieved by complexing one of the double bonds directly to a transition metal. For example, like in this zirconocene-hexapentaene complex, where $$\ce{C=C=C}$$ angle is reduced to approx. $$130^\circ$$ by complexation with $$\ce{Zr}$$ [7]:

Unfortunally, up to date there are only mononuclear metal-cumulene complexes isolated in a large scale, and none of them are cyclo[n]carbons, but theoretically polynuclear transition metal complexation can significantly reduce the ring size.

References

1. Carbon Rich Compounds II, Macrocyclic Oligoacetylenes and Other Linearly Conjugated Systems; de Meijere, A., Ed.; Topics in Current Chemistry; Springer Berlin Heidelberg: Berlin, Heidelberg, 1999; Vol. 201; ISBN 3540653015.
2. Stang, P. J.; Diederich, F. Modern Acetylene Chemistry; Wiley-VCH: Weinheim, 2008; ISBN 9783527615261.
3. Hoffmann, R. Tetrahedron 1966, 22 (2), 521–538. DOI 10.1016/0040-4020(66)80020-0.
4. Diederich, F.; Rubin, Y.; Knobler, C. B.; Whetten, R. L.; Schriver, K. E.; Houk, K. N.; Li, Y. Science 1989, 245 (4922), 1088–1090. DOI 10.1126/science.245.4922.1088.
5. Hutter, J.; Luethi, H. P.; Diederich, F. J. Am. Chem. Soc. 1994, 116 (2), 750–756. DOI 10.1021/ja00081a041.
6. Stone, F. G. A.; West, R. Advances in organometallic chemistry; 1998; Vol. 42. ISBN 0120311429.
7. Suzuki, N.; Hashizume, D.; Yoshida, H.; Tezuka, M.; Ida, K.; Nagashima, S.; Chihara, T. J. Am. Chem. Soc. 2009, 131 (6), 2050–2051. DOI 10.1021/ja8077472.