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Could cyclotetrazane — a compound with nitrogen atoms arranged in a square configuration — be synthesized? A hydrogen atom would be bonded to each vertex (nitrogen atom):

Tetrazetidine

Could cyclotetrazane be made by removing two hydrogen atoms from hydrazine to form diazene, but before the nitrogen forms double bonds, make the nitrogen form single bonds with another molecule of diazene?

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What might formally be considered a magnesium salt, $\ce{Mg2[N4]}$, has been formed from the elements at 50 GPa pressure[1]. The salt is recovered as a metastable compound at atmospheric pressure.

The $\ce{N4^{4-}}$ anion in this compound is not exactly a deprotonated cyclo-$\ce{N4H4}$ molecule. Rather, one of the proposed sigma bonds is lost; but the atoms that would have been joined by the lost bond remain cis to each other, making the ion a trapezoid. Possibly this trapezoidal structure is stabilized by a homoaromatic coupling, since breaking the sigma bond leads to having six conjugated pi-symmetry electrons.

Reference

  1. Laniel, D., Winkler, B., Koemets, E. et al. "Synthesis of magnesium-nitrogen salts of polynitrogen anions". Nat Commun 10, 4515 (2019). https://doi.org/10.1038/s41467-019-12530-w
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Tl:DR: These sorts of compound are extremely unstable largely due to ring strain, so it is unlikely that it will be synthesized.

Long Answer:

The study of the chemistry of experimentally known nitrogen homocycle was initially limited to two classes of compounds: 1.saturated three membered rings (triaziridines) 2. pentazoles (N,H). Later, the study was expanded to three nitrocycles: cyclotetrazene dianion ($\ce{N4^2-}$), the neutral 1,3-dihydrocyclotetrazene ($\ce{N2H2N2}$), and the cyclotetrazane dication ($\ce{N4H4^{2-}}$).

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These species were experimentally studied and because they are highly unstable, they study is confined to pure theory only. For more details, refer to the paper:

Zandwijk, van, G., Janssen, R. A. J., & Buck, H. M. (1990). 6pi Aromaticity in four-membered rings. Journal of the American Chemical Society, 112(11), 4155-4164. DOI: 10.1021/ja00167a009

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    $\begingroup$ We also have some high pressure species that may be stabilized by homoaromatic coupling; see my answer here and also this one, which includes a polymeric version of the tetranitrogen dianion. $\endgroup$ Commented Apr 20 at 11:47

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