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Can the ground state of an organic molecule be a triplet. This would imply something like a "HOMO" formed by 2 degenerate levels and less than 4 electrons to fill them. If yes, what are the conditions?

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    $\begingroup$ There are molecules like cyclic butadiene having two degenrate level with single eletron in each with parallel spins (triplet) even molecular oxygen which we breathe is triplet oxygen. $\endgroup$ – Kushal Tripathi Jul 6 '17 at 8:47
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    $\begingroup$ Cyclobutadiene has a singlet ground state. And $\ce{O2}$ is definitely triplet, but isn't an organic molecule. $\endgroup$ – orthocresol Jul 6 '17 at 9:14
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As Martin has mentioned, carbenes are a good starting point if you are looking for organic compounds with a triplet ground state.

In these carbenes, the HOMO is not twofold degenerate, as your question suggests. Instead, there are two singly occupied orbitals with different energy. The energy gap, though, is typically small, and since there is an energy cost associated with pairing electrons in the same orbital, the electrons prefer to stay apart.

$\ce{CH2}$ itself has a triplet ground state, but is very difficult to observe; most methods of generating carbenes actually produce singlet carbenes, and $\ce{CH2}$ is so reactive that singlet $\ce{CH2}$ reacts with other stuff before it decays. Not all that interesting, eh.

There are some examples of persistent triplet carbenes, though.1 This one, reported by Hirai and Tomioka in 1999,2 has a half-life of $\pu{940 s}$ in degassed benzene at $\pu{20 ^\circ C}$:

2,6-dibromo-4-tert-butyl-2′,6′-bis(trifluoromethyl)-4′-isopropyldiphenylcarbene

In general, simple alkyl and aryl carbenes adopt a triplet ground state. These particular aryl groups on the carbene provide a large degree of kinetic stabilisation, which prevents simple dimerisation of the carbene (to form an alkene); the para alkyl groups also hinder reaction at that position. A further computational study of this molecule has been carried out by the Schaefer group, which supports the analysis above.3


References

  1. Hirai, K.; Itoh, T.; Tomioka, H. Persistent Triplet Carbenes. Chem. Rev. 2009, 109 (8), 3275–3332. DOI: 10.1021/cr800518t.

  2. Hirai, K.; Tomioka, H. A Triplet Carbene That Can Almost Be Bottled. J. Am. Chem. Soc. 1999, 121 (43), 10213–10214. DOI: 10.1021/ja991387c.

  3. Woodcock, H. L.; Moran, D.; Schleyer, P. v. R.; Schaefer, H. F., III. The Almost Bottleable Triplet Carbene:  2,6-dibromo-4-tert-butyl-2′,6′-bis(trifluoromethyl)-4′-isopropyldiphenylcarbene. J. Am. Chem. Soc. 2001, 123 (18), 4331–4335. DOI: 10.1021/ja003552q.

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There are various examples of molecules in triplet ground state. The smallest (organic) is possibly methylene, $\ce{:\!CH2}$.

Carbenes in general may adopt triplet state as their ground states.

Other possibilities are biradicals (From the IUPAC gold book):

An even-electron molecular entity with two (possibly delocalized) radical centres which act nearly independently of each other, e.g.
B00671

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