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Order  Stability and Resonance Energy:-  Aromatic > Homoaromatic > Antiaromatic > Nonaromatic

   Aromaticity:- in Organic Chemistry, the Aromaticity is a property of Cyclic, planar structure with a ring of resonance bonds that gives the increased stability compared to other geometric with the same set of atoms. Aromatic compounds are very stable, and they do not break apart easily to react with other compounds. 



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 Antiaromatic:- it is also a Cyclic molecule with a pi-electron system but there has higher energy due to the presence of 4n delocalized electrons in it. And Antiaromatic compounds are highly unstable and highly reactive in nature.

 Nonaromatic:- nonaromatic are every other molecules that fails one of these conditions (i.e., Cyclic, planar, conjugation and 4n or (4n+2) number of delocalized electrons, so anyone of these conditions is not fulfill, then the molecule is called non Aromatic compound).

Quasi-aromatic:- Quasi-aromatic compounds are those in which the charges are present on the ring/molecule, and they take a part of Aromaticity of the compound.

Homoaromaticity:- in Organic Chemistry, it refers to a special type of Aromaticity in which conjugation is interrupted by a single sp³ hybridized carbon atom. 

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๐Ÿ“š✔️  In the given molecule Pyrene , the ฯ€ bond shown in green is in cross conjugation (shown in green in figure 1). This bond( figure 2) is excluded from interaction with rest of conjugated system. Hence , the central C=C isn't hosted by the largest p orbital loop, so it's not counted towards Hรผckel's rule for aromaticity.
          The outer periphery (shown in blue in figure 3), is a loop of 14ฯ€ electrons , therefore aromatic. This may be possible hรผckel double bond you are referring to.
          The ฯ€ bond in cross-conjugation (shown in green) maybe non-hรผckel double bond. 


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