Chemistry of Aromaticity. What's non hückel double bond ? What are the difference between hückel and non-hückel double bond ?enter image description here [ Here, in figure the green one is non hückel double bond ]

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    $\begingroup$ There is no intrinsic difference. Hückel's rule counts the number of pi electrons in a closed ring. It says nothing about other double bonds. $\endgroup$ – Karl Jul 14 '19 at 23:26

In the given molecule Pyrene , the π 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 $\ce{C=C}$ isn't hosted by the largest p orbital loop, so it's not counted towards Hückel's rule for aromaticity.

The outer periphery (shown in blue in figure 3), is a loop of $\ce{14 π}$ electrons , therefore aromatic. This may be possible hückel double bond you are referring to.

The π bond in cross-conjugation (shown in green) maybe non-hückel double bond.

Pyrene structure

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    $\begingroup$ Beware: The central pi bond does not count towards the Hückel rule, however that does NOT mean that bond is not part of the aromatic system! $\endgroup$ – Karl Jul 14 '19 at 23:23

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