What is three dimensional aromaticity?

Question

I have recently come across a statement which states:

The relative stability of the 1,3-dehydro-5,7-adamantanediyl dication is ascribed to its three-dimensional aromaticity. [1]

My understanding of aromatic nature is, that the molecule must be cyclic and planar, every atom in the ring must be $\ce{sp^2}$-hybridized and the molecule must have [4n+2] $\pi$ electrons, where $n\in\mathbb{N}$ (Hückel’s rule). The molecular orbitals also have to have the correct orientation and phase for aromaticity (2).

In my understanding 1,3-dehydro-5,7-adamantanediyl dication does not follow these rules for Y-aromaticity (3).

So my question is: What is three-dimensional aromaticity and how does it stabilise the 1,3-dehydro-5,7-adamantanediyl dication?

References

1. Firme, C. L.; Antunes, O. A. C.; Esteves, P. M. Electronic nature of the aromatic adamantanediyl ions and its analogues. J. Braz. Chem. Soc. 2008, 19 (1), 140–149. DOI: 10.1590/S0103-50532008000100020.

2. LibreTexts

3. What is Y-aromaticity? Is the trinitromethanide anion aromatic?

Answer 1

Aromaticity in Organic chemistry textbooks covers the Huckel version, which abide by the rules you have correctly listed above. However, these rules are mainly a tool to identify an organic molecule which has a high delocalization of electrons across a ring structure (a valuable tool in identifying a molecule's stability/reactivity/etc...).

So, for 3-dimensional aromaticity, the molecule must reach a certain threshold of delocalization in the ring structure (which, in 3D, becomes more of a sphere). Rules such as trigonal planar hybridization in 3D are no longer useful, and thus different approaches are taken to measure the resonance state of a system, and thus its level of aromaticity.

Nozawa et al. (2016, cited below) discusses use of "[t]he bond length alternation (BLA) in cyclic compounds" as "a good indicator of aromaticity, as it allows an evaluation of the degree of effective $\pi$-electron delocalization." These values, using a "harmonic oscillator model of aromaticity (HOMA)," are close to 1 for aromatic molecules by definition, and Nozawa obtains these from a variety of experimental and computational techniques to compare aromaticity in normally anti-aromatic structures.

For your above structure, as some comments pointed out as well, they are mesomeric, or resonance structures of the molecule illustrating the ability for the electrons to delocalize to an aromatic state. Firme et al's article mentions "delocalization indexes among bridged atoms," what I assume to be yet another similar measure of aromaticity related to the BLA using HOMA quantification.

Hope this helps! Please let me know if there is any confusion, as this response is a bit packed with summary detail.

References: 1. Nozawa, R. et al. Stacked antiaromatic porphyrins. Nature Communications 7, 13620 (2016).