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While studying about hyperconjugation, I came across the following compound which posseses one α-hydrogen (with respect to the $\mathrm{sp^2}$ hybridised carbon atom with an empty $\mathrm{p}$-orbital), however, apparently does not show a hyperconjugation effect.

3,3‐dimethylbicyclo[2.2.1]heptan‐2‐ylium

This led me to draw an ‘assumption’ that an alpha hydrogen on a bridged carbon atom may not get involved in hyperconjugation when placed next to a carbocation, alkene, free radical, or an arene. Is this true? If so, then why?

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Hyperconjugation is indeed very difficult to practically in cases involving bridgehead protons. The underlying reason is essentially the same that also explains why these protons cannot be enolised if there is a carbonyl group adjacent to them. This is best explained by drawing out the compound in a three-dimensional manner, which I have done for you in the figure below.

approximate 3D representation of 1,1-dimethylbicyclo[2.2.1]heptan-2-yl cation
Figure 1: approximate three-dimensional representation of the cation in question; cation represented by the p orbital.

If you look at figure 1 closely, you can see that the empty p orbital and the bridgehead hydrogen σ-bond are almost at a $90^\circ$ angle towards each other. For hyperconjugation, you want approximately parallel arrangements, so this obviously fails.

However, you should always be aware that this type of compounds, essentially a norbonyl cation, is essentially the example species for non-classical cations. In this case, instead of a hyperconjugation involving the $\ce{C-H}$ bond one essentially expects the $\ce{C-C}$ bond of the ‘back half’ of the six-membered ring to provide that hyperconjugation. For a more accurate and detailed explanation, please consult the Wikipedia page or the textbook of your choice.

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