Will bridged compounds the undergo SN1 reaction? $\mathrm{S}_{\mathrm{N}}1$ reaction involves only one molecule in the rate determining step. So, the molecule which undergoes $\mathrm{S}_{\mathrm{N}}1$ reaction should be stable when it forms a positive ion. $\ce{(CH3)3C^+}$ is considered stable due to inductive effect from the three carbon atoms and also hyper-conjugation. But, what about bridged compounds like 1-bromobicyclo[2.2.1]heptane:
  

When $\ce{Br^-}$ is abstracted the positive ion formed will be stable and therefore this will undergo $\mathrm{S}_{\mathrm{N}}1$ reaction. Am I thinking right? Please help me correct myself.
Very interesting question… I would give two answers, at different levels:


At undergrad level, I think the carbenium ion formed at the bridge head is less stabilized that “regular” ternary carbenium ions, because it cannot adopt the planar trigonal geometry which would be ideal for it.
At higher level, this is much more complicated: the norboryl cation is an example of non-classical ion…