  We present a theoretical study on the behavior under pressure of the two
hypothetical C$_{46}$ and Li$_8$C$_{46}$ type-I carbon clathrates in order to
bring new informations concerning their synthesis. Using \textit{ab initio}
calculations, we have explored the energetic and structural properties under
pressure of these two carbon based cage-like materials. These low-density
meta-stable phases show large negative pressure transitions compared to diamond
which represent a serious obstacle for their synthesis. However, we evidence
that a minimum energy barrier can be reached close to 40 GPa, suggesting that
the synthesis of the Li-clathrate under extreme conditions of pressure and
temperature may be possible. Electronic band structure with related density of
states behavior under pressure as well as the dependence of the active Raman
modes with pressure are also examined.
