  We report first-principles calculations of the frequency dependent linear and
second-order optical properties of the two probable extended-solid phases of
CO$_2$--V, i.e. $I\bar42d$ and $P2_12_12_1$. Compared to the parent $Cmca$
phase the linear optical susceptibility of both phases is much smaller. We find
that $I\bar42d$ and $P2_12_12_1$ differ substantially in their linear optical
response in the higher energy regime. The nonlinear optical responses of the
two possible crystal structures differ by roughly a factor of five. Since the
differences in the nonlinear optical spectra are pronounced in the low energy
regime, i.e. below the band gap of diamond, measurements with the sample inside
the diamond anvil cell are feasible. We therefore suggest optical experiments
in comparison with our calculated data as a tool for the unambiguous
identification of the high pressure phase of CO$_2$.
