  Experimental progress finally reached the metallic solid hydrogen phase,
which was predicted by Wigner and Huntington over 80 years ago. However, the
different structures in the phase diagram are still been debated due to the
difficulty of diffraction experiments for high-pressured hydrogen. The
determination of crystal structures under extreme condition is both of the
basic condensed matter physics, and in planetary science: the behavior of giant
gaseous planets (e.g. Jupiter, Saturn...) strongly depends on the properties of
inner high-pressured hydrogen. This work describes new possible structures
appearing under high pressures of 400$\sim$600 GPa. We applied a structural
search using particle swarm optimization with density functional theory (DFT)
to propose several candidate structures. For these structures, we performed
fixed-node diffusion Monte Carlo simulations combined with DFT zero-point
energy corrections to confirm their relative stability. We found $P2_{1}/c$-8
as a promising candidate structure for the H$_2$-PRE phase. $P2_{1}/c$-8 is
predicted the most stable at 400 and 500~GPa. $P2_{1}/c$-8 reproduces
qualitatively the IR spectrum peaks observed in the H$_2$-PRE phase.
