  We present an accurate computational study of the electronic structure and
lattice dynamics of solid molecular hydrogen at high pressure. The band-gap
energies of the $C2/c$, $Pc$, and $P6_3/m$ structures at pressures of 250, 300,
and 350 GPa are calculated using the diffusion quantum Monte Carlo (DMC)
method. The atomic configurations are obtained from ab-initio path-integral
molecular dynamics (PIMD) simulations at 300 K and 300 GPa to investigate the
impact of zero-point energy and temperature-induced motion of the protons
including anharmonic effects. We find that finite temperature and nuclear
quantum effects reduce the band-gaps substantially, leading to metallization of
the $C2/c$ and $Pc$ phases via band overlap; the effect on the band-gap of the
$P6_3/m$ structure is less pronounced. Our combined DMC-PIMD simulations
predict that there are no excitonic or quasiparticle energy gaps for the $C2/c$
and $Pc$ phases at 300 GPa and 300 K. Our results also indicate a strong
correlation between the band-gap energy and vibron modes. This strong coupling
induces a band-gap reduction of more than 2.46 eV in high-pressure solid
molecular hydrogen. Comparing our DMC-PIMD with experimental results available,
we conclude that none of the structures proposed is a good candidate for phases
III and IV of solid hydrogen.
