  We present an accurate study of the static-nucleus electronic energy band gap
of solid molecular hydrogen at high pressure. The excitonic and quasiparticle
gaps of the $C2/c$, $Pc$, $Pbcn$, and $P6_3/m$ structures at pressures of 250,
300, and 350~GPa are calculated using the fixed-node diffusion quantum Monte
Carlo (DMC) method. The difference between the mean-field and many-body band
gaps at the same density is found to be almost independent of system size and
can therefore be applied as a scissor correction to the mean-field gap of an
infinite system to obtain an estimate of the many-body gap in the thermodynamic
limit. By comparing our static-nucleus DMC energy gaps with available
experimental results, we demonstrate the important role played by nuclear
quantum effects in the electronic structure of solid hydrogen. Our DMC results
suggest that the metallization of high-pressure solid hydrogen occurs via a
structural phase transition rather than band gap closure.
