Quasiparticle energies and excitonic effects of solid hydrogen under ultrahigh pressures

We investigate the insulator-to-metal transition in the crucial $Cmca$-12 phase of solid hydrogen employing the many-body perturbation theory with Green's functions. In particular quasiparticle energies are calculated within the $GW$ approximation to accurately determine the insulator-to-metal transition pressure. We consider the effects of self-consistency, plasmon-pole models to the dielectric function, off-diagonal elements of the self-energy, and vertex corrections on $GW$ calculations, and our results show that the band gap of the $Cmca$-12 phase of solid hydrogen is sensitive to the choice of $GW$ procedures and approximations involved, leading to a change of $\sim 20$ GPa in transition pressure. We also compute the optical absorption and electron-hole binding energy by solving the Bethe-Salpeter equation, and the resulting optical absorption shows a redshift and enhancement of absorption peaks compared to the GW-RPA absorption with excitonic effects omitted.