Ground state of metallic hydrogen

A proposed liquid state of metallic hydrogen at zero temperature is explored, and a variational upper bound to the ground-state energy is calculated. It is shown that the possibility that metallic hydrogen is a liquid around the metastable point ($r_s=1.64\mathrm{}$) cannot be ruled out. This conclusion crucially hinges on the contribution to the energy arising from the third order in the electron-proton interaction, which is shown here to be more significant in the liquid phase than in crystals.