  Being the simplest element with just one electron and proton the electronic
structure of the Hydrogen atom is known exactly. However, this does not hold
for the complex interplay between them in a solid and in particular not at high
pressure that is known to alter the crystal as well as the electronic
structure. Back in 1935 Wigner and Huntington predicted that at very high
pressure solid molecular hydrogen would dissociate and form an atomic solid
that is metallic. In spite of intense research efforts the experimental
realization, as well as the theoretical determination of the crystal structure
has remained elusive. Here we present a computational study showing that the
distorted hexagonal P6$_3$/m structure is the most likely candidate for Phase
III of solid hydrogen. We find that the pairing structure is very persistent
and insulating over the whole pressure range, which suggests that metallization
due to dissociation may precede eventual bandgap closure. Due to the fact that
this not only resolve one of major disagreement between theory and experiment,
but also excludes the conjectured existence of phonon-driven superconductivity
in solid molecular hydrogen, our results involve a complete revision of the
zero-temperature phase diagram of Phase III.
