  A hexagonal structure of solid molecular hydrogen with $P6_122$ symmetry is
calculated to be more stable below about 200 GPa than the monoclinic $C2/c$
structure identified previously as the best candidate for phase III. We find
that the effects of nuclear quantum and thermal vibrations play a central role
in the stabilization of $P6_122$. The $P6_122$ and $C2/c$ structures are very
similar and their Raman and infra-red data are in good agreement with
experiment. However, our calculations show that the hexagonal $P6_122$
structure provides better agreement with the available x-ray diffraction data
than the $C2/c$ structure at pressures below about 200 GPa. We suggest that two
phase-III-like structures may be formed at high pressures, hexagonal $P6_122$
below about 200 GPa and monoclinic $C2/c$ at higher pressures.
