  We performed simulations for solid molecular hydrogen at high pressures
(250GPa$\leq$P$\leq$500GPa) along two isotherms at T=200 K (phases III and VI)
and at T=414 K (phase IV). At T=200K we considered likely candidates for phase
III, the C2c and Cmca12 structures, while at T=414K in phase IV we studied the
Pc48 structure. We employed both Coupled Electron-Ion Monte Carlo (CEIMC) and
Path Integral Molecular Dynamics (PIMD) based on Density Functional Theory
(DFT) using the vdW-DF approximation. The comparison between the two methods
allows us to address the question of the accuracy of the xc approximation of
DFT for thermal and quantum protons without recurring to perturbation theories.
In general, we find that atomic and molecular fluctuations in PIMD are larger
than in CEIMC which suggests that the potential energy surface from vdW-DF is
less structured than the one from Quantum Monte Carlo. We find qualitatively
different behaviors for systems prepared in the C2c structure for increasing
pressure. Within PIMD the C2c structure is dynamically partially stable for
P$\leq$250GPa only: it retains the symmetry of the molecular centers but not
the molecular orientation; at intermediate pressures it develops layered
structures like Pbcn or Ibam and transforms to the metallic Cmca-4 structure at
P$\geq$450GPa. Instead, within CEIMC, the C2c structure is found to be
dynamically stable at least up to 450GPa; at increasing pressure the molecular
bond length increases and the nuclear correlation decreases. For the other two
structures the two methods are in qualitative agreement although quantitative
differences remain. We discuss various structural properties and the electrical
conductivity. We find these structures become conducting around 350GPa but the
metallic Drude-like behavior is reached only at around 500GPa, consistent with
recent experimental claims.
