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We performed simulations for solid molecular hydrogen at high pressures(250GPa$\leq$P$\leq$500GPa) along two isotherms at T=200 K (phases III and VI)and at T=414 K (phase IV). At T=200K we considered likely candidates for phaseIII, the C2c and Cmca12 structures, while at T=414K in phase IV we studied thePc48 structure. We employed both Coupled Electron-Ion Monte Carlo (CEIMC) andPath Integral Molecular Dynamics (PIMD) based on Density Functional Theory(DFT) using the vdW-DF approximation. The comparison between the two methodsallows us to address the question of the accuracy of the xc approximation ofDFT for thermal and quantum protons without recurring to perturbation theories.In general, we find that atomic and molecular fluctuations in PIMD are largerthan in CEIMC which suggests that the potential energy surface from vdW-DF isless structured than the one from Quantum Monte Carlo. We find qualitativelydifferent behaviors for systems prepared in the C2c structure for increasingpressure. Within PIMD the C2c structure is dynamically partially stable forP$\leq$250GPa only: it retains the symmetry of the molecular centers but notthe molecular orientation; at intermediate pressures it develops layeredstructures like Pbcn or Ibam and transforms to the metallic Cmca-4 structure atP$\geq$450GPa. Instead, within CEIMC, the C2c structure is found to bedynamically stable at least up to 450GPa; at increasing pressure the molecularbond length increases and the nuclear correlation decreases. For the other twostructures the two methods are in qualitative agreement although quantitativedifferences remain. We discuss various structural properties and the electricalconductivity. We find these structures become conducting around 350GPa but themetallic Drude-like behavior is reached only at around 500GPa, consistent withrecent experimental claims.

Coupled Electron-Ion Monte Carlo simulation of hydrogen molecular crystals