The structure of high pressure hydrogen has attracted enormous attention, but convincing agreement between theory and experiment about the crystal structures remain elusive.  Experimental data comes from X-ray, which cannot resolve molecular orientation, and spectroscopy which has no direct to structures.  Calculations can find low energy atomic positions, but these produce equivocal results which are sensitive to methodological details.  Here, I describe our efforts to determine the phases observed at room temperature (I, III, IV, IV' V).  Using molecular dynamics and path integral molecular dynamics to describe quantum nuclear effects, it is shown that all plausible non-metallic phases involve rotating molecules, and their experimental signature will not be that of a conventional harmonic solid.  We calculate measurable Raman and X-ray signatures for these phases.   With increasing pressure, we find that in Phase I there is an isosymmetric shift from 3-D rotor to 2-D rotor behaviour.   Long-lived local ordering typical of Phase III appears before the full long ranged symmetry breaking.  The local ordering breaks the symmtry such that each molecular acquires a dipole moment. For the system sizes we look at, it is difficult to be sure whether long-ranged symmetry breaking occurs at all.  Phase IV entails the breaking of the C-glide symmetry from hcp, a transformation from P63/mmm to P6/mmm.  Nevertheless, the unit cell remains haxagonal, and the transformation is predicted to have a only subtle effect on the X-ray pattern.   Higher pressure phases are all hexagonal, involving various symmetry-breakings which, remarkably, conspire to remain in the   P6/mmm space group by combined with unit-cell-doubling.   Throughout all this process, the DFT band gap is closing but all phases remain molecular and non-metallic.

Path Integral molecular dynamics show relatively little qualitative change from the classical picture, the main effect being a wider variation in molecular length due to zero-point energy.  There is some small effect on the phase boundaries.