  The electric quadrupole-quadrupole ($\mathcal{E}_{qq}$) interaction is
believed to play an important role in the broken symmetry transition from Phase
I to II in solid hydrogen. To evaluate this, we study structures adopted by
purely classical quadrupoles using Markov Chain Monte Carlo simulations of fcc
and hcp quadrupolar lattices. Both undergo first-order phase transitions from
rotationally ordered to disordered structures, as indicated by a discontinuity
in both quadrupole interaction energy ($\mathcal{E}_{qq}$) and its heat
capacity. Cooling fcc reliably induced a transition to the P$a3$ structure,
whereas cooling hcp gave inconsistent, frustrated and $c/a$-ratio-dependent
broken symmetry states. Analysing the lowest-energy hcp states using simulated
annealing, we found P$6_3/m$ and P$ca2_1$ structures found previously as
minimum-energy structures in full electronic structure calculations. The
candidate structures for hydrogen Phases III-V were not observed. This
demonstrates that $\mathcal{E}_{qq}$ is the dominant interaction determining
the symmetry breaking in Phase II. The disorder transition occurs at
significantly lower temperature in hcp than fcc, showing that the
$\mathcal{E}_{qq}$ cannot be responsible for hydrogen Phase II being based on
hcp.

Materials Science;Statistical Mechanics