Quadrupole arrangements and the ground state of solid hydrogen

The electric quadrupole-quadrupole $\left({\mathcal{E}}_{qq}\right)$ 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 on face centered cubic (fcc) and hexagonal close packed (hcp) lattices. Both undergo first-order phase transitions from rotationally ordered to disordered structures, as indicated by a discontinuity in both quadrupole interaction energy $\left({\mathcal{E}}_{qq}\right)$ and its heat capacity. Cooling fcc reliably induced a transition to the $Pa3$ structure, whereas cooling hcp gave inconsistent, frustrated, and $c/a$-ratio-dependent broken symmetry states. Analyzing the lowest-energy hcp states using simulated annealing, we found $P{6}_3/m$ and $Pca{2}_1$ 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.

Figure 1

A selection of favorable and unfavorable orientations of two-point quadrupoles (with positive quadrupole moment $Q$) adapted from Ref. [25, p. 53]. Numbers correspond to the orientational factor $\mathrm{\Gamma }$.

Figure 2

Ideal unit cell arrangement of quadrupole rotors on a triangular lattice. The main motif consists of a central vertical rotor surrounded by six planar rotors with no frustration. This structure has P6 symmetry.

Figure 3

$T^{*}\sim 0$ structure upon cooling of disordered fcc in the conventional unit cell representation. Red dashed lines are a guide to the eye. The rotors do not lie in the close-packed plane like in the ground-state triangular lattice.

Figure 4

$T^{*}\sim 0$ energies per rotor obtained by simulated annealing of disordered hcp for all three supercells. The lines indicate the lowest obtained energies for each supercell, from which intersections distinguish the various phases. The 6 x 6  x 3 data did not produce any lower energy structures.

Figure 5

Ground states for low to intermediate $c/a$ ratios. Left is at $c/a=1.47$ and space group $Pbca$, while right is at $c/a=1.55$ and spacegroup $Pca{2}_1$. Note that these structures are not planar: Arrows indicate the direction of the rotor that points out of the page. Red arrows correspond to the next layer stacked in the $\mathbf{c}$ direction.

Figure 6

Histograms of $cos\theta$ and $\varphi$ for all supercells, where $\theta$ and $\varphi$ are polar and azimuthal angles, as a function $c/a$ ratio.

Figure 7

Two possible stackings of triangular layers observed at $c/a>1.61$. Left is stacking in $P{6}_3/m$, which is only stable at very high $c/a$. Right is another possible stacking where the distorted layers still resemble those of $P{6}_3/m$. While the structure as a whole loses all symmetry, the individual layers possess P1 symmetry. Circles represent vertical rotors, while single triangles are near vertical. Red arrows correspond to the next layer stacked in the $\mathbf{c}$ direction.

Figure 8

Heating of the $Pa3$ fcc ground state and the four hcp ground state structures. Runs for $Pbca$, $Pca{2}_1$, P1, and $P{6}_3/m$ were performed at $c/a$ ratios of 1.4, 1.6, 1.75, and 2.1 respectively. All are observed to undergo a first-order phase transition. While the critical temperature varies among the structures, the low- and high-temperature heat capacities are similar. Because the autocorrelation time diverges around the critical point, the error bars in that region are likely underestimated.