Stable Solid Molecular Hydrogen above 900 K from a Machine-Learned Potential Trained with Diffusion Quantum Monte Carlo

We survey the phase diagram of high-pressure molecular hydrogen with path integral molecular dynamics using a machine-learned interatomic potential trained with quantum Monte Carlo forces and energies. Besides the HCP and $C2/c\text{−}24$ phases, we find two new stable phases both with molecular centers in the $Fmmm\text{−}4$ structure, separated by a molecular orientation transition with temperature. The high temperature isotropic $Fmmm\text{−}4$ phase has a reentrant melting line with a maximum at higher temperature (1450 K at 150 GPa) than previously estimated and crosses the liquid-liquid transition line around 1200 K and 200 GPa.

Figure 1

Phase diagram of dense hydrogen. The dots indicate the ($P$, $T$) values where we ran PIMD simulations using a DMC-trained DPMD interatomic potential. Colors indicate the identified phase: dark blue (liquid), green (HCP), purple ($Fmmm\text{−}4$ isotropic), orange ($Fmmm\text{−}4$ oriented), and cyan ($C2/c\text{−}24$). The vertical green dashed line at 105 GPa indicates a crossover within HCP to an in-plane orientation. The thick blue line is the estimate of the transition from liquid ${\mathrm{H}}_2$ to liquid H from Ref. [10]. Red bars are estimates of the melting temperature from two-phase coexistence simulation. The black lines are experimental estimates of phase boundaries, solid lines for the melting, dashed line for $\mathrm{I}\text{−}{\mathrm{I}}^{\prime }$ transition [39].

Figure 2

(a) Unit cell volume and (b) $c/a$ ratio, vs pressure in GPa both at 300 K. Blue dots are the results of PIMD simulation. The crosses are experimental XRD measurements Ref. [40] and the lines are fits to experimental data. (c) The distribution of the molecular angle with the basal plane at various pressures within HCP at 300 K from PIMD.

Figure 3

(a) The oriented $Fmmm\text{−}4$ structure with layer 1 (green) and layer 2 (orange). The primitive cell (shown in black) contains one molecule per layer all lying in the basal plane. Layer 2 is shifted by half a primitive lattice vector from the HCP site. Also the hexagonal symmetry within the basal plane is broken. (b) Histograms of molecular orientation at $P=200\mathrm{GPa}$, where $\theta$ is the angle with respect to the basal plane and $\varphi$ is the azimuthal angle. There are three distinct in-plane orientations in the $C2/c\text{−}24$ structure at 100 K, one orientation in the $Fmmm\text{−}4$ structure at 200 K. Orientation order is absent in the isotropic $Fmmm\text{−}4$ phase at 400 K and above.

Figure 4

Comparison of melting lines from various calculations. Circles are our two-phase MD results using ML constrained by data from either PBE forces (blue) or QMC forces (purple). Open symbols represent classical protons, closed symbols quantum protons. Previous PBE-DFT predictions, all using classical protons, are reported: gold downward triangles Ref. [18], light purple upward triangles Ref. [17], orange squares Ref. [46]). Lines (either continuous or dashed) are guides for the eyes. The shaded region is the uncertainty estimated in Cheng et al. [18].