Towards a predictive first-principles description of solid molecular hydrogen with density functional theory

We examine the influence of the main approximations employed in density functional theory descriptions of the solid phase of molecular hydrogen near dissociation. We consider the importance of nuclear quantum effects on equilibrium properties and find that they strongly influence intramolecular properties, such as bond fluctuations and stability. We demonstrate that the combination of both thermal and quantum effects make a drastic change to the predicted optical properties of the molecular solid, suggesting a limited value to dynamical, e.g., finite-temperature predictions based on classical ions and static crystals. We also consider the influence of the chosen exchange-correlation density functional on the predicted properties of hydrogen, in particular, the pressure dependence of the band gap and the zero-point energy. Finally, we use our simulations to make an assessment of the accuracy of typically employed approximations to the calculation of the Gibbs free energy of the solid, namely the quasi-harmonic approximation for solids. We find that, while the approximation is capable of producing free energies with an accuracy of $\approx 10$ meV, this is not enough to make reliable predictions of the phase diagram of hydrogen from first principles due to the small free energy differences seen between several potential structures for the solid; direct free energy calculations for quantum protons are required in order to make definite predictions.

Figure 1

The proton-proton PCFs for the $C2c$ (left), $Cmca$-12 (center), and $Pbcn$ (right) structures of hydrogen for BOMD (dashed blue) and PIMD (solid red) simulations, using the PBE DF. From top to bottom the PCFs correspond to pressures of $p\sim 350$, 300, 250, and 200 GPa. All simulations were performed at $T=200$ K.

Figure 2

Lindemann ratio (top) and orientational order parameter (bottom) of various structures of solid molecular hydrogen from BOMD (solid symbols) and PIMD (open symbols) simulations, using the PBE DF. All the simulations were performed at a temperature of $T=200$ K. Squares correspond to $C2c$, circles to $Cmca$-12, and triangles to $Pbcn$.

Figure 3

PCFs for the $C2c$ (left), $Cmca$-12 (center), and $Pbcn$ (right) structures of hydrogen from PIMD simulations using PBE (solid red), HSE (dotted blue), and vdW-DF2 (dashed-dotted black). From top to bottom, the PCFs correspond to pressures of approximately $p\sim 350$, 300, 250, and 200 GPa. All simulations were performed at $T=200$ K.

Figure 4

Lindemann ratio (top) and orientational order parameter (bottom) for solid molecular hydrogen from PIMD simulations using vdW-DF2. Note that all the simulations were performed at $T=200$ K. Squares correspond to $C2c$, circles to $Cmca$-12, and triangles to $Pbcn$.

Figure 5

Band gap of $C2c$ using PBE. Red squares are the band gaps for the static crystal, green circles are band gaps for classical nuclei at 200 K, and blue triangles are band gaps for quantum nuclei at 200 K. Cyan crosses are recent experimental measurements from Goncharov et al.[40] Only one set of experimental results is shown, in order to provide a scale that allows a clear comparison between the BOMD and PIMD simulations. See Fig. 7 for a more complete set of experimental results.

Figure 6

Band gap as a function of pressure. Filled symbols correspond to PIMD simulations with HSE, while empty symbols correspond to experimental results; lines represent guides to the eye. Red squares, green circles, and blue triangles are theoretical results for the $C2c$, $Cmca$-12, and $Pbcn$ structures, respectively. The experimental results correspond to: orange pentagrams, Loubeyre et al.;[38] brown diamonds, Zha et al.;[39] cyan asterisks, Goncharov et al.;[40] and downward black triangles, Howie et al.[3]

Figure 7

Band gap as a function of pressure. Filled symbols correspond to PIMD simulations with vdW-DF2. See the caption of Fig. 6 for additional details.

Figure 8

Internal energy of $Cmca$-12, as a function of temperature, at a density of $r_s\approx 1.38$. Red squares correspond to PIMD simulations with vdW-DF2, the solid blue curve corresponds to QHA results (also using vdW-DF2) and the dashed black curve corresponds to the energy of an isolated quantum rotor, displaced to match the PIMD energy at 200 K and with a rotational constant of an isolated hydrogen molecule of ${\Theta }_{\mathrm{Rot}}\sim 87$ K.

Figure 9

Comparison between PIMD and the QHA of the thermal and quantum contribution to the internal energy of hydrogen in $C2c$ and $Cmca$-12 at 0 K (top) and 200 K (bottom). Symbols represent results with PIMD and lines the QHA. Note that in both cases, PBE was used.

Figure 10

Comparison between PIMD and the QHA of the thermal and quantum contribution to the pressure component of the enthalpy in $C2c$ and $Cmca$-12. Symbols represent results with PIMD and lines with the QHA. Note that in both cases, PBE was used.

Figure 11

Thermal and quantum contribution to the energy of hydrogen from PIMD simulations. From top to bottom the results correspond to: HSE, vdW-DF2, and PBE. Red squares, black circles, and blue triangles correspond to results for $C2c$, $Cmca$-12, and $Pbcn$, respectively. Insets show the error of the QHA, if it were to be employed.

Figure 12

Enthalpy of several structures of hydrogen at 200 K, relative to a polynomial fit to that of $C2c$. Symbols represent PIMD simulations with vdW-DF2 and lines results from the QHA.