Benchmarking density functionals for hydrogen-helium mixtures with quantum Monte Carlo: Energetics, pressures, and forces

An accurate understanding of the phase diagram of dense hydrogen and helium mixtures is a crucial component in the construction of accurate models of Jupiter, Saturn, and Jovian extrasolar planets. Though density-functional-theory-based first-principles methods have the potential to provide the accuracy and computational efficiency required for this task, recent benchmarking in hydrogen has shown that achieving this accuracy requires a judicious choice of functional, and a quantification of the errors introduced. In this work, we present a quantum Monte Carlo (QMC) -based benchmarking study of a wide range of density functionals for use in hydrogen-helium mixtures at thermodynamic conditions relevant for Jovian planets. Not only do we continue our program of benchmarking energetics and pressures, but we deploy QMC-based force estimators and use them to gain insight into how well the local liquid structure is captured by different density functionals. We find that TPSS, BLYP, and vdW-DF are the most accurate functionals by most metrics, and that the enthalpy, energy, and pressure errors are very well behaved as a function of helium concentration. Beyond this, we highlight and analyze the major error trends and relative differences exhibited by the major classes of functionals, and we estimate the magnitudes of these effects when possible.

Figure 1

$\langle |\widetilde{\delta E^{\text{DF}}}{|\rangle }_{\text{global}}$ averaged over all helium concentrations for all considered functionals. The different bar colors/patterns denote the different densities.

Figure 2

$\langle \delta [E\left(x_{\text{He}}\right)-E(x_{\text{He}}=1)]\rangle$ vs $x_{\text{He}}$ for all considered functionals at $r_s=1.10$ (left), $r_s=1.25$ (middle), and $r_s=1.34$ (right). All energies are measured relative to the average energy of all pure helium configurations at the specified density.

Figure 3

$\langle |\widetilde{\delta E^{\text{DF}}}{|\rangle }_{\text{local}}$ averaged over all helium concentrations for all considered functionals. The different bar colors/patterns denote the different densities.

Figure 4

$\langle \delta P^{\text{DF}}\rangle /\langle P^{\text{QMC}}\rangle$ in units of (%) averaged over all helium concentrations for all considered functionals. The different bar colors/patterns denote the different densities. Not shown: M06-L and TPSS.

Figure 5

$\langle \delta P^{\text{DF}}\rangle /\langle P^{\text{QMC}}\rangle$ in units of (%) vs $x_{\text{He}}$ for all considered functionals at $r_s=1.10$ (left), $r_s=1.25$ (middle), and $r_s=1.34$ (right). The different colors/shapes are given in the legend and denote the density functional. Note the broken axis, which shows the mean relative pressure error for the pure helium configurations.

Figure 6

$\langle \delta [H^{\text{DF}}\left(x_{\text{He}}\right)-H^{\text{DF}}(x_{\text{He}}=1)]\rangle$ vs $x_{\text{He}}$ for all considered functionals at $r_s=1.10$ (left), $r_s=1.25$ (middle), and $r_s=1.34$ (right). All energies are measured relative to the average energy of all pure helium configurations at the specified density. Note that the reference enthalpy for each density is taken to be the mean enthalpy of the pure helium configurations at that density.

Figure 7

$\langle \left|\delta {\mathbf{f}}_H^{\text{DF}}\right|\rangle$ aggregated over all helium concentrations for all considered functionals. The different colors denote different densities.

Figure 8

$\langle \delta f_{\mu -\nu }^{\text{PBE}}\left(r\right)\rangle$ vs $r/r_s$ as density is changed. The different marker colors/styles represent different densities. Top: $\langle \delta f_{\text{H-H}}^{\text{PBE}}\left(r\right)\rangle$ calculated at $x_{\mathrm{He}}=1.6\%$, middle: $\langle \delta f_{\text{H-He}}^{\text{PBE}}\left(r\right)\rangle$ calculated at $x_{\text{He}}=20.7\%$, and bottom: $\langle \delta f_{\text{He-He}}^{\text{PBE}}\left(r\right)\rangle$ calculated at $x_{\text{He}}=100\%$

Figure 9

$\langle \delta f_{\mu -\nu }^{\text{PBE}}\left(r\right)\rangle$ vs $r/r_s$ as helium concentration is changed. The different marker colors/styles represent different helium concentrations. Top: $\langle \delta f_{\text{H-H}}^{\text{PBE}}\left(r\right)\rangle$, middle: $\langle \delta f_{\text{H-He}}^{\text{PBE}}\left(r\right)\rangle$, and bottom: $\langle \delta f_{\text{He-He}}^{\text{PBE}}\left(r\right)\rangle$. All configurations are at a density of $r_s=1.25$.

Figure 10

$\langle \delta f_{\mu -\nu }^{\text{DF}}\left(r\right)\rangle$ vs $r/r_s$ as the functional is changed. The different marker colors/styles represent different density functionals. Top: $\langle \delta f_{\text{H-H}}^{\text{PBE}}\left(r\right)\rangle$ at $x_{\text{He}}=1.6\%$, middle: $\langle \delta f_{\text{H-He}}^{\text{PBE}}\left(r\right)\rangle$ at $x_{\text{He}}=20.7\%$, and bottom: $\langle \delta f_{\text{He-He}}^{\text{PBE}}\left(r\right)\rangle$ at $x_{\text{He}}=100\%$. All configurations are at a density of $r_s=1.25$.

Figure 11

Plot of enhancement factors $F_x\left(s\right)$ for all the best performing functionals.