Quantum Monte Carlo determination of the principal Hugoniot of deuterium

We present coupled electron-ion Monte Carlo results for the principal Hugoniot of deuterium together with an accurate study of the initial reference state of shock-wave experiments. We discuss the influence of nuclear quantum effects, thermal electronic excitations, and the convergence of the potential energy surface by wave-function optimization within variational Monte Carlo and projection quantum Monte Carlo methods. Compared to a previous study, our calculations also include low pressure-temperature ($P,T$) conditions resulting in close agreement with experimental data, while our revised results at higher ($P,T$) conditions still predict a more compressible Hugoniot than experimentally observed.

Figure 1

Comparison between ground-state (red squares, this work) and thermal electrons (blue circles, [5]) for thermodynamic points along the principal Hugoniot as obtained with DFT vdW-DF1 based simulations. None of the points included nuclear quantum effects in the calculations.

Figure 2

QMC Hugoniot for $T=2000$, 4000, and 8000 as a function of the ratio between density $\rho$ and density at the reference point ${\rho }_0$. The shaded regions represent the uncertainties in the interpolations of the QMC data. Uncertainties due to changes in the reference energy are within these regions.

Figure 3

VMC excitation energies versus Kohn-Sham excitation (top) and difference in pressure from VMC and DFT computations of excited states (bottom). The black line indicates equality between the VMC and DFT estimates.

Figure 4

Difference in energy per atom, pressure, and Hugoniot between a classical and a quantum system of 54 deuterium atoms at $T=2000$ K at different densities.

Figure 5

QMC Hugoniot compared with previous results. We show experimental data [5, 9] and theoretical DFT-BOMC and QMC data [12, 13]; (a)–(c) Label different data sets, corresponding to computations performed at different temperatures, with $T=2000,4000$, and 8000 K, respectively. Open symbols at $T=2000$ K represent our QMC and DFT-BOMC results including nuclear quantum effects (NQE); all other results are obtained with classical nuclei.

Figure 6

Proton-proton pair correlation function from CEIMC and from DFT-BOMC along the $T=2000$ K isotherm.

Figure 7

Proton-proton pair correlation function from CEIMC and from DFT-BOMC along the $T=4000$ K isotherm.

Figure 8

Proton-proton pair correlation function from CEIMC and from DFT-BOMC along the $T=8000$ K isotherm.

Figure 9

Molecular fraction from three different estimators for the CEIMC trajectories along the $T=4000$ K isotherm (upper panel) and the $T=8000$ K isotherm (lower panel). $P_p$ is the probability for an atom to be paired to the same partner along the entire trajectory, $N_{av}$ the average number of pairs found within a cutoff distance corresponding to the first minimum of $g_{pp}\left(r\right)$, and Holst indicates the estimator based on the coordination number at the molecular peak of $g_{pp}\left(r\right)$.

Figure 10

Normalized, reweighted Boltzmann weight of 136 nuclear configurations for deuterium at $T=8000$ K and $r_s=1.9$. We can see how the weights of the single configurations generated with CEIMC span several orders of magnitude.

Figure 11

RMC convergence with respect to ${\tau }_e$ and ${\beta }_e$ for total and potential energy.