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Equation of state of metallic hydrogen from Coupled Electron-Ion Monte Carlo simulations

We present a study of hydrogen at pressures higher than molecular dissociation using the Coupled Electron-Ion Monte Carlo method. These calculations use the accurate Reptation Quantum Monte Carlo method to estimate the electronic energy and pressure while doing a Monte Carlo simulation of the protons. In addition to presenting simulation results for the equation of state over a large region of phase space, we report the free energy obtained by thermodynamic integration. We find very good agreement with DFT calculations for pressures beyond 600 GPa and densities above $ρ=1.4 g/cm^3$. Both thermodynamic as well as structural properties are accurately reproduced by DFT calculations. This agreement gives a strong support to the different approximations employed in DFT, specifically the approximate exchange-correlation potential and the use of pseudopotentials for the range of densities considered. We find disagreement with chemical models, which suggests a reinvestigation of planetary models, previously constructed using the Saumon-Chabrier-Van Horn equations of state.