Thermodynamics of atomic and ionized hydrogen : analytical results versus OPAL tables and Monte Carlo data

We compute thermodynamical properties of a low-density hydrogen gas within the physical picture, in which the system is described as a quantum electron-proton plasma interacting via the Coulomb potential. Our calculations are done using the exact Scaled Low-Temperature (SLT) expansion, which provides a rigorous extension of the well known virial expansion -- valid in the fully ionized phase -- into the Saha regime where the system is partially or fully recombined into hydrogen atoms. After recalling the SLT expansion of the pressure [A. Alastuey et al, J. Stat. Phys. {\bf 130}, 1119 (2008)], we obtain the SLT expansions of the chemical potential and of the internal energy, up to to order $\exp(|E_H|/kT)$ included ($E_H \simeq -13.6$ eV). Those truncated expansions describe the first five non-ideal corrections to the ideal Saha law. They account exactly, up to the considered order, for all effects of interactions and thermal excitations, including the formation of bound states (atom $H$, ions $H^-$ and $H_2^+$, molecule $H_2$, ...) and atom-charge and atom-atom interactions. Among the five leading corrections, three are easy to evaluate, while the remaining ones involve well-defined internal partition functions for molecule $H_2$ and ions $H^-$ and $H_2^+$, for which no closed-form analytical formula exist currently. We provide accurate low-temperature approximations for those partition functions by using known values of rotational and vibrational energies. We compare then the predictions of the SLT expansion with the OPAL EOS and data of path integral quantum Monte Carlo (PIMC) simulations. In general, a good agreement is found. At low densities, the simple analytical SLT formulae reproduce the values of the OPAL tables up to the last digit in a large range of temperatures, while at higher densities ($\rho\sim10^{-2}$ g/cm$^3$), some discrepancies between the SLT, OPAL and PIMC results are observed