  Hot rocky super-Earths are thought to be sufficiently irradiated by their
host star to melt their surface and thus allow for long-lasting magma oceans.
Some processes have been proposed for such planets to have retained primordial
hydrogen captured during their formation while moving inward in the planetary
system. The new generation of space telescopes such as the JWST may provide
observations precise enough to characterize the atmospheres and perhaps the
interiors of such exoplanets. We use a vaporization model that calculates the
gas-liquid equilibrium between the atmosphere (including hydrogen) and the
magma ocean, to compute the elemental composition of a variety of atmospheres
for different quantities of hydrogen. The elemental composition is then used in
a steady-state atmospheric model to compute the atmospheric structure and
generate synthetic emission spectra. With this method, we confirm previous
results showing that silicate atmospheres exhibit a thermal inversion, with
notably an emission peak of SiO at 9~$\mu m$. We compare our method to the
literature on the inclusion of hydrogen in the atmosphere, and show hydrogen
reduces the thermal inversion, because of the formation of H2O which has a
strong greenhouse potential. However planets that are significantly irradiated
by their host star are sufficiently hot to dissociate H2O and thus also
maintain a thermal inversion. The observational implications are twofold: 1)
H2O is more likely to be detected in colder atmospheres; 2) Detecting a thermal
inversion in hotter atmospheres does not a priori exclude the presence of H (in
its atomic form). Due to the impact of H on the overall chemistry and
atmospheric structure, and therefore observations, we emphasize the importance
of including volatiles in the calculation of the gas-liquid equilibrium.
Finally, we provide a criterion to determine potential targets for observation.
