  The extent of hydrogen coverage of the Si(001)c(4x2) surface in the presence
of hydrogen gas has been studied with dispersion corrected density functional
theory. Electronic energy contributions are well described using a hybrid
functional. The temperature dependence of the coverage in thermodynamic
equilibrium was studied computing the phonon spectrum in a supercell approach.
As an approximation to these demanding computations, an interpolated phonon
approach was found to give comparable accuracy. The simpler ab initio
thermodynamic approach is not accurate enough for the system studied, even if
corrections by the Einstein model for surface vibrations are considered. The
on-set of H2 desorption from the fully hydrogenated surface is predicted to
occur at temperatures around 750 K. Strong changes in hydrogen coverage are
found between 1000 and 1200 K in good agreement with previous reflectance
anisotropy spectroscopy experiments. These findings allow a rational choice for
the surface state in the computational treatment of chemical reactions under
typical metal organic vapor phase epitaxy conditions on Si(001).
