  We investigate the rotational properties of molecular hydrogen and its
isotopes physisorbed on the surfaces of graphene and hexagonal boron nitride
($h$-BN), grown on Ni(111), Ru(0001), and Rh(111), using rotational excitation
spectroscopy (RES) with the scanning tunneling microscope. The rotational
thresholds are in good agreement with $\Delta J=2$ transitions of freely
spinning para-H$_2$ and ortho-D$_2$ molecules. The line shape variations in RES
for H$_2$ among the different surfaces can be traced back and naturally
explained by a resonance mediated tunneling mechanism. RES data for
H$_2$/$h$-BN/Rh(111) suggests a local intrinsic gating on this surface due to
lateral variations in the surface potential. An RES inspection of H$_2$, HD,
and D$_2$ mixtures finally points to a multi molecule excitation, since either
of the three $J=0\rightarrow2$ rotational transitions are simultaneously
present, irrespective of where the spectra were recorded in the mixed
monolayer.
