  We investigate the adsorption of graphene sheets on h-BN substrates by means
of first-principles calculations in the framework of adiabatic connection
fluctuation-dissipation theory in the random phase approximation. We obtain
adhesion energies for different crystallographic stacking configurations and
show that the interlayer bonding is due to long-range van der Waals forces. The
interplay of elastic and adhesion energies is shown to lead to stacking
disorder and moir\'e structures. Band structure calculations reveal substrate
induced mass terms in graphene which change their sign with the stacking
configuration. The dispersion, absolute band gaps and the real space shape of
the low energy electronic states in the moir\'e structures are discussed. We
find that the absolute band gaps in the moir\'e structures are at least an
order of magnitude smaller than the maximum local values of the mass term. Our
results are in agreement with recent STM experiments.
