  Recent x-ray Compton scattering experiments in ice have provided useful
information about the quantum nature of the interaction between H$_2$O
monomers. The hydrogen bond is characterized by a certain amount of charge
transfer which could be determined in a Compton experiment. We use ab-initio
simulations to investigate the hydrogen bond in H$_2$O structures by
calculating the Compton profile and related quantities in three different
systems, namely the water dimer, a cluster containing 12 water molecules and
the ice crystal. We show how to extract estimates of the charge transfer from
the Compton profiles.
