  By analyzing the momentum distribution obtained from path integral and phonon
calculations we find that the protons in hexagonal ice experience an
anisotropic quasi-harmonic effective potential with three distinct principal
frequencies that reflect molecular orientation. Due to the importance of
anisotropy, anharmonic features of the environment cannot be extracted from
existing experimental distributions that involve the spherical average. The
full directional distribution is required, and we give a theoretical prediction
for this quantity that could be verified in future experiments. Within the
quasi-harmonic context, anharmonicity in the ground state dynamics of the
proton is substantial and has quantal origin, a finding that impacts the
interpretation of several spectroscopies.
