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.
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