Surface energies of hexagonal and cubic water ice are calculated using first-principles quantum mechanical methods, including an accurate description of anharmonic nuclear vibrations. We consider two proton-orderings of the hexagonal and cubic ice basal surfaces and three proton-orderings of hexagonal ice prism surfaces, finding that vibrations reduce the surface energies by more than 10%. We compare our vibrational densities of states to recent sum frequency generation absorption measurements and identify surface proton-orderings of experimental ice samples and the origins of characteristic absorption peaks. We also calculate zero point quantum vibrational corrections to the surface electronic band gaps, which range from -1.2 eV for the cubic ice basal surface up to -1.4 eV for the hexagonal ice prism surface. The vibrational corrections to the surface band gaps are up to 12% smaller than for bulk ice. Published by AIP Publishing.
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