Electron-phonon coupling in hexagonal and cubic water ice is studied usingfirst-principles quantum mechanical methods. We consider 29 distinct hexagonaland cubic ice proton-orderings with up to 192 molecules in the simulation cellto account for proton-disorder. We find quantum zero-point vibrationalcorrections to the minimum electronic band gaps ranging from -1.5 to -1.7 eV,which leads to improved agreement between calculated and experimental bandgaps. Anharmonic nuclear vibrations play a negligible role in determining thegaps. Deuterated ice has a smaller band-gap correction at zero-temperature of-1.2 to -1.4eV. Vibrations reduce the differences between the electronic bandgaps of different proton-orderings from around 0.17 eV to less than 0.05 eV, sothat the electronic band gaps of hexagonal and cubic ice are almost independentof the proton-ordering when quantum nuclear vibrations are taken into account.The comparatively small reduction in the band gap over the temperature range0-240 K of around 0.1 eV does not depend on the proton ordering, or whether theice is protiated or deuterated, or hexagonal or cubic. We explain this in termsof the atomistic origin of the strong electron-phonon coupling in ice.
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