  Electron-phonon coupling in hexagonal and cubic water ice is studied using
first-principles quantum mechanical methods. We consider 29 distinct hexagonal
and cubic ice proton-orderings with up to 192 molecules in the simulation cell
to account for proton-disorder. We find quantum zero-point vibrational
corrections to the minimum electronic band gaps ranging from -1.5 to -1.7 eV,
which leads to improved agreement between calculated and experimental band
gaps. Anharmonic nuclear vibrations play a negligible role in determining the
gaps. Deuterated ice has a smaller band-gap correction at zero-temperature of
-1.2 to -1.4eV. Vibrations reduce the differences between the electronic band
gaps of different proton-orderings from around 0.17 eV to less than 0.05 eV, so
that the electronic band gaps of hexagonal and cubic ice are almost independent
of the proton-ordering when quantum nuclear vibrations are taken into account.
The comparatively small reduction in the band gap over the temperature range
0-240 K of around 0.1 eV does not depend on the proton ordering, or whether the
ice is protiated or deuterated, or hexagonal or cubic. We explain this in terms
of the atomistic origin of the strong electron-phonon coupling in ice.
