Vibrational renormalisation of the electronic band gap in hexagonal and cubic ice

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 -1.4 eV. 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.