Dynamical electron-phonon coupling, GW self-consistency, and vertex effect on the electronic band gap of ice and liquid water

Abstract

We study the impact of dynamical electron-phonon (el-ph) effects on the electronic band gap of ice and liquid water by accounting for frequency-dependent Fan contributions in the el-ph mediated self-energy within the many-body perturbation theory (MBPT). We find that the dynamical el-ph coupling effects greatly reduce the static el-ph band-gap correction of the hydrogen-rich molecular ice crystal from$\ensuremath{-}2.46$ to $\ensuremath{-}0.23$ eV in great contrast to the result of Monserrat et al. [Phys. Rev. B 92, 140302 (2015)]. This is of particular importance as otherwise the static el-ph gap correction would considerably reduce the electronic band gap, leading to considerable underestimation of the intense peaks of optical absorption spectra of ice which would be in great disagreement to experimental references. By contrast, the static el-ph gap correction of liquid water is very moderate ($\ensuremath{-}0.32$ eV), and inclusion of dynamical effects slightly reduces the gap correction to $\ensuremath{-}0.19$ eV. Further, we determine the diverse sensitivity of ice and liquid water to the $GW$ self-consistency and show that the energy-only self-consistent approach (${G}_{n}{W}_{n}$) exhibits large implicit vertex character in comparison to the quasiparticle self-consistent approach, for which an explicit calculation of vertex corrections is necessary for good agreement with experiment.