In the first part of the present paper, the two-photon absorption spectrum of liquid water is calculated using an ab initio real-time many-body approach. Correlation effects, such as single-particle and excitonic effects, are included, with the latter showing large impact on the shape and peak positions of the two-photon spectrum. A broad peak at 10.0 eV is calculated, in excellent agreement with the experimental reference. In the second part, we show the impact of the on- and off-diagonal elements of the self-energy operator upon $GW$ self-consistency (depending on the number of quasiparticle-corrected unoccupied bands) on the electronic band gap of liquid water at the $\mathrm{\Gamma }$ point. The off-diagonal self-energy elements increase the electronic gap, correcting previous results reported by W. Chen et al. [Phys. Rev. Lett. 117, 186401 (2016)] by 0.52 eV, and putting at the same time the non-self-consistent bootstrap approximation for the vertex function into question. Furthermore, depending on the calculated band gap using different $GW$ flavors, the absolute position of the absorption band of the nonlinear spectrum is affected.

• Received 29 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.245109