Finite electric-field approach to evaluate the vertex correction for the screened Coulomb interaction in the quasiparticle self-consistent GW method

We apply the quasiparticle self-consistent GW method (QSGW) to slab models of ionic materials—LiF, KF, NaCl, MgO, and CaO—under electric field. Then we obtain the optical dielectric constants ε∞(Slab) from the differences of the slopes of the electrostatic potential in the bulk and vacuum regions. Calculated ε∞(Slab) show very good agreement with experiments. For example, we have ε∞(Slab)=2.91 for MgO, in agreement with the experimental value ε∞(Experiment)=2.96. This is in contrast to ε∞(RPA)=2.37, which is calculated in the random-phase approximation for the bulk MgO in QSGW. After we explain the difference between the quasiparticle-based perturbation theory and the Green's-function-based perturbation theory, we interpret the large difference ε∞(Slab)−ε∞(RPA)=2.91−2.37 as the contribution from the vertex correction of the proper polarization, which determines the screened Coulomb interaction W. Our result encourages the theoretical development of the self-consistent G0W approximation along the line of QSGW self-consistency, as was performed by Shishkin, Marsman, and Kresse [Phys. Rev. Lett. 99, 246403 (2007)].