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 epsilon(infinity)(Slab) from the differences of the slopes of the electrostatic potential in the bulk and vacuum regions. Calculated epsilon(infinity)(Slab) show very good agreement with experiments. For example, we have epsilon(infinity)(Slab) = 2.91 for MgO, in agreement with the experimental value epsilon(infinity)(Experiment) = 2.96. This is in contrast to epsilon(infinity)(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 epsilon(infinity)(Slab) - epsilon(infinity)(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 G(0)W approximation along the line of QSGW self-consistency, as was performed by Shishkin, Marsman, and Kresse [Phys. Rev. Lett. 99, 246403 (2007)].