  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 E(Slab) from the differences of the
slopes of the electrostatic potential in the bulk and vacuum regions.
Calculated E(Slab) show very good agreements with experiments. For example, we
have E(Slab)=2.91 for MgO, in agreement with the experimental value
E(Experiment)=2.96. This is in contrast to E(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 Greens
function based perturbation theory, we interpret the large difference
E(Slab)-E(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 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)].
