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 ${\epsilon }_{\infty }\left(\mathrm{Slab}\right)$ from the differences of the slopes of the electrostatic potential in the bulk and vacuum regions. Calculated ${\epsilon }_{\infty }\left(\mathrm{Slab}\right)$ show very good agreement with experiments. For example, we have ${\epsilon }_{\infty }\left(\mathrm{Slab}\right)=2.91$ for MgO, in agreement with the experimental value ${\epsilon }_{\infty }\left(\mathrm{Experiment}\right)=2.96$. This is in contrast to ${\epsilon }_{\infty }\left(\mathrm{RPA}\right)=2.37$, 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 ${\epsilon }_{\infty }\left(\mathrm{Slab}\right)-{\epsilon }_{\infty }\left(\mathrm{RPA}\right)=2.91-2.37$ 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)].

Figure 1

Band structure of a narrowband model to illustrate the quasiparticle-based perturbation. CBM and VBM denote conduction-band minimum and valence-band maximum. The widths of bands $B_{\mathrm{C}}$ and $B_{\mathrm{V}}$ are smaller than the band gap $E_{\mathrm{G}}$. As in the text, we expect well-defined QPs in this model.

Figure 2

A slab (18 atoms per cell) is placed in the middle of a supercell (60 a.u. width along the $z$ axis, which is perpendicular to the slab), with electrodes at the left and right ends. In the top panel, we show $V_z^{\mathrm{es}}(z,E)$ for $E=0.2$ and 0.0 Ry. In the bottom panel, we show their difference $\mathrm{\Delta }{V}_z^{\mathrm{es}}\left(z\right)$. From the ratio of two slopes of $\mathrm{\Delta }{V}_z^{\mathrm{es}}$ in the slab region (green) and in the vacuum region (violet), we obtain ${\epsilon }_{\infty }\left(\mathrm{Slab}\right)$. We achieve better numerical accuracy by using $\mathrm{\Delta }{V}_z^{\mathrm{es}}\left(z\right)$ instead of $V_z^{\mathrm{es}}(z,0.2\mathrm{Ry})$ directly.