Efficient $GW$ calculations for SnO${}_2$, ZnO, and rubrene: The effective-energy technique

In a recent Rapid Communication [J. A. Berger, L. Reining, and F. Sottile, Phys. Rev. B 82, 041103(R) (2010)], we presented the effective-energy technique to evaluate, in an accurate and numerically efficient manner, electronic excitations by reformulating spectral sum-over-states expressions such that only occupied states appear. In our approach all the empty states are accounted for by a single effective energy that can be obtained from first principles. In this work we provide further details of the effective-energy technique, in particular, when combined with the $GW$ method, in which a huge summation over empty states appears in the calculation of both the screened Coulomb interaction and the self-energy. We also give further evidence of the numerical accuracy of the effective-energy technique by applying it to the technological important materials SnO${}_2$ and ZnO. Finally, we use this technique to predict the band gap of bulk rubrene, an organic molecular crystal with a 140-atom unit cell.

Figure 1

Convergence behavior of the calculated energies (in eV) of the VBM, CBM, and fundamental gap ($E_g$) of SnO${}_2$ with the number of empty states in both the screening and the self-energy calculcations. Triangles (black), SOS approach; open (red) circles, $\mathrm{SOS}+\mathrm{EET}$ approach using ${\delta }^{\prime \left(0\right)}$ and ${\delta }^{\left(0\right)}$; filled (red) circles, EET using ${\delta }^{\prime \left(2\right)}$ and ${\delta }^{\left(2\right)}$ without empty states.

Figure 2

Convergence behavior of the calculated energies (in eV) of the VBM, CBM, and fundamental gap ($E_g$) of ZnO with the number of empty states in both the screening and the self-energy calculations. Triangles (black), SOS approach; open (red) circles, $\mathrm{SOS}+\mathrm{EET}$ approach using ${\delta }^{\prime \left(0\right)}$ and ${\delta }^{\left(0\right)}$; filled (red) circles, EET using ${\delta }^{\prime \left(2\right)}$ and ${\delta }^{\left(2\right)}$ without empty states.