Predictive $GW$ calculations using plane waves and pseudopotentials

We derive a finite-basis-set correction for quasiparticle (QP) energies in the $GW$ approximation and many-body correlation energies in the random phase approximation. Since the correction requires only knowledge of the ground-state density distribution, it is straightforward to implement in any plane-wave code and significantly improves convergence at negligible computational cost. The expression also indicates that QP energies might converge to the wrong value using the projector augmented wave (PAW) method since the overlap densities of occupied orbitals and high-energy, plane-wave-like orbitals are inaccurately described. The error is shown to be related to the incompleteness of the partial waves inside the atomic spheres. It can be avoided by adopting norm-conserving partial waves. $G_0{W}_0$ and $G{W}_0$ results based on such norm-conserving PAW potentials are presented for a large set of semiconductors and insulators. Accurate extrapolation procedures to the infinite-basis-set limit and infinite-$k$-point limit are discussed in detail.

Figure 1

QP corrections to Kohn-Sham eigenvalues for ZnO $G_0{W}_0$ calculations. The number of bands varies between 50 (right most point) and 2400 (left most point). To improve the presentation, an upward shift of 1 eV was added to the VB and $3d$ QP corrections (i.e., the true corrections can be obtained by subtracting 1 eV, moving the lines down by 1 eV).

Figure 2

QP corrections to Kohn-Sham eigenvalues for GaAs $G_0{W}_0$ calculations. The number of bands varies between 50 (right most point) and 1600 (left most point). To the $3d$ QP corrections, 1.5 eV has been added.

Figure 3

QP corrections to Kohn-Sham eigenvalues for AlAs $G_0{W}_0$ calculations. The number of bands varies between 50 (right most point) and 1600 (left most point). To the VB and $3d$ QP corrections, 0.5 and 3 eV have been added, respectively.

Figure 4

Dependence of (a) the unscaled QP correction (without $Z)$ and (b) of the quasiparticle band gap of wurzite ZnO on the basis set used for the response function. Bare data $(+$ sign) and data corrected for the leading error according to Eq. (27), marked with $\times$ sign, are shown. Horizontal lines are extrapolated values. In (a) the data for the $3d$ and VBM states were shifted to a higher energy by 1.5 eV.

Figure 5

Band gaps predicted using $G{W}_0$@LDA compared to experimental values. The theoretical band gaps have been corrected for spin-orbit coupling given in the Supplemental Material of Ref. [35]. Experimental data are also collected in Ref. [35].