Accurate $GW$ self-energies in a plane-wave basis using only a few empty states: Towards large systems

The $GW$ approximation to the electronic self-energy yields band structures in excellent agreement with experimental data. Unfortunately, this type of calculation is extremely cumbersome even for present-day computers. The huge number of empty states required both in the calculation of the polarizability and of the self-energy is a major bottleneck in $GW$ calculations. We propose an almost costless scheme, which allows us to divide the number of empty states by about a factor of 5 to reach the same accuracy. The computational cost and the memory requirements are decreased by the same amount, accelerating all calculations from small primitive cells to large supercells.

Figure 1

(Color online) Convergence study of the correlation part of the self-energy at top valence (upper panel) and at bottom conduction (middle panel) and of the band gap (lower panel) of $\beta \text{-SiC}$ as a function of the number of unoccupied states explicitly included in the calculation of the polarizability. The solid curve shows the usual $GW$ result with no correction. The other curves include the correction of Eq. (8) with different values for the energy parameter ${\overline{ϵ}}_{{\chi }_0}$: 0.5 Ha , 1.0 Ha , 2.0 Ha, and 3.0 Ha above the last explicitly calculated band.

Figure 2

(Color online) Upper panel: Value of the integral in Eq. (10) as a function of the transferred momentum $\left|\mathbf{q}+\mathbf{G}\right|$ without any correction using 10, 20, 50, 100, 200, and 550 empty bands in $\beta \text{-SiC}$. Lower panel: Value of the integral in Eq. (10) as a function of the transferred momentum $\left|\mathbf{q}+\mathbf{G}\right|$ with 20 empty bands using no correction or a correction with an average energy ${\overline{ϵ}}_{{\chi }_0}$ of 0.5, 1.0, 2.0, and 3.0 Ha above the last explicitly calculated band in $\beta \text{-SiC}$.

Figure 3

(Color online) Convergence study of the correlation self-energy at top valence (upper panel) and at bottom conduction (middle panel) and of the band gap (lower panel) of $\beta \text{-SiC}$ as a function of the number of unoccupied states explicitly included in the calculation of the self-energy.

Figure 4

(Color online) Convergence study of the correlation self-energy at top valence (upper panel) and at bottom conduction (middle panel) and of the band gap (lower panel) of $\beta \text{-SiC}$ in a 64-atom cubic supercell as a function of the number of unoccupied states explicitly included in the calculation of the polarizability and in the self-energy.

Figure 5

(Color online) Convergence study of the correlation self-energy at top valence (upper panel) and bottom conduction (middle panel) and of the band gap (lower panel) of solid argon as a function of the number of unoccupied states explicitly included in the calculation of the polarizability and in the self-energy.

Figure 6

(Color online) Convergence study of the correlation self-energy at HOMO (upper panel) and at LUMO (middle panel) and of the band gap (lower panel) of the benzene molecule $\left({\text{C}}_6{\text{H}}_6\right)$ as a function of the number of unoccupied states explicitly included in the calculation of the polarizability and in the self-energy.