Coulomb-hole summations and energies for $GW$ calculations with limited number of empty orbitals: A modified static remainder approach

Ab initio $GW$ calculations are a standard method for computing the spectroscopic properties of many materials. The most computationally expensive part in conventional implementations of the method is the generation and summation over the large number of empty orbitals required to converge the electron self-energy. We propose a scheme to reduce the summation over empty states by the use of a modified static remainder approximation, which is simple to implement and yields accurate self-energies for both bulk and molecular systems requiring a small fraction of the typical number of empty orbitals.

Figure 1

(top) The convergence of the Coulomb-hole contribution to the self-energy [Eq. (2)], with respect to the number of orbitals included in the summation $N$, using a dielectric matrix calculated with 1000 empty bands. For all calculations on ZnO, a $5\times 5\times 4$ k-point grid is used. (bottom) The convergence of the quasiparticle energy $E_{\text{QP}}$ with respect to empty states in the polarizability sum [Eq. (3)] and with respect to empty states in the Coulomb-hole sum [Eq. (2)]. The red curve shows the valence band maximum (VBM) $E_{\text{QP}}$ in ZnO using a fixed 3000 bands in the Coulomb-hole summation and varying the number of bands included in the polarizability summation. The black curve shows the VBM $E_{\text{QP}}$ in ZnO using a fixed 1000 bands in the polarizability summation and varying the number of bands included in the Coulomb-hole summation. ZnO calculations were carried out within the GPP approximation.

Figure 2

Comparison between the contributions to the Coulomb-hole sum for the full $GW$ operator vs results from one half the static COHSEX Coulomb-hole operator for orbitals beyond the number of real DFT bands/orbitals used: 12 in silicon and 100 in silane. A $5\times 5\times 5$ k-point grid is used in Si. The plotted quantity is ${\sum }_{n^{\prime \prime }=n_{DFT}+1}^N{\sum }_{{\mathbf{qGG}}^{\prime }}\langle n\mathbf{k}|e^{i(\mathbf{q}+\mathbf{G})·\mathbf{r}}|n^{\prime \prime }\mathbf{k}-\mathbf{q}\rangle \langle n^{\prime \prime }\mathbf{k}-\mathbf{q}|e^{-i(\mathbf{q}+{\mathbf{G}}^{\prime })·{\mathbf{r}}^{\prime }}|n^{\prime }\mathbf{k}\rangle \times I_{{\mathbf{GG}}^{\prime }}^{\text{CH}}(\mathbf{q},n,n^{\prime },n^{\prime \prime })$, where $I^{\text{CH}}$ is the term in braces in Eqs. (2) and (5).

Figure 3

Coulomb-hole energies of the valence band maximum in (left) Si and (right) ZnO in the modified static remainder approach compared to the energies from the standard approach of truncating the Coulomb-hole summation in Eq. (2) as a function of the number of DFT bands. In the static remainder approach the summation is also truncated at the same number of bands, but the modified static remainder is added to the sum. A $5\times 5\times 5$ and a $5\times 5\times 4$ k-point grid is used in Si and ZnO, respectively. The gray lines represent the result using the maximum number of bands and having the static remainder included.

Figure 4

Coulomb-hole part of the self-energy, with and without the static remainder, for the highest occupied molecular orbital of the BND (shown in the inset) molecule as a function of the number of DFT orbitals included in the Coulomb-hole sum. The gray line represents the result using the maximum number of bands and having the static remainder included.