Static subspace approximation for the evaluation of $G_0{W}_0$ quasiparticle energies within a sum-over-bands approach

Many-body perturbation theory within the $GW$ approach has been established as a quantitatively accurate approach for predicting the quasiparticle and excited-state properties of a wide variety of materials. However, the successful application of the method is often complicated by the computational complexity associated with the evaluation and inversion of the frequency-dependent dielectric matrix $\varepsilon \left(\omega \right)$. Here, we describe an approach to speed up the evaluation of the frequency-dependent part of $\varepsilon \left(\omega \right)$ in the traditional sum-over-states $GW$ framework based on the low-rank approximation of the static dielectric matrix, a technique often used in $GW$ implementations that are based on a starting mean field within density-functional perturbation theory. We show that the overall accuracy of the approach, independently from other calculation parameters, is solely determined by the threshold on the eigenvalues of the static dielectric matrix, $\varepsilon (\omega =0)$, and that it can yield orders-of-magnitude speed-ups in full-frequency $GW$ calculations. We validate our implementation with several benchmark calculations ranging from bulk materials to systems with reduced dimensionality, and show that this technique allows one not only to study larger systems, but also to carefully consider the convergence of computationally demanding systems, such as ZnO, without relying on plasmon-pole models.

Figure 1

Eigenvalue spectrum (logarithmic scale) for the static dielectric and symmetrized susceptibility matrices for $\beta \text{−}\mathrm{SiC}$ at $\mathrm{\Gamma }$.

Figure 2

Band structures of (a) silicon and (b) $\beta$-SiC, along with the error introduced by the static subspace approximation. The solid blue line is obtained without approximation and the red dashed employs the static subspace approximation with an eigenvalue screening threshold $t_{\text{eig}}=0.01$. The zero reference in both cases has been set to the valence band maximum (VBM). The inset blow-ups show the band structure around the ${\mathrm{\Gamma }}_{15c}$ point to show the difference between the reference and approximate results.

Figure 3

For silicon the mean absolute error between the reference and approximate results calculated for 196 quasiparticle energies is reported as a function of $t_{\text{eig}}$, in this case, the error is reported with (relative) and without (absolute) shift with respect to the Fermi level.

Figure 4

For silicon (a) and $\beta$-SiC (b), blue columns show the number of eigenvectors in percentage of the total that are retained in the computation of the inverse dielectric matrix $\left(\varepsilon \right)$ as a function of the eigenvalue screening threshold $\left(t_{\text{eig}}\right)$. Red columns report the percentage reduction of the time to solution for the evaluation of $\varepsilon$ with respect to the reference calculation (no approximation).

Figure 5

The frequency dependence of the matrix elements of the time-ordered self-energy operator for the first eight bands of silicon evaluated at the $\mathrm{\Gamma }$ point. The zero of the frequency axis is set to the center of the gap. (a) and (b) show the real and imaginary part of $\langle {\psi }_{n\mathbf{k}}|\mathrm{\Sigma }\left(\omega \right)|{\psi }_{n\mathbf{k}}\rangle$, respectively. The reported calculations are in excellent agreement with previous works [39, 51].

Figure 6

Mean absolute error between the reference and approximate results for a total of 50 quasiparticle energies as a function of the number of eigenvectors included in the subspace basis. The error is reported with (relative) and without (absolute) shift with respect to the Fermi level. The calculations have been performed with a cutoff of 32 Ry and including 300 and 1000 bands in the calculation of the inverse dielectric matrix and self-energy, respectively. The considered number of eigenvectors ranges between 10%–30% of the full basis.

Figure 7

Band structure for copper obtained by employing a cutoff of 45 Ry and including 400 and 1000 bands in the calculation of the inverse dielectric matrix and self-energy, respectively. The size of the subspace basis has been fixed to 100 eigenvectors ($\sim 25\%$ of the total). A uniform $\mathbf{k}\mathrm{grid}$ [52] of $16\times 16\times 16$ and $32\times 32\times 32$ have been used for the evaluation of ${\varepsilon }^{-1}$ for $\mathbf{q}\ne 0$ and $\mathbf{q}\to 0$ respectively. Calculations performed with a norm-conserving scalar-relativistic pseudopotential [53, 54] including semicore $s$ and $p$ states.

Figure 8

Convergence study of monolayer ${\mathrm{MoS}}_2$ with the static subspace approximation. (a) $G_0{W}_0$ quasiparticle band structure (red lines) of monolayer ${\mathrm{MoS}}_2$ obtained with $N_b=400$ eigenvectors, and the corresponding LDA band structure (blue dashed lines). (b) Convergence of the error of the quasiparticle evaluated at the highest valence and lowest conduction band states at $K$ and $\mathrm{\Gamma }$ with respect to the screening threshold on the eigenvalues $t_{\text{eig}}$. We show the absolute error on the $\mathrm{\Gamma }\to K$ and $K\to K$ band gaps (green and blue triangles respectively), the mean absolute error (red circles) and the maximum absolute error (black squares). The percentage labels in the plot shows the fraction of eigenvectors retained in the calculations for each $t_{\text{eig}}$.

Figure 9

Convergence of the HOMO quasiparticle energy of water with respect to the number of bands included in the calculation and the dielectric matrix $\epsilon$ and the screened Coulomb cutoff.

Figure 10

(a) Convergence of the mean absolute error for the quasiparticle states with respect to the truncation threshold on the eigenvalues $t_{\text{eig}}$. Each curve was evaluated with a different cutoff for the dielectric matrix, but with a fixed number of bands (1600). The error is computed with respect of the calculation performed without the subspace approximation, but for the same cutoff of the dielectric matrix. (b) Similar plot as in (a), but where each curve was computed with a different number of bands, but with a fixed cutoff for the dielectric matrix of 20 Ry.

Figure 11

Convergence study of the $GW$ quasiparticle band gap of ZnO with respect to three interdependent parameters (see the text). The convergence function is plotted for two fixed parameters (dashed lines) used to produce the data points (discs), and for the extrapolated parameters (solid lines).