Frequency dependence in $GW$ made simple using a multipole approximation

In the $GW$ approximation, the screened interaction $W$ is a nonlocal and dynamical potential that usually has a complex frequency dependence. A full description of such a dependence is possible but often computationally demanding. For this reason, it is still common practice to approximate $W\left(\omega \right)$ using a plasmon pole (PP) model. Such an approach, however, may deliver an accuracy limited by its simplistic description of the frequency dependence of the polarizability, i.e., of $W$. In this work, we explore a multipole approach (MPA) and develop an effective representation of the frequency dependence of $W$. We show that an appropriate sampling of the polarizability in the frequency complex plane and a multipole interpolation can lead to a level of accuracy comparable with full-frequency methods at a much lower computational cost. Moreover, both accuracy and cost are controllable by the number of poles used in MPA. Eventually, we validate the MPA approach in selected prototype systems, showing that full-frequency quality results can be obtained with a limited number of poles.

Figure 1

An illustration of the double parallel sampling with a nine points semihomogeneous grid along the real axis with ${\varpi }_2=1\mathrm{Ha}$, similar to the imaginary frequency used in the GN PPM and ${\varpi }_1=0.1\mathrm{Ha}$, except for the origin of coordinates. The isolines in the background correspond to a toy polarizability function with 200 poles on the real axis.

Figure 2

Selected Si $X$ matrix elements computed within MPA with 1, 2, 3, and 8 poles and compared with the corresponding FF real-axis results. Although the real and imaginary parts of the function are plotted using different arbitrary units, the different matrix elements can be compared since their scale is consistent and indicated in each plot.

Figure 3

Selected hBN $X$ matrix elements computed within MPA with 1, 2, 3, and 8 poles and compared with the corresponding FF real-axis results. The same scheme from Fig. 2 is used for the units.

Figure 4

Selected ${\mathrm{TiO}}_{2}X$ matrix elements computed within MPA with 1, 3, 4, and 9 poles and compared with the corresponding FF real-axis results. The same scheme from Fig. 2 is used for the units.

Figure 5

(Top) $\langle N_F\rangle$, mean number of matrix elements for which the position of the poles was corrected according to Eq. (20) for MPA and Eq. (19) for PPA, and (bottom) $\langle RSD\rangle$, average deviation as defined in Eq. (28), as a function of the number of poles used in the MPA approach. Due to the high computational cost of the FF calculations, in case of hBN the comparison is made with a non fully converged dimension of the polarizability matrix, of 10 Ry. The plots correspond to the AA stacking, but similar values are seen also for the AA${\mathrm{AA}}^{\prime }$ stacking with 10 Ry of X matrix.

Figure 6

Deviations of the fundamental gap calculated via the MPA with respect to FF results as a function of number of poles for Si, hBN and ${\mathrm{TiO}}_2$. As in Fig. 5, in case of hBN we use 10 Ry of X matrix and plot only the results of the AA stacking.

Figure 7

Frequency dependence of the Si valence (v) and conduction (c) real part of the self-energy (top) and spectral function (bottom) of the quasiparticles involved in the fundamental gap computed with PPA, MPA, and FF.