Frequency dependence in G W 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 $W(\ensuremath{\omega})$ 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.