Static subspace approximation for the evaluation of G0W0 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 $\ensuremath{\varepsilon}(\ensuremath{\omega})$. Here, we describe an approach to speed up the evaluation of the frequency-dependent part of $\ensuremath{\varepsilon}(\ensuremath{\omega})$ 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, $\ensuremath{\varepsilon}(\ensuremath{\omega}=0)$, 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.