Solving multipole challenges in the $GW100$ benchmark enables precise low-scaling $GW$ calculations

The $GW$ approximation is a widely used method for computing electron addition and removal energies of molecules and solids. The computational effort of conventional $GW$ algorithms increases as $O\left(N^4\right)$ with the system size $N$, hindering the application of $GW$ to large and complex systems. Low-scaling $GW$ algorithms are currently very actively developed. Benchmark studies at the single-shot $G_0{W}_0$ level indicate excellent numerical precision for frontier quasiparticle energies, with mean absolute deviations $<10$ meV between low-scaling and standard implementations for the widely used $\mathrm{GW}100$ test set. A notable challenge for low-scaling $GW$ algorithms remains in achieving high precision for five molecules within the $\mathrm{GW}100$ test set, namely $O_3, \mathrm{BeO}, \mathrm{MgO}, \mathrm{BN}$, and $\mathrm{CuCN}$, for which the deviations are in the range of several hundred meV at the $G_0{W}_0$ level. This is because of a spurious transfer of spectral weight from the quasiparticle to the satellite spectrum in $G_0{W}_0$ calculations, resulting in multipole features in the self-energy and spectral function, which low-scaling algorithms fail to describe. We show in this paper that including eigenvalue self-consistency in the Green's function ($\mathrm{ev}G{W}_0$) achieves a proper separation between satellite and quasiparticle peak, leading to a single solution of the quasiparticle equation with spectral weight close to one. $\mathrm{ev}G{W}_0$ quasiparticles energies from low-scaling $GW$ closely align with reference calculations; the mean absolute error is only 12 meV for the five molecules. We thus demonstrate that low-scaling $GW$ with self-consistency in $G$ is well suited for computing frontier quasiparticle energies.