Low-Scaling GW Calculations of Quasi-Particle Energies for Extended Systems within the Numerical Atomic Orbital...

The many-body perturbation theory within the GW approximation is a widely used method for describing the electronic band structures in real materials. Its application to large-scale systems is, however, impeded by its high computational cost. The rate-limiting steps in a typical GW implementation are the evaluation of the polarization function under the random phase approximation (RPA) and the evaluation of the GW self-energy, both of which have a canonical $O\left(N^4\right)$ scaling with N being the system size. The conventional space-time algorithm within the plane-wave basis sets reduces the scaling from $O\left(N^4\right)$ to $O\left(N^3\right)$, albeit with a large prefactor and increased memory cost. Here, we present a space-time algorithm within the numerical atomic orbital (NAO) basis-set framework, for which the evaluation of the polarization function and self-energy is formally reduced to $O\left(N^2\right)$ or better with respect to system size. This is achieved by computing these quantities in real space, where low-scaling algorithms can be formulated by leveraging the localized resolution of identity (LRI) technique. The resulting NAO-based, LRI-enhanced space-time GW algorithm has been implemented in the LibRPA library interfaced with the FHI-aims code package. Benchmark calculations for crystalline solids show that the low-scaling implementation yields quasi-particle energies in close agreement with the conventional $O\left(N^4\right)$ k-space formalism previously implemented in FHI-aims. For the systems studied here, the observed overall scaling is substantially reduced relative to the canonical approach, and the low-scaling implementation becomes advantageous already for systems containing fewer than 100 atoms.