Real-Space Stochastic GW Calculations Benchmark on GW20

The stochastic G₀W₀ method, which was developed in [Neuhauser, D. . Phys. Rev. Lett. 2014, 113, 076402], scales quadratically and is well suited for computing quasiparticle energies of large systems. It uses a stochastic resolution of identity to express the Green's function as a product of random orbitals at different times. The Green's function is further divided into a stochastic part and an embedded part, the latter computed deterministically from a small set of Kohn-Sham orbitals. We have implemented the stochastic G₀W₀ method within the real-space finite-difference DFT code PARSEC and benchmarked it on a subset of GW100 data set, including GW20 set, obtaining excellent agreement with established implementations. A key achievement of this work is the demonstration that incorporating a small set of embedded Kohn-Sham orbitals dramatically reduces stochastic error─by up to a factor of 5─while preserving efficiency. By systematically analyzing the SiH₄ molecule, we further identify which orbitals are most effective as embedded states, providing practical insight into optimizing accuracy in stochastic G₀W₀ calculations.