Embedding Vertex Correction in Stochastic GW Self-Energy

I will present a stochastic vertex-corrected $GWGamma$ method that embeds vertex correction in the one-shot GW (G₀W₀) self-energy. The embedding scheme starts with partitioning the single-particle space into an active space and the its orthogonal complement (denoted the environment). The active space is defined by energetically-active states that can be eigenstates of a mean-field Hamiltonian, chemical bonds of molecular systems, or Wannier functions of an energy band. I will introduce our latest algorithmic development in Pipek-Mezey localization, which renders a linear scaling approach for obtaining regionally localized states on a subsystem. The core step of our embedding scheme is to perform the "separation-propagation-recombination" treatment on the random vectors that sample the Green's function and the induced density and density matrix fluctuations. The active component of a random vector is treated specially with vertex, while the environment is not. The proposed embedding method is applied to various chemical systems, including donor-acceptor (D-A) complexes, D-A copolymer, and D-A double layers with up to 2000 electrons. Single-particle states featuring low-energy charge-transfer excitation are chosen to form the active space. Computational results with embedded vertex correction significantly improve the fundamental gap prediction upon the G₀W₀ approximation. Active spaces with varied sizes and basis sets are tested. The size effect is critical to the prediction of electron affinity. The change of basis for the active space can simplify the screening calculation. Furthermore, I will demonstrate the separation of the vertex correction into two terms: the correction to the polarizability and the Gamma term. The first term is found to destabilize the quasiparticle energies consistently, while the second term contributes differently to occupied and unoccupied states.