From Full Dynamic to Pure Static: A Family of $GW$-Based Approximations

We introduce a systematic hierarchy of one-body Green's function methods derived from the $GW$ approximation, constructed by progressively reducing the dynamical content of the self-energy. Starting from the fully dynamical Dyson formulation, we generate a family of approximations that interpolates between the standard $GW$ approximation to purely static effective single-particle Hamiltonians. This framework enables a controlled investigation of the role of dynamical effects and particle-hole coupling in the description of ionization potentials. Within this unified formalism, the hole and particle branches can be selectively decoupled through downfolding strategies into reduced one-particle spaces. By benchmarking the different members of this hierarchy on molecular ionization energies, we assess their accuracy, numerical robustness, and algorithmic complexity. We demonstrate that consistently derived partially static schemes can yield reliable quasiparticle energies while significantly simplifying the underlying eigenvalue problem. We further introduce a novel static Hermitian self-energy obtained as the static limit of this hierarchy. Despite its conceptually distinct origin, it produces results remarkably close to those of qs$GW$, thereby providing an alternative static route toward partial self-consistency.