An Algebraic-Diagrammatic Construction for Vertex Corrections to the GW Self-Energy

The $G3W2$ approximation -- the second-order self-energy beyond $GW$ -- is known to violate some fundamental analytic properties of the self-energy. In particular, its lack of positive semi-definiteness leads to unphysical features such as negative spectral functions. In this work, we reformulate the $G3W2$ approximation within the algebraic-diagrammatic construction (ADC) framework. The resulting ADC-$G3W2$ formalism enforces the same analytic form as the exact self-energy, namely a sum-over-state representation, and, consequently, guarantees positive semi-definiteness. Starting from the $GW$ self-energy, we construct a hierarchy of ADC-based approximations of increasing sophistication, including ADC-2SOSEX, ADC(3)-$G3W2$, and a full ADC-$G3W2$ scheme. These methods can be interpreted as nonperturbative resummations of vertex corrections to the self-energy, yielding Hermitian effective Hamiltonians whose diagonalization provides quasiparticle and satellite energies. This establishes a formal bridge between many-body perturbation theory formulated in terms of the screened interaction $W$ and conventional ADC schemes based on the bare Coulomb interaction. The performance of these ADC-based approximations is gauged for valence ionization potentials and benchmarked against their parent method.