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 semidefiniteness 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 spectral representation, and, consequently, guarantees positive semidefiniteness. 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 infinite 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.