  We present a self-consistent approach for computing the correlated
quasiparticle spectrum of charged excitations in iterative $\mathcal{O}[N^5]$
computational time. This is based on the auxiliary second-order Green's
function approach [O. Backhouse \textit{et al.}, JCTC (2020)], in which a
self-consistent effective Hamiltonian is constructed by systematically
renormalizing the dynamical effects of the self-energy at second-order
perturbation theory. From extensive benchmarking across the W4-11 molecular
test set, we show that the iterative renormalization and truncation of the
effective dynamical resolution arising from the $2h1p$ and $1h2p$ spaces can
substantially improve the quality of the resulting ionization potential and
electron affinity predictions compared to benchmark values. The resulting
method is shown to be superior in accuracy to similarly scaling quantum
chemical methods for charged excitations in EOM-CC2 and ADC(2), across this
test set, while the self-consistency also removes the dependence on the
underlying mean-field reference. The approach also allows for single-shot
computation of the entire quasiparticle spectrum, which is applied to the
benzoquinone molecule and demonstrates the reduction in the single-particle gap
due to the correlated physics, and gives direct access to the localization of
the Dyson orbitals.
