  We present an approach to renormalized second-order Green's function
perturbation theory (GF2) which avoids all dependency on continuous variables,
grids or explicit Green's functions, and is instead formulated entirely in
terms of static quantities and wave functions. Correlation effects from MP2
diagrams are iteratively incorporated to modify the underlying spectrum of
excitations by coupling the physical system to fictitious auxiliary degrees of
freedom, allowing for the single-particle orbitals to delocalize into this
additional space. The overall approach is shown to be rigorously
$\mathcal{O}[N^5]$, after an appropriate compression of this auxiliary space.
This is achieved via a novel scheme which ensures that a desired number of
moments of the underlying occupied and virtual spectra are conserved in the
compression, allowing a rapid and systematically improvable convergence to the
limit of the effective dynamical resolution. The approach is found to then
allow for the qualitative description of stronger correlation effects, avoiding
the divergences of MP2, as well as its orbital-optimized version. On
application to the G1 test set, we find that modifications to only up to the
third spectral moment of the underlying spectrum from which the double
excitations are built is required for accurate energetics, even in strongly
correlated regimes. This is beyond simple self-consistent changes to the
density matrix of the system, but far from requiring a description of the full
dynamics of the frequency-dependent self-energy.
