  Due to non-linear structure, iterative Green's function methods can result in
multiple different solutions even for simple molecular systems. In contrast to
the wave-function methods, a detailed and careful analysis of such molecular
solutions was not performed before. In this work, we use two-particle density
matrices to investigate local spin and charge correlators that quantify the
charge-resonance and covalent characters of these solutions. When applied
within unrestricted orbital set, spin correlators elucidate the broken symmetry
of the solutions, containing necessary information for building effective
magnetic Hamiltonians. Based on GW and GF2 calculations of simple molecules and
transition metal complexes, we construct Heisenberg Hamiltonians,
four-spin-four-center corrections, as well as biquadratic spin-spin
interactions. These Hamiltonian parametrizations are compared to prior
wave-function calculations.
