Due to non-linear structure, iterative Greens 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.
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