  We introduce a new computational method to study porphyrin-like transition
metal complexes, bridging density functional theory and exact many-body
techniques, such as the density matrix renormalization group (DMRG). We first
derive a multi-orbital Anderson impurity Hamiltonian starting from first
principles considerations that qualitatively reproduce GGA+U results when
ignoring inter-orbital Coulomb repulsion $U'$ and Hund exchange $J$. An exact
canonical transformation is used to reduce the dimensionality of the problem
and make it amenable to DMRG calculations, including all many-body terms (both
intra, and inter-orbital), which are treated in a numerically exact way. We
apply this technique to FeN$_4$ centers in graphene and show that the inclusion
of these terms has dramatic effects: as the iron orbitals become single
occupied due to the Coulomb repulsion, the inter-orbital interaction further
reduces the occupation yielding a non-monotonic behavior of the magnetic moment
as a function of the interactions, with maximum polarization only in a small
window at intermediate values of the parameters. Furthermore, $U'$ changes the
relative position of the peaks in the density of states, particularly on the
iron $d_{z^2}$ orbital, which is expected to greatly affect the binding of
ligands.
