# Many-Body Effects in FeN\({}_{4}\) Center Embedded in Graphene
## 2 Model Hamiltonian  
In order to derive a simplified model Hamiltonian, we focus on taking into account only the most relevant features that determine the electronic structure and occupation of the transition metal atom. Correlation effects in the carbon and nitrogen atoms are neglected, and the organic backbone of the molecule is constructed by using an LCAO-based tight-binding Hamiltonian. Exact numerical methods are used to account for the many-body physics introduced by the central atom, which we model as a multi-orbital Anderson-like impurity. Generalized Anderson impurity models have been applied previously to porphyrin-like molecules [32; 34; 37; 38; 74; 75] to predict potential energy surfaces and electronic coupling factors [63; 76] for various transition-metal complexes. The advantages of our model Hamiltonian approach are that in this way, we can account for the many-body correlation effects in a numerically exact way, and also, the computations can be scaled relatively easily to handle systems with multiple impurities and more complex geometries. The transition-metal center is created by removing six carbon atoms from the graphene sheet and replacing these with four nitrogens and a single iron atom, as shown in Figure 1a. For simplicity, we assume the structure to be planar with point-group symmetry \(D_{2h}\). In order to identify the active orbitals, we start by considering the carbon atoms to build a model for graphene. The carbon atoms are connected by \(\sigma\)-bonds of hybrid \(sp^{2}\) orbitals, formed from linear combinations of carbon \(2p_{x}\), \(2p_{y}\), and \(2s\) orbitals. The \(\pi\)-bonds are weaker than the \(\sigma\)-bonds, formed by the remaining \(p_{z}\) orbitals. The graphene sheet is then functionalized by adding hydrogen atoms bonded with each dangling \(sp^{2}\) orbital of the carbons at the edges. We approximated the values of the hopping parameters to approximate the results for bulk graphene. Modeling the \(p_{z}\) orbitals then only requires a simple nearest-neighbor hybridization \(t\), resulting in the well-known two-band model of graphene. Graphene's \(\sigma\) bands require a more sophisticated approach; see Appendix A for details. As an approximation, the N-C hopping is set equal to the C-C hopping. The nitrogens however have a remaining orbital that points toward iron. Looking at the coordination of iron and nitrogen atoms, the complex is approximately square planar with \(D_{4h}\) symmetry, which is useful in determining the bonding orbitals and the crystal-field splitting. Using only the \(3d\)-orbitals of iron, \(d_{x^{2}-y^{2}}\) will form a \(\sigma\) bond with the dangling \(sp^{2}\) orbital from the nitrogen, while \(\pi\)-bonds will form from the iron \(d_{xz}\), \(d_{yz}\) and nitrogen \(2p_{z}\) orbitals, where we define the \(x\) and \(y\) axes to point along the lines connecting the Fe and the N atoms. The five dimensional \((l=2)\) irreducible representation of \(K\) (the continuous rotation group) becomes reduced to four irreducible representations when the symmetry is decreased to \(D_{4h}\). If only electrostatic effects are considered, the crystal field splitting is predicted to be as seen in Figure 2a. However, this does not capture higher order effects or Jahn-Teller distortions [77]. For a more accurate description, the levels in a graphitic structure are arranged as in [47], as shown in Figure 2b. As discussed below, the positions of the energy levels are adjusted to match the occupations of the \(3d\) orbitals approximately given by the DFT at \(U=4.0\)eV. They are then held fixed in all other calculations. Figure 1: (**a**) A FeN4 complex embedded in a graphene sheet. (**b**) The FeC10N4 complex functionalized by 10 hydrogen atoms. Figure 2: (**a**) Splitting of \(d\) levels under \(D_{4h}\) symmetry. \(\Delta_{0}\) is the splitting in an octahedral field. (**b**) Arrangement of energy levels used throughout this work following a similar convention adopted by [PERSON] _et al._[47].