# Many-Body Effects in FeN\({}_{4}\) Center Embedded in Graphene
## 2 Model Hamiltonian
The difference between the \(\pi\) and \(z^{2}\) orbitals is small and has been reversed to match the occupation of the orbitals according to DFT calculations. For iron, we include Coulomb and Hund interactions. The resulting general form of the Hamiltonian can be written as [78; 79]:  
\[H_{int}=\sum_{ \begin{subarray}{c}m_{1}=m_{2},m^{\prime}_{1},m^{\prime}_{2}\\ \sigma_{1},\sigma_{2},\sigma_{1}^{\prime},\sigma_{2}^{\prime}\end{subarray} } \left\langle m_{1}m_{2}\sigma_{1}\sigma_{2}\right|V\left|m^{\prime}_{1}m^{ \prime}_{2}\sigma^{\prime}_{1}\sigma^{\prime}_{2}\right\rangle\times d^{ \dagger}_{m_{1}\sigma_{1}}d^{\dagger}_{m_{2}\sigma_{2}}d_{m^{\prime}_{2}\sigma ^{\prime}_{2}}d_{m^{\prime}_{1}\sigma^{\prime}_{1}} \tag{1}\]  
In the preceding expression, \(m\) labels the \(d\)-orbitals and \(V\left(\mathbf{r}\right)\) is the screened Coulomb potential. This can be simplified to four matrix elements known as Kanamori parameters [80]: the intra-band Coulomb interaction \(U\), the inter-band Coulomb interaction \(U^{\prime}\), the inter-band exchange interaction \(J\), and the pair-hopping amplitude \(J^{\prime}\) (Notably, in quantum chemistry codes such as Gaussian, Jaguar, and DMol, Coulomb integrals are denoted by the symbol \(J\) and not \(U\). The symbol \(J\) is used here for magnetic exchange or Hund couplings, following common practice in the physics literature.). It can be shown [81] that \(J=J^{\prime}\) due to the symmetry of the orbitals involved and the fact that all coefficients are just integrals of the Coulomb term over the radial part of the wave functions. In order to ensure rotational invariance in orbital space, the condition \(U=U^{\prime}+2J\) must be satisfied [78] for all combinations of orbitals. The effects of the crystal field on the Coulomb interactions are ignored and are assumed to be relatively small. The interaction part of the Hamiltonian now takes the form:  
\[H_{int}=U\sum_{m}n_{\uparrow m}n_{\downarrow m}+\sum_{m>m^{\prime}}\left(U^{ \prime}_{mm^{\prime}}-\frac{l_{mm^{\prime}}}{2}\right)n_{m}n_{m^{\prime}}-2 \sum_{m<m^{\prime}}J_{mm^{\prime}}S_{m^{\prime}}S_{m^{\prime}}-\sum_{m>m^{ \prime}}J_{mm^{\prime}}d^{\dagger}_{\uparrow m}d^{\dagger}_{\downarrow m}d_{ \downarrow m^{\prime}} \tag{2}\]  
The interactions \(U\), \(U^{\prime}\), and \(J\) can be expressed in terms of the Racah parameters \(A\), \(B\), and \(C\), as shown in Table 1. The values of \(B\) and \(C\) were given in [82] for Fe\({}^{3+}\) and Fe\({}^{2+}\) and are restated in Table 2. While \(U\) and \(J\) take different values for different pairs of orbitals, the interaction \(U=A+4B+3C\) is orbital independent. Our calculations assumed that there were approximately six electrons in the iron orbitals so that we were close to Fe\({}^{2+}\). This leaves only one free parameter for the interactions, i.e., \(A\), or equivalently \(U\), since all \(U\) and \(U^{\prime}\) have the same dependence on \(A\) (note that \(J\) is independent of \(A\)). In general, it is the competition between \(U\) and \(J\) that determines if the complex is high or low spin. In addition, spin-orbit coupling can be introduced as it becomes important for heavier elements that could substitute the iron in similar molecules. However, this interaction for iron is relatively small (\(\approx\)0.05 eV) [83], and we have neglected it in our calculations, although it has been shown to be relevant in iron-based superconductors [84]. Notably, multi-orbital models for transition metals have also been used in dynamical-mean-field calculations of (bulk) correlated materials to discuss orbital-selective Mott transitions in iron-based superconductors, where the results can be sensitive to