# Many-Body Effects in FeN\({}_{4}\) Center Embedded in Graphene
## 4 Results  
DFT is known to have difficulties in describing correlated electrons in open \(d\) and \(f\) shell systems. In particular, DFT can fail in predicting if the ground state has low, intermediate, or high total spin polarization (this failure can however be mitigated if we allow spin contamination (spurious mixing of different spin-states) in the calculations [55]). Conventional DFT methods, such as LDA or GGA, also fail to account for the effects of the Coulomb interaction between localized electrons properly. For these reasons, the calculations presented here were performed using GGA+U within the VASP package [91; 92]. One main source of error in the GGA+U scheme arises from the fact that the \(U\) and \(J\) terms in the Hamiltonian (or functional) are handled in a mean-field fashion, resulting in the single \(U_{eff}=U-J\) parameter. With this in mind, we first compared our exact DMRG results with DFT+U calculations, where we ignored all many-body terms except for the intra-orbital Coulomb repulsion and the Hund coupling of the spins. In other words, \(H_{int}=U\sum_{m}n_{\uparrow m}n_{\downarrow m}-2\sum_{m\
eq m^{\prime}}J_{ nm^{\prime}}\vec{S}_{m}\vec{S}_{m^{\prime}}\). Figure 3: Chain geometry of FeC\({}_{10}\)N\({}_{4}\) after the Lanczos mapping. Red sites correspond to carbon atoms. Orange sites correspond to the orbitals of the seed \(|\alpha_{0}\rangle\) and \(|\beta_{0}\rangle\) on the nitrogen atom (see text). Blue sites represent the 5 \(d\)-orbitals of iron. The box (dotted lines) indicates the region where interactions are included, and black lines denote hopping. In order to benchmark our approach, we began with the iron complex, FeC\({}_{10}\)N\({}_{4}\), depicted in Figure 1. Here, we used level splittings between the \(d\) orbitals that matched the occupations obtained from GGA+U using the physically relevant value of \(U=4\) eV, as shown in Table 3 (Note that obtaining exact agreement between the DFT results and our model Hamiltonian was not possible due to the complex interplay between various terms in the Hamiltonian. The results shown were as close as we could get by searching within the multi-dimensional parameter space and fixing the value of \(U\). Our formulation was \(SU(2)\)-invariant, so that any high-spin ground state would be a \((2S+1)\)-fold degenerate multiplet.). The ground-state occupation and magnetic moment of iron (total spin) as a function of \(U\) is shown in Figure 4. Calculations were carried out by varying the number of electrons and the value of the total spin \(S\). The ground state was then obtained by minimizing the energy. The spin was seen to remain zero until the Coulomb interaction reached a value of \(U\approx 3.0\) eV. As the value of \(U\) increased beyond this point, the occupation of the iron levels decreased as the magnetic moment increased until it saturated at \(S=2\) (\(\langle N\rangle=4\)). For large values of \(U\), Coulomb repulsion prevented double-occupation of orbitals. Over the range of \(3.5\leq U\leq 4.5\), the ground state has \(S=1\), which is the physically interesting range (Note that different Racah parameters should be used, in principle, for various occupations of the iron atom, for simplicity, we used a fixed value of these parameters, which corresponded to Fe\({}^{2+}\)). When all the interaction terms were included in the Hamiltonian, the ground state changed drastically. In order to gain insight into this result, we considered a rigid shift, \(-V_{Fe}\sum_{m}n_{m}\), in the position of the Fe energy levels. This ad hoc potential could be physically thought of as representing the screened interaction with the nucleus, so that it could be used as a parameter to mimic the occupation of the levels for various transition metals. In Figure 5, we show the overall occupation of the molecule, the Fe atom, and the total spin \(S\), as a function of \(U\) and \(V_{Fe}\).