# Many-Body Effects in FeN\({}_{4}\) Center Embedded in Graphene
## Appendix A Tight-Binding Model for [PERSON]'s \(\sigma\)-Bands  
Before discussing the electronic structure of the \(\sigma\) bands, we briefly present the general approach for determining the vanishing and non-vanishing matrix elements between various orbitals. For the \(s\) and \(p\) orbitals, there are just four non-zero overlap integrals to consider: \(ss\sigma\), \(sp\sigma\), \(pp\sigma\), and \(pp\pi\). Due to the radial symmetry of the \(s\)-orbitals, the \(ss\sigma\) bond has no angular dependence. \(\sigma\) and \(\pi\) bonds are classified by whether the interatomic separation direction is parallel or perpendicular to the orbital axis. Since the \(p\)-orbitals can orient with any angle between them (Figure 11), we project these orbitals into their normal and parallel (\(\sigma\) and \(\pi\)) components as follows.  
\[\left|p\right\rangle=\text{cos}\theta\left|p_{\sigma}\right\rangle+\text{sin} \theta\left|p_{\pi}\right\rangle.\]  
Matrix elements between neighboring \(s\) and \(p\) states can then be written as:  
\[\left\langle s\right|H\left|p\right\rangle=H_{sp\sigma}\text{cos}\theta,\]where the definition \(\left\langle s\right|H\left|p_{\sigma}\right\rangle=H_{spr}\) has been used, and \(\left\langle s\right|H\left|p_{\pi}\right\rangle=0\) by symmetry. The angle \(\theta\) is defined in Figure 11. Similarly, the matrix elements between \(p\) states are:  
\[\left\langle p_{1}\right|H\left|p_{2}\right\rangle=H_{ppc}\text{cos}\theta_{1} \text{cos}\theta_{2}+H_{pp\pi}\text{sin}\theta_{1}\text{sin}\theta_{2}.\]  
In general, \(sp^{2}\) orbitals are planar and form angles of \(120\deg\). Since the unit cell of graphene has two atoms and each atom contributes three \(sp^{2}\) states, we obtain six \(\sigma\)-bands, which are split into two groups of three bands each where one group lies above and the other below the Fermi energy. Following [103], the \(6\times 6\) matrix takes the following form:  
\[H=\begin{array}{c}2s^{a}\\ 2p_{x}^{a}\\ 2p_{y}^{b}\\ 2p_{x}^{b}\\ 2p_{y}^{b}\\ \end{array}\begin{pmatrix}2s^{a}&2p_{x}^{a}&2p_{y}^{a}&2s^{b}&2p_{x}^{b}&2p_{ y}^{b}&2p_{y}^{b}\\ h_{11}&h_{12}&h_{13}&h_{14}&h_{15}&h_{16}\\ h_{21}&h_{22}&h_{23}&h_{24}&h_{25}&h_{26}\\ h_{31}&h_{32}&h_{33}&h_{34}&h_{35}&h_{36}\\ h_{41}&h_{42}&h_{43}&h_{44}&h_{45}&h_{46}\\ h_{51}&h_{52}&h_{53}&h_{54}&h_{55}&h_{56}\\ h_{61}&h_{62}&h_{63}&h_{64}&h_{65}&h_{66}\\ \end{pmatrix}. \tag{11}\]  
The values of the matrix elements are given by:  
\[h_{11}=h_{44}=\epsilon_{s} \tag{12}\]  
\[h_{22}=h_{33}=h_{55}=h_{66}=\epsilon_{p} \tag{13}\]  
\[h_{14}=H_{ss\sigma} \tag{14}\]  
\[h_{15}=H_{spc}\text{cos}\theta \tag{15}\]  
\[h_{16}=H_{spc}\text{sin}\theta \tag{16}\]  
\[h_{25}=H_{ppc}\text{cos}^{2}\theta+H_{pp\pi}\text{sin}^{2}\theta \tag{17}\]  
\[h_{26}=(H_{pp\sigma}-H_{pp\pi})\text{cos}\theta\text{sin}\theta \tag{18}\]  
\[h_{36}=H_{ppo}\text{sin}^{2}\theta+H_{pp\pi}\text{cos}^{2}\theta, \tag{19}\]  
with all the remaining elements being zero. The numerical values for the hopping are (in \(eV\)): \(\epsilon_{s}=-8.7\), \(\epsilon_{p}=0\), \(H_{ss\sigma}=-6.7\), \(H_{sp\sigma}=5.5\), \(H_{pp\sigma}=5.1\), and \(H_{pp\pi}=-3.1\).