## Introduction

Single-atom Fe-N-C catalysts are considered a candidate to replace the platinum group metals in oxygen reduction reaction (ORR), due to its high activity, anti-toxicity, and metal atom utilization1,2,3,4. Advanced Fe-N-C materials show comparable ORR performance to that of benchmark Pt/C3,5,6,7. Both theoretical and experimental studies indicate that the Fe-N4 site is the active species of Fe-N-C catalysts3,4,6,8,9,10. However, the ORR activity of Fe-N4 site on different carbon supports varies greatly, and Fe-N4 ORR activity on curved carbon supports is usually orientation-dependence11,12,13,14,15. The essential of the difference and orientation-dependence of Fe-N4 ORR activity remains unclear. Therefore, understanding the ORR activity of the Fe-N4 site is crucial to further improve the ORR performance of Fe-N-C catalysts.

The intrinsic ORR activity of the Fe-N4 site relates to the electronic structure of carbon supports7,16. Controlling the electronic structure of carbon supports alters the interaction between the supports and the Fe-N4 site, resulting in the activity change of the Fe-N4 site6,7,13,16,17,18. However, the essential of the change in Fe-N4 ORR activity is still puzzling due to the complex factors affecting it. Some works reported that the electronic structure of carbon supports (electron-withdrawing/donating property) can affect the d-band center or d-orbital level of Fe site17,19,20,21. While, others claimed that it regulates the charge state of Fe site6,22,23. To understand the Fe-N4 ORR activity, many efforts have been made to explore the descriptors of Fe-N4 ORR activity13,16,20,24,25,26. For example, the full width at half-maximum of C 1s photoemission spectra13, d-band center20, and the electronegativity of the nearest neighbor atoms24, are considered to describe the ORR activity of the Fe-N4 site. However, these descriptors are difficult to understand the orientation-dependence and differences of ORR activity at the same Fe-N4 site on different carbon supports11,12,14. Therefore, elucidating the origin of ORR activity and exploring the universal descriptors of ORR activity on the Fe-N4 site are urgent.

In this work, we demonstrate that the hybridization among Fe 3dz2, 3dyz (3dxz), and O2 π* orbital is the origin of ORR activity on the Fe-N4 site by density functional theory (DFT) calculations. We established the relationship between the geometric, electronic structure, and ORR activity of the Fe-N4 site. From the geometric structure, the Fe–O bond length (LFe-O) can accurately describe the ORR activity of the Fe-N4 site. From the electronic structure, the d-band center gap of spin states (Δd), the magnetic moment of the Fe site (MFe), and *O2 (M*O2) are nearly linear in relationship to the ORR activity of the Fe-N4 site. We found that these descriptors and Fe-N4 ORR activity are located in two regions. We provided a model to elucidate the origin and correlation of these descriptors from the spatial structure and level of the frontier orbital. This model reveals that descriptors essentially reflect hybridization between Fe 3dz2, 3dyz (3dxz), and O2 π* orbital, which is related to the height of 3d orbital spatial distribution. This study offers new insights into the origin and descriptor of ORR activity on M-N-C catalysts and provides guidance for the design of high-performance ORR catalysts.

## Results

### Structural stability and ORR activity of Fe-N4 sites

Here, we built the model of the Fe-N4 site embedded in the graphene slab (Fig. 1a). We introduced various divacancy defects around the Fe-N4 site to obtain the various sample of Fe-N4 ORR activity, since divacancy defects can modify the electronic structure of the carbon supports and also affect the ORR activity of the Fe-N4 site13,16,22,27,28,29. Then pentagon-octagon-pentagon (5-8-5) defects were constructed by selectively removing C–C bonds around the Fe-N4 site. Triple pentagon-triple heptagon (555-777) and pentagon-heptagon-heptagon-pentagon (5-7-7-5) defects were obtained by rotating the different C–C bonds (Fig. 1b). A total of 42 types of divacancy defects were constructed around the Fe-N4 site, including ten types of 5-8-5 (Supplementary Fig. 1), nine types of 555-777 (Supplementary Fig. 2), and 23 types of 5-7-7-5 (Supplementary Fig. 3) defects.

To evaluate the stability of single-atom Fe sites on carbon supports, we calculated the binding energy (Ebind) of single-atom Fe sites (Fig. 1c). For the perfect Fe-N4 configuration, the binding energy of the single-atom Fe site is –1.01 eV, indicating that the configuration of the Fe-N4 site on carbon supports is stable. Under the control of divacancy defects, the binding energies of Fe sites range from –1.88 ~0.63 eV. Obviously, divacancy defects on the carbon supports can affect the stability of a single-atom Fe site. Among them, the relatively stable systems (Ebind <–1.01 eV) were selected as examples to study the ORR activity of Fe-N4 sites, which include 5-8-5 (6, 8, 9), 555-777 (5a, 6a, 7b, 8a, 9a), and 5-7-7-5 defect (6I, 7I, 8I, 8II, 8III, 9I, 10I). The ORR activity on the upper (blue) and lower (red) sides of Fe-N4 configurations were investigated due to surface asymmetry (Fig. 1d). We found the hydrogenation of *O2 is a potential-determining step (PDS) at all Fe-N4 configurations. Divacancy defect can regulate the free energy difference of this step on the Fe-N4 site (Supplementary Table 1). The maximal free energy difference (ΔGmax) of ORR at the perfect FeN4 site is –0.45 eV, which is consistent with previous studies3,6,30. Among them, divacancy defects, such as 6, 5a, 6a, 6I, 7I, and 8I, can enhance the activity of the Fe-N4 site, with ∆Gmax less than –0.60 eV (Fig. 1d). For the asymmetric surfaces, the ORR activity of the Fe-N4 site showed orientation-dependence and difference, such as 6I, 7I, 8I, etc. Besides, divacancy defects can control the electron-donating capability of the carbon support, ultimately affecting the charge state and the d-band center of the Fe site (Supplementary Fig. 4). However, these factors are difficult to elucidate the orientation-dependence of the ORR activity of the Fe-N4 site on an asymmetric surface.

### Factors of ORR activity at the Fe-N4 site

We investigated the ORR process of Fe-N4 configurations from energy, geometry, and electronic structure to explore the origin of Fe-N4 ORR activity. Here, we highlight the results of 6 (5-8-5), 6a (555-777), and 6I (5-7-7-5) models because they can improve the ORR activity of the Fe-N4 site with good thermodynamic stability (Supplementary Fig. 5). The free energy diagram of ORR shows that the ΔG value of *OOH formation on 6, 6a, and 6I are –0.62, –0.65, and –0.66 eV (Fig. 2a and Supplementary Fig. 6), respectively. Compared with the perfect FeN4 site (–0.45 eV), divacancy defects (6, 6a, and 6I) can facilitate *O2 conversion to *OOH intermediate. We found that the adsorption energy of *O2 with end-on1 and end-on2 configurations on the FeN4 site are –0.53 and –0.41 eV, respectively, and the adsorption energy of *O2 with side-on configuration on the FeN4 site is –0.01 eV (Supplementary Fig. 7). This result shows that the end-on adsorption is more favorable than the side-on adsorption at the FeN4 site. Among them, the adsorption energies of *O2 on 6, 6a, and 6I are –0.34, –0.33, and –0.38 eV (Fig. 2b), respectively, which are larger than that of the perfect FeN4 site (–0.53 eV). These results suggest the weak adsorption of *O2 on the Fe-N4 site is the key to enhance the ORR activity of the Fe-N4 site. Meanwhile, the O–O bond lengths on the O2@FeN4, O2@6, O2@6a, and O2@6I are 1.284, 1.297, 1.294, and 1.296 Å (Fig. 2b), respectively, which are longer than the bond length of O2 molecule (1.234 Å). Besides, the Fe–O bond lengths on the O2@FeN4, O2@6, O2@6a, and O2@6I are 1.974, 1.745, 1.724, and 1.731 Å (Fig. 2b), respectively. These results reveal that divacancy defects (6, 6a, and 6I) can enhance the interaction between *O2 and Fe site and promote the activation of *O2, thus increasing the Fe-N4 ORR activity.

We studied the electronic structure of the Fe-N4 site and O2 to understand the activation mechanism of *O2. Before the adsorption of O2, the projected density of state (PDOS) of the Fe site indicates that the Fe site has spin polarization with a spin magnetic moment of 2 μB. The occupied states of the Fe site near the Fermi level consist of localized 3dxz, 3dx2y2, and 3dz2 (Supplementary Fig. 8). Compared to the PDOS of the FeN4 site, divacancy defects (6, 6a, and 6I) modified the intensity and distribution of Fe 3d states (especially 3dxz and 3dz2). For the O2 molecule, the PDOS of O2 also shows spin polarization with a spin magnetic moment of 2 μB. The anti-bonding orbitals (π*) of O2 are mainly composed of the O2 2py and 2pz orbitals (Supplementary Fig. 8). When O2 adsorb on the Fe-N4 site, the 2pz orbital in the O2 π* orbital split into the discrete level, and the intensity of localized Fe 3dz2 states is significantly reduced (Supplementary Fig. 9). A new hybrid state formed near the Fermi level, due to the hybridization between O2 2pz and Fe 3dz2 orbital. In the O2@6, O2@6a, and O2@6I, the strong hybridization between Fe 3dz2, and O 2pz orbital lead to the significant discrete of the O2 2pz orbital state (Fig. 2c), suggesting that the O2 molecules are better activated. Besides, there is also hybridization between the 2py orbital of O2 and the 3dyz orbital of Fe (Supplementary Fig. 9). The d-band center of Fe spin-up and spin-down are close to each other due to hybridization (Fig. 2c and Supplementary Fig. 4). The d-band center gap of Fe spin state (Δd) of O2@FeN4, O2@6, O2@6a, and O2@6I are 2.45, 1.76, 1.67, and 1.70 eV (Fig. 2c), respectively, meaning the Δd is related to the activation of O2.

We also investigated the spatial distribution of Fe 3dz2 (3dyz) and O2 2pz (2py) orbital by the maximally localized Wannier functions (MLWF)31. The Wannier function of Fe 3dz2 and Fe 3dyz are located vertically and slant (Supplementary Figs. 10 and 11). O2 2pz orbital Wannier function shows high mirror symmetry (Supplementary Fig. 12). When O2 is adsorbed on the Fe-N4 site, the Wannier function of Fe 3dz2 (3dyz) and O2 2pz (2py) shows the trend of orbital matching (Fig. 2d and Supplementary Figs. 13, 14). Wannier function of Fe 3dz2 orbital decreases with that of O2 2pz orbital increases (Fig. 2d and Supplementary Fig. 13). Especially, the changes of Fe 3dz2 (3dyz) and O2 2pz (2py) orbital Wannier functions on O2@6, O2@6a, and O2@6I are larger than that on O2@FeN4. These results demonstrate that divacancy defects (6, 6a, and 6I) can enhance the hybridization between Fe and O2 to facilitate the activation of O2.

Generally, the filling of electrons in the O2 π* orbital is the beginning of the activation of O2. Charge density difference and Bader charge analysis reveal *O2 gain 0.34, 0.38, 0.40, and 0.39 e from perfect FeN4, 6, 6a, and 6I (Fig. 2e), respectively. The electron-gaining region (yellow region) of the *O2 mainly distributes in the π* orbitals of O2, which is consistent with the distribution region of the Wannier function (Fig. 2d and Supplementary Figs. 13, 14), reflecting that the electron transfer between Fe 3dz2 (3dyz) and O2 π* orbitals. Furthermore, due to electrons transfer between Fe sites and O2, the Fe 3dz2 (3dyz) and O2 π* orbitals are broken, thus changing the magnetic moment of the Fe site and O2. Spin density analysis show divacancy defects (6, 6a, and 6I) hardly change the spin density of Fe sites before O2 adsorption (Supplementary Fig. 15). After O2 adsorption, the spin density is mainly localized at the Fe site and *O2 has the opposite direction (Fig. 2f). The spin density of the Fe site and *O2 in the O2@6, O2@6a, and O2@6I is weaker than that in O2@FeN4 (Fig. 2f). Therefore, the origin of O2 activation is the hybridization of the Fe 3dz2 (3dyz) and *O2 2pz (2py) orbitals, which results in electron transfer from the Fe site to *O2, breaking the Fe 3dz2 (3dyz) and O2 2pz (2py) orbitals and reducing Δd and magnetic moments.

### Descriptors of ORR activity at Fe-N4 sites

As mentioned above, the adsorption energy of *O2, the bond length, the charge state, Δd, and the magnetic moment may be related to the ORR activity of the Fe-N4 site (Supplementary Table 2). To gain accurate descriptors for the ORR activity of the Fe-N4 site, we investigated the correlation between these factors and ORR activity (Fig. 3 and Supplementary Figs. 1618). According to the free energy diagram, the adsorption energy of *O2 may be related to ORR activity. The adsorption energy of *O2 at the Fe-N4 site ranges from –0.6 to –0.1 eV under the regulation of divacancy defects (Supplementary Fig. 16). The linear fitting results reveal that the coefficient of determination R-square between the adsorption energy of *O2 and the ORR activity of the Fe-N4 site is 0.15. No direct correlation between the adsorption energy of *O2 and ORR activity at the Fe-N4 site. From the geometric structure, the O–O bond length (LO-O) is related to the activation of O2 molecules, and the Fe–O bond length (LFe-O) can reveal the strength of the interaction between *O2 and Fe-N4 site. Herein, we found that the O–O bond length is distributed between 1.264–1.298 Å by regulating divacancy defects (Supplementary Fig. 17). When the O–O bond length is larger than 1.290 Å, and the free energy difference for hydrogenation of *O2 is less than –0.6 eV. The coefficient of determination R-square between the O–O bond length and Fe-N4 ORR activity is 0.81, with the relevant formula being ΔG = –9.93LO–O + 12.3 (Supplementary Fig. 17). Besides, the length of the Fe–O bond is distributed in stages (Fig. 3a). For the active Fe-N4 site (golden circle), the average Fe–O bond length is around 1.737 Å. For the inactive Fe-N4 site (blue circle), the average Fe–O bond length is near 2.013 Å. The coefficient of determination R-square between the Fe–O bond length and Fe-N4 ORR activity is 0.95 (Fig. 3a), indicating the Fe–O bond length can accurately describe the ORR activity of the Fe-N4 site with the relevant formula ΔG = 0.75LFe–O + 1.94.

From the electronic structure, the charge state of the Fe site (*O2) can reflect the electrons transfer between Fe and O2, which may have a potential relationship with Fe-N4 ORR activity (Supplementary Fig. 18). The Bader charge state of Fe and *O2 are –1.22 to –1.13 e and 0.24 to 0.41 e (Supplementary Fig. 18), respectively, due to the regulation of divacancy defect. The linear fitting results show that the coefficient of determination R-square between the charge state of the Fe site (*O2) and the ORR activity of the Fe-N4 site is 0.10 (0.58), indicating that they are difficult to describe the ORR activity of Fe-N4 site (Supplementary Fig. 18). In addition, Δd (Fig. 3b), MFe (Fig. 3c), and M*O2 (Fig. 3d) are also characteristic symbols for the change of Fe and *O2 electronic structure. We found that Δd, MFe, and M*O2 are also distributed in stages (Fig. 3b–d). The smaller their values, the higher the ORR activity of the Fe-N4 site. After the adsorption of O2, the average values of Δd, MFe, and M*O2 on the active Fe-N4 site (golden circle) are 1.755 eV, 1.190 μB, and 0.881 μB, respectively. Correspondingly, the average of ΔG is –0.638 eV. For the inactive Fe-N4 sites (blue circle), the average of ΔG, Δd, MFe, and M*O2 are –0.430 eV, 2.570 eV, 1.756 μB, and 1.177 μB, respectively. These fitting results show that the correlation between Δd, MFe, M*O2, and ΔG is nearly linear. The R-squares are 0.95, 0.94, and 0.91 (Fig. 3b–d), respectively, indicating Δd, MFe, M*O2 are promising ORR activity descriptors for the Fe-N4 site. The relevant formulas are ΔG = 0.25Δd – 1.08, ΔG = 0.37MFe – 1.08, ΔG = 0.67 M*O2 – 1.22, respectively.

To generalize the correlations between electronic structure and Fe-N4 ORR activity, we calculated the electronic structure of O2@Fe-N4 by using other theoretical approximations, including local density approximation (LDA (CA))32,33, Perdew–Wang 91 (GGA (PW91))34, strongly constrained and appropriately normed semilocal density functional (meta-GGA (SCAN))35, and Heyd–Scuseria–Ernzerhof screened hybrid density functional (HSE06)36,37. The results show that the type of DFT exchange-correlation functional can affect the absolute value of the spin magnetic moment of Fe (MFe), the spin magnetic moment of *O2 (M*O2), and the d-band center gap of spin state (∆d), but the variation tendency of the value is consistent (Supplementary Figs. 1922 and Supplementary Tables 35). There is still a nearly linear correlation between electronic structure and Fe-N4 ORR activity in the theoretical framework of LDA (CA) + U, GGA (PW91) + U, meta-GGA (SCAN), and HSE06 (Supplementary Figs. 2022). MFe, M*O2, and Δd are still distributed in stages. For the LDA (CA) + U, GGA (PW91) + U, meta-GGA (SCAN), and HSE06, the R-Square between MFe and ΔG are 0.93, 0.95, 0.91, and 0.91 (Supplementary Fig. 20), respectively. In addition, M*O2 and Δd were also highly correlated with ΔG (Supplementary Figs. 21 and 22). Therefore, the correlation between the electronic structure and the Fe-N4 ORR activity remains valid in other theoretical approximations.

### Correlations between the descriptors of ORR activity at Fe-N4 sites

We explored the correlation between these descriptors. Obviously, there is a nearly linear relationship between Δd, MFe, M*O2, and the Fe–O bond length, with the R-squares of 0.98, 0.99, and 0.94 (Fig. 4a and Supplementary Fig. 23), respectively, indicating that these descriptors can reflect the origin of Fe-N4 ORR activity. To reveal the essence of descriptors, we analyzed the projected magnetic moment of Fe, the spatial structure of Fe 3d and O2 π* orbital, and the schematic frontier orbital level diagram.

The projected magnetic moment of Fe fluctuates obviously, due to the regulation of the divacancy defects (Fig. 4b). Only when the magnetic moments of 3dz2 and 3dyz + 3dxz decrease at the same time, the Fe-N4 site shows higher ORR activity. This result suggests that the ORR activity at the active Fe-N4 site is attributed to the strong hybridization of Fe 3dz2–*O2 2pz and Fe 3dyz–*O2 2py (Fe 3dxz–*O2 2px).

The spatial distribution of Fe 3d and O2 π* orbital can elucidate the correlation between the Fe–O bond length and MFe. As shown in Fig. 4c, the 3dxy (3dx2y2), 3dyz (3dxz) and 3dz2 are different heights in the Z direction. When the Fe–O bond is long, the activation of O2 attribute to the hybridization between Fe 3dz2 and O2 2pz orbital, which leads to the decrease of the projected magnetic moments of Fe 3dz2. When the Fe–O bond is short, the activation of O2 is additionally improved by the hybridization of Fe 3dyz (3dxz) and 2py (2px) orbitals. Meanwhile, the projected magnetic moments of Fe 3dz2 and 3dyz (3dxz) are significantly reduced. The Fe 3dxy and 3dx2y2 have weak hybridization with O2 π* orbital due to their low spatial distribution in the Z direction. Thereby the projected magnetic moments of 3dxy and 3dx2y2 vary only slightly.

The schematic frontier orbital diagram of O2@Fe-N4 can reveal the differences in O2 activation (Fig. 4d)38. Before the adsorption of O2, the Fe 3d orbital splits into an empty 3dxy, singly occupied 3dyz, 3dz2 type orbitals, and occupied 3dxz, 3dx2y2 orbitals, due to Fe 3d orbital hybridization with the carbon support. For the O2@Fe-N4, the differences in O2 activation is attributed to the degree of hybridization of Fe 3dz2 and 3dyz (or 3dxz) with O2 π* orbital. For the active Fe-N4 site, the strong hybridization of 3dz2 and 3dyz (or 3dxz) with O2 π* orbital promote the O2 activation. For the inactive Fe-N4 site, the strong hybridization of 3dz2 with O2 π* orbital and negligible hybridization of 3dyz (or 3dxz) with O2 π* orbital lead to the poor O2 activation. The change of Δd, MFe, and M*O2 are also due to the above factors. Therefore, the Fe-N4 ORR activity is origin from the activation of O2, that is, the hybridization between Fe 3dz2 (3dyz or 3dxz) and *O2 π* orbital. The Fe–O bond length, Δd, MFe, and M*O2 are the specific symbols describing hybridization.

## Discussion

In this work, we investigated the origin and descriptors of ORR activity on the Fe-N4 site using DFT calculation. We explored divacancy defects on carbon supports that could modulate the ORR activity of the Fe-N4 site. The first electron step (*O2 + H+ + e → *OOH) is the potential-determining step (PDS) on all Fe-N4 configurations. Fe-N4 site on 6, 5a, 6a, 6I, 7I, and 8I showed high ORR activity with ΔGmax less than –0.60 eV. The ORR activity of Fe-N4 sites on the asymmetric surface showed orientation-dependence. We demonstrated that the hybridization between Fe 3dz2, 3dyz (3dxz), and O2 π* orbitals is the origin of Fe-N4 ORR activity by studying the electronic structure. The Fe–O bond length, Δd, MFe, and M*O2 show staged differences, which can be used as descriptors to accurately describe the ORR activity of the Fe-N4 site. There is a nearly linear correlation between Δd, MFe, M*O2, and the Fe–O bond length from geometry and electron structure. Importantly, we elucidate that the Fe–O bond length, Δd, MFe, and M*O2 are essentially hybrids between Fe 3dz2 (3dyz) and O2 π* orbitals from the spatial distribution and energy level of the frontier orbital. The height of the Fe 3d projected orbital is related to the activation of O2. For the active Fe-N4 sites, the strong hybridization of 3dz2 and 3dyz (or 3dxz) with O2 π* orbital results in the short Fe–O bond length (1.737 Å), low Δd (1.755 eV), MFe (1.190 μB), and M*O2 (0.881 μB). For the inactive Fe-N4 site, the strong hybridization of 3dz2 with O2 π* orbital and the negligible hybridization of 3dyz (3dxz) with O2 π* orbital lead to the long Fe–O bond length (2.013 Å), high Δd (2.570 eV), MFe (1.756 μB), and M*O2 (1.177 μB). This work offers insights into the Fe-N4 ORR activity and provides guidelines for designing the high-performance Fe-N-C catalysts.

## Methods

### DFT calculations

Our spin-polarized density functional theory simulation was calculated by using the Vienna ab initio simulation package (VASP)39. The PAW potentials and the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) were employed to describe the interaction of electron-ion and the electron-electron exchange and correlation functional, respectively40,41. After the test, the cutoff energy was set at 400 eV, and Monkhorst–Pack mesh with a 2 × 2 × 1 grid was used in our calculations after testing (Supplementary Fig. 24). van der Waals (VDW) forces were corrected with the D2 method of Grimme42. The convergence criterion was set at 0.02 eV Å−1 for the force and 1 × 10−5 eV per atom for energy. We used the correlation energy (U) of 4 eV and the exchange energy (J) of 1 eV for Fe 3d orbitals43,44. We built a super-cell (7 × 8) carbon substrate as a model, which includes 106 C atoms, 4 N atoms, and 1 Fe atom. The lattice parameters of this slab are a = 17.31 Å and b = 16.98 Å after optimization. The lattice constants are optimized again due to the introduction of divacancy defects in the system. The vacuum layer was set at 15 Å. Maximally localized Wannier functions (MLWFs) methods are applied by Wannier90 to calculate the spatial distribution of Fe 3d and O2 2p orbital31,45,46,47. We also considered the effect of solvent on the ORR activity of the Fe-N4 site by employing an implicit solvent model48,49. To confirm the generalization validity of the correlation between electronic structure and Fe-N4 ORR activity, local density approximation with the Ceperly–Alder functional (LDA (CA))32,33, Perdew–Wang 91 (GGA (PW91))34, strongly constrained and appropriately normed semilocal density functional (meta-GGA (SCAN))35, and Heyd–Scuseria–Ernzerhof screened hybrid density functional (HSE06) were used to evaluate the electronic structure36,37. In the framework of LDA (CA) and GGA (PW91), we also consider the Coulomb interaction between Fe 3d orbital electrons, and the Hubbard U value is 3 eV. First-principles Born–Oppenheimer molecular dynamics (BOMD) simulation was performed in the canonical ensemble with Nosé–Hoover heat bath schemes50,51. During the simulation, spin polarization is not considered, and the Brillouin-zone integrations were performed using the Gamma-point grid. The total simulation time is 5 ps with a time step of 1 fs at 300 K. We also used the VESTA package to visualize the structures and charge density differences52.

The binding energy of the Fe site (Ebind) on the carbon supports can be calculated by

$${{{{{{\rm{E}}}}}}}_{bind}={{{{{{\rm{E}}}}}}}_{{{{{{\rm{Fe}}}}}}@{{{{{\rm{sub}}}}}}}-{{{{{{\rm{E}}}}}}}_{{{{{{\rm{sub}}}}}}}-{{{{{{\rm{E}}}}}}}_{{{{{{\rm{Fe}}}}}}-{{{{{\rm{bulk}}}}}}}$$
(1)

where EFe@sub is the total energy of the Fe atom embedded in the carbon supports; Esub and EFe are the energies of the substrate and single Fe atom in the Fe bulk, respectively.

The Gibbs free energy can be expressed as

$$\Delta {{{{{\rm{G}}}}}}=\Delta {{{{{\rm{E}}}}}}+\Delta {{{{{\rm{ZPE}}}}}}-{{{{{\rm{T}}}}}}\cdot\Delta {{{{{\rm{S}}}}}}$$
(2)

where ∆E, ∆ZPE, and ∆S are the reaction energy calculated by the DFT methods, the changes in zero-point energies, and the entropy during the reaction, respectively53,54,55. T is the temperature (298.15 K, in our work).

The d-band center gap of spin state (∆d) can be defined as

$$\Delta {{{{{\rm{d}}}}}}=|{\varepsilon }_{{{{{{\rm{up}}}}}}}-{\varepsilon }_{{{{{{\rm{dn}}}}}}}|=\left|\frac{{\int }_{-\infty }^{+\infty }{n}_{{{{{{\rm{up}}}}}}}(\varepsilon )\varepsilon d\varepsilon }{{\int }_{-\infty }^{+\infty }{n}_{{{{{{\rm{up}}}}}}}(\varepsilon )d\varepsilon }-\frac{{\int }_{-\infty }^{+\infty }{n}_{{{{{{\rm{dn}}}}}}}(\varepsilon )\varepsilon d\varepsilon }{{\int }_{-\infty }^{+\infty }{n}_{{{{{{\rm{dn}}}}}}}(\varepsilon )d\varepsilon }\right|$$
(3)

where εup is the d-band center of the Fe 3d spin-up projected density of states. And εdn is the d-band center of the Fe 3d spin-down projected density of states56.

The magnetic moment (M) is defined as

$${{{{{\rm{M}}}}}}={{{{{\rm{N}}}}}}({{{{{\rm{spin}}}}}}\mbox{-}{{{{{\rm{up}}}}}})-{{{{{\rm{N}}}}}}({{{{{\rm{spin}}}}}}\mbox{-}{{{{{\rm{down}}}}}})$$
(4)
$${{{{{\rm{M}}}}}}={{{{{\rm{M}}}}}}({{{{{\rm{3\it dxy}}}}}})+{{{{{\rm{M}}}}}}({{{{{{\rm{3\it dx}}}}}}}^{2}-{{{{{\it{y}}}}}}^{2})+{{{{{\rm{M}}}}}}({{{{{\rm{3\it dyz}}}}}})+{{{{{\rm{M}}}}}}({{{{{\rm{3\it dxz}}}}}})+{{{{{\rm{M}}}}}}({{{{{{\rm{3\it dz}}}}}}}^{2})$$
(5)

where N(spin-up) and N(spin-down) represent the number of electrons in the spin-up and spin-down occupied state, respectively57,58. M(3dxy), M(3dx2 – y2), M(3dyz), M(3dxz), and M(3dz2) represent the projected of magnetic moments on Fe 3dxy, 3dx2y2, 3dyz, 3dxz, and 3dz2, respectively.