Figure 3. (a) Gibbs free energy changes for the N2 adsorbed (*N2 → *N2H), (b) Gibbs free energy changes for the first protonate step (*N2 → *N2H), (c) Gibbs free energy changes for the last protonate step (*NH2 → *NH3), (d)-(g) Features importance of the GBR model precited for Gibbs free energy changes of ΔG*N2 (end on), ΔG*N2 (side on), ΔG*N2→*N2H, and ΔG*NH2→*NH3, respectively, where Ne, ra, θd, rTM, χ, Ei, rd, m, EA are the outmost electron number, the average bond length between TM atom and the coordinated C and N atom, d electron count, bond length between TM atoms, electronegativity, first ionization energy, radus of TM atoms, mass of TM atoms, electron affinity, the values of the parameters are summed in Table S8 and Table S9 .
In brief, several potential candidates are selected to show excellent electrocatalytic performance due to their proper N2capture energy small Gibbs free energy changes of the first and last hydrogenate steps. It is worth noting that the relatively unimportant structural properties, such as rTM, ra, and rd, play the predominant role in the N2 capture hydrogenation process even though the structural variety is entirely removed. In fact, those factors do not contradict the proposed atomic diversity effect since the objects of rd are the same. Hypothetically, we think the distance between TM atoms may play an essential role during the hydrogenation process since both ra and rd are highly related to the rTM, which should be further investigated.
To accurately obtain certain steps determining the NRR process, the whole Gibbs free energy changes are systematically investigated for the selected homonuclear systems. The complete reaction contains six proton-couple electron transfer steps (N2 + 3H2 → 2NH3), making several possible intermediates (*N2H, *N2H2, *N2H3, *N2H4, *N, *NH, *NH2, *NH3) existed under the electrocatalytic process. The difference in the N2 configurations leads to various reaction pathways for the eNRR. To obtain detail information of the NRR on TM2@sitex (x = 1, 2, 3, 4), the traditional distal, alternative, and enzymatic reaction pathways are selected for limiting potential searching. Besides, we expand a new reaction pathway for the end-on configuration systems for the confirmed ’capture-charge distribution-recapture-charge redistribution’ mechanism and name it Distal-1, as depicted in Figure 4 (a) . For the distal and alternative mechanisms, we do not have much discussion on them since a single TM atom participated predominately in the electrocatalytic NRR process, which does not efficiently reflect the synergistic effect for DAC. Among the screened systems, Ni2@site4 and Re2@site4 follow the distal and alternative reaction pathway, and the limiting potentials are -2.69 V/-2.31 V and -0.61 V/-1.47 V, respectively. For the highlighted distal-1 and enzymatic reaction pathway, the synergistic effect between dual TM atoms varies with each other. But both reaction pathways enlarge the charge distribution asymmetry under the ’capture-charge distribution-recapture-charge redistribution’ mechanism, thus accelerating the hydrogenation process. Among the rest systems, Mn2@site1 and Co2@site2 follow the Distal-1 reaction pathway, while Ru2@site1, W2@site2, Os2@site3, V2@site4, Cr2@site4, and W2@site4 go through the enzymatic reaction pathway. Briefly, Co2@site2, V2@site4, and Cr2@site4 are selected as examples for analysis. For the Co2@site2 system, Gibbs free energy change is -0.32 eV after N2 capture, within N-N bond length elongated to 1.15 Å; continuously, the *N2 to *N2H need an energy input of 0.44 eV, within the N-N bond increase to 1.25 Å and the second TM atom participated to bond with the proximal N atom. Then, the distal N atom keeps being attacked by the second proton-electron pair to form *N2H2 with an energy input of 0.08 eV, and the N-N bond expands into 1.33 Å, demonstrating the further activation of the N2 molecule. After the third proton hydrogenation, the first NH3 is formed and desorbed away from the adsorbed system, leaving a bald N atom adsorbed on the active site, and the energy change of this process is -0.87 eV. Subsequently, the proton-electron pair continues to attack the isolate N atom until the second NH3 form, within energy changes of -0.30 eV, -0.70 eV, and +0.35 eV for *N → *NH, *NH → *NH2, and *NH2 → *NH3, respectively. Therefore, among the six electron-participated intermediates, the maximum values of the input energy are 0.44 eV during the first protonation process; thus, the *N2 to *N2H is the potential determining step for the Mn2@site1system via end-on configuration, and the limiting potential is -0.44 V, as depicted in Figure 4(b) and Figure S8 .
For the enzymatic mechanism under the cooperation effect between two TM atoms in atomic diversity free DAC, as depicted in Figure 4(a) , the H+ + e- pair initially protonates the distal-like adsorbed N atom as confirmed former, then the second H+ + e- pair points out to protonate the other N atom. Gradually, the additional H+ + e- pairs from the solution continue to hydrogenate the protonated N atoms until all the NH3 desorption. As shown in Figure 4 for V2@site4 and Cr2@site4 systems, N2 is spontaneously captured by the active site with side-on configuration within adsorption energy changes of -0.83 eV and -0.49 eV. Meantime, the N-N bonds increase to 1.22 Å and 1.21 Å, respectively. Then the initially activated N2 is attacked by the first H+ + e- pair, forming *N-*NH spices, where the Gibbs free energy climbs to -0.55 eV and -0.19 eV, and the N-N bonds increase to 1.33 Å and 1.30 Å. The second H+ + e- pair successfully turns to hydrogenate the other side N atom, generating *NH-*NH with the Gibbs free energy changes of 0.91 eV and 0.23 eV.
Consequently, the third and fourth H+ + e- pair continue to attack the formed *NH-*NH to generate *NH-*NH2 and *NH2-*NH2, causing the breakdown of the N-N triple bond. At the same time, the changes of Gibbs free energy are 0.18 eV and -0.58 eV, meaning that a 0.18 eV energy input is enough for the N-N bond breaking for V2@site4. Excitingly, the N-N broken energy of the Cr2@site4 system is -0.58 eV, suggesting the N-N bond’s spontaneous breaking. Finally, for the fifth and sixth protonation, the H+ + e- pair keeps attacking both isolated *NH2, causing the formation of *NH3. For this process, the Gibbs free energy climb step by step for both V2@site4 and Cr2@site4 systems, where the energy inputs are 0.19 eV and 0.05 eV for fifth step and 0.18 eV and 0.19 eV for sixth step, respectively. For the desorption of *NH3, the energy changes of this process usually do not conclude for limiting potential finding since any H+ + e- pair is participating. Therefore, among the six H+ + e- pair protonation processes, the max Gibbs free energy changes of V2@site4 and Cr2@site4 systems are 0.28 eV and 0.29 eV for Gibbs energy changes of *N-*N → *N-*NH in both V2@site4 and Cr2@site4 systems, thus the limiting potential of them are -0.28 V and 0.29 V vs CHE (Computational Hydrogen Electrode71) , respectively. Similarly, the Gibbs free energy is -0.36 V, -0.25V, and -0.48 V for the rest of Ru2@site1, W2@site2, and Os2@site3 systems, respectively.
It is noted that for all the selected candidates, the Mn2@site1 and Co2@site2 exhibit end-on configuration while Ru2@site1, W2@site2, Os2@site3, V2@site4, and Cr2@site4 show side-on configuration. But for the hydrogenation process, especially for the *N2 to *N2H, *N2H to *N2H2, *N-*N to *N-*NH, *N-*NH to *NH-*NH, *NH-*NH to *NH-*NH2, *NH-*NH2to *NH2-*NH2, *N2H2 to *N, *N to *NH, and *NH to *NH2 steps, nearly all of them adsorbed on both TM active sites (see in Figure S9 ), thus lower the activation energy during the protonation process. For the Ni2@site4, W2@site4, and Re2@site4 systems, the isolated active site participated in the initial N-N activation and hydrogenation process reaction steps. In contrast, the second TM site does not participate in it significantly, thus leading to their hard hydrogenation of them. Hence, dual TM atoms must take part in the eNRR process to enhance the cooperation effect between dual atoms. At the same time, the adsorption behavior, activation behavior, and desorption behavior are significantly changed since the heteronuclear of dual TM exhibits differential properties in heteronuclear DAC, thus breaking the limitation of homonuclear DAC.