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Page 6 of 14 Zhang et al. J Mater Inf 2024;4:1 https://dx.doi.org/10.20517/jmi.2023.34

MN -gra/MMN -gra/M1M2N -gra
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Theoretically, the HER performance of the above catalysts could be improved by reducing the number of
coordinating N atoms because it will endow the central metal with a stronger coordination ability to
enhance its H adsorption. We then constructed a series of SACs and DACs with three N atoms around each
metal: MN -gra, MMN -gra, and M1M2N -gra [50,51] . However, the metal atoms in MN -gra (M = Fe, Co, Ni)
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SACs all protrude from the catalyst surface upon structural optimization [Supplementary Figure 1]. The
formation energies of MN -gra are also positive, suggesting that they may be unstable and unsuitable for
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electrocatalytic reactions.
By comparison, the optimized structures of MMN -gra/M1M2N -gra DACs are all basically planar
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[Figure 3], with each metal atom well incorporated in the graphene plane. The metal atom and three nearby
N atoms have average distances between 2.03~2.14 Å, all close to the sum of the covalent atomic radii
[Supplementary Table 11]. This structural feature suggests that M-N covalent bonds may be formed, which
are supported by the electron localization functions (ELFs), where substantial electron accumulation was
found in the M-N regions [Supplementary Figure 2]. Consistently, the formation energies of these DACs
are all negative (-0.61 to -0.06 eV), further confirming their energetic stabilities [Figure 3]. Moreover, the
distances between two metal neighbors are all smaller than their covalent radii sum [Supplementary Table
11], indicating that M-M covalent bonds are formed, which are consistent with the ELF results in
Supplementary Figure 2. The crystal orbital Hamiltonian population (COHP, Supplementary Figure 3)
reveals that there are strong M-M bonding interactions under the Fermi level, which are largely due to the
σ-type overlap of the metal 3d x2-y2 orbitals [Supplementary Table 12]. The large and negative integral-COHP
(ICOHP) values of -5.03, -3.70, -2.74, -4.36, -3.15, and -2.97 for Fe-Fe, Co-Co, Ni-Ni, Fe-Co, Fe-Ni, and
Co-Ni, respectively, further support the formation of M-M bonds. The metal atoms all donate some
electrons to the N-doped graphene, as indicated by the charge density difference and Bader charges
[Supplementary Figure 4].

The electronic properties were then investigated by the density of states (DOSs) and electronic band
structures. As shown in Supplementary Figure 5, substantial hybridization was observed between the N-2p
state and the M-3d state. Compared with MMN -gra/M1M2N -gra, the metal d-band centers (ε ) in
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d
MMN -gra/M1M2N -gra all shift to higher energy levels [Supplementary Figure 5]. This difference can be
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understood from the lifted energy level of the metal 3d x2-y2 orbitals in projected DOSs (PDOS,
Supplementary Figure 6). Meanwhile, the metals also exhibit lower spin states [Supplementary Figure 6]. In
addition, there are rich electronic states around the Fermi level with small band gaps of 0 to 0.04 eV
(Supplementary Figure 7; note that the gaps are generally underestimated at the PBE level). Thus, rapid
electron transfer between the catalyst surface and the adsorbates is expected to speed up the electrocatalytic
reaction.

The changed electronic structure caused by the reduced metal coordination number will inevitably affect
the adsorption of *H. Supplementary Tables 13-18 summarize the adsorption structures and corresponding
ΔG on different active sites of all these low-coordinated DACs. For the metal sites, the H atom prefers to
*H
adsorb on the bridge site between two metal atoms of MMN -gra/M1M2N -gra as compared to a single
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metal site on MMN -gra/M1M2N -gra. Owing to such a novel dual-atom synergy, the ΔG values decrease
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*H
significantly [Figure 4] and are very close to 0 eV on the metal sites of FeFeN -gra (-0.16 eV), FeCoN -gra
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(-0.18 eV), FeNiN -gra (-0.04 eV), and CoNiN -gra (-0.06 eV). Thus, they, especially FeNiN -gra (ΔG :
*H
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-0.04 eV) and CoNiN -gra (ΔG : -0.06 eV), may exhibit high catalytic activity.
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*H