Screening Electrocatalysts at the Level of Kinetic Barriers under Realistic Potential and Solvation

Abstract

Kinetic barriers under realistic solvation and potential conditions, known as critical in electrochemistry in recent years, have not been widely applied in the screening of electrocatalysts, mainly due to the high computational cost. Here, we demonstrate the establishment of quantitative relations between thermodynamics and kinetic barriers, which guides electrocatalyst screening from 51 candidates, taking single-atom@coinage-metal (M₁@CM) alloys catalyzing electrochemical nitrogen reduction reaction (eNRR) as an example. For CM = Cu, Ag, and Au, separated linear relations are found between the free energy changes (ΔG) based on the computational hydrogen-electrode model and the kinetic barriers (ΔG ^# ) calculated from enhanced sampling of constant-potential ab initio molecular dynamics (cp-AIMD). The variations among Cu, Ag, and Au can be primarily attributed to differences in interfacial water orientation and surface charge under the calculated potential, properties governed by their respective work functions. Furthermore, a unified mapping from ΔG to ΔG ^# is found with a prediction error of about 0.05 eV across the three hosts using machine learning regression methods. Based on these relations, the high-active zone is identified, while the full path is calculated for the representative case Re₁@Ag. Indeed, all barriers are no higher than 0.85 eV, significantly lower than other reported systems if barriers of all steps are examined. This work not only presents a screening strategy to quickly identify an eNRR catalyst with all-low kinetic barriers along the full path but also demonstrates how to establish and apply the quantitative relation between thermodynamics and cp-AIMD barriers, to significantly accelerate accurate screening of electrocatalysts.

Introduction

Electrocatalysis plays a pivotal role in a sustainable society, particularly in renewable energy and green chemistry. Designing electrocatalysts typically requires an understanding of the reaction, e.g., binding prediction, Brønsted–Evans–Polanyi (BEP) correlation and d band center theory, and a massive screening of electrode materials. In the past decade, there have been tremendous works devoted to rational screening of electrocatalysts. , In order to understand the reaction mechanism and accelerate the catalyst screening, descriptors are developed to predict the binding energies of intermediate species. − For instance, Xu et al. proposed a descriptor φ′ that incorporates the valence electrons (d-orbital of the metal center), the local coordination environment (coordination number and electronegativity), and the periodic number of the metal center in the periodic table, correlating these features with the activity (based on the free energy change, ΔG) of graphene-based single-atom catalysts (SACs) for various electrochemical reactions. Ren et al. established the “single-atom saturation” descriptor, which governs the intensity of intermediate adsorption on single-atom alloys (SAAs), to identify optimal candidates for five distinct electrochemical reactions, demonstrating broad extensibility to graphene-based SACs. Using first-principles calculations coupled with the SISSO data analytics methodology, Han et al. identified low-dimensional descriptors and multiproperty models to describe hydrogen binding energy, dissociation barriers, and guest-atom segregation energetics in SAAs, enabling high-throughput screening in an immensity of SAA candidates for hydrogenation catalysis. We note that almost all of the screening schemes are based on the free energy change calculated through the computational hydrogen electrode model (CHEM). CHEM is a conventional thermodynamic model − that counts pH and potential effects through the proton with approximations, while neglecting potential and solvation effects on the adsorbate and substrate. Moreover, using CHEM results to predict activity/selectivity also assumes that the kinetic barrier of each step can be approximated as the free energy change.

In recent years, however, it has been found in many cases that explicit kinetic barriers, − as well as electrode potential, − pH, and explicit solvation (hydrogen bonds), , are critical information to explain experimental observations. A prominent example of the divergence between thermodynamics and kinetics is the origin of the outstanding HER activity of Pt. Lindgren et al. found that *H on the hollow site (ΔG ∼ 0 eV, thermodynamically stable) is kinetically inactive, whereas *H on the top site (ΔG ∼ 0.4 eV) is actually responsible for the HER activity through a top–top Tafel mechanism. Qian et al. analyzed the electrochemical nitrogen reduction reaction (eNRR) thermodynamic results of traditional CHEM with and without explicit solution model, revealing the significant impact of an explicit solution effect on the free energy profile, and further proved the dependence of key steps on electrode potential. Zhao and Liu showed that surface charges and the dynamic hydrogen bonding under realistic potential and solvation play a critical role in CO₂ reduction. Notably, an optimized thermodynamic descriptor, G _max(U) based on free-energy span model, incorporates partial kinetic information, thereby substantially improving the generality of activity predictions. Nevertheless, although thermodynamics-based approaches have made significant progress toward rapid and accurate electrocatalyst screening, precise simulation of kinetic barriers under realistic potentials and solvation conditions remain essential, as they provide atomistic insights into explicit solvent effects and more realistic potential dependence. − In last few decades, the BEP correlation has been well established for thermal heterogeneous catalysts and established a conceptual framework for correlating reaction enthalpies ΔG with activation energies ΔG ^# , typically derived via nudged elastic band (NEB) calculations for gas–solid interface reactions. For electrocatalysis, typically with solvation, however, a similar correlation is difficult to obtain due to the dynamical hydrogen bond network as well as constant-potential requirements in reality and simulations. The recently developed grand canonical “constant-potential hybrid-solvation dynamic model” (CP-HS-DM) approach allows the number of electrons to change on-the-fly to keep the potential constant and accounts for the dynamic hydrogen bonding. , Combined with the enhanced sampling method, reaction barriers ΔG ^# under realistic potential and solvation conditions can be evaluated, while the details of charge transfer can be simulated. Nevertheless, the computational cost of obtaining one ΔG ^# using CP-HS-DM is estimated to be over 200 times that of obtaining a ΔG, hindering its use as routinely as ΔG calculations. Even though methods like CP-HS-DM are being more and more used in study key steps in different electrochemical processes, such as the clustering and recover of Cu single-atom catalyst and CO–CO coupling with alkaline cations, it is hardly affordable to be directly used in high-throughput screening for highly active and selective electrocatalysts.

eNRR, as a sustainable alternative to the conventional Haber–Bosch process with tremendous carbon emission, is an outstanding example with respect to the challenge of electrocatalyst screening. , Nevertheless, eNRR catalysts generally suffer from low activity and poor selectivity, primarily due to high activation energy and low competitivity against HER. Despite extensive theoretical efforts devoted to eNRR catalyst discovery, experimentally validated catalysts with high performance remain scarce. Many theoretical studies predict high activity for N₂ activation but neglect the constraint of NH₃ desorption, − whereas materials that allow facile NH₃ desorption often suffer from poor N₂ activation. , Conventional thermodynamic study of free energy profile may limit the accuracy of catalyst screening, while the thermodynamic descriptor G _max(U) proposed by Exner enables moving beyond the theoretical description of a single reaction mechanism. , In addition to thermodynamic descriptors, discrepancies between thermodynamic and realistic kinetic barriers may also contribute to inaccurate predictions. Recent kinetic studies on eNRR have revealed that many unremarkable elementary steps on metal surfaces involve non-negligible energy barriers. , Furthermore, kinetic simulations can capture dynamic effects such as cation involvement and alternative reaction pathways, providing a more comprehensive description of the complete eNRR reaction network. Therefore, leveraging the CP-HS-DM approach to combine thermodynamic and kinetic information allows for a more accurate screening of eNRR catalysts under realistic potential and solvation conditions.

SAAs, which typically consist of active metal atoms atomically dispersed on the surface of a less reactive host metal, offer two distinct types of sitesstrong-binding guest sites and weak-binding host sites. , Such a unique configuration enables fine-tuning of the electronic structure, endowing the dopant metal with free-atom-like d-states , that significantly enhance the capabilities of activating adsorbate, offering the potential for breaking linear scaling relations established on pure metals ,− and thus emerging as a promising strategy to address the challenge of eNRR. Moreover, SAAs, as a specialized class of SACs, also serve as a well-defined model system for elucidating fundamental thermodynamic principles and evaluation criteria. ,,− For instance, based on the first-principles calculations, Schumann et al. proposed a 10-electron count rule to elucidate the binding of adsorbates on the dopant site of SAA surfaces, enabling the precise prediction of binding intensities for targeted reactions. They also systematically investigated the coadsorption preference of adsorbates on the active dopant site of SAA, where the coadsorption behavior was found to facilitate product desorption and rationalize the catalytic candidacy of early transition metal-based SAAs. Although these studies offer valuable theoretical guidance for SAA catalysts, the screening methodologies are still limited at the thermodynamics level, while the screening with kinetic barrier information remains scarce.

In this work, taking eNRR catalyzed by SAA as an example, we demonstrate an accurate screening framework based on kinetic barriers, as depicted in Scheme . Considering the Sabatier principle, which states that the adsorption and formation of intermediates and the desorption of products should be balanced, the two steps of *NNH formation and NH₃ desorption were selected as two key steps in our screening. Our screening space includes all plausible transition metals as the single-atom dopant, while the host is one of the coinage metals (M₁@CM). First, by exploring 17 potential transition metal-doped (3d: V, Cr, Mn, Fe, Co, Ni; 4d: Nb, Mo, Ru, Rh, Pd; 5d: Ta, W, Re, Os, Ir, Pt) Ag-based SAAs, we identify a linear BEP relation between thermodynamic free energy changes (ΔG) and kinetic barriers (ΔG ^# , calculated by enhanced sampling methods) under typical operating conditions. Moreover, when it is extended to Cu- and Au-based SAAs, similar BEP relations with different slopes and intercept values are also found, indicating the generalizability of BEP relations. Furthermore, machine learning regression analysis using various models reveals the strong unified correlations (R ² ∼ 0.93, RMSE ∼ 0.05 eV) across different hosts, enabling accurate predictions of ΔG ^# from physical descriptors such as ΔG, work function of host metal (W _F), valence electron number of guest metal (N _v), and lattice constant of host metal (Lat). Among all M₁@CM systems, representative Re₁@Ag is selected and found to exhibit extraordinary eNRR activity and selectivity in more detailed studies, with kinetic barriers no higher than 0.85 eV for each elementary step, suggesting its extraordinary eNRR performance when evaluated from kinetic insights under realistic potential and solvation. This study not only demonstrates the BEP relations in electrocatalysis and the finding of an extraordinary eNRR catalyst, but most importantly, provides a reliable and affordable strategy to massively screen electrocatalysts at the level of kinetic barriers under operational potential and solvation conditions.

1. Workflow of Thermodynamics-Kinetic Barriers Relation Guided eNRR Electrocatalyst Design.

Results and Discussion

Constant Charge and Constant Potential Thermodynamics

For the model systems, we chose single-atom transition metal-doped coinage metals (M₁@CM, CM = Cu, Ag, and Au) for the following reasons: (1) coinage metals generally have low activity for the competing HER; , (2) coin metals are in general more inert than the doping atom so that the reaction mechanism is easier to explore; and (3) transition metals such as Fe and Ru are known as good thermal catalyst for eNRR, and Ru SACs have been reported to have eNRR activity. Moreover, the most compact (111) facet of these coinage metals is selected as the representative surface. As shown in Figure S1, Supporting Information, projected density of states (PDOS) reveals that the surface Ag atom on the pure Ag (111) facet occupies at low energy levels and presents a degenerate and inactive status, whereas construction of the SAA introduces a nondegenerate state across the Fermi level, , which has the potential to trigger eNRR. Furthermore, the Ag (111) facet has the weakest H binding intensity among the three CMs, so we first investigated the thermodynamic performance of eNRR on Ag-based SAA surfaces. Notably, for eNRR, the *NNH formation step has been identified as one of the potential rate-determining steps on Au surfaces , and Ru SACs. Besides *NNH formation, NH₃ desorption has been reported to influence eNRR kinetics, but more studies still overlook NH₃ desorption in pursuit of enhanced N₂ activation performance. − Analogous to CO poisoning on Pt-based catalysts, difficult ammonia desorption may disrupt the trade-off between reactant and product, thus violating the Sabatier principle and showing low activity. On the basis of the thermodynamic balance, our screening primarily targets the *NNH formation and NH₃ desorption steps to identify catalysts that achieve this essential equilibrium for eNRR.

Figure a illustrates the free energy change of N₂ adsorption (ΔG _N2‑ads), NH₃ desorption (ΔG _NH3‑des), and *NNH formation (ΔG _*NNH‑form) for 17 transition metal dopants (3d: V, Cr, Mn, Fe, Co, Ni; 4d: Nb, Mo, Ru, Rh, Pd; 5d: Ta, W, Re, Os, Ir, and Pt) embedded as SAA on the Ag (111) surface (M₁@Ag). The constant-charge model (CCM) results indicate that 3d metals generally exhibit weaker N₂ adsorption compared to their 4d and 5d counterparts, consistent with the previous reports. In contrast, the 4d and 5d metals clearly exhibit a V-shaped adsorption trend, with Group 8 metals (Ru and Os) showing the strongest N₂ adsorption capabilities, which aligns with the ten-electron count rule (eight valence electrons from the dopant metal + two valence electrons from the adsorbate). Meanwhile, the difficulty of NH₃ desorption gradually weakens from left to right across the periodic table. This trend arises because NH₃ adsorption is predominantly governed by electrostatic contributions (lone-pair electron donation), resulting in weaker NH₃ binding on dopants with more valence electron counts. Therefore, Nb and Taboth exhibiting high NH₃ desorption energieswere excluded from subsequent high-cost barrier calculations owing to inhibited NH₃ desorption. In addition, *NNH formation exhibits an opposite periodic trend from NH₃ desorption, with the difficulty of *NNH formation progressively increasing across each period from left to right.

1.

(a) Calculated ΔG (eV) values of N₂ adsorption, NH₃ desorption, and *NNH formation on M₁@Ag (111) surfaces. The CCM and CPM (pH 7, U _RHE = −0.1 V) results are depicted by blue and red lines, respectively. (b) Illustration of constrained cp-AIMD for *NNH formation and NH₃ desorption steps. BEP relation between thermodynamics (ΔG of CPM) and kinetic barriers (activation energies ΔG ^# obtained from cp-AIMD). (c) Negative correlation between kinetic barriers of *NNH formation and NH₃ desorption.

Notably, CCM calculations sometimes fail to accurately describe adsorption behaviors under constant-potential conditions due to d-electron rearrangement of the metal active centers induced by applied potential, so we performed a constant-potential model (CPM) to calculate the ΔG under a neutral condition (pH = 7) and a small overpotential (U _RHE = −0.1 V). One can see that the major contribution of CPM on thermodynamic results is the adsorption behaviors, i.e., enhanced N₂ adsorption and weakened NH₃ desorption, which implies a favorable trend to proceed with eNRR. , Such positive influences are mainly observed on 4d and 5d dopants of Groups 6–8.

Based on the CCM or CPM thermodynamics results above, with information on ΔG _*NNH‑form and ΔG _NH3‑des, one might identify Cr₁@Ag, Mn₁@Ag, Ru₁@Ag, and Os₁@Ag as the four most active systems. However, as will be discussed in the next session, this screening method may be misleading, since the ΔG ^# can deviate significantly from the corresponding ΔG, and the ΔG∼ΔG ^# BEP relations are evidently different for these two steps.

Thermodynamics-Kinetic Barriers Relation

To explore the thermodynamics-kinetic barriers relation among electrocatalysts, we started with representative M₁@Ag systems and calculated adequate kinetic barriers. We employed the grand canonical CP-HS-DM approach (see computational details) to investigate the eNRR dynamic behavior under realistic solvation and potential conditions (pH = 7, U _RHE = – 0.1 V), where 4d and 5d dopants in Groups 6–8 (Mo, Ru, W, Re, and Os) and 3d dopants (Cr and Co) are selected. As illustrated in Figure b, we examined the kinetic process of *NNH formation (Collective variables, CV = d ₁ – d ₂, water is the exclusive hydrogen source and proton donor) and NH₃ direct desorption (CV = d ₁) on the selected M₁@Ag systems. All of the free energy profiles and the structures of slow-growth calculations are depicted in Figure S2. Interestingly, the obtained kinetic barriers (i.e., activation energy ΔG ^# ) demonstrated good BEP relations with CPM-derived thermodynamic data ΔG _CPM (Figure b, R ² = 0.99 and 0.94, RMSE = 0.04 and 0.07 for the *NNH formation and the NH₃ desorption, respectively). These results clearly indicate the BEP relations between intermediate reaction energies ΔG and corresponding ΔG ^# for SAAs sharing a common host. Consequently, accurate ΔG ^# obtained from expensive cp-AIMD simulations can be reliably inferred from much more available thermodynamic data ΔG alone, significantly enhancing predictive efficiency and accuracy. Considering the high correlation coefficient obtained from the linear fitting and the nondeviation of upper and lower bound cases (Co and W), establishing a relatively reliable linear relation for inferring requires rather fewer data points than that in Figure b, thus computational cost can be further reduced during massive catalyst screening. A plausible plot involving minimal cp-AIMD calculation could be constructed using only two extreme cases (Co and W), according to ΔG, supplemented by an extra intermediate data point for validation.

For the *NNH formation step, the derived BEP relation is given by the equation: ΔG ^# = 0.49·ΔG _*NNH‑form + 0.79, reflecting a smaller dependence on the free energy change. This suggests that the contribution of thermodynamic energies to kinetic barriers for the *NNH formation is relatively minor, while the higher intercept is attributed to the significant involvement of N–N bond activation and hydrolysis kinetics. In contrast, the scaling equation for the NH₃ desorption is ΔG ^# = 0.87·ΔG _NH3‑des + 0.39, exhibiting a slope much closer to 1. This indicates a larger correlation between NH₃ desorption kinetics and the associated free energy change, while the intercept primarily accounts for the constant part irrelevant to the binding of NH₃, likely attributed to breaking the hydrogen bonding network.

In addition to thermodynamic balance, maintaining a comparable balance in the kinetic barriers of *NNH formation and NH₃ desorption is also critical for an ideal catalyst. Interestingly, as shown in Figure c, a negative correlation among different dopants in M₁@Ag exists between the activation energies of *NNH formation and NH₃ desorption. Constrained by this correlation, the optimal M₁@Ag catalyst for eNRR exhibits a theoretical minimum activation energy of 0.97 eV (where the ΔG ^# of *NNH formation and NH₃ desorption is equal). In this scenario, Re₁@Ag emerges as the most favorable catalyst candidate due to its proximity to the optimal case, showing ΔG ^# of 0.85 and 1.06 eV for the *NNH formation and the NH₃ direct desorption, respectively. Nevertheless, a diversity of NH₃ desorption pathways is found: several alternative routes exist to reduce the final kinetic barrier significantly. Particularly, N₂-assisted desorption can notably reduce the activation barrier from 1.06 to 0.78 eV (∼0.3 eV), showing a nonlinear BEP relation and rendering Re₁@Ag a catalyst with exceptionally enhanced eNRR catalytic performance. We note that such reactant-assisted desorption of product has also been reported in the study of the oxygen reduction reaction. These alternative desorption mechanisms are further elaborated in Section 2.5.

BEP Relations on Coinage Metals

With ΔG∼ΔG ^# BEP relations established on M₁@Ag SAAs, we further extend the study to the other coinage metals Cu and Au to explore more BEP relations and potential eNRR electrocatalysts. Mo, W, Re, and Os, which have been demonstrated to have superior catalytic potential when utilized as dopants on the Ag host (Figure c), were selected to construct SAAs on the Cu and Au hosts. Then we computed ΔG and ΔG ^# for *NNH formation and the NH₃ desorption steps on selected Cu- and Au-based SAAs. Corresponding free energy profiles and final structures of slow-growth calculations for all M₁@Cu and M₁@Au are presented in Figures S3 and S4, respectively. Our results clearly illustrate similar thermodynamics-kinetic barriers BEP relations occurring on Cu and Au hosts (Figure a,b). Notably, a pronounced mismatch between thermodynamic predictions and kinetic behavior is observed, particularly on the Au host, where steps predicted to be thermodynamically favorable are kinetically hindered. For instance, the *NNH formation step on W₁@Au is thermodynamically facile (ΔG = 0.20 eV), but it involves a prohibitively high barrier (ΔG ^# = 1.51 eV), far beyond the thermodynamic expectation. In contrast, the corresponding barriers on W₁@Cu (ΔG ^# = 0.90 eV) and W₁@Ag (ΔG ^# = 0.74 eV) are substantially lower, indicating kinetically accessible pathways. The thermodynamic picture fails to primarily originate from significant differences in the interfacial solvent structure among different electrode materials, as revealed by detailed analyses of cp-AIMD simulations (discussed below).

2.

(a,b) General BEP relations between the CPM thermodynamic results and the kinetic barriers for (a) *NNH formation and (b) NH₃ desorption steps. (c) Illustrations of different BEP relations for Cu, Ag, and Au hosts. (d) Concentration distributions of O and H in the water layer during cp-AIMD simulations of *N₂ on Re₁@CM. (e) Distribution of cos φ between the H–O–H bisector and the surface normal during cp-AIMD simulations of *NH₃ on Re₁@CM. (f) Screening optimal candidates based on all the kinetic results. (g) Working screening criteria obtained from the backward inference to screen candidates based on all the CPM thermodynamic results.

In the case of M₁@Cu, the BEP relation for the *NNH formation step (ΔG ^# = 0.57·ΔG _*NNH‑form + 0.81) is analogous to and positioned slightly above that of M₁@Ag (ΔG ^# = 0.49·ΔG _*NNH‑form + 0.79). Conversely, the scaling equation for the NH₃ desorption on Cu (ΔG ^# = 0.70·ΔG _NH3‑des + 0.45) lies below that of Ag (ΔG ^# = 0.87·ΔG _NH3‑des + 0.37) and exhibits a slightly smaller slope. These BEP relations indicate the disadvantage for Cu over Ag in the *NNH formation step, similar to the thermodynamic findings (ΔG _*NNH‑form: Ag < Cu), while the advantage is found for Cu in the NH₃ desorption step, contradicting the thermodynamic trends (ΔG _NH3‑des: Ag < Cu). Therefore, it highlights again that the sole reliance upon thermodynamic metrics ΔG to assess catalytic performance ΔG ^# on different hosts might be misleading.

In the case of M₁@Au, notably, our kinetic results unveil an appearance of “thermodynamic deception”: although M₁@Au exhibits favorable thermodynamics, their kinetic barriers are significantly larger than those suggested by ΔG values alone. Although thermodynamics-kinetic barriers BEP relations are also observed on the Au host, the slope of the *NNH formation scaling equation (ΔG ^# = 0.08·ΔG _*NNH‑form + 1.50) is exceedingly small, implying minimal thermodynamic influence on kinetic barriers and suggesting an inherent “kinetic shield” for eNRR. The scaling equation for the NH₃ desorption (ΔG ^# = 0.51·ΔG _NH3‑des + 1.07) has both a smaller slope and a higher intercept compared to Ag and Cu, again indicating the existence of considerable “kinetic shield” during eNRR proceeding on the Au host. Although thermodynamic results indicate that the property of high electronegativity of Au is favorable for the NH₃ desorption due to its electron-rich status under constant-potential conditions, the reaction kinetics remain highly unfavorable. As schematically illustrated in Figure c, Au is kinetically unsuitable as a host metal due to the “kinetic shield”, whereas Ag and Cu, exhibiting similar BEP relations, allow more tailored catalyst design according to specific reaction requirements.

For the *NNH formation step, it is shown that the “kinetic shield” appearance primarily arises from combined effects: first, the activation of N₂ adsorption on M₁@Au is obviously weaker compared to M₁@Ag and M₁@Cu, as evidenced by the shortest N–N bond length observed for M₁@Au (Figure S5). Additionally, the uneven distribution of the water layer induced by the surface charge of the Au host further contributes to the difficulty of *NNH formation. As illustrated in Figures d and S6, the local H concentration near the top N atom of *N₂ on M₁@Au (3.07 Å) is significantly lower than that on M₁@Cu and M₁@Ag (3.02 Å). On M₁@Cu and M₁@Ag, the first water layer (∼2 Å) lies closer to the SAA surface than on M₁@Au (∼2.7 Å), allowing it to envelop *N₂ and favor hydrogenation.

For the NH₃ desorption step, the uneven distribution of the water layer suppresses the kinetics of NH₃ desorption and dominates the “kinetic shield” appearance. From the distribution of cos φwhere φ is the angle between the H–O–H bisector and the surface normalthe first water layer on Re₁@Cu and Re₁@Ag shows positive values, whereas M₁@Au is negative, indicating that the first-layer waters on Re₁@Au orient toward the metal surface (Figure e). This is consistent with the fact that Au has a significantly higher work function, resulting in a much more charged status (1.24 e^– on Au vs 0.06 e^– on Ag) under the same potential. Therefore, the inverted, surface-directed hydrogen-bonding network induced by the surface charge of the Au host forms above the Re₁@Au surface, hindering the kinetics of NH₃ desorption. Besides, the first water layer that envelops *NH₃ also facilitates NH₃ desorption on Re₁@Cu and Re₁@Ag.

For comparison, we also tested predicting kinetic barriers from CCM thermodynamic results, as ΔG _CCM is the most readily available thermodynamic data. Similar BEP relations still exist for Cu, Ag, and Au, but worse linear correlations are observed (Figure S7a,b). The scaling equations generally follow trends analogous to those of ΔG _CPM, except that the linear correlation of *NNH formation on M₁@Cu intersects with M₁@Ag (Figure S7c). Notably, the BEP relation of NH₃ desorption on M₁@Au exhibits a pronounced rightward horizontal shift, which is attributed to the larger ΔG _CPM – ΔG _CPM difference (Figure S7d) caused by the electron addition under applied potential, considering that the work function of Au (5.18 eV) is larger than that of Cu (4.75 eV) and Ag (4.48 eV).

To proceed with the screening, all kinetic barriers obtained from slow-growth calculations are summarized in Figure f, and we set the favorable barrier for the two critical steps to be lower than 1.0 eV as the screening criterion. Given that alternative reaction pathways could reduce the barrier of NH₃ desorption by approximately 0.3 eV (see previous session and Figure c), the kinetic barrier criteria are as follows: 1 eV for the *NNH formation and 1.3 eV for the NH₃ direct desorption. The screening based on kinetic barriers emphasizes Re₁@Ag, Mo₁@Ag, and Re₁@Cu as the most promising eNRR candidates due to their low barriers for both *NNH formation (ΔG ^# < 0.9 eV) and NH₃ desorption (ΔG ^# < 1.1 eV). We note that this list is significantly different from what barely thermodynamic calculations suggested, highlighting the importance of screening based on barriers with solvation and potential effects taken into account. To evaluate the thermodynamic stability of the three optimal catalysts identified by kinetic screening (Re₁@Ag, Mo₁@Ag, and Re₁@Cu), we calculated the replacement energy of M₁ and CM. As shown in Figure S8, the M₁@CM sites are not favorable to be replaced by isolated CM atoms, indicating strong bonding between the guest atom and the host metal support.

Furthermore, we realized that BEP relations derived from various coinage metal hosts could inversely generate a working thermodynamic screening criterion. By backward inference, favorable kinetic barrier thresholds can be translated into precise thermodynamic upper bounds, enabling the accurate definition of the thermodynamic screening criteria. Here, the kinetic criteria of 1.0 and 1.3 eV (Figure f) are set to favorable barriers for the *NNH formation and the NH₃ desorption, respectively. As detailed in Table , for Cu, Ag, and Au, strict ΔG _*NNH‑form upper bounds of 0.33, 0.43, and −6.25 eV were established for the *NNH formation, whereas looser upper bounds of 1.21, 1.07, and 0.45 eV were identified for ΔG _NH3‑des. To prove refined thermodynamic screening criteria, we calculated ΔG _N2‑ads, ΔG _NH3‑des, and ΔG _*NNH‑form for the rest of the M₁@Cu and M₁@Au systems (Figures S9 and S10). As depicted in Figure g, several promising catalysts are identified according to the working criterion (Cu host: Re₁@Cu, Mo₁@Cu, and W₁@Cu; Ag host: Re₁@Ag, Mo₁@Ag, W₁@Ag; Au host: none), where all suggested candidates are identical to those found through kinetic calculations. This procedure substantially reduces the computational effort required for kinetic simulations.

1. Equations Obtained from BEP Relation on Different SAA Hosts and Refined Thermodynamic Upper Bound from Backward Inference.

step	slab	equation	favorable barrier (ΔG # )	ΔG upper bound
*NNH formation	Cu	ΔG # = 0.57·ΔG *NNH‑form + 0.81	<1.0	<0.33
	Ag	ΔG # = 0.49·ΔG *NNH‑form + 0.79		<0.43
	Au	ΔG # = 0.08·ΔG *NNH‑form + 1.50		<−6.25
NH3 desorption	Cu	ΔG # = 0.70·ΔG NH3‑des + 0.45	<1.3	<1.21
	Ag	ΔG # = 0.87·ΔG NH3‑des + 0.37		<1.07
	Au	ΔG # = 0.51·ΔG NH3‑des + 1.07		<0.45

Machine Learning Regression

In order to unify the quantitative relations between ΔG and ΔG ^# for all of the calculated systems, i.e., on three hosts, we first analyze the possible influencing factors on ΔG ^# and then employ machine learning regression methods to account for these factors.

The factors leading to the distinct BEP relations observed among Cu, Ag, and Au hosts likely include their electronic properties (substrate electron number change relative to electric neutrality (Δq), work function (W _F), valence electron number of guest metal (N _v), etc.) and geometric structures (lattice constants (Lat), etc.). Regarding electronic structure, we analyzed substrate Δq under constant-potential conditions (pH = 7, U _RHE = −0.1 V) and found Δq values for pure Cu, Ag, and Au (111) substrates were 0.36, 0.06, and 1.24 e^–, respectively. Under these conditions, the Ag (111) surface approaches its potential of zero charge (PZC), resulting in the smallest Δq, consistent with the benchmark. With single-atom doping, Δq of all of the SAA substrates followed the same trend: Au > Cu > Ag (Figure a). Notably, excessive substrate charge in Au may induce the downward orientation of interface water molecules (Figure e) and cause the appearance of “kinetic shield”, whereas the moderate charge in Cu and Ag could favor catalytic reactions, analogous to how PZC modification tunes the hydrogen bonding network structure within the electric double layer. Note that substrate electron numbers under constant-potential conditions are governed by W _F, where W _F exhibits a linear correlation with Δq. Considering that W _F is easier to obtain than Δq, W _F is included in the regression analysis. Furthermore, N _v represents a dopant atom property that has been proposed as an essential descriptor of SAA reactivity in the literature, such as the ten-electron count rule. , N _v is also included in the regression analysis to describe the guest atom.

3.

(a) Electron number change (Δq) relative to electric neutrality for three host substrates constructed SAAs. (b) Four descriptors representing the properties of host, guest, and adsorbate in M₁@CM. (c,f) Heat map of Pearson’s correlation coefficient for (c) *NNH formation and (f) NH₃ desorption. (d,e,g,h) ΔG ^# prediction by (d,g) LR and (e,h) SVR models for (d,e) *NNH formation and (g,h) NH₃ desorption, with evaluation metrics of the test set.

Moreover, geometric structures might also affect the BEP relations. First, the host lattice determines the local environment around the doping atom, thus influencing its reactivity; second, the lattice is likely to disturb the hydrogen bonding network. Hence, the lattice constant is included in the regression analysis. As illustrated in Figure b, ΔG (reflecting complex host–guest–adsorbate interactions), W_F and Lat (host properties), and N_v (guest property) serve as promising descriptors for the regression modeling to predict ΔG ^# . Due to the limitations of the data size of cp-AIMD barriers, the current regression analysis only involves data from 15 points (Cr₁,Co₁,Ru₁,Mo₁,W₁,Re₁,Os₁@Ag; Mo₁,W₁,Re₁,Os₁@Cu; Mo₁,W₁,Re₁,Os₁@Au) and discusses the two steps of *NNH formation and NH₃ desorption separately.

As shown in Figure c,f, Pearson correlation heat maps indicate strong correlations between ΔG ^# and W_F for both *NNH formation (correlation coefficient |r| = 0.77) and NH₃ desorption (|r| = 0.78), confirming W _F as the primary factor driving different BEP relations. Notably, Δq also displays a strong correlation with ΔG ^# (|r| = 0.82 and 0.75), but its correlation |r| with W _F is as high as 0.99, indicating collinearity between them, thus excluding Δq from the regression is justified. For the *NNH formation, ΔG ^# shows a moderate correlation with ΔG (|r| = 0.57), whereas N_v and Lat demonstrate weaker correlations. For the NH₃ desorption, ΔG ^# exhibits a moderate correlation with N _v (|r| = 0.55), while ΔG ^# shows nearly no correlation with Lat, which itself correlates with ΔG (|r| = 0.63).

Using these four physically meaningful descriptors, four ML algorithms were employed to predict activation energies, including linear regression (LR), kernel ridge regression (KRR), support vector regression (SVR), and gradient boosting regression (GBR). Detailed training results and Leave-One-Out Cross-Validation (LOOCV) analyses for the *NNH formation and the NH₃ desorption steps are shown in Figures S11 and S12, respectively. Here, we first discuss the LR results, as the data size used in this work is small, and LR is the least prone to overfitting. Notably, LR demonstrates robust regression performance and consistent LOOCV outcomes for both *NNH formation (Figure d, R ² = 0.87) and NH₃ desorption (Figure g, R ² = 0.86) steps, indicating a strong linear correlation within the data set. The derived LR equations are as follows

$$\begin{array}{l}*\mathrm{N}\mathrm{N}\mathrm{H}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}:\mathrm{\Delta }{G}^{\#}=-3.609+0.495·\mathrm{\Delta }G+0.689·{\mathrm{W}}_{\mathrm{F}}+(-0.039)·{\mathrm{N}}_{\mathrm{v}}+0.137·\mathrm{L}\mathrm{a}\mathrm{t},\\ {\mathrm{N}\mathrm{H}}_3\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}:\mathrm{\Delta }{G}^{\#}=-5.435+0.870·\mathrm{\Delta }G+0.715·{\mathrm{W}}_{\mathrm{F}}+0.000·{\mathrm{N}}_{\mathrm{v}}+0.227·\mathrm{L}\mathrm{a}\mathrm{t},\end{array}$$

These equations highlight that the work function significantly contributes to predicting ΔG ^# , consistent with the analysis described above. SVR, employing a linear kernel, exhibits excellent test scores for both *NNH formation (Figure e, R ² = 0.94, RMSE = 0.05) and NH₃ desorption (Figure h, R ² = 0.92, RMSE = 0.06) steps, further confirmed by the strong generalization capability evidenced by the LOOCV results (R ² = 0.91 and 0.92, respectively). Although GBR and KRR also show reasonable regression accuracy, neither method describes both reaction steps simultaneously as accurately as SVR. Overall, these ML methods enable a unified mapping from ΔG to ΔG ^# across different host metals using only four easily accessible descriptors. In the regression analysis, we propose LR and SVR (linear kernel) to minimize the risk of overfitting for a small data set. Nevertheless, it should be noted that the limited data size may restrict the generality of the regression results to all metal hosts.

eNRR Kinetic Cycle on Representative M₁@CM

To evaluate whether the promising candidates identified based on *NNH formation and NH₃ desorption barriers are active, we selected the optimal model, Re₁@Ag, as a representative to comprehensively explore its thermodynamic and kinetic eNRR pathways under operating potential and solvation conditions. Our calculations focused specifically on neutral conditions of pH = 7 and U _RHE = −0.1 V (with a 0.2 V overpotential), employing the explicit solvation model, and considering water as the exclusive proton source to induce protonation hydrogenation. The thermodynamic free energy profile (Figure a, blue lines) suggests that eNRR proceeds readily (the potential-determining step, *NH₂ formation, has a maximum free energy change of 0.28 eV under specified constant-potential conditions) except that NH₃ desorption is comparatively difficult (0.76 eV). However, such thermodynamic pictures can be misleading due to the lack of critical kinetic information, which has been found to be significantly influenced by the factors including applied potential, pH, and dynamic hydrogen bonding network. − Indeed, each elementary step is closely associated with a kinetic energy barrier (as depicted by the orange curves in Figure a).

4.

(a) Full free energy profile of eNRR occurring on the Re₁@Ag SAA surface (pH = 7, U _RHE = −0.1 and −0.5 V) with the corresponding kinetic barriers. Blue lines represent the thermodynamic profile calculated by CPM, and yellow curves represent the kinetic barriers calculated by the slow-growth approach. (b) Slow-growth calculated free energy profile and electron number change (Δq) relative to electric neutrality vary with reaction coordinate for the *NNH₃ formation and the cleavage of N–N bond in *NNH₃. (c) Alternative routes of NH₃ desorption pathways. (d) Structural snapshots of *NH₃/H₂O and *NH₃/N₂ coadsorbed states during cp-AIMD simulation. (e) Free energy profile for the formation of *NH₃/N₂ coadsorbed state and the corresponding NH₃ desorption.

The calculated kinetic barriers indicate that the highest-barrier step, *NNH formation, involves the most challenging hydrogenation of stable adsorbed *N₂ with a kinetic barrier of 0.85 eV. Subsequently, activated *NNH can be hydrogenated to form *NNH₂ readily, with a barrier of 0.47 eV. The subsequent *N and NH₃ formation is relatively difficult due to the breaking of the N–N bond, where direct N–N cleavage produces a significantly higher barrier of 1.43 eV (Figure S13a). The alternative pathways of *N direct formation and the case of *NHNH₂ formation are also excluded due to high kinetic barriers exceeding 1.2 eV (Figures S13b,c). However, it is found that *NNH₂ can first undergo protonation to yield an metastable *NNH₃ species like in the literature, with a kinetic barrier of 0.79 eV (Figure b). Interestingly, there is no variation in the electron number (Δq = −0.15 e^–) between the initial and final states, indicating that the extra H (from water) in *NNH₃ species remains protonic, resembling an ammonia–water-like complex, −NH₂–H–OH. Figure S14 further shows that the extra H oscillates rapidly between *NNH₂ and H₂O within 2 ps cp-AIMD simulations, supporting its protonic property and the existence of the *NNH₃ intermediate. Then the N–N cleavage occurs with a low barrier of 0.49 eV and releases NH₃ into the aqueous phase, accompanied by electron redistribution and proton recombination with −NH₂ (Figure b, where Δq varies from ∼−0.4 e^– to ∼0.0 e^–). To check the thermodynamic stability of *NNH₃ species, we optimized its ground state by CCM and obtained only *N and physically adsorbed NH₃ (Figure S15a). Using CPM, however, one can see that the *NNH₃ species exists stably with Δq of −0.76 e^– under relevant conditions (Figure S15b, pH = 7, U _RHE = −0.1 V), implying that the extra H atom fluctuates between protonic and atomic states. The ΔG of *NNH₂ forming *NNH₃ is calculated to be 0.85 eV, slightly higher than the kinetic barrier (0.79 eV) derived from cp-AIMD simulations, owing to the enhanced stabilization via hydrogen bonding interactions with explicit H₂O molecules. After the N–N cleavage step, the remaining *N intermediate undergoes successive facile hydrogenation steps to generate *NH₃, with low kinetic barriers of 0.47, 0.60, and 0.38 eV for *NH, *NH₂, and *NH₃ formation steps, respectively. All of the slow-growth results from *N₂ to *NH₃ are provided in Figure S16.

From thermodynamic and kinetic perspectives, direct NH₃ desorption constitutes the major bottleneck for eNRR. Nevertheless, it is found that NH₃ desorption can undergo several alternative pathways, as reported in the literature for oxygen reduction reaction on single-atom iron catalyst. Given the thermodynamically favorable adsorption of H₂O (ΔG _H2O‑ads = −0.08 eV), we first hypothesized that H₂O might facilitate NH₃ desorption. However, kinetic results in Figure c indicate that H₂O provides minor assistance with slightly lower barriers of 0.98 and 0.95 eV for coadsorption and replacement pathways, respectively. This arises from the electrostatic-dominated adsorption interactions and significant steric repulsion, which prevent a stable *NH₃/H₂O coadsorption state (Figure d, upper panel). Interestingly, the N₂ from the water layer replacing *NH₃ pathway emerged as a more feasible strategy since N₂ adsorption (ΔG _N2‑ads = −1.25 eV) is much stronger than that of *NH₃ (ΔG _NH3‑ads = −0.72 eV). The coadsorption state, *NH₃/N₂, is formed with a low barrier of 0.78 eV (Figure e, left panel) and remains stable for over 2 ps under operational conditions (Figures d and S17). Such a coadsorption state is critical for the NH₃ desorption, facilitating easy subsequent NH₃ release with a barrier of 0.33 eV (Figure e, right panel). Without *NH₃, the barrier for direct N₂ adsorption on a bare­(*) site is highly likely to be lower than 0.78 eV, indicating that the direct adsorption scenario is unlikely to be the bottleneck for eNRR. Moreover, the potential migration mechanism of *NH₃ from the dopant site to adjacent Ag sites prior to desorption was hardly able to occur with a high barrier of 1.33 eV (Figure c). This N₂-assisted coadsorption mechanism results in a much lower kinetic barrier compared to the direct desorption in this work and literature. The coadsorption mechanism can also be elucidated by the thermodynamics, primarily arising from the disparate adsorption strengths of *H₂O < *NH₃ < *N₂. The *NH₃/N₂ coadsorption state is feasible under both CCM and CPM calculations (Figure S18), with ΔG of 0.08 and −0.08 eV for CCM and CPM, respectively. In contrast, the *H₂O/N₂ configuration is achievable only under CPM and exhibits a markedly high ΔG _CPM of 1.02 eV (Figure S18b).

With regard to the competing reaction HER, Re₁@Ag also demonstrates exceptional HER suppression under neutral conditions. From the perspective of thermodynamics, the adsorption energy of H (ΔG _H‑ads) is −0.62 eV at the setting constant-potential conditions, which meets all screening criteria of both eNRR activation and selectivity proposed by Chen et al. (ΔG _N2‑ads = −1.25 < −0.40, U _eNRR = −0.38 > −0.50, ΔG _N2‑ads – ΔG _H‑ads = −0.84 eV < 0) for a variety of systems. As revealed by cp-AIMD simulations of bare Re₁@Ag with an explicit water layer, a hydrogen-bonded water molecule spontaneously adsorbs onto the dopant site within 0.32 ps (Figure S19a), which is attributed to the favorable water adsorption energy compared with the pure Ag (111) surface (−0.08 eV vs 0.28 eV). Subsequent proton transfer from the adsorbed water to the dopant site is hard, with an overall energy barrier exceeding 1.9 eV (Figure S19b). This barrier encompasses the cumulative costs of water desorption, steric hindrance for molecular reorientation, and bond dissociation during hydrolysis. We also considered an alternative Volmer pathway (* + H₂O + e^– → *H + OH^–) in which a second water molecule (not the adsorbed *H₂O) donates the proton to the Re site, but this route was found to be infeasible due to the occupation of the active site by *H₂O. Furthermore, the recombination of preadsorbed *H species with protons from water to form H₂ (Heyrovsky reaction, *H + H₂O + e^– → H₂ + * + OH^–) requires an energy barrier of 0.88 eV (Figure S19c), surpassing the highest barrier of 0.85 eV for eNRR in our proposed mechanism.

To quantitatively assess the competition between eNRR and HER, we computed the cp-AIMD barriers for *NNH formation and Heyrovsky steps at potentials of −0.1 and −0.5 V. The *NNH formation barrier decreases from 0.85 to 0.74 eV (Figure S20a), while the Heyrovsky barrier decreases from 0.88 to 0.81 eV (Figure S20b), indicating the *NNH formation step has a higher coefficient of the potential-dependent barrier than the HER Heyrovsky steps. Although the kinetic barrier of the Volmer reaction is already sufficiently inhibited to suppress HER, the results based on the assumed linear potential dependence further confirm that eNRR dominates over HER in the competitive process. Based on the currently available potential-dependent cp-AIMD barriers (Figure S20c), we performed preliminary microkinetic modeling for both eNRR and HER (see Section S2, Tables S1 and S2 for details). In consistency with the high barrier of *H formation, HER has a significantly lower selectivity, suggesting high NH₃ selectivity of Re₁@Ag (Figure S21). Please note that the present microkinetic modeling relies on several approximations due to the limited amount of potential-dependent data; therefore, the results should be interpreted as qualitative rather than quantitative predictions.

The summary of the catalytic cycle for eNRR on Re₁@Ag, based on our kinetic barriers, is depicted in Scheme . It begins with N₂ adsorption via an end-on configuration, required to overcome the rate-limiting *NNH formation barrier (0.85 eV). Sequential protonation yields a transient ammonia–water-like *NNH₃ intermediate, readily facilitating N–N bond cleavage to generate adsorbed *N. Further hydrogenation leads to *NH₃ formation, but direct NH₃ desorption is challenging. Adsorption of the reactant N₂ in the aqueous phase to form a stable *N₂/NH₃ coadsorption state significantly reduces the barrier of NH₃ desorption, allowing subsequent catalytic cycles to continue efficiently.

2. Full eNRR Catalytic Cycle Occurring on Re₁@Ag SAA System with the Constant-Potential Conditions of pH = 7, U _RHE = −0.1 V.

Compared to similar studies on eNRR, Mao et al. reported a *NNH formation barrier of 0.81 eV (−0.3 V) on Fe surface without cation assistance. Jiang et al. found the *NNH formation barrier on Ru surface in pure water at −0.2 V to be 0.94 eV. Qian et al. reported an *NNH formation barrier of ∼1.0 eV on graphene-based Fe–N₄ SAC at −0.23 V. From the kinetic barrier perspective and considering the entire eNRR pathway, our proposed Re₁@Ag system shows more promising eNRR activity and selectivity.

Conclusions

In this work, we proposed an efficient workflow to guide the high-throughput screening of electrocatalysts on the basis of the free energy change–kinetic barriers BEP relations. Taking eNRR as an example, the first hydrogenation step and the last product desorption step are calculated as they need to be balanced according to the Sabatier principle. Using constant-potential molecular dynamics and the enhanced sampling method, the kinetic barriers (i.e., activation energies) were then accurately evaluated under realistic solvation and potential conditions. A clear linear BEP relation between free energy change and kinetic barriers was observed in Ag-based SAAs, enabling reliable prediction of activation energies from accessible thermodynamic data and thus allowing identification of the optimal eNRR candidates.

When extended to the other coinage-metal-based SAAs (Cu and Au), it is found that different host metals resulted in distinct BEP relations, highlighting that thermodynamics alone may fail to reflect the true catalytic performance under operating conditions. Cu exhibited slightly inferior *NNH formation but superior NH₃ desorption kinetics compared to Ag, whereas Au showed an appearance of “thermodynamic deception”, displaying deceptively favorable thermodynamics but significantly high activation barriers. It is shown later that the abnormal behavior of Au arises from changes in the orientation of interfacial water molecules and the hydrogen network, caused by the most negatively charged surface under the same potential, owing to its significantly larger work function than Ag and Cu. Based on the host-specific BEP relations, the kinetic performance of various SAAs was quantitatively assessed, where Re₁@Ag, Mo₁@Ag, and Re₁@Cu emerged as the most promising eNRR electrocatalyst candidates. These BEP relations also enabled the backward inference of a precise thermodynamic upper bound that balances both *NNH formation and NH₃ desorption, facilitating large-scale screening with greatly reduced computational cost.

Moreover, machine learning regression analysis was conducted for all of the systems using three hosts. Feature analysis revealed that the work function plays a dominant role in defining the scaling behavior across the different host metals. By employing four physical descriptorsfree energy change (ΔG reflecting complex host–guest–adsorbate interactions), work function of host metal (W _F), valence electron number of guest metal (N_v), and lattice constant of host metal (Lat)the activation barriers (ΔG ^# ) under realistic conditions could be accurately predicted. Among the four models tested, support vector regression shows a more balanced and robust performance for both steps. Even though we have not conducted a systematic search for better descriptors, we demonstrate that a unified mapping from ΔG to ΔG ^# via machine learning regression is accessible across different host metals.

Finally, a detailed mechanistic investigation of a representative, Re₁@Ag, confirmed its outstanding catalytic performance. Following a distal pathway, the *N₂ intermediate undergoes stepwise hydrogenation to *NNH₂, which then forms an ammonia-like *NNH₃ species. A subsequent facile N–N bond cleavage generates *N, which is further hydrogenated to *NH₃. Notably, NH₃ direct desorption shows a high barrier (1.06 eV), but the critical coadsorption state of *NH₃/N₂ provides a favorable pathway with a critical formation barrier of 0.78 eV and reduces NH₃ desorption barrier to 0.49 eV, allowing remaining *N₂ to re-enter the catalytic cycle. The highest-barrier step remains *NNH formation, with a barrier of 0.85 eV, and all steps exhibit barriers no higher than 0.85 eVindicating an extraordinary eNRR performance if kinetic barriers of all steps (including NH₃ desorption) are examined. ,,

This generalizable free energy change–kinetic barriers BEP relation, as well as the machine learning regression framework, bridges the thermodynamics and kinetic barriers under realistic potential and solvation, thus offering a significantly more accurate yet still efficient strategy for rational electrocatalyst design at the level of kinetic barriers and providing a new paradigm for accurate and accelerated high-throughput electrocatalyst screening for other important reactions and catalytic materials.

Methods

Computational Parameters

The Vienna ab initio Simulation Package (VASP) was used for DFT calculations and ab initio molecular dynamics (AIMD) simulations. Electron–ion interactions were described by the projector augmented wave (PAW) potentials with a 400 eV cutoff energy. The Perdew–Burke–Ernzerhof (PBE) functional was employed within the generalized gradient approximation (GGA), where the DFT-D3 method was considered for the van der Waals dispersion correction. Spin polarization was considered for all of the calculations. For structural optimization and AIMD simulations, the Brillouin zone was sampled with a 3 × 3 × 1 k-mesh. The convergence criteria were set to 1.0 × 10^–5 eV and 0.01 eV/Å for energies and atomic forces, respectively.

Gibbs Free Energy Correction

To correct the Gibbs free energy (G), frequencies were further determined by solving the Hessian matrix. All the corrections were performed by the VASPKIT toolkit. For gaseous species, the Gibbs free energy of gas, G _gas(T,p), was calculated by, G _gas(T,p) = E _VASP + ZPE + ∫C _pdT – TS, where the E _VASP, ZPE, ∫C _pdT, and TS are calculated electronic energy, zero-point energies, and enthalpic and entropic contributions. Note that translational, rotational, and vibrational contributions were taken into consideration for gaseous species. For adsorbed species, the pV term could be negligible so that the ∫C _pdT term turns into the ∫C _vdT term. G _ads(T) was calculated by, G _ads(T) = E _VASP + ZPE + ∫C _vdT – TS. Note that only the vibrational contribution was taken into consideration for adsorbed species.

Considering the potential impact of gas-phase error corrections on eNRR predictions, we compared our thermodynamic values with the Calle-Vallejo correction values, in which the gas-phase N₂ energy is corrected by −0.14 eV and the NH₃ energy is reduced by 0.10 eV per NH₃ molecule after the overall reaction free energy is fixed at −0.888 eV. Based on these corrections, for the currently optimal Re₁@Ag system, free energies of two steps under the CPM conditions (U _RHE = −0.1 V, pH = 7) shift in a more favorable direction (ΔG _*NNH‑form, 0.08→ −0.06 eV; ΔG _NH3‑des, 0.76→ 0.66 eV). Notably, although the strength of N₂ adsorption is slightly weakened (ΔG _N2‑ads, – 1.25→ −1.11 eV), it remains highly favorable.

Constant Potential Electrochemical Simulations

To maintain the Fermi energy E _F fluctuating around a constant “electrode potential”, the implicit solution implemented by VASPsol , was employed, where the electron chemical potential μ_e coincides with the E _F. Then μ_e = μ_SHE – |e|·U _SHE, where the U _SHE is the electrode potential referenced to the standard hydrogen electrode (SHE), and μ_SHE is −4.6 eV according to VASPsol. To simulate the condition of the reversible hydrogen electrode (RHE) with a neutral environment, the relation between E _F and U _RHE (the electrode potential referenced to RHE) can be calculated by, U _RHE = U _SHE + 0.059*pH = – 4.6 V – (E _F + E _Fermishift)/e + 0.059*pH, where the E _Fermishift is the correction of the Fermi energy shift printed by the VASPsol code.

Computational hydrogen electrode models (CHEMs) within the constant-charge model (CCM) and the constant-potential model (CPM) were performed to compare the free energy change of N₂ adsorption (ΔG _N2‑ads), *NNH formation (ΔG _*NNH‑form), and NH₃ desorption (ΔG _NH3‑des). Herein, taking *NNH formation step for example, H⁺ _aq + e^– + *N₂ → *NNH. The Gibbs free energy change of *NNH formation under CCM was calculated by, ΔG _*NNH(CCM) = G _*NNH – G _*N2 – G _H2/2 + |e|U _RHE, where the |e| and U _RHE are the number of electrons and electrode potential referred to the RHE. During the calculations under CPM, the number of electrons was adjusted to a constant value in the relaxation, where the Fermi level converged to the target potential. So the equation was written by H⁺ _aq + (Q1 – Q2 + 1)­e^– + *N₂ ^Q1 → *NNH^Q2. Thus, the Gibbs free energy change of *NNH formation under CPM was calculated by, ΔG _*NNH(CPM) = G _*NNH ^Q2 – G _*N2 ^Q1 – G _H2/2 + |e|U _RHE – (Q1 – Q2)­μ_e, where the Q1 and Q2 are the charge in the systems of *N₂ and *NNH, and μ_e is the chemical potential of the electron referenced to the RHE.

AIMD Simulations

During AIMD simulations, the Nose–Hoover thermostat was performed to maintain the fluctuation of system temperature at 300 K. The time step in AIMD was set to 1.0 fs, where the H mass was set to that of deuterium. With both implicit solution and explicit solvent molecules included, the grand canonical “constant-potential hybrid-solvation dynamic model” (CP-HS-DM) ,,, approach was used to simulate the dynamics under the operating conditions, and the enhanced sampling method “slow-growth” approach was performed to evaluate the reaction barriers. During the “slow-growth” simulations, the transformation velocity was set as 0.0004 Å/fs for the individual variable (NH₃ desorption step) and 0.0008 Å/fs for the collective variable (2 variables combined in *NNH formation step).

Regression

Machine learning was implemented using the scikit-learn library in Python. Prior to model training, the data set was partitioned into training and testing subsets (4:1 ratio) with stratified sampling to preserve population distribution. The hyperparameters used are provided in Table S3.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.5c01411.

• Details of used models, microkinetic modeling, used regression hyperparameters, PDOS analysis, details of slow-growth simulations, bond length analysis, concentration distributions of H₂O, free energy changes from CCM and CPM calculations, detailed regression results, and cp-AIMD simulations (PDF)

X. Z. conceived the project, C. J. performed the theoretical calculations, and C. J. and X. Z. conducted the results analysis and manuscript preparation.

The authors declare no competing financial interest.