Data-driven structural angle mining elucidates hidden design rules for hydrogen evolution single-atom electrocatalysts

Abstract

Structural distortions in modified two-dimensional transition metal dichalcogenides (MX₂) influence electrocatalytic activity, yet quantitative and predictive structure–property relationships remain underdeveloped. To bridge this gap, we perform data-driven structural angle mining across hundreds of thousands of single-atom doped configurations (TM₁@MX₂) and establish geometrically defined angular descriptors. These descriptors exhibit high predictive accuracy for hydrogen evolution electrocatalysis. Crucially, our analysis reveals that catalytic activity correlates more strongly with long-range angular parameters describing peripheral geometric effects than with the local coordination environment. Guided by these descriptors, we identify specific angular signatures as quantitative predictors for high-performance catalysts: an outer-shell S-centered angle indicates optimal hydrogen evolution reaction (HER) activity for Ir₁@MoS₂ (S-vacancy), while a distinct Mo-centered angle identifies V₁@MoS₂ (Mo-vacancy) as a promising earth-abundant candidate. Experimental verification confirms these predictions: synthesized Ir₁@MoS₂, with an ultralow loading of 0.1 wt%, achieves performance comparable to commercial Pt/C on a mass-activity basis, while V₁@MoS₂ enhances HER performance relative to pristine MoS₂. The framework also shows strong computational correlations with oxygen evolution activity, though experimental validation for OER remains an important direction for future investigation. The angular descriptor framework introduced here provides a geometrically intuitive and electronically grounded strategy for the rational design and accelerated discovery of advanced energy materials.

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Introduction

The transition to a sustainable energy future is contingent on the development of efficient electrocatalysts for water splitting, a key technology for green hydrogen production. However, widespread deployment of electrolyzers remains constrained by their reliance on scarce platinum-group metals, spurring the search for earth-abundant alternatives without compromising activity.^1–4 While single-atom catalysts (SACs) embedded in two-dimensional transition metal dichalcogenides (TM₁@MX₂) present a promising platform to reduce reliance on platinum-group metals, their rational design remains fundamentally challenging.^5–8 The catalytic performance is exquisitely sensitive to the local atomic environment, where subtle structural distortions can dramatically modulate activity.^9–14 However, a critical gap persists between observing these structural effects and quantitatively predicting them, as current paradigms struggle to decouple the intricate interplay between geometry and electronic structure.

The descriptor-based approach, central to modern catalyst design, has long been dominated by electronic structure parameters, such as adsorption energies and d-band centers.^15–20 While powerful, these descriptors often serve as computational proxies that are notoriously difficult to translate into actionable synthesis guidelines. There exists a critical gap between the abstract electronic parameters and the tangible geometric structures that chemists manipulate. They answer the “what” but not the “how”, failing to provide experimentalists with clear geometric blueprints for catalyst creation. Structural descriptors present a compelling alternative,^21–23 with concepts such as “surface distortion” successfully bridging structure–activity relationships across nanocatalysts.²⁴ However, such ensemble-averaged structural probes, while insightful, lack the atomic-scale resolution to pinpoint the specific geometric motifs that govern function. This gap between macroscopic observation and atomic-scale design underscores the critical lack of an intuitive, quantitative language to link synthesis to performance.

Efforts to establish such a language by directly linking three-dimensional atomic structure to activity face significant challenges. Pioneering work has highlighted the role of local geometry, using parameters like bond lengths.²⁵ Despite their utility, such one-dimensional metrics are inherently limited in capturing the multidimensional nature of structural distortion.^26–28 Similarly, coordination numbers often oversimplify the complex, anisotropic coordination environment, failing to resolve critical features beyond the first shell.^29–33 Thus, the pressing need is for multidimensional geometric descriptors that are synthetically interpretable yet rich enough to encode the complexity of atomic-scale environments and to predict their catalytic function.

Here, we introduce data-driven angular descriptors to meet this pressing need. Unlike conventional approaches, these geometric parameters serve as a direct quantitative bridge between the synthesis-induced lattice strain and the resulting catalytic function. We demonstrate that catalytic activity is governed not merely by the local dopant identity, but by the precise angular distortions in the coordination shells. As outlined in Scheme 1, our integrated workflow progresses from data-driven descriptor mining to experimental validation. Through high-throughput density functional theory (DFT) screening of hundreds of thousands of TM₁@MX₂ configurations, we establish that catalytic activity is governed not by the local coordination, but by long-range peripheral angles, geometric parameters that are both highly predictive and synthetically interpretable. Crucially, we decode how macroscopic lattice strain manifests in these specific angular motifs, a relationship that shows good transferability across reactions (hydrogen evolution reaction, HER, and oxygen evolution reaction, OER) and materials. Guided by this principle, we discover and experimentally validate two high-performance catalysts: an ultralow-loading Ir₁@MoS₂ competitive with Pt/C and an earth-abundant V₁@MoS₂. Thereby, we establish angular descriptors as a transformative tool, providing a quantitative, geometry-driven blueprint for rational catalyst design.

Scheme 1 Integrated workflow from data-driven angular descriptor mining to experimental validation of high-performance water splitting electrocatalyst.

Results

Structural angular features

Guided by established experimental frameworks,^34,35 and motivated by the well-recognized catalytic activity of MoS₂ for water electrolysis,^36–39 we systematically investigate two distinct classes of TM₁@MX₂ systems classified by vacancy type: chalcogen-vacancy (TM₁@MX₂–X) and metal-vacancy (TM₁@MX₂–M). Our study encompasses 22 transition metal dopants spanning the 3d, 4d, and 5d series (Fig. 1a), providing broad chemical space exploration while excluding Tc and Hg for safety considerations and Cu, Ag, and Au due to their known catalytic inertness.^40–42 In TM₁@MX₂–X systems, each dopant atom occupies an S/Se vacancy site, forming three-fold coordination with three adjacent metal atoms (Fig. 1b and S1). Conversely, for TM₁@MX₂–M, each dopant atom occupies an Mo/W vacancy site and adopts six-fold coordination with surrounding chalcogen atoms (Fig. 1c and S2).

Fig. 1 (a) Periodic table highlighting the 22 transition metal dopants (3d/4d/5d) investigated in this study. (b) Atomic labeling scheme for TM₁@MX₂–X: top-layer chalcogens (S/Se, atoms 0–23), bottom-layer chalcogens (24–48), host metal atoms (M = Mo/W, 49–73), and dopant transition metal (TM, atom 74). (c) Atomic labeling for TM₁@MX₂–M: top-layer chalcogens (0–23, 49), bottom-layer chalcogens (24–48), host metals (50–73), and dopant (74). (d) Three characteristic distortion modes: surface distortion (angle between vertex and top S/Se atom), vertical distortion (angle between vertex and interlayer S/Se axis), and internal distortion (angle between central Mo/W atom and vertex). (e) Active site geometry in TM₁@MX₂–X, showing the doped TM center coordinating key reaction intermediates (*H, *O, *OH, *OOH) for HER/OER processes. The corresponding HER active site configuration for TM₁@MX₂–M is provided in Fig. S3. Color scheme: blue = Mo/W atoms, pink = S/Se atoms, gold = dopant atoms.

The critical role of structural distortions in modulating catalytic performance is well recognized, and dopant-induced lattice distortions are known to perturb electronic structure and influence activity.^43,44 While structural distortion descriptors are gaining attention for accelerating structure–performance understanding,^24,45 more intuitive geometric parameters are still needed. Given that all macroscopic properties originate from atomic arrangements, we identify angular distortions as particularly sensitive metrics for quantifying structure–activity relationships.

Our analysis reveals three fundamental angular distortion modes that define the structural landscape (Fig. 1d): surface distortion, characterizing top-layer geometry; vertical distortion, capturing interlayer interactions; and internal distortion, describing core structural perturbations. Doping-induced variations generate over 190 000 angular configurations per catalyst model, enabling thorough quantification of local geometry.

The microstructure of metal-doped catalysts exhibits well-defined shell environments (Fig. 1e). In TM₁@MX₂–X systems, the active site resides at the doped transition metal center, where ligand atoms form concentric shell environments, including the first shell, the second shell, and peripheral shells that coordinate key intermediates (*H, *O, *OH, *OOH) during HER and OER. Different metal dopants induce characteristic distortions across these shell structures, ultimately governing catalytic performance. A similar shell distortion mechanism operates in TM₁@MX₂–M systems, though the active center shifts to adjacent chalcogen sites (Fig. S3).

Data-driven development of angular descriptor for HER

To establish quantitative structure–activity relationships for HER, we initially focus on TM₁@MoS₂–X as a model system. We start by systematically extracting all structural angles centered at the dopant metal site, defined as the angle formed by the dopant and any two adjacent atoms, which yielded 2631 unique angular parameters. These geometric parameters are rigorously correlated with hydrogen adsorption Gibbs free energy (ΔG_*H) using Pearson correlation analysis, with color mapping indicating correlation strength (dark red: strong positive, r ≈ +1; dark blue: strong negative, r ≈ −1). Among these, the ∠S₁₇M₇₄S₂₂ angle exhibits the strongest correlation (r = 0.75, Fig. 2a, S4a and b), suggesting its potential as a HER activity descriptor. However, the moderate correlation coefficient also reveals a fundamental limitation: dopant-centered angles, being spatially localized, fail to fully capture the longer-range structural perturbations that significantly influence catalytic behavior. This finding motivates us to explore the role of extended coordination environments.

Fig. 2 Correlation analysis of structural distortions with HER activity in TM₁@MoS₂–X. (a–d) Pearson correlation coefficient (r) between ΔG_*H and angles centered on (a) dopant atoms, (b) first-shell S atoms, (c) second-shell S atoms, and (d) Mo atoms. The angle exhibiting the strongest linear correlation is highlighted in each panel. (e–h) Linear relationship between the optimal descriptor φ_HER and ΔG_*H across different substrates: (e) MoS₂, (f) WSe₂, (g) WS₂, and (h) MoSe₂. Color scheme: (a–c) structural representations showing the doped transition metal center with first-shell S atoms in orange and second-shell S atoms in yellow. (d) Structural representation showing the doped metal center with first-shell Mo atoms in light purple, second-shell Mo in dark purple, third-shell Mo in green, and outermost-shell Mo in yellow.

The influence of peripheral atomic arrangements on single-atom catalysts has been proposed to arise from microstress variations that propagate across multiple coordination shells.^29–31 Indeed, metal doping not only modifies the electronic structure of the central active site but also induces long-range microstress fields that critically affect HER activity.^46,47 To quantitatively test this hypothesis and develop more accurate structural descriptors, we extend our angle-based analysis to atoms residing in outer coordination shells surrounding the TM₁ centers.

To intuitively visualize the coordination environment, sulfur atoms in equivalent positions are assigned identical colors, with gradation in shading representing increasing shell distance from the active center (Fig. 2a–d). Our systematic analysis reveals a significant trend: the correlation with ΔG_*H strengthens as we move from the dopant center toward peripheral atomic shells. The internal angle ∠S₂₃S₁₂S₄₂ exhibits a substantially improved correlation (r = 0.82; Fig. 2b, S5a and b), compared to dopant-centered descriptors. The trend becomes even more pronounced in the second shell, where surface angle ∠S₁S₂₃S₈ shows an even stronger correlation (r = 0.85; Fig. 2c, S6a and b), underscoring the critical role of peripheral effects in modulating catalytic performance.

To determine whether this phenomenon extends beyond sulfur-centered coordination, we expand our analysis to angles centered on molybdenum atoms in outer shells. By applying crystallographic symmetry principles, we systematically map Mo coordination environments (Fig. 2d, S7a and b) while maintaining consistent color-coding. Among these, the Mo-centered internal angle ∠S₂₁Mo₆₅Mo₆₇ also demonstrates a strong correlation with HER activity (r = 0.84, Fig. 2d). Through a comprehensive comparison of all candidate descriptors, we identify the second-shell surface angle ∠S₁S₂₃S₈, which exhibits the strongest overall correlation, as the optimal angular descriptor for HER, designated φ_HER (Fig. 2e). The consistent superiority of peripheral-shell-centered angles over dopant-site-centered descriptors confirms that long-range structural effects, rather than local metal-centered geometry, predominantly govern HER performance in these SACs.

We further evaluate the transferability of φ_HER across different chalcogen-vacancy systems. Impressively, this descriptor maintains strong linear correlations with ΔG_*H in TM₁@WS₂–S (r = −0.83), TM₁@WSe₂–Se (r = −0.83), and TM₁@MoSe₂–Se (r = −0.81) (Fig. 2f–h). Within the TM₁@MoS₂–S system, Ir₁@MoS₂–S emerges as the optimal catalyst, with φ_HER = 26.99° corresponding to a near-ideal ΔG_*H of −0.01 eV (Fig. 2e). Similarly, Hf₁@WSe₂–Se, Ru₁@WS₂–S, and Rh₁@MoSe₂–Se show superior HER activity in their respective systems, with φ_HER values of 26.99°, 27.00°, and 26.96°, predicting ΔG_*H values of 0.06, −0.04, and 0.41 eV, respectively. This consistent predictive accuracy across diverse material systems establishes φ_HER as a robust and generalizable descriptor for structure–activity relationships. These results collectively reveal the intrinsic mechanism of “periphery over core”: HER activity is more strongly correlated with the geometric distortion of the peripheral coordination shell than with the locally doped center. Additionally, they validate the good transferability of this type of angular descriptor across different MX₂ hosts. This indicates that the doped atom acts merely as a trigger, while the long-range structural response of the host lattice—quantified by specific peripheral angular configurations—is the key determinant of catalytic performance.

Transferability of angular descriptors from HER to OER

We then aim to determine whether this angular descriptor approach could be extended to a more complex reaction (OER). To evaluate the transferability of our descriptor framework, we systematically analyze angle–overpotential relationships across the same family of chalcogen-vacancy systems.

Initial screening of dopant-centered angles identifies ∠S₁M₇₄S₆ as the most promising candidate (r = 0.85, Fig. S8a–c). However, as observed with HER, extending our analysis to peripheral coordination shells substantially enhances correlation strengths. First-shell sulfur-centered angles, particularly the surface angle ∠S₁₅S₁₂S₁₉, exhibit a strong negative correlation with OER overpotential (r = −0.90, Fig. S9a–c), while second-shell sulfur-centered angles such as ∠S₁₁S₂₃S₂₂ demonstrate an equally significant positive correlation (r = 0.90, Fig. S10a–c), indicating that smaller angles in this descriptor family favor enhanced OER activity.

Extending our analysis to Mo-centered angles in the outer coordination shells reveals that the internal angle ∠S₈Mo₅₂S₃₆, centered on the outermost Mo₅₂ atom, exhibits a good correlation (r = 0.91, Fig. 3a), slightly exceeding all S-centered descriptors. We therefore establish this Mo-centered angle as the optimal OER descriptor, designated φ_OER (Fig. 3b). This descriptor identifies Pt₁@MoS₂–S as the most promising catalyst within the TM₁@MoS₂–S system, with φ_OER = 67.98° corresponding to a predicted overpotential of 0.51 V.

Fig. 3 Angular descriptors for OER activity in TM₁@MX₂–X systems. (a) Correlation matrix of angle–overpotential relationships, identifying the internal angle of ∠S₈Mo₅₂S₃₆ as exhibiting the strongest correlation. (b) Atomic configuration of the optimal descriptor ∠S₈Mo₅₂S₃₆ (denoted as φ_OER). (c–f) Linear regression analysis demonstrating the robust correlation between φ_OER and OER overpotential across different substrates: (c) MoS₂, (d) WSe₂, (e) WS₂, and (f) MoSe₂.

The robustness of φ_OER is confirmed through rigorous validation across four distinct material systems though we note that experimental OER validation is not performed in the present work: TM₁@MoS₂–S (r = 0.91), TM₁@WSe₂–Se (r = 0.89), TM₁@WS₂–S (r = 0.85), and TM₁@MoSe₂–Se (r = 0.86) (Fig. 3c–f). Among these, Pt₁@WSe₂–Se exhibits the highest predicted activity with φ_OER = 68.92° corresponding to an overpotential of 0.36 V, while Pt₁@WS₂–S (φ_OER = 68.21°) and Pt₁@MoSe₂–Se (φ_OER = 71.02°) show predicted overpotentials of 0.56 V and 0.60 V, respectively.

Our results demonstrate that the angular descriptor framework successfully transfers from HER to OER, revealing both universal principles and reaction-specific characteristics. While both optimal descriptors, φ_HER (∠S₁S₂₃S₈) for HER and φ_OER (∠S₈Mo₅₂S₃₆) for OER, exhibit strong correlations with their respective activities (r = 0.85 and 0.91), confirming that coordination shell geometry governs catalytic performance across different electrochemical reactions, OER displays significantly greater angular sensitivity. This is evidenced by both higher maximum correlation coefficients and a greater number of angles exhibiting |r| > 0.8. We attribute this enhanced sensitivity to the more complex OER mechanism, which involves multiple adsorbed intermediates along a convoluted reaction pathway, making it particularly susceptible to subtle geometric variations. Crucially, despite these mechanistic differences, both reactions adhere to a unified design principle in which dopants modulate catalytic activity primarily through long-range structural perturbations in peripheral environments rather than through local metal-centered effects. This fundamental understanding establishes angular descriptors as a universal strategy for geometry-informed catalyst design across diverse electrochemical processes.

Transferability of angular descriptors from chalcogen-vacancy to metal-vacancy systems

After establishing the efficacy of angular descriptors in chalcogen-vacancy systems, we move on to investigate their applicability to the fundamentally distinct metal-vacancy configurations. MX₂ materials offer diverse doping sites,^48,49 with surface chalcogen vacancies and subsurface metal vacancies representing two predominant configurations that yield substantially different reactive sites and catalytic behaviors.^50,51 To assess the universal applicability of our descriptor framework, we extend our investigation to transition metal SACs formed by substituting Mo/W sites within the MX₂ lattice, which maintain the characteristic sandwich-like coordination geometry but position the active site differently (Fig. 4a for MoS₂ and Fig. 4d for WSe₂; atomic labeling in Fig. S7). In these metal-vacancy systems, reaction intermediates adsorb at surface chalcogen atoms rather than at the dopant site.

Fig. 4 Angular descriptors for HER activity in TM₁@MX₂ (metal-vacancy) systems. (a) Atomic configuration of the TM₁@MoS₂–Mo showing the optimal angular descriptor ∠S₇S₁₉Mo₆₉, with blue, pink, and gold spheres representing Mo, S, and dopant atoms, respectively. (b) Correlation matrix of angle–ΔG_*H correlation, identifying the internal angle of ∠S₇S₁₉Mo₆₉ exhibiting the strongest negative correlation. (c) Linear regression analysis of the correlation between ∠S₇S₁₉Mo₆₉ and ΔG_*H. (d) Atomic configuration of the TM₁@WSe₂–W featuring the optimal descriptor ∠Se₃₉W₅₆W₆₆, with purple, light red, and gold spheres representing W, Se, and dopant atoms, respectively. (e) Correlation matrix identifying the internal angle of ∠Se₃₉W₅₆W₆₆ as showing the strongest positive correlation. (f) Linear regression analysis of the correlation between ∠Se₃₉W₅₆W₆₆ and ΔG_*H.

Our descriptor mining methodology proves equally effective in identifying key angular parameters within these metal-vacancy systems. Through systematic coordination shell analysis, we visualize the correlation landscape using a color-coded scheme (Fig. 4b and e), where dark red (r = −1 to −0.5) and dark blue (r = 0.5 to 1) represent the strongest correlations. This analysis reveals two highly predictive angular descriptors: φ_HER-Mo (an internal angle of ∠S₇S₁₉Mo₆₉ in MoS₂ systems, Fig. 4b) and φ_HER-W (an internal angle of ∠Se₃₉W₅₆W₆₆ in WSe₂, Fig. 4e), collectively designated as φ_HER-M. These descriptors exhibit robust yet fundamentally distinct correlations with ΔG_*H. Specifically, φ_HER-Mo displays a significant negative correlation (r = −0.81), where larger angles correspond to enhanced HER activity (Fig. 4c). Accordingly, V₁@MoS₂–Mo emerges as the most active catalyst in this series with φ_HER-Mo = 139.31° and a near-optimal ΔG_*H = 0.01 eV, while the least active Pt₁@MoS₂–Mo shows φ_HER-Mo = 139.41° with a strongly binding ΔG_*H of −1.65 eV. Conversely, φ_HER-W demonstrates a strong positive correlation (r = 0.86), where increasing angular values improve catalytic performance (Fig. 4f). The optimal Ir₁@WSe₂–W displays φ_HER-W = 47.06° with ΔG_*H = −0.08 eV, whereas Mo₁@WSe₂–W, which shows the lowest HER activity, exhibits φ_HER-W = 47.70° with ΔG_*H = 1.26 eV.

The consistently high correlation strengths (average |r| > 0.8) across both vacancy types establish the universal predictive power and good transferability of angular descriptors in TM₁@MX₂ catalysts. However, we observe a fundamental distinction in descriptor behavior between the two systems: chalcogen-vacancy systems employ a universal descriptor (φ_HER = ∠S₁S₂₃S₈) across all catalysts, whereas in metal-vacancy systems, different atomic angles serve as the key descriptors, such as ∠S₇S₁₉Mo₆₉ and ∠Se₃₉W₅₆W₆₆. This divergence in correlation patterns stems from fundamental differences in how the two vacancy types perturb the electronic structure, a phenomenon we quantitatively explore in the following electronic structure investigation.

Electronic origins of angle–activity relationships

The established correlations between angular descriptors and catalytic activity prompt us to investigate their underlying electronic origins. While the connection between structural distortions and catalytic performance is known to be mediated by electronic structure,^52–54 conventional electronic descriptors show limited effectiveness in single-atom systems. Our systematic density of states (DOS) analysis of TM₁@MoS₂–S reveals the inadequacy of the conventional d-band center (ε_d) theory for SACs. As shown in Fig. S11 and S12, ε_d shows only weak correlations with both HER (ΔG_*H, r = −0.30) and OER (overpotential η, r = 0.32) activity. This limitation likely originates from fundamental characteristics of SACs: (i) anisotropic d-orbital/adsorbate interactions create orientation-dependent binding energies that are averaged out in ε_d analysis; (ii) the localized nature of catalytic activity in SACs depends on specific orbital contributions rather than global d-band properties. These observations underscore the need for descriptors that can bridge atomic-scale geometry with electronic structure—not as a replacement for all existing descriptors, but as an additional perspective.

To address these limitations, we develop a multidimensional descriptor framework integrating d-band and p-band center theory with our angular descriptors (φ_HER and φ_OER).^55,56 Inspired by the binary descriptor approach,³² we apply the SISSO algorithm⁵⁷ to derive band-structure descriptors I_band-HER and I_band-OER. These incorporate s-, p-, and d-band contributions from both dopant TM₁ (ε_M-s, ε_M-d) and surrounding atoms (ε_23-p, ε_1-s, ε_36-p, ε_52-s).

The I_band-HER descriptor shows an outstanding correlation with ΔG_*H (r = 0.88, Fig. 5a) and a strong anti-correlation with φ_HER (r = −0.85, Fig. 5b). These dual correlations reveal a coherent structure–activity relationship: while individual orbital bands provide incomplete activity descriptions, the integrated I_band-HER descriptor captures essential electronic features governing HER performance, including band-center positions, bandwidth distributions, and multi-orbital coupling. More importantly, these results establish a fundamental mechanistic link: geometric distortions directly modulate the catalyst's electronic band structure, which in turn regulates ΔG_*H through coordinated s-, p-, and d-orbital interactions. This provides an atomic-level understanding of how structural modifications drive catalytic enhancements in TM₁@MX₂–X systems.

where ε_M-s and ε_M-d represent the s and d-band center values of the doped metal TM₁, respectively; ε_23-p and ε_1-s represent the p-band center value of atom S₂₃ and the s-band center value of atom S₁ in angle ∠S₁S₂₃S₈, respectively (Tables S1–4).

Fig. 5 Correlations between composite electronic descriptors and catalytic activities for TM₁@MoS₂–S. (a) Linear relationship between I_band-HER and ΔG_*H. (b) Linear correlation between I_band-HER and the optimal angular descriptor ∠S₁S₂₃S₈ (φ_HER). (c) Linear relationship between I_band-OER and OER overpotential. (d) Linear correlation between I_band-OER and the optimal angular descriptor ∠S₈Mo₅₂S₃₆ (φ_OER).

Extending our descriptor framework to OER, the composite descriptor I_band-OER integrates key electronic features: ε_M-s and ε_M-d represent the s- and d-band centers of the dopant TM₁, while ε_36-p and ε_52-s correspond to the p-band center of S₃₆ and s-band center of Mo₅₂ within the ∠S₈Mo₅₂S₃₆ angular configuration. Remarkably, I_band-OER demonstrates good dual linear correlations with both φ_OER and OER overpotential, achieving identical correlation coefficients of r = 0.91 (Fig. 5c and d). These robust relationships establish that dopant-induced structural modifications systematically alter catalyst geometry, which linearly modulates OER activity through well-defined electronic structure changes.

I_band-OER = (ε_M-d − ε_36-p) × ε_M-s × ε_52-s, (2)

The versatility of our electronic descriptor framework is further demonstrated by its successful application to metal-vacancy systems. For TM₁@MoS₂–Mo, I_band-Mo incorporates ε_M-s and ε_M-d from the dopant TM₁, along with ε_19-p and ε_69-s (p-band center of S₁₉ and s-band center of S₆₉) derived from the key angle ∠S₇S₁₉Mo₆₉ (Table S5). I_band-Mo exhibits a significant correlation with ΔG_*H (r = 0.71, Fig. S13) and a strong anti-correlation with φ_HER-Mo (r = −0.67, Fig. S14). Similarly, for TM₁@WS₂–W, I_band-W integrates ε_M-p (p-band center of TM₁) and ε_56-d (d-band center of W₅₆) from ∠Se₃₉W₅₆W₆₆ (Table S6), demonstrating strong anti-correlations with both ΔG_*H (r = −0.85, Fig. S15) and φ_HER-W (r = −0.80, Fig. S16).

Our electronic structure analysis provides a unified mechanistic understanding of the observed descriptor behaviors across different vacancy types. Quantitative analysis reveals that metal doping induces markedly different electronic perturbations in the two systems: d-band center shifts of 0.008–0.640 eV in chalcogen-vacancy systems (Table S7), compared to substantially larger shifts of 0.013 to 1.760 eV in metal-vacancy systems (Table S8). This order-of-magnitude difference in electronic perturbation directly manifests in the geometric structure, explaining why angular descriptors exhibit limited transferability between vacancy types while maintaining strong predictive power within each category. Importantly, these results successfully resolve the “structural distortion → electronic structure → catalytic activity” causal chain in TM₁@MX₂ catalysts through electronic descriptors that surpass conventional single-parameter approaches.

Reliability assessment

To rigorously evaluate the predictive accuracy of our angular descriptors, we perform systematic comparisons between descriptor-estimated catalytic activities and their corresponding DFT-computed values across all descriptor families: φ_HER (Fig. S17–20), φ_OER (Fig. S21–24), and φ_HER-M (Fig. S25 and 26). The predictive performance is quantified by analyzing the slopes between predicted and computed values, where slopes approaching unity indicate superior accuracy. Our analysis demonstrates that all angular descriptors exhibit remarkable agreement with DFT benchmarks, confirming their reliability for catalytic activity prediction.

We further evaluate the practical utility of our framework by comparing descriptor-predicted activity trends with experimentally measured performances from published studies. Experimental overpotentials, compiled as averaged values from previous studies, are benchmarked against theoretical predictions obtained by converting ΔG_*H to overpotential via η = |ΔG_*H|/e. As shown in Fig. 6a, theoretical predictions for Co₁@MoS₂–S,^34,58,59 Pt₁@MoS₂–S,^60–62 V₁@MoS₂–S,⁶³ and Ir₁@MoS₂–S align well with experimental trends. Notably, Ir₁@MoS₂–S is predicted to exhibit good HER activity with an overpotential of only 0.11 V. Correspondingly, as shown in Fig. S27, Ru₁@MoS₂–S,⁶⁴ Co₁@MoS₂–S,⁶⁵ and Ni₁@MoS₂–S⁶⁶ show good agreement with the OER performance trends experimentally measured in the literature. Among them, Pt₁@MoS₂–S demonstrates optimal OER performance at 0.29 V, which aligns with experimental report of Pt SAC as efficient OER materials,⁶⁷ providing independent validation of our descriptor approach. These results validate the predictive capacity of our angular descriptors and establish a rational foundation for dopant selection in catalyst synthesis.

Fig. 6 Reliability assessment of angular descriptors. (a) Comparison between experimental measurements and φ_HER-predicted HER overpotential in chalcogen-vacancy systems. Blue and red lines represent fits to experimental and predicted values, respectively. (b) Correlation distributions for angles centered on different atomic sites (from left to right): dopant atom (M₇₄), first-shell sulfur (S₁₂), second-shell sulfur (S₂₃), first-shell molybdenum (Mo₅₆), and second-shell molybdenum (Mo₅₅). Box plots show statistical variations while overlaid curves indicate normal distributions. (c) Comparison between experimental measurements and φ_HER-Mo-predicted HER overpotential in metal-vacancy systems, with blue and red lines representing experimental and predicted fits, respectively. (d) Correlation distributions for angles centered on different atomic sites in metal-vacancy systems (from left to right): first-shell sulfur (S₄₉), second-shell sulfur (S₆), further-shell sulfur (S₁₉), first-shell molybdenum (Mo₅₆), and second-shell molybdenum (Mo₇₃).

To investigate the spatial dependence of descriptor effectiveness, we systematically analyze angular descriptors centered on five representative atomic sites across coordination shells: the dopant M₇₄, first-shell S₁₂, second-shell S₂₃, first-shell Mo₅₆, and second-shell Mo₅₅. Considering only angles with meaningful correlations (|r| > 0.6, Fig. 6b), we observe that second-shell-centered angles are substantially more abundant than those from dopant or first-shell sites and exhibit higher average correlation coefficients. This spatial analysis definitively demonstrates that structural perturbations beyond the first coordination shell exert more substantial influence on HER activity than local metal-centered geometries.

The predictive power of angular descriptors is further validated in metal-vacancy systems through comparison of φ_HER-Mo-derived overpotentials with experimental values. The strong agreement between predictions and measurements for Pt₁@MoS₂–Mo,^35,68 Ni₁@MoS₂–Mo,^69,70 Rh₁@MoS₂–Mo,⁷¹ and Ru₁@MoS₂–Mo^70,72 confirms descriptor transferability across vacancy types, with V₁@MoS₂–Mo exhibiting solid predicted HER activity (Fig. 6c). Supplementary analysis (Fig. 6d) filtering for angles with overpotential correlation |r| > 0.6 reveals that descriptors centered on outer-shell atoms are not only more numerous but also show stronger correlations, further highlighting the dominance of peripheral environments. While there is an optimal descriptor for HER, the results also indicate that some other angles exhibit good descriptive performance. The relatively reduced number of effective descriptors (|r| > 0.6) in Mo-vacancy systems compared to S-vacancy systems may originate from smaller atomic size mismatch between dopant and host metal atoms, resulting in less pronounced lattice distortions than those generated at chalcogen vacancy sites.

Experimental verification by case studies over Ir₁@MoS₂ and V₁@MoS₂

Guided by computational predictions identifying Ir₁@MoS₂ (S-vacancy) and V₁@MoS₂ (Mo-vacancy) as notable candidates, we synthesized these SACs with controlled atomic dispersion. X-ray diffraction (XRD) patterns (Fig. S28) confirm the phase purity of all samples. Notably, the characteristic diffraction peaks for V₁@MoS₂ and Ir₁@MoS₂ exhibit a slight but consistent shift toward lower angles compared to pristine MoS₂, which is indicative of lattice expansion induced by the doping of single-atom metal species into the MoS₂ matrix.^35,73 Aberration-corrected high-angle annular dark-field scanning transmission electron microscopy (AC HAADF-STEM) images provide direct visual evidence of single-atom dispersion state, which are consistent with Ir atoms occupying S-vacancy sites (Fig. 7a) and V atoms substituting the Mo sites (Fig. 7b). In addition, X-ray photoelectron spectroscopy (XPS) was performed to identify the electronic states. As shown in Fig. S29, the Ir 4f_7/2 binding energy for Ir₁@MoS₂ is 62.70 eV, consistent with the oxidation state of Ir species. These configurations are consistent with our theoretically modelled TM₁@MX₂–X and TM₁@MX₂–M structures (Fig. 1b and c). The corresponding intensity line profiles (Fig. 7a and b insets) quantitatively corroborate the atomic-scale substitution, showing contrast variations consistent with the atomic numbers of the dopants (Ir, V) relative to the host atoms (S, Mo). We note that while AC HAADF-STEM provides direct visualization of single-atom dispersion and allows site assignment based on atomic number contrast, the precise local coordination remains inferred from microscopy contrast and image analysis.

Fig. 7 Structural characterization and electrocatalytic HER performance of Ir₁@MoS₂ and V₁@MoS₂. (a) Aberration-corrected HAADF-STEM image of Ir₁@MoS₂, with the corresponding linear intensity profile demonstrating Ir single-atom substitution at S sites (inset: the atom distribution of Ir, Mo, and S in panel (a)). (b) Aberration-corrected HAADF-STEM image of the V₁@MoS₂, with the corresponding linear intensity profile demonstrating V single-atom substitution at Mo sites (inset: the atom distribution of V, Mo, and S in panel (b)). (c) HER polarization curves for the Ir₁@MoS₂ (0.1 wt% Ir) and commercial 20% Pt/C samples. (d) Comparison of current densities for Ir₁@MoS₂ and 20% Pt/C at different overpotentials. (e) Stability test of Ir₁@MoS₂ before and after 1000 CV cycles. (f) HER polarization curves for V₁@MoS₂ and pristine MoS₂. (g) Comparison of current densities for V₁@MoS₂ and pristine MoS₂ at different overpotentials. (h) Stability test of V₁@MoS₂ before and after 1000 CV cycles.

The HER performance of Ir₁@MoS₂ was evaluated in an acidic electrolyte. Despite an ultralow loading of only 0.1 wt% Ir, this catalyst exhibits good activity, achieving a low overpotential of ∼0.22 V at a current density of 10 mA cm⁻² (Fig. 7c). With increasing content of Ir, the overpotential at 10 mA cm⁻² decreases from 220 mV at 0.05% Ir to 216 mV at 0.1% Ir, then increases to 227 mV at a higher Ir content of 0.3% (Fig. S30). At the same time, the current densities at identical overpotentials present a volcanic trend with increasing Ir content. The optimum Ir content of 0.1% (Ir₁@MoS₂) leads to the highest HER activity. This performance is competitive with commercial Pt catalysts under identical conditions. To account for the minute precious metal content, we compared mass activities. As shown in Fig. 7d, the mass activity of Ir₁@MoS₂ at −0.05 V vs. RHE compares favorably with that of a commercial Pt/C catalyst with >20 wt% Pt loading. The electrochemically active surface area (ECSA), estimated via double-layer capacitance (C_dl) measurements (Fig. S31), increased from 16.4 mF cm⁻² for pristine MoS₂ to 25.6 mF cm⁻² for Ir₁@MoS₂, indicating a greater density of accessible active sites. Furthermore, Ir₁@MoS₂ demonstrates good stability, maintaining its activity through 1000 continuous cyclic voltammetry cycles (Fig. 7e). Moreover, electrochemical impedance spectroscopy (EIS) was employed to gain further insight into reaction kinetics and mechanisms. The charge transfer resistance (R_ct) was related to the electrocatalytic kinetics and its lower value corresponding to a faster reaction rate. Compared with MoS₂, Ir₁@MoS₂ possesses the lowest charge transfer resistance (Fig. S32). These results demonstrate that an ultralow content of Ir is essential to induce a high HER activity. The reaction step diagram shows that in Ir₁@MoS₂–S, the doped Ir₁ metal center site becomes the core active center for HER compared to other non-central surface sites (Fig. S33). Differential charge density and Bader charge analyses reveal that Ir₁ doping induces electron transfer from the surrounding Mo atoms to the Ir₁–H region, reconstructing the local electronic environment of MoS₂ and forming a strongly polarized adsorption site, thereby facilitating *H activation and enhancing HER performance (Fig. S34).

The earth-abundant V₁@MoS₂ catalyst also shows a substantial enhancement in HER activity compared to pristine MoS₂. Linear sweep voltammetry (LSV) in Ar-saturated 0.5 M H₂SO₄ shows that the overpotential required to achieve a current density of 10 mA cm⁻² decreases significantly from ∼0.27 V for pristine MoS₂ to ∼0.22 V for V₁@MoS₂ (Fig. 7f). Across the tested potential range, V₁@MoS₂ consistently delivers current densities that are 2 to 3 times higher than those of the undoped baseline (Fig. 7g). This activity boost is supported by an increased C_dl value (Fig. S31), suggesting a more favorable surface structure or improved conductivity. V₁@MoS₂ also maintains good stability over 1000 CV cycles (Fig. 7h), demonstrating the inherent stability of the single-atom architecture within the MoS₂ lattice.

The exceptional experimental performance of both Ir₁@MoS₂ and V₁@MoS₂ provides strong validation for our computational framework. We compare the HER performance of Ir₁@MoS₂ and V₁@MoS₂ with currently reported excellent catalysts. Our catalysts exhibit good activity under ultra-low loading conditions, of which the intrinsic activity is superior to commercial Pt/C, demonstrating competitive performance under ultralow loading conditions.^{58,62,74–77} These results emphasize that the key outcome is the descriptor-guided discovery, as our angular descriptors successfully led to experimentally active candidates and validated the predictive framework. Together, these case studies complete a rigorous design cycle, from data-driven descriptor discovery and theoretical screening to targeted synthesis and performance validation.

Discussion and conclusions

The rational design of advanced catalysts is fundamentally limited by a translational gap: we lack descriptors that are both predictive of activity and prescriptive for synthesis. While electronic-structure-based parameters have advanced our understanding, they often fail to translate into actionable geometric guidelines for experimentalists.^15–19 Critically, even promising structural descriptors, such as ensemble-averaged distortion metrics,²⁴ typically operate at a macroscopic level. They capture the existence of lattice strain or distortion, long recognized as crucial for catalysis, but lack the atomic-scale resolution to reveal which specific, local geometric motifs are functionally decisive. This gap between observing macroscopic strain and deciphering its atomically precise origins has hindered the development of a true “synthesis-to-function” blueprint.

Our work bridges this gap by introducing angular descriptors as the missing link between macroscopic lattice distortion and atomic-scale catalytic function. We propose that the doping-induced macroscopic strains, often probed by techniques like HAADF-STEM and XRD, are fundamentally manifested and quantifiable as precise changes in key structural angles within the coordination network. Through systematic data mining of single-atom doped transition metal dichalcogenides, we decode this relationship, developing interpretable angular parameters that directly quantify how atomic-scale geometry governs catalytic activity. We demonstrate that activity correlates more strongly with long-range angular parameters, such as the surface-oriented angle ∠S₁S₂₃S₈ (φ_HER) and the internal torsion angle ∠S₈Mo₅₂S₃₆ (φ_OER), than with local coordination environments. These structural descriptors exhibit both remarkable predictive accuracy (|r| > 0.81 for HER, |r| > 0.85 for OER) and good transferability, precisely because they capture the essential geometric signatures of functional strain.

A fundamental insight emerging from this decoding process is the dominant role of peripheral structural environments over first-shell coordination spheres. This finding challenges conventional design paradigms and provides a new geometric principle: catalytic activity is tuned not merely by the immediate neighbors of the active site, but by how the dopant atom reconfigures the entire local lattice. This geometric dominance is electronically rooted in multi-orbital interaction mechanisms, which are quantitatively captured by our developed multidimensional electronic descriptor (I_band). This descriptor demonstrates superior predictive power (|r| = 0.88 for HER; |r| = 0.91 for OER) compared to conventional d-band theory, while providing a mechanistic understanding of how geometric distortions modulate electronic states and adsorption energetics.

The practical utility of our framework is demonstrated through experimental validation. Guided by our angular descriptors, we identified and synthesized Ir₁@MoS₂ (S-vacancy), which achieves a low overpotential of 0.22 V at an ultralow loading of 0.1 wt%. When normalized by precious metal content, the intrinsic activity surpasses that of commercial Pt/C catalysts requiring >20 wt% Pt. Simultaneously, the earth-abundant V₁@MoS₂ (Mo-vacancy) exhibits significantly enhanced HER activity compared to pristine MoS₂. These successes are not isolated discoveries; they are direct validations of the predictive framework that connects synthesizable atomic-scale geometry (specific doping-induced angles) to targeted high performance.

In conclusion, this work establishes a generalizable geometric framework for single-atom catalysis, demonstrating that “peripheral geometric regulation” applies across different reactions and vacancy types. By decoding macroscopic lattice distortion into key angular descriptors, we bridge the long-standing gap between structural effects and atomic-scale catalytic function. Unlike previous electronic-model – confined studies, our angular signatures can serve as a generalizable blueprint and predictors that show good transferability for the discovery of high-performance single-atom electrocatalysts for sustainable hydrogen production. Nevertheless, its applicability to other catalyst classes (e.g., alloy nanoparticles or metal surfaces) and to more complex reaction networks remains to be explored.

Author contributions

B. G.: data curation, methodology, writing original draft, writing – review & editing; Y. C.: data curation, methodology, writing original draft, writing – review & editing; Y. W.: data curation, methodology, writing original draft, writing – review & editing; F. W.: visualization; F. L.: visualization; L. C.: visualization; J. L.: conceptualization, funding acquisition, supervision, project administration, writing – review & editing; X. F.: supervision; S. L.: conceptualization, funding acquisition, supervision, project administration, writing – review& editing.

Conflicts of interest

The authors declare no conflict of interest.

Data availability

The data supporting this article have been included as part of the article and supplementary information (SI). Supplementary information: the detailed information for the theoretical calculations and additional experiments. See DOI: https://doi.org/10.1039/d6sc01740a.

Acknowledgements

The present work was supported by the National Natural Science Foundation of China (22461160253, 22373017, 22502083, 22508379 and 22378379), the National Natural Science Foundation of Fujian Province of China (2024I0004), the “Chuying Program” for the Top Young Talents of Fujian Province, the China Postdoctoral Science Foundation (2024M751372) and the Jiangsu Funding Program for Excellent Postdoctoral Talent (2024ZB761). Part of DFT computations were conducted at the Supercomputing Centre of Fujian.

References

1. L.-M. Cao and C.-T. He, Chin. J. Struct. Chem., 2026, 45, 100814.
2. W. Shi, T. Shen, C. Xing, K. Sun, Q. Yan, W. Niu, X. Yang, J. Li, C. Wei, R. Wang, S. Fu, Y. Yang, L. Xue, J. Chen, S. Cui, X. Hu, K. Xie, X. Xu, S. Duan, Y. Xu and B. Zhang, Science, 2025, 387, 791–796.
3. H. Chen, L. Wang, M. Na and X. Zou, Chem. Sci., 2025, 16, 20662–20676.
4. H. Zeng, Z. Chen, Q. Jiang, Q. Zhong, Y. Ji, Y. Chen, J. Li, C. Liu, R. Zhang, J. Tang, X. Xiong, Z. Zhang, Z. Chen, Y. Dai, C. Li, Y. Chen, D. Zhao, X. Li, T. Zheng, X. Xu and C. Xia, Nat. Commun., 2025, 16, 4314.
5. Q. Deng, Z. Li, R. Huang, P. Li, H. Gomaa, S. Wu, C. An and N. Hu, J. Mater. Chem. A, 2023, 11, 24434–24453.
6. L. Chang, Z. Sun and Y. H. Hu, Electrochem. Energy Rev., 2021, 4, 194–218.
7. Y. Liu, M. Han, Q. Xiong, S. Zhang, C. Zhao, W. Gong, G. Wang, H. Zhang and H. Zhao, Adv. Energy Mater., 2019, 9, 1803935.
8. M. Urso, X. Ju, R. Nittoor-Veedu, H. Lee, D. Zaoralová, M. Otyepka and M. Pumera, ACS Catal., 2025, 15, 11617–11663.
9. S. Yuan, Y. Zheng, Y. Chu, C. Xia, R. Dong, J. Xu, B. Teng, Y. Wu and Y. He, Green Energy Environ., 2025, S246802572500189X.
10. J. Han, H. Wang, Y. Wang, H. Zhang, J. Li, Y. Xia, J. Zhou, Z. Wang, M. Luo, Y. Wang, N. Wang, E. Cortés, Z. Wang, A. Vomiero, Z. Huang, H. Ren, X. Yuan, S. Chen, D. Feng, X. Sun, Y. Liu and H. Liang, Angew. Chem., Int. Ed., 2024, 63, e202405839.
11. X. Li, J. Wang, H. Xue, L. Zhao, J. Lu, H. Zhang, M. Yan, F. Deng and C. Hu, Adv. Funct. Mater., 2025, 35, 2503360.
12. T. Sun, T. Yang, W. Zang, J. Li, X. Sheng, E. Liu, J. Li, X. Hai, H. Lin, C. H. Chuang, C. Su, M. Fan, M. Yang, M. Lin, S. Xi, R. Zou and J. Lu, ACS Nano, 2025, 19, 5447–5459.
13. P. Cao, X. Mu, F. Chen, S. Wang, Y. Liao, H. Liu, Y. Du, Y. Li, Y. Peng, M. Gao, S. Liu, D. Wang and Z. Dai, Chem. Soc. Rev., 2025, 54, 3848–3905.
14. M. Li, J. Li, X. Zheng and Y. Zhou, Rare Met., 2024, 43, 4019–4037.
15. A. A. Latimer, A. R. Kulkarni, H. Aljama, J. H. Montoya, J. S. Yoo, C. Tsai, F. Abild-Pedersen, F. Studt and J. K. Nørskov, Nat. Mater., 2017, 16, 225–229.
16. F. Studt, F. Abild-Pedersen, T. Bligaard, R. Z. Sørensen, C. H. Christensen and J. K. Nørskov, Science, 2008, 320, 1320–1322.
17. J. Noh, S. Back, J. Kim and Y. Jung, Chem. Sci., 2018, 9, 5152–5159.
18. G. Cao, S. Yang, J.-C. Ren and W. Liu, Nat. Commun., 2025, 16, 1251.
19. J. Li, M. Chen, P. Wang, X. Li, Y. Huang, K. Ding, L. Mao, X. Xiang and X. Chen, Chem. Sci., 2025, 16, 22071–22083.
20. X. Sun, R. B. Araujo, E. C. Dos Santos, Y. Sang, H. Liu and X. Yu, Chem. Soc. Rev., 2024, 53, 7392–7425.
21. H. Xu, J. Wang, J. Liu and D. Cheng, Acc. Chem. Res., 2025, 58, 2535–2549.
22. J. Wang, M. Zheng, X. Zhao and W. Fan, ACS Catal., 2022, 12, 5441–5454.
23. M. Li, Y. Cheng, C. Shao and Y. Li, Chin. J. Chem. Phys., 2025, 38, 199–211.
24. R. Chattot, O. L. Bacq, V. Beermann, S. Kühl, J. Herranz, S. Henning, L. Kühn, T. Asset, L. Guétaz, G. Renou, J. Drnec, P. Bordet, A. Pasturel, A. Eychmüller, T. J. Schmidt, P. Strasser, L. Dubau and F. Maillard, Nat. Mater., 2018, 17, 827–833.
25. R. B. Wexler, J. M. P. Martirez and A. M. Rappe, J. Am. Chem. Soc., 2018, 140, 4678–4683.
26. C. Ren, Y. Cui, Q. Li, C. Ling and J. Wang, J. Am. Chem. Soc., 2025, 147, 13610–13617.
27. H. Xu, D. Cheng, D. Cao and X. C. Zeng, Nat. Catal., 2024, 7, 207–218.
28. W. Shu, J. Li, J. X. Liu, C. Zhu, T. Wang, L. Feng, R. Ouyang and W. X. Li, J. Am. Chem. Soc., 2024, 146, 8737–8745.
29. F. Wei, L. Cao, B. Ge, Y. Chen, X. Pan, Y. Chai, R. Jing, X. Hu, X. Wang, J. Lin and S. Lin, Angew. Chem., Int. Ed., 2025, 64, e202416912.
30. B. Ge, F. Wei, Q. Wan, H. Zhang, P. Yuan and S. Lin, Precis. Chem., 2023, 1, 429–436.
31. Y. Chai, F. Wei, L. Cao, X. Wang, S. Lin, J. Lin and T. Zhang, Coord. Chem. Rev., 2025, 536, 216649.
32. Z. Han, R. Gao, T. Wang, S. Tao, Y. Jia, Z. Lao, M. Zhang, J. Zhou, C. Li, Z. Piao, X. Zhang and G. Zhou, Nat. Catal., 2023, 6, 1073–1086.
33. C. Tang, L. Chen, H. Li, L. Li, Y. Jiao, Y. Zheng, H. Xu, K. Davey and S. Z. Qiao, J. Am. Chem. Soc., 2021, 143, 7819–7827.
34. M. Pan, X. Zhang, C. Pan, J. Wang and B. Pan, ACS Appl. Mater. Interfaces, 2023, 15, 19695–19704.
35. J. Xu, G. Shao, X. Tang, F. Lv, H. Xiang, C. Jing, S. Liu, S. Dai, Y. Li, J. Luo and Z. Zhou, Nat. Commun., 2022, 13, 2193.
36. S. Wang, X. Ning, Y. Cao, R. Chen, Z. Lu, J. Hu, J. Xie and A. Hao, Inorg. Chem., 2023, 62, 6428–6438.
37. G. Liu, L. Ding, Y. Meng, A. Ali, G. Zuo, X. Meng, K. Chang, O. L. Li and J. Ye, Carbon Energy, 2024, 6, e521.
38. Z. Duan, H. Xia, H. Li, G. Shao, Y. Ren, X. Tang, Q. Liu, J. Hong, S. Dai, Y. Lin, K. Suenaga, Y. He and S. Liu, Rare Met., 2025, 44, 3130–3140.
39. Z. Jiang, W. Zhou, C. Hu, X. Luo, W. Zeng, X. Gong, Y. Yang, T. Yu, W. Lei and C. Yuan, Adv. Mater., 2023, 35, 2300505.
40. T. W. Clarkson and L. Magos, Crit. Rev. Toxicol., 2006, 36, 609–662.
41. L. Järup, Br. Med. Bull., 2003, 68, 167–182.
42. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
43. C. Guan, X. Yue and Q. Xiang, Adv. Mater., 2025, 37, 2501209.
44. B. Zhou, B. Bai, X. Zhu, J. Guo, Y. Wang, J. Chen, Y. Peng, W. Si, S. Ji and J. Li, J. Colloid Interface Sci., 2024, 653, 1177–1187.
45. X. B. Han, C. Q. Jing, H. Y. Zu and W. Zhang, J. Am. Chem. Soc., 2022, 144, 18595–18606.
46. Y. Liu, L. Xing, Y. Liu, D. Lian, M. Chen, W. Zhang, K. Wu, H. Zhu, Z. Sun, W. Chen, P. Wu, D. Wang and Y. Ji, Appl. Catal., B, 2024, 353, 124088.
47. T. Zhang, Q. Ye, Y. Liu, Q. Liu, Z. Han, D. Wu, Z. Chen, Y. Li and H. J. Fan, Nat. Commun., 2025, 16, 3644.
48. Z. J. Duan, H. Xia, H. Z. Li, G. L. Shao, Y. Z. Ren, X. Tang, Q. N. Liu, J. H. Hong, S. Dai, Y. C. Lin, K. Suenaga, Y. M. He and S. Liu, Rare Met., 2025, 44, 3130–3140.
49. Q. Jiang, H. Xu, K. S. Hui, Y. Wei, L. Liu, Z. Ye, C. Zha, M. Zheng, J. Lu and K. N. Hui, Adv. Mater., 2025, 37, 2415986.
50. Z. Shi, X. Zhang, X. Lin, G. Liu, C. Ling, S. Xi, B. Chen, Y. Ge, C. Tan, Z. Lai, Z. Huang, X. Ruan, L. Zhai, L. Li, Z. Li, X. Wang, G.-H. Nam, J. Liu, Q. He, Z. Guan, J. Wang, C. S. Lee, A. R. J. Kucernak and H. Zhang, Nature, 2023, 621, 300–305.
51. Y. Jia, Y. Zhang, H. Xu, J. Li, M. Gao and X. Yang, ACS Catal., 2024, 14, 4601–4637.
52. A. Gaur, J. Sharma, G. Kaur, S. Mhin and H. Han, Adv. Funct. Mater., 2025, e16674.
53. C. Yin, F. Mo, X. Li, F. Li, W. Xue, Q. Wang, R. Shi, Y. Zhao, A. Zeb, J. Wang, W. Liu, S. Zhan and Q. Zhou, Appl. Catal., B, 2025, 376, 125470.
54. R. Wang, L. Zhang, J. Shan, Y. Yang, J. Lee, T. Chen, J. Mao, Y. Zhao, L. Yang, Z. Hu and T. Ling, Adv. Sci., 2022, 9, 2203917.
55. J. Wang, X. Li, B. Wei, R. Sun, W. Yu, H. Y. Hoh, H. Xu, J. Li, X. Ge, Z. Chen, C. Su and Z. Wang, Adv. Funct. Mater., 2020, 30, 1908708.
56. Y. Wang, D. Wang, J. Gao, X. Hao, Z. Li, J. Zhou and F. Gao, Phys. Chem. Chem. Phys., 2020, 22, 14537–14543.
57. R. Ouyang, S. Curtarolo, E. Ahmetcik, M. Scheffler and L. M. Ghiringhelli, Phys. Rev. Mater., 2018, 2, 083802.
58. T. H. M. Lau, X. Lu, J. Kulhavý, S. Wu, L. Lu, T. S. Wu, R. Kato, J. S. Foord, Y. L. Soo, K. Suenaga and S. C. E. Tsang, Chem. Sci., 2018, 9, 4769–4776.
59. S. Park, J. Park, H. Abroshan, L. Zhang, J. K. Kim, J. Zhang, J. Guo, S. Siahrostami and X. Zheng, ACS Energy Lett., 2018, 3, 2685–2693.
60. W. Yan, Z. Zhao, Z. Xin, J. Hu, J. Xia, L. Zhou, Y. Xu, Y. Zhang, K. Liu, R. Wang and Y. Sun, Adv. Funct. Mater., 2025, 35, 2423262.
61. M. Li, J. Xu, D. Liu, J. Yang, J. Lin, X. Xiao, Z. Wang, X. Liu, L. Jia, Y. Liu, C. Yao, Y. Li, Z. Lian and W. Yang, Appl. Surf. Sci., 2024, 667, 160384.
62. T. H. M. Lau, S. Wu, R. Kato, T. S. Wu, J. Kulhavý, J. Mo, J. Zheng, J. S. Foord, Y. L. Soo, K. Suenaga, M. T. Darby and S. C. E. Tsang, ACS Catal., 2019, 9, 7527–7534.
63. S. Bolar, S. Shit, N. C. Murmu, P. Samanta and T. Kuila, ACS Appl. Mater. Interfaces, 2021, 13, 765–780.
64. V. H. Hoa, D. T. Tran, S. Prabhakaran, D. H. Kim, N. Hameed, H. Wang, N. H. Kim and J. H. Lee, Nano Energy, 2021, 88, 106277.
65. X. Gong, Z. Jiang, W. Zeng, C. Hu, X. Luo, W. Lei and C. Yuan, Nano Lett., 2022, 22, 9411–9417.
66. G. Wang, G. Zhang, X. Ke, X. Chen, X. Chen, Y. Wang, G. Huang, J. Dong, S. Chu and M. Sui, Small, 2022, 18, 2107238.
67. X. Xu, H. Xu and D. Cheng, Nanoscale, 2019, 11, 20228–20237.
68. J. Deng, H. Li, J. Xiao, Y. Tu, D. Deng, H. Yang, H. Tian, J. Li, P. Ren and X. Bao, Energy Environ. Sci., 2015, 8, 1594–1601.
69. J. Ge, Y. Chen, Y. Zhao, Y. Wang, F. Zhang and X. Lei, ACS Appl. Mater. Interfaces, 2022, 14, 26846–26857.
70. J. Ge, D. Zhang, Y. Qin, T. Dou, M. Jiang, F. Zhang and X. Lei, Appl. Catal., B, 2021, 298, 120557.
71. X. Meng, C. Ma, L. Jiang, R. Si, X. Meng, Y. Tu, L. Yu, X. Bao and D. Deng, Angew. Chem., Int. Ed., 2020, 132, 10588–10593.
72. J. Wang, W. Fang, Y. Hu, Y. Zhang, J. Dang, Y. Wu, B. Chen, H. Zhao and Z. Li, Appl. Catal., B, 2021, 298, 120490.
73. J. Zhu, Z. Wang, H. Yu, N. Li, J. Zhang, J. Meng, M. Liao, J. Zhao, X. Lu, L. Du, R. Yang, D. Shi, Y. Jiang and G. Zhang, J. Am. Chem. Soc., 2022, 139, 10216–10219.
74. Y. Qu, B. Chen, Z. Li, X. Duan, L. Wang, Y. Lin, T. Yuan, F. Zhou, Y. Hu, Z. Yang, C. Zhao, J. Wang, C. Zhao, Y. Hu, G. Wu, Q. Zhang, Q. Xu, B. Liu, P. Gao, R. You, W. Huang, L. Zheng, L. Gu, Y. Wu and Y. Li, J. Am. Chem. Soc., 2019, 141, 4505–4509.
75. R. Zhang, Y. Li, X. Zhou, A. Yu, Q. Huang, T. Xu, L. Zhu, P. Peng, S. Song, L. Echegoyen and F.-F. Li, Nat. Commun., 2023, 14, 2460.
76. H. Jiang, W. Yang, M. Xu, E. Wang, Y. Wei, W. Liu, X. Gu, L. Liu, Q. Chen, P. Zhai, X. Zou, P. M. Ajayan, W. Zhou and Y. Gong, Nat. Commun., 2022, 13, 6863.
77. Y. Shi, W.-M. Huang, J. Li, Y. Zhou, Z.-Q. Li, Y.-C. Yin and X.-H. Xia, Nat. Commun., 2020, 11, 4558.

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Footnote

† These authors contributed equally to this work.

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