General workflow for localizing hydrides in metal nanoclusters by combining stochastic surface walking with neural-network potentials

Abstract

Ligand-protected metal hydride nanoclusters are crucial for applications in catalysis, luminescence, and energy technologies. However, accurately locating hydrogen atoms (hydrides) within these complex structures remains a significant challenge, hindering the full exploitation of their properties. Developing a universal and accessible method for hydride localization is essential. Here, we present a general computational workflow that combines global structural search algorithms with machine learning-based neural-network potentials to efficiently locate hydrides. We validate this approach across 93 experimentally reported systems, including coinage-metal, transition-metal, and multimetallic polyoxometalates. Using this framework, here we show the generalized rules governing hydride positioning and their preferred coordination environments. Furthermore, we reveal the atomic-level dynamics of hydride movement, discovering that surface migration is the predominant pathway. Practically, our approach provides a reliable theoretical supplement to resolve uncertainties in experimental hydride quantification, such as those from mass spectrometry. Overall, this study advances the fundamental understanding of hydride behavior in nanoclusters and offers a robust, predictive tool to guide the synthesis and structural characterization of nanomaterials.

Accurately determining the positions of hydrides in metal nanoclusters is challenging. Here, the authors present a machine-learning-based workflow that efficiently identifies hydrides and reveals their positioning rules and migration dynamics at the atomic level.

Introduction

Ligand-protected coinage metal nanoclusters, along with their alloy analogues, provide an ideal platform for atomic-precision engineering, offering insights into structure-property relationships at the nanoscale¹. Among these, metal hydride nanoclusters, a subclass of metal nanoclusters, have attracted increasing research interest due to their demonstrated applications in catalysis^2,3, luminescence^4,5, and hypergolic fuel technologies⁶. With the continuous advances of synthetic methodologies and structural modulation techniques, metal hydride nanoclusters are emerging as versatile functional materials with promising applications across an expanding range of scientific and technological fields. Despite numerous reported successes in synthesizing metal hydride nanoclusters, the scarcity of neutron sources fundamentally restricts the routinely employing neutron diffraction for precise hydride localization, thereby severely limiting their practical applications⁷.

To address this critical bottleneck in hydride localization, Jiang et al. pioneered a deep-learning approach utilizing three-dimensional convolutional neural-networks (3D-CNN)^8,9. Their methodology trained on copper hydride (CuH) clusters with known hydride positions from neutron diffraction data, enabling probabilistic prediction of hydride occupancy in X-ray structures with considerable accuracy (≥94%). While the 3D-CNN method demonstrated the effectiveness for predicting hydride positions in copper clusters using neutron diffraction training data, this approach faces inherent limitations for silver (AgH), gold (AuH), and alloy hydride clusters where neutron diffraction data remain scarce. We recently introduced an alternative strategy that neural network potentials (NN potentials) with the Stochastic Surface Walking (SSW) global optimization algorithm^10–14. This methodology has located hydrides in multiple CuH clusters without requiring neutron diffraction inputs¹⁵. More importantly, the approach demonstrates generalizability, extending its predictive capability to AgH, AuH, and alloy hydride systems, thereby providing a viable solution to this long-standing structural determination challenge.

To broaden coverage to essentially all reported classes of metal hydride nanoclusters, we adopt a structural-partitioning strategy that is generally acknowledged as effective⁷. In this strategy, metal hydride clusters are divided into three functional components: (1) the ligand shell, which undergoes a simplification protocol to preserve local coordination geometry while enhancing NN potentials transferability; (2) the metal-ligand (M-L) interface, designed to integrate a comprehensive library of established coordination motifs; and (3) specialized NN potentials separately trained for each metal core (Cu, Ag, and Au) in pure metal hydride clusters. Given the increased complexity from diverse metal combinations and coordination environments, system-specific NN potentials are required for alloy hydride clusters.

Following this strategy, we demonstrate that ligand simplification can enhance the generality of the potentials with reduced computational cost. Subsequently, utilizing a recently characterized copper hydride cluster as a model, the comprehensive process of hydride localization in pure metal hydride systems is elucidated. Importantly, general trends in hydride localization within coinage-metal hydride clusters are identified, providing valuable guidance for making approximate predictions of hydride positions across various metal hydride clusters. Moreover, this methodology functions as an auxiliary tool for hydride quantification employing mass spectrometry¹⁶, thereby reducing potential measurement biases. Finally, the method is applied to investigate the energy barriers associated with hydride dynamics on metal hydride clusters, contributing to a better understanding of hydride storage and catalysis processes. This methodology has the potential to develop into a more broadly accessible workflow for hydride localization that can be readily adopted by experimental researchers without specialized computational expertise, compared with the recently developed efficient HALOM algorithm by Tang et al.¹⁷.

Results

General workflow for hydride localization

In ligand-protected metal nanoclusters, the structure comprises a metal core, protective ligands, and the M-L interface (Fig. 1)⁷. Metal hydride clusters share this organization, allowing a structural partition strategy to develop NN potentials for precise hydride localization. Starting with the metal core, The SSW method effectively captures all coordination environments in both pure and alloyed clusters, forming a reliable basis for NN-based hydride determination^13,15. Depending on their metallic composition, pure metal hydride clusters can utilize specialized NN potentials parameterized for their respective metallic elements (e.g., Cu, Ag, Au; Fig. 1a), while alloyed hydrides require tailored training to incorporate multiple metals. Subsequently, the diversity of ligands results in significant variations in M-L coordination (Fig. 1b). Analysis of reported structures, including those in CuH, AgH, and AuH clusters identifies six principal M-L interaction types: M-SR, M-NHC, M-C ≡ CR, M-PR₃ (including M-DPM, Ph₂P(CH₂)_nPPh₂), M-halide, and M-OR (Fig. 1c). CuH clusters display the most versatile M-L bonding, whereas AgH and AuH systems remain less diverse due to their higher reduction potentials. As synthetic methods advance, iterative self-learning cycles integrating DFT refinement and NN retraining will enable more complete representation of these chemical environments.

Fig. 1. Structural diversity of metal hydride nanoclusters.

a Metal core. b Metal-ligand interface. c Protecting ligand. The abbreviation DPM in b and c denotes the diphosphine ligand Ph₂P(CH₂)_nPPh₂. M, light blue; C, grey; N, blue; O, red; P, pink; S, yellow; Cl, light green; I, brown; Br, dark red.

Ligand simplification is employed to address the challenge posed by the diversity of ligands in constructing generalized NN potentials for hydride localization. For a specific metal hydride cluster, constructing a tailored NN potential by incorporating only the ligands present in that system is feasible. However, incorporating all possible ligands into the training dataset for a transferable model would incur prohibitive computational costs, requiring thousands of data points per ligand. Even subtle differences in functional groups can render the NN potential function ineffective. As the hydride position is predominantly governed by interactions within the chemical environment formed by the metal core and M-L interface, rather than the ligand shell, the adoption of a fixed heavy-atom framework during bias-potential perturbation is justified. To simplify this, we group ligands into representative categories based on coordination chemistry (Fig. 1c, Supplementary Table 1). As illustrated in Fig. 2a, representative functionalized alkynyl ligands (-C ≡ C-C₆H⁴OMe, -C ≡ C-C₆H⁶F, -C≡CPh)¹⁸ were further simplified to their fundamental -C ≡ CCH₃ fragment. This approach ensures that once the NN potentials captures the simplified coordination environment of a given ligand, hydride positions can be accurately predicted without retraining, thereby greatly enhancing the generality of the potentials. Systematic validation was performed on four hydrogen-rich clusters (PtCu₂₃H₃¹⁹, Au₂₄H₃², Ag₁₀H₈²⁰, Cu₂₄H₆⁵; Fig. 2b), which demonstrated that ligand simplification has negligible impact on hydride localization accuracy. In the case of Au₂₄H₃¹⁶, replacing the benzyl substituent on the N-donor with a methyl group yielded nearly identical results. The full-ligand model required four SSW steps, sampling 1,326 structures and converging within 7 min, while the simplified model required five iterations and 761 structures, converging in just 2 min (Fig. 2c,d). Both approaches produced identical hydrogen positions, with deviations in metal core geometry confined to less than 0.22% (Fig. 2e). The validation results for the remaining four hydrogen-poor clusters (PdAu₁₀H²¹, Au₁₁H²², Ag₈H²³, and Cu₁₁H²⁴; Supplementary Fig. 1) aligned with the representative example, indicating that ligand simplification preserves the accuracy of hydride localization in metal hydride clusters, enabling the effective transfer of trained NN potentials to diverse metal hydride clusters without requiring retraining to accommodate varying functional groups. In addition to enhancing transferability, ligand simplification significantly minimizes the training dataset size, improves computational efficiency, and reduces overall costs. Following these strategies, we systematically collected a set of 35 representative CuH, AgH, AuH, and alloy hydride clusters (Supplementary Table 2, Supplementary Figs. 2, 3). The corresponding computational times were summarized to provide a practical time-scale reference (Supplementary Table 3).

Fig. 2. Validation of the ligand simplification strategy.

a Schematic of simplifying three distinct alkynyl ligands (-C ≡ C-C₆H⁴OMe, -C ≡ C-C₆H⁶F, -C≡CPh) into a methyl-terminated model (-C ≡ CCH₃) with a fixed Cu₁₁H₂ core for hydride localization using NN potential. b Representative metal hydride nanoclusters with simplified and complete ligands: alloy (PtCu₂₃H₃), Au (Au₂₄H₃), Ag (Ag₁₀H₈), and Cu (Cu₂₄H₆). Potential energy surface (PES) exploration for Au₂₄H₃ using c simplified and d complete ligands. e Structural comparison between simplified and complete ligand models for Au₂₄H₃ optimized by DFT. In panels c and d, “Str-#” denotes the structure index of the candidate structures. Cu, orange; Au, gold; Ag, light blue; H, white; P, pink; S, yellow; Cl, green; N, blue; C, grey; O, red. Source data are provided with this paper.

Building upon this strategy, we next demonstrate the hydride localization workflow for monometallic hydride clusters using the recently reported Cu₄₀H₃₈ as a representative case study (Fig. 3)²⁵. Beginning with the SC-XRD-determined crystal structure of the copper cluster framework, an appropriate Cu-containing NN potential was first selected. To ensure its suitability, we verified that all elemental components of the simplified cluster (Cu, P, C, H) were included in the trained NN potential (Cu-H-C-N-O-P-S-Cl)¹⁵ and that the training dataset encompassed all relevant M-L coordination environments, specifically the Cu-P-C motifs of the phosphine ligands. For the Cu₄₀H₃₈ cluster, this NN potential fulfilled both criteria, enabling direct hydride localization and global structural optimization. This systematic search evaluated 220,000 distinct configurations and identified 1,900 local minima before converging to the global minimum structure (Supplementary Fig. 4). The final structure was found to be consistent with previously reported results. This highly efficient computational approach, which requires no prior knowledge of hydride positions, employs an identical workflow for both AuH and AgH systems.

Fig. 3. Workflow for hydride localization in monometallic clusters.

a Schematic of the systematic protocol for locating hydride positions. The specialized machine-learning potential, customized for CuH clusters, is denoted as CuHCNOPSCl.pot. b Representative snapshots of a Cu₄₀H₃₈ cluster during global optimization, illustrating the structural evolution from a random initial state to the identified hydride-position global minimum. Cu, orange; P, pink; C, gray; H, white. Source data are provided with this paper.

To establish the robustness of the monometallic hydride-localization workflow beyond the Cu₄₀H₃₈ case study, we next examined whether the predicted hydride sites are reproducible with respect to both the initial hydride arrangement and the choice of NN potential. For representative Ag-H and Au-H clusters (Ag₈H, Ag₁₀H₈, Au₂₄H₃, Au₁₁H) SSW searches starting from three independently randomized hydride placements all converged to the same final hydride binding sites (Supplementary Figs. 5-8). indicating that the localization results are insensitive to initialization. We further repeated these searches using both general-purpose and system-specific NN potentials and obtained identical hydride distributions in all cases examined (Supplementary Figs. 9-12). showing that the predictions are not artifacts of a particular trained potential. These results demonstrate that the workflow yields reproducible and model-robust hydride localization for monometallic clusters, thereby providing a reliable basis for extending the strategy to more complex alloy systems.

Although the hydride positions in CuH, AgH, and AuH clusters have been elucidated, the accurate localization of hydrides in alloy hydride clusters remains considerably more challenging due to the increased structural complexity introduced by heterogeneous metal compositions and diverse coordination environments. To overcome this limitation, we developed a series of tailored NN potentials capable of explicitly capturing the elemental dependence, coordination characteristics, and M-L interactions intrinsic to alloy systems. For instance, the IrAg₂₄H and OsAg₂₄H₂ alloy clusters were obtained by incorporating the metal hydride superatoms IrH and OsH₂ into the central silver site of Ag₂₅ clusters through a co-reduction process²⁶. The detailed procedures for constructing the alloy NN potentials are provided in Supplementary Fig. 13. Following this method, 14 NN potentials were trained and subsequently employed to predict the structures of 19 alloy hydride clusters covering broad compositional space and interfacial configurations (Supplementary Table 4, Supplementary Fig. 3). The resulting predictions underscore the reliability, robustness, and transferability of the developed NN potentials, thus providing a general and practical solution to the long-standing challenge of hydride localization in alloy hydride clusters.

Beyond ligand-protected metal nanoclusters, we next considered two additional classes of chemically distinct systems: transition-metal hydride clusters and polyoxometalates (POMs). For transition-metal hydride clusters, a total of 28 literature-reported examples were collected and used to further evaluate the applicability of the SSW combined with the NN potential (SSW-NN) global structure optimization in locating hydrides. Among them, 27 clusters were predicted with hydride configurations fully consistent with the literature-reported structures, including those supported by neutron diffraction data (highlighted in bold in Supplementary Fig. 14 and Supplementary Table 5). The only exception was Ru₃H₃, for which the hydride configuration located by SSW-NN converged, after DFT optimization, to a lower-energy structure than that reported in the literature, suggesting a more energetically favorable arrangement of hydrides (Supplementary Fig. 15)²⁷.

Proton localization is likewise challenging in oxygen-rich POMs, where the large number of chemically similar oxygen sites and the presence of extended hydrogen-bonding networks often preclude an unambiguous experimental assignment of hydrogen positions. For example, the literature-reported Te₄Mo₂₄ cage contains eight protons, yet proton transport and water-assisted dynamics were mainly inferred from proton conductivity measurements, ¹H solid-state nuclear magnetic resonance (NMR), PXRD characterization under different humidity conditions, and molecular dynamics simulations, rather than direct localization of all hydrogen atoms²⁸. Using Te₄Mo₂₄ as a representative case, our SSW-NN global structure optimization shows that these eight hydrogen atoms consistently migrate to bridging oxygen sites and converge to a reproducible 3:2:2:1 distribution among the four TeMo₆ subunits, irrespective of their initial placements (Supplementary Fig. 16). Furthermore, alternative distributions such as 4:3:1:0 and 3:3:1:1 were not obtained in any of the independent searches, providing theoretical insight into proton localization in this POM framework. The computational time required for each system is summarized in Supplementary Table 6. All these global structure optimizations were performed using recently developed generalized global machine-learning potentials as implemented in the LASPAI platform²⁹.

General patterns and characteristics of hydride positions

Examination of single-crystal structures and relevant literature indicates that in silver-based alloy hydride clusters, doping the Ag₂₅ cluster with Ir, Rh, Os, or Ru yields IrAg₂₄H, RhAg₂₄H, OsAg₂₄H₂, and RuAg₂₄H₂, respectively^26,30. Despite the introduction of different heterometals, the overall metal framework remains largely unaltered, and the positions of the hydrides show no significant changes (Fig. 4a). The observed differences in hydride content can be attributed to the fact that group 8 metals, through dihydrogen coordination, more readily form a stable d¹⁰ closed-shell configuration, whereas group 9 metals tend to achieve a closed-shell structure via monohydrogen coordination. Similar trends are observed in gold-based systems. In the Au₁₁Cl₂³¹ framework, doping with Pd, Cu, or Ag results in PdAu₁₀H²¹, Cu₂PdAu₈H³², and Ag₂PdAu₈H³², respectively. Owing to the local metal framework being largely preserved and the rigid protection provided by the phosphine ligands, hydrides remain stably localized at the bridging sites between Pd and surface Au atoms, highlighting the stabilizing influence of both the local framework and the ligands on hydride positions (Fig. 4b).

Fig. 4. Summary of hydride coordination trends.

a Alloy hydride clusters derived from Ag₂₅ doped with Ir, Rh, Os, and Ru: IrAg₂₄H, RhAg₂₄H, OsAg₂₄H₂, and RuAg₂₄H₂. b Alloy hydride clusters derived from Au₁₁Cl₂ doped with Pd, Cu, and Ag: PdAu₁₀H, Cu₂PdAu₈H, and Ag₂PdAu₈H. c Distribution of average Cu-H bond lengths and d Cu-Cu distances for different hydride coordination numbers. e Overview of various hydride coordination modes within Cu, Ag, and Au hydride clusters. In the box plots, the center line indicates the median, the bounds of box represent the 25th and 75th percentiles, the whiskers extend to 1.5× the interquartile range, and the individual open circles represent outliers. The statistics were derived from a total of n = 314 local Cu-H environments analyzed across the clusters. H, white; P, pink; S, yellow; Cl, light green; Cu, orange; Ag, light blue; Au, gold; Pd, teal; Ir/Rh, purple; Os/Ru, blue. Source data are provided with this paper.

Further comparison reveals that when Cu, Ag, or Au atoms are introduced as dopants into pure Cu, Ag, or Au clusters, hydrides do not necessarily coordinate directly to these heterometal centers. In contrast, doping with late transition metals, such as Os, Ir, Rh, Ru, Pd, or Pt, promotes the formation of direct metal-hydrie bonds^{19,21,26,30,32–35}. This tendency likely arises from the electronic configuration of the late transition metals, whose accessible d-orbitals allow favorable overlap with the hydride 1 s orbital, thereby facilitating the formation of stable metal-hydride bonds.

Taken together, the consistent structural behavior observed in the icosahedral Ag₂₅ and the bicapped-prism Au₁₁ frameworks indicates that heterometal doping perturbs hydride positions only minimally, provided the overall metal framework remains intact. Although further studies are required to establish the generality of this behavior across broader systems, the present results provide a preliminary structural basis for predicting hydride positions in alloy hydride clusters and offer insights into the effects of alloying on hydride localization.

To obtain deeper insight into the structural determinants of hydride localization, we performed a statistical analysis of the Cu-H configurations in CuH clusters from both our previous and current studies (Fig. 4c,d). The results show that both the average Cu-H bond length and the mean Cu-Cu distance within each local H environment increase proportionally with the hydride coordination number. In the box plots, outliers in three-coordinate (μ₃-H) at longer distances are primarily found in unique local motifs within clusters (Cu₂₄H₆⁵, Cu₄₀H₃₈²⁵, Cu₅₇H₂₀³⁶, Cu₅₈H₂₀³⁷). This trend is consistent with geometric constraints: as the number of atoms or ligands coordinated to a Cu center increases, steric repulsion leads to a moderate elongation of both Cu-H and Cu-Cu distances, helping to maintain structural stability.

Furthermore, a comprehensive summary and analysis of 345 local hydride-metal coordination environments identified in pure coinage-metal hydride clusters were conducted (Fig. 4e), based on both our previous and current studies¹⁵. For the Cu-based clusters, which contain a relatively large number of hydrides, the proportions of different hydride coordination modes within each cluster were further visualized (Supplementary Fig. 17). The results reveal that in copper hydride clusters, μ₃-H and four-coordinate (μ₄-H) hydride species are predominant. In contrast, five-coordinate (μ₅-H) hydride sites are relatively rare, and six-coordinate (μ₆-H) hydrides are even less frequent. AgH clusters are primarily characterized by μ₃-H coordination, whereas AuH clusters preferentially adopt the two-coordinate (μ₂-H) configuration, with occasional occurrences of terminal (μ₁-H) or ligand-bound hydrides. Beyond revealing coordination trends, this summary and analysis also provide a practical reference: when the localization of H atoms remains uncertain in existing investigations, cross-comparison with the aggregated data can facilitate further confirmation of their positions.

Auxiliary method for determining hydride quantity

Mass spectrometry (MS) coupled with NMR is instrumental in determining hydride counts. However, molecular formulas deduced from MS may occasionally exhibit minor deviations from the true composition, particularly when employing low-resolution MS, where discrepancies of one or two hydrides can arise. Moreover, direct hydride quantification in neutral metal hydride clusters via MS poses significant challenges due to ionization difficulties. Unstable clusters may also experience signal loss under MS conditions, complicating data interpretation. While NMR serves as a qualitative tool for hydride detection, its utility is restricted for low-abundance hydride signals, often limiting analysis to binary detection (presence or absence) rather than precise quantification. In such cases, the SSW-NN method emerges as a robust auxiliary approach for validating hydride content.

Taking Au₇H₅³⁸, whose hydride count was accurately determined experimentally, as a representative example (Fig. 5), we demonstrate how the SSW-NN method can assist in confirming the correct number of hydrides. To explore potential discrepancies, structural searches comprising 1000 SSW steps were performed for models containing three, four, five, and six hydrides, respectively. Owing to the high efficiency of the SSW-NN method, all four cases can be explored rapidly. The results revealed that the structure with five hydrides converges to the lowest-energy configuration, which satisfies two critical criteria upon subsequent DFT refinement. First, the optimized structure retains the integrity of the Au core, with Au-Au bond lengths and overall framework essentially unchanged relative to the pre-optimization geometry and the SC-XRD data. In contrast, the three, four and six hydrides configuration undergoes bond cleavage within the Au-Au framework, yielding a configuration inconsistent with the SC-XRD data. Second, symmetry analysis demonstrates that the Au₇H₅ structure preserves the expected D_5h symmetry, whereas the Au₇H₃, Au₇H₄, and Au₇H₆ model breaks this symmetry.

Fig. 5. Stoichiometry verification for the Au₇H₅ cluster.

Structural comparison between global minimum structures generated by SSW-NN global search (LASP code) with fixed heavy-atom frameworks and corresponding structures after DFT refinement (CP2K) for a Au₇H₃, b Au₇H₄, c Au₇H₅, and d an incorrectly assigned Au₇H₅H_error. Red dashed lines and “×” symbols indicate structural deformation or hydride dissociation during DFT optimization in a, b, and d. Color code: Au, gold; P, pink; H, white (H_error in red).

Consistent results are obtained for different oxidation states of Au hydride clusters, including Au₇H₅ and Au₁₁H (Supplementary Tables 7,8), the prototypical silver hydride cluster Ag₈H, and the alloy hydride cluster Ag₂₄OsH₂, regardless of whether CP2K/PBE or Gaussian/PBE0 was used for DFT refinement (Supplementary Figs. 18-21). Cross-functional comparisons further indicate that the relative energy ordering of low-energy candidate structures is preserved across Gaussian (PBE, PBE0, B3LYP), VASP-PBE, and the corresponding NN potentials (Supplementary Figs. 22-27). These observations demonstrate that, provided the underlying DFT functional reasonably describes the PES and the NN potential reproduces it within ≈10 meV per atom, initial model randomization does not compromise the reliable localization of hydrides or the validation of their number.

Exploration of the dynamics of hydride migration

Through the application of the SSW-NN approach to locate hydrides in metal hydride clusters, we observed that near the global minimum configurations, one or two structures with energies equivalent to the global minimum frequently emerge. This observation suggests that, even at the global minimum energy, the positions of hydrides are not strictly fixed but may undergo dynamic exchanges between different sites. To further validate this hypothesis, we systematically explored the transition states (TS) between various global minimum configurations using stochastic double-ended surface walking (DESW)³⁹.

Taking the [AuCu₂₄H₂₂(PPh₃)₁₂]⁺ system as an example, we first treated the global minimum configuration as the initial state (IS) and generated a series of reaction coordinate pairs through SSW global sampling⁴⁰. After screening, final states (FS) consistent with the IS in terms of geometry were selected, and the reaction pathways between the IS and FS were constructed using the DESW method. From these pathways, we randomly identified two representative hydride -exchange modes: (i) hydrides exchange positions via surface migration, and (ii) hydrides exchange positions by shuttling through the cluster interior. For each type of IS/FS pathway, a representative reaction path was selected, and transition states along these paths were calculated using the DESW method (Fig. 6a-c). The results indicate that the energy barrier for hydride exchange via surface migration is ≈1.04 eV, while the barrier for interior shuttling reaches as high as 1.57 eV. In comparison, the surface migration pathway requires significantly less energy, making it thermodynamically more favorable.

Fig. 6. Pathways and energetics of hydride migration.

a Schematic of representative hydride migration modes generated by the reaction sampling of SSW, illustrating surface migration and internal shuttling pathways in [AuCu₂₄H₂₂(PPh₃)₁₂]⁺, the dashed line and solid line represent the surface migration and internal shuttling pathways, respectively. b Energy profiles along the reaction coordinates for the respective migration modes. c Representative snapshots of transition-state structures identified along the migration pathways. d Statistical distribution of energy barriers for surface migration and internal shuttling modes. In the box plots, the center line indicates the median, the bounds of box represent the 25th and 75th percentiles, the whiskers extend to 1.5× the interquartile range, and the individual open circles represent outliers. The statistics were derived from n = 10 independent measurements. Cu, orange; H in white, yellow, and green. Source data are provided with this paper.

To establish a general energy threshold for the two types of hydride exchange, we further additional analyzed 10 surface migration pathways and 10 interior shuttling pathways, examining the distribution of their energy barriers (Fig. 6d). The results show that the energy barriers for surface migration are relatively narrow, with a median value around 1.1 eV, whereas those for interior shuttling are higher (median ≈1.8 eV) and exhibit a broader distribution (≈1.5-2.5 eV), reflecting the more complex intracluster atomic rearrangements involved. Based on these observations, we propose a general energy-threshold criterion: surface migration typically dominates hydride exchange processes, while interior shuttling occurs only under higher-energy conditions. These findings suggest that even in the global minimum configuration, hydrides remain dynamically exchanged, with surface-migration-dominated mechanisms being the more physically plausible pathway

Recently, Liu and co-workers reported an experimentally supported visualization of hydride migration within the enclosed icosahedral core of an alloy hydride cluster, RhH₂Ag₁₉, through combined neutron diffraction, single-crystal X-ray diffraction, and DFT analyses, with the lattice disorder consistently observed by both diffraction techniques providing direct experimental evidence for this dynamic migration process⁴¹. Using the same computational level and reported configurations as in the literature, we employed the DESW method to connect the relevant states and search for the corresponding transition states, thereby reproducing part of the interconversion pathway among the reported hydride configurations. As shown in Supplementary Fig. 28, the calculated transition-state geometries are nearly identical to the reported ones, and the barrier for the 1 → 2 migration is about 4 kcal mol⁻¹. This low barrier is consistent with the variable-temperature NMR results reported in the literature⁴¹, which confirmed the dynamic behavior of hydrides in solution.

These results have two important implications. First, the comparison between RhH₂Ag₁₉ and AuCu₂₄H₂₂ indicates that different metal hydride clusters can exhibit substantially different propensities for hydride migration, which may be closely associated with their structural stability and potential functional properties. Second, for fluxional hydride systems, especially those involving complex or multiple-hydride rearrangements, efficient location of transition states remains a challenge. The present results suggest that DESW can serve as a useful strategy for exploring such migration pathways.

Discussion

In this study, we developed a general workflow combining the SSW method with NN potentials and a structural partition strategy to achieve precise and transferable hydride localization in diverse metal hydride clusters. Our analysis uncovered key patterns in hydride positioning and coordination environments across different clusters, enhancing the understanding of hydride localization. The method also proven effective in supplementing mass spectrometry to resolve uncertainties in hydride quantification. Furthermore, we explored the dynamics of hydride migration, revealing surface migration as the predominant pathway. These findings offer a solution for hydride localization, establishing a foundation for synthesis and diverse applications.

Methods

SSW-NN method

All simulations based on the global neural-network (G-NN) potential^10,11 were performed using the LASP code⁴² (Large-Scale Atomic Simulation with Neural Network Potential, www.lasphub.com) developed by Liu and colleagues. The SSW method^13,14 utilizes small structural perturbations driven by bias potentials to move a structure from one minimum to another, enabling automatic exploration of the multidimensional PES and identification of key structures, as demonstrated in previous work^10,11,43. The G-NN potential adheres to the atom-centered NN potential framework proposed by Behler and Parrinello^44,45, where power-type structure descriptors (PTSD) are used for the input layer¹⁰. The G-NN potential is derived by iteratively fitting DFT global PES data, which is initially sampled using the SSW method. Compared with DFT, SSW combined with the NN potential (SSW-NN) accelerates PES computation by 3-4 orders of magnitude, facilitating efficient PES exploration at comparable accuracy levels to DFT calculations.

The G-NN was constructed using an iterative self-learning scheme based on the SSW-NN dataset¹¹. These models were trained on a diverse range of compositions and configurations, including bulk phases, surface structures, and ligand-protected clusters. Structural data were incrementally incorporated into the training set, selected either randomly, based on distinctive structural descriptors, or when significant energy deviations between G-NN predictions and DFT results were observed during the SSW-NN global optimization trajectories. The neural network architecture consists of four hidden layers with a structure of 80-80-80-6. The training dataset was normalized using min–max scaling. Hyperbolic tangent (tanh) activation functions were employed for all hidden layers, while a linear transformation was applied to the output layer. The limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization algorithm was used to minimize the loss function, ensuring consistency between G-NN predictions and DFT-derived energies, forces, and stresses. Further details on the generation of G-NN potentials and the implementation of the SSW-NN approach can be found in the referenced literature. Information regarding the number of training structures, PTSD descriptor parameters, total number of network parameters, and the root-mean-square errors (RMSE) is provided in Supplementary Table 3. The atomic configurations and potential energy data for the local minima that were collected with progressively decreasing energies during the search, were generated using the LASP software and are provided as Supplementary Data 1.

DESW method

The DESW method³⁹ is a transition state (TSs) search approach designed to construct a pseudo-reaction pathway connecting two local minima, typically representing a reactant and a product. Starting from both endpoint structures, DESW iteratively adds Gaussian bias potentials from each side in an alternating fashion, allowing the two configurations to progressively approach one another and establish a continuous path. The introduction of Gaussian functions effectively smooths the PES, enabling rapid generation of a plausible reaction trajectory. The highest energy point (E_max) along this pseudo-path is taken as the initial guess for the TSs, which is subsequently refined using single-ended TSs search algorithms, such as the constrained Broyden dimer (CBD) method⁴⁶. The DESW method is applicable to both fixed-cell and variable-cell systems, and is broadly suitable for locating TSs in molecular and solid-state reactions.

Hydrogen migration pathway calculation

The global minimum-energy structure was selected as the IS, and a large number of reaction coordinate pairs were generated using the stochastic surface walking based reaction sampling (SSW-RS) method⁴⁷. The generated FS were screened based on their energy difference relative to the IS, and the product with the lowest energy was selected as the FS. A pseudo-reaction pathway connecting the IS and FS and the TSs along the pathway were then constructed using the DESW method.

Extending NN potentials to alloy hydrides

To extend the NN potentials for alloy hydride clusters, PTSD with two-body, three-body, and four-body functional groups were employed to comprehensively capture the diversity of atomic coordination environments. The two-body terms (S¹, S²) encodes radial distance distributions via power functions; the three-body terms (S³, S⁴, S⁵) resolves angular environments by combining bond angles and spherical harmonics; the four-body terms (S⁶) captures torsional degrees of freedom critical for molecular conformations¹⁰. Through a self-training protocol, the dataset is sampled via SSW and subsequently refined using high-precision DFT calculations to obtain critical quantities, including energies, forces, and stresses. By iteratively fitting the potential to this dataset, the NN potential is optimized to yield a robust representation of the PES, thereby ensuring accurate localization of hydrides within the targeted alloy hydride clusters.

The geometry optimization of the full ligand-protected nanoclusters was performed using the CP2K package⁴⁸. To minimize basis set superposition errors, the wave functions were expanded in the DZVP-MOLOPT-SR-GTH basis set^49,50, along with a cutoff energy of 400 Ry for the auxiliary plane-wave basis set. Core electrons were modeled by Geodecker-Teter-Hutter (GTH) pseudopotentials^51–53. Only the Γ-point approximation was employed to sample the Brillouin zone⁵⁴.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Author contributions

Z.W. performed the calculations, analyzed the data, and wrote the manuscript. C.F. conceived the study, analyzed the data, and wrote the manuscript. L.Z., W.Z., and Y.D. discussed the calculation results and commented on the paper. X.S. and S.M. acquired the funding and were involved in the review and editing of the manuscript. All authors have read and approved the final version of the manuscript.

Peer review

Peer review information

Nature Communications thanks Kevin Rossi, Shinjiro Takano, and the other, anonymous, reviewer for their contribution to the peer review of this work. A peer review file is available.

Data availability

The data that support the findings of this study are available from the corresponding authors upon request. Unprocessed raw data are provided as Supplementary Data 1. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-72966-9.