Thermal dynamics and coalescence of Au_144(SR)_60 clusters from a machine-learned potential

Domain: physics
Published: 2025-12-18
Editorial outcome: major_revision

Abstract
Ligand-protected metal nanoclusters have gained significant attention due to their diverse applications in catalysis, bioimaging, and nanomedicine. While their ground-state electronic structure and optical properties have been extensively studied via density functional theory (DFT) methods, theoretical insights into their dynamic behavior, particularly for larger clusters, remain scarce due to the prohibitive numerical cost of long-time scale simulations using forces calculated from DFT. Here we investigate, using molecular dynamics (MD) simulations up to 0.12 μs timescale, thermal dynamics of the well-known Au144(SR)60 at 300-550 K using the recently parametrized atomic cluster expansion (ACE) potential, trained from DFT data for thiolate-protected gold clusters. Our findings reveal that thermal effects induce increased mobility in a layer-by-layer fashion, leading to formation of polymeric gold-thiolate units and rings which may fragment from the cluster at high temperatures. The remaining smaller clusters resemble experimentally observed cluster compositions. Close interaction of two Au144(SR)60 clusters leads to coalescence, resulting in a cluster composition and structure of the inner metal core close to ones identified in previous experiments. This work reveals mechanisms for thermal effects in ligand-protected gold clusters and larger nanoparticles that are instrumental for understanding their catalytic activity and inter-particle reactions at the atomistic level.

Introduction
Monolayer-protected metal clusters (MPCs) are hybrid nanomaterials consisting of a metal core and a protecting shell of organic ligands. MPCs exhibit the electronic, optical, catalytic and biocompatible properties, which are highly dependent on their atomic and electronic structures1–4. Experimental work over the last 20–30 years has yielded numerous precise atomic structures of MPCs with noble metals and various organic ligands, which has made it possible to investigate their size-structure-property relations in detail via computational methods. External factors such as temperature, pressure and ligand/chemical environment can induce dramatic changes in the size, shape and properties of the clusters. In particular, thermal effects can drive critical structural transformations in MPCs, including melting, fragmentation, and coalescence, altering their stability and functional behavior. Understanding these phenomena is fundamental to advancing our knowledge of nanoscale physics and rational design of nanoclusters with tailored functionalities. However, investigations of long-term MPC dynamics at elevated temperatures have been scarce5 due to numerical demands in accurate DFT-based simulations and lack of more efficient, reactive classical force fields.

Machine learning (ML) has become a valuable tool in computational science, offering solutions to complex problems through data-driven models6–12. ML-derived force fields are currently accelerating material simulations across various domains, with applications to MPCs enabling structural optimizations and MD simulations for thiolate-protected gold clusters13–16. In this work, we focus on the thermal dynamics of the Au144(SR)60 clusters, which are one of the most studied MPCs. It was initially characterized in 1990 by Whetten and collaborators17 as an “ubiquitous 29 kDa Au:SR cluster compound” due to the ease of synthesis and its high ambient stability, but its exact chemical composition remained a mystery until Jin’s and Murray’s groups determined the chemical formula from high-resolution mass spectrometry in 200918,19. At the same time, DFT calculations20 by Lopez-Acevedo et al. predicted its atomic structure (Fig. 1) that was experimentally confirmed nine years later by Wu and collaborators via X-ray single-crystal diffraction21.

Unlike smaller MPCs, the stability of which can often be selected by considering electronic “magic numbers” (electron shell closures of superatoms)3,22–24, Au144(SR)60 does not conform to an electronic shell closing with 84 superatom electrons. Instead, its chemical stability is assigned to an icosahedral metal core and an extended metal framework from the protected gold-thiolate units. Other studies have also explored the bonding hierarchy and ligand effects in Au144. Yamazoe et al.25 revealed hierarchical bond stiffness within the cluster, showing that interior Au−Au bonds are stiffer than those at the surface, in line with the icosahedral core-shell organization. This structural integrity was further confirmed when Lei et al.26 demonstrated that replacing thiolate ligands with alkynyl groups still preserved the icosahedral Au114 core, underscoring the structural resilience of Au144 across ligand modifications. The electronic and optical properties of Au144 distinguish it from smaller nanoclusters. Knappenberger et al.27 studied its electronic relaxation dynamics by femtosecond spectroscopy and observed plasmon-like behavior in the metal core, marking the transition from molecular-like to metallic properties, in agreement with the theoretical studies by Malola et al.28. Kaappa et al. have pointed out that the symmetry of the metal core plays a crucial role in the optical response for clusters in this size range, where molecular-to-metallic transition takes place29. The chemical reactivity of Au144 has been studied as well, particularly in ligand exchange and cluster growth. Ghosh et al.30 found that bidentate ligands promote cluster aggregation, suggesting that ligand chemistry dictates the stability and fusion behavior of Au144. Meanwhile, Wijesinghe and Dass31 demonstrated that ligand-mediated coalescence can drive Au144 to grow into Au279, a process likely influenced by surface mobility and ligand-induced restructuring. These properties make Au144(SR)60 an ideal model system for investigating how thermal dynamics influence the cluster stability, surface atom diffusion, ligand mobility, and the fundamental mechanisms governing structural rearrangements, fragmentation, and coalescence.

In this work, we explore these phenomena in detail, providing insights into the dynamic behavior of Au144(SCH3)60 under varying thermal conditions. Here, we employ the 144-driven ACE potential15 to model interatomic interactions within Au144(SR)60 cluster using MD simulations. We highlight the ability of the ACE force field to understand the dynamical and structural behavior at elevated temperatures and longer timescales of up to 0.12 μs. Furthermore, we observe and explain the coalescence process of two Au144(SR)60 clusters, producing a realistic chemical composition and metal core symmetry of a larger cluster, similar to a compound that has been identified from earlier experiments.

Results
Au144(SR)60 clusters exhibit a hierarchical shell-like structure, as illustrated in Fig. 1a–e. The DFT-relaxed structure of Au144(SCH3)60 is shown in Fig. 1a. At the innermost core of the cluster, there is an icosahedral Au12 shell (L1) as shown in Fig. 1b. The second icosahedral layer (L2), composed of 42 gold atoms, is depicted in green and forms a Mackay icosahedral structure with L1, Au54 (Au12 + Au42), as shown in Fig. 1c. The third layer (L3), consists of 60 gold atoms (orange), completing the Au golden cage (Au12 + Au42 + Au60) as shown in Fig. 1d. This robust core is further stabilized by 30 monomeric staple motifs (-SR-Au-SR-) arranged in a rhombicosidodecahedral geometry as illustrated in Fig. 1e, with an inset highlighting the individual monomeric staple motif with sulfur atoms shown in yellow.

The MD simulations using the ACE potential provide a comprehensive analysis of the structural stability, fragmentation dynamics, inter-cluster interactions, and cluster-cluster dynamics of Au144(SR)60. In this work, we studied the cluster across temperatures from 300 K to 450 K, with simulations running up to 120 ns. We collected data from four replicas, each running for 120 ns, for each temperature. The replicas at a given temperature were started from an identical atomic structure but from different initial velocities of atoms, which drives the replica runs to explore different areas of the phase space, improving statistical sampling. The replicas are labeled as I–IV. Cluster-cluster interactions were further studied through simulations at temperatures ranging from 400 K to 550 K.

A number of snapshots from the MD simulations of the single cluster with the ACE potential were selected, and the corresponding DFT energies were calculated. The correlation between the ACE and DFT energies is shown in Fig. 2 where the data is color-coded for temperatures ranging from 300 K to 450 K. The snapshots in this plot were taken from the first replica (I) of the four conducted simulations at each temperature. The analysis reveals a strong linear correlation (R2=0.9362) between ACE-derived and DFT-calculated energies, further confirming the good performance of the ACE potential even at high temperatures.

Structural dynamics of Au144(SCH3)60
Figure 3 present the pair correlation functions g(r) for Au-Au and Au-S interactions at different temperatures, ranging from 300 K to 450 K averaged over all the four replicas at each temperature. The first dominant Au-Au peak appears at ≈2.97 Å, corresponding to the nearest-neighbor Au-Au bond lengths, and remains a distinct feature across all the temperatures. The first Au-S peak is positioned at a shorter distance of ≈2.3 Å. The green vertical dashed lines at the first two peaks of each Au-Au and Au-S bond distribution mark the position of the first major g(r) peak, serving as a visual guide to track shifts in the bond length distribution with increasing temperature. At lower temperatures (300–350 K), the g(r) profiles exhibit well-defined, sharp peaks, indicating structurally ordered and more stable clusters. As the temperature reaches 375–400 K, the Au–Au g(r) peaks broaden, and their maxima shift slightly towards shorter bond distances. This trend becomes even more pronounced at 450 K, with a significant shift in peak maxima observed.

Figure 4a–f shows a layer-by-layer average root mean square deviation (RMSD) for gold atoms as a function of the simulation time at six temperatures from 300 K to 450 K. At each temperature, the data is combined from all four replicas (I–IV). Supplementary Figs. 1 and 2 present the RMSD data of the four individual replicas for each of the six temperatures. Moreover, Supplementary Figs. 3–26 provide detailed insights into each MD simulation replica, showing time evolution of RMSDs of each replica run and selected snapshots with timestamps visualizing the atom dynamics.

At 300 K (Fig. 4a), the average RMSD values are relatively low, reflecting primarily the vibrational motion of gold atoms. The average RMSD values for the layers are ≈0.12–0.24 Å for the 1st layer, 0.18–0.28 Å for the 2nd layer, 0.23–0.41 Å for the 3rd layer, and 0.26-0.74 Å for the 4th layer. Interestingly, the data shows systematic stepwise increase of RMSD for the 4th layer gold atoms, which originates from two of the four replica runs (Supplementary Figs. 3 and 4). Analysis of atomic configurations shows that the stepwise increase is induced by formation of elongated ligand motifs on the surface that however remain bound to the cluster.

As the temperature increases, the average RMSD rises gradually at 325 K (Fig. 4b) and 350 K (Fig. 4c) for all layers. This trend is consistently observed across the four replicas, as shown in Supplementary Fig. 1e–h and Supplementary Fig. 1i–l for 325 K (I–IV) and 350 K (I–IV), respectively. The increase in RMSD at 325 K and 350 K compared to 300 K is attributed to structural changes, as illustrated by the selected snapshots in Supplementary Figs. 7b–10b for 325 K (I–IV) and Supplementary Figs. 11b–14b for 350 K (I–IV). These changes are due to the formation of longer gold-thiolate chains (2–3 units) that are still bound to the surface as visualized in the corresponding snapshots. Interestingly, we also observe formation of a gold-thiolate ring in three replica runs at 350 K (Supplementary Figs. 12–14). These processes at the cluster surface induce increased mobility of gold atoms also in deeper layers, leading to exchange of gold atoms between the layers.

As the temperature increases to 375 K and higher, the average RMSD, as shown in Fig. 4d–f for 375 K, 400 K, and 450 K, respectively, exhibits a significant increase and qualitative change as compared to the average RMSD calculated for 300 K, 325 K, and 350 K (Fig. 4a–c, respectively). The RMSD values for the 4th gold layer fluctuate in the range of several tens of Å, and values for the deeper layers show a steady increase during the simulation time, qualitatively indicating diffusive or “liquid-like" behavior. The high values and fluctuations for the 4th layer (in the range of 16–56 Å) are due to fragmentation of gold-thiolate chains or rings from the cluster, as visualized in Supplementary Figs. 15–26. In some cases, the fragments can return to bind back to the cluster surface, which decreases the RMSD momentarily. The surface of the cluster in these runs remains quite open and less stabilized than the thiolates, which give rise to a significant increase in mobility of atoms in the deeper layers. In some cases, we have been able to follow the diffusive motion of gold atoms from the 1st layer to the surface during one replica run. We may note that at the highest temperature of 450 K, gold atoms, even (originally) in the first layer, reach the RMSD value exceeding 10 Å, which means that they have traveled (diffused) at least one cluster diameter.

Table 1 provides an overview of the fragmentation events observed at 375 K (I–IV), 400 K (II–IV), and 450 K (II–IV), highlighting the detached fragments and the molecular compositions of the remaining clusters. We observe two types of fragments, ring-like moieties in the range of Au4(SCH3)4 to Au7(SCH3)7 and open chains in the range of Au2(SCH3)3 to Au4(SCH3)5. In this data, we observe up to three fragments detaching from the cluster. We did not investigate the structure of the remaining clusters Au132−142 (SR)47−57 in more detail, but we note here that experiments have identified a few stable compositions in this size range, such as Au137(SR)56 and Au133(SR)52.

One of the main applications of metal nanoclusters is in catalytic reactions, where their efficiency depends on the availability of surface-active sites for molecular adsorption and activation. A common method to activate a ligand-stabilized gold nanocluster to function as a catalyst is by thermal treatment to 150–250 °C35,36, which coincides with the temperature range in our simulations. Therefore, it is interesting to analyze from our MD data how the temperature-driven transformations of the surface Au-ligand structure expose bare gold sites. We performed this analysis using the solvent-accessible surface area (SASA) method, which maps the contact area between a spherical probe of a chosen radius and the gold atoms in the cluster. Figure 5 presents the SASA results calculated for various probe sphere sizes (1.0, 1.2, and 1.4 Å) across temperatures ranging from 300 to 450 K. These results, derived from selected snapshots taken of the first replica run at different temperatures, show increased solvent exposure at higher temperatures due to enhanced atomic fluctuations and conformational changes.

Figure 6 illustrates the solvent-accessible regions at the cluster surface for various snapshots. The snapshots were chosen to be representative of four distinct structural deformations: (i) partial opening of the ligand shell on the surface selected from 300 K (I) and shown in Fig. 6a, (ii) elongated polymer-like gold-thiolate units selected from 450 K (I) and shown in Fig. 6b, (iii) formation of a large ring-like structure on the cluster surface selected from 400 K (I) and shown in Fig. 6c, and (iv) detachment of a fragment from the cluster selected from 375 K (I) and shown in Fig. 6d. For clarity, each structure is displayed from two different orientations.

Cluster–cluster interaction and coalescence
Understanding the atomistic mechanisms of coalescence is essential for guiding the rational design of nanomaterials for applications in catalysis, optoelectronics, and nanoscale device engineering, to name a few. In this section, we discuss a series of MD simulations at 400–550 K where two Au144(SCH3)60 clusters were initially positioned either 3.08 or 1.6 Å (measured as the shortest distance between the outermost H atoms of each cluster) apart from each other, aiming to investigate their propensity to coalesce.

Figure 7a–k presents a series of snapshots depicting the structural evolution and interactions between the two Au144(SCH3)60 nanoclusters during MD simulations at 400 K. The clusters are color-coded for clarity, with Au atoms in purple and S atoms in blue for one cluster, and Au atoms in red and S atoms in yellow for the other. The selected snapshots are color-coded in red and purple for clear distinction of more detailed local atomistic-level processes. Figure 7a present the initial configuration at 0 ns, where the two clusters are positioned 3.08 Å apart. As the simulation progresses, Fig. 7b illustrates the early thermal fluctuations, where some ligands begin to elongate slightly, but the clusters remain structurally relatively intact. Moving to Fig. 7c at ≈12 ns, ligand dynamics become more pronounced as several thiolate ligands extend outward, forming ring-like structures that remain attached to the cluster surface. By Fig. 7d at 17.40 ns, ligand-mediated bridging between the clusters initiates. Simultaneously, a second structural fragment, an Au4(SCH3)4, detaches from the red-colored cluster. From Fig. 7d at 17.40 ns onward, the number and composition of the fragmented structures remain consistent, with no change in their atom counts across subsequent frames. These fragments exhibit dynamic movement around the clusters, suggesting that the system reaches an approximate dynamic equilibrium where the fragments continuously move without undergoing further structural decomposition or aggregation. Between Fig. 7e at 47.68 ns and Fig. 7f at 50.73 ns, the ligand bridge extended. By the end of the simulation at ≈88 ns, the clusters maintain their partially connected state without undergoing full coalescence. The formation of an inter-cluster bridge at 400 K without complete coalescence can be attributed to the behavior of nanoparticles as governed by their size and atomic coordination37–40. The relatively large size of Au144 results in a significant fraction of interior atoms with high coordination numbers (CN), reducing their reactivity compared to surface atoms. In contrast, surface atoms with lower CN values exhibit higher mobility, facilitating ligand-shell restructuring and the formation of molecular bridges under elevated thermal conditions. However, at 400 K, the energy barrier for complete coalescence is not fully overcome. Instead, the system undergoes partial restructuring, where surface atoms migrate, form transient bridges, or contribute to ligand-shell fragmentation.

The lack of full coalescence at 400 K contrasts with the behavior of smaller clusters, where a higher surface-to-volume ratio results in a greater proportion of low-CN atoms, enhancing their reactivity. Our previous study of coalescence of smaller Au25(SR)18 clusters15 demonstrated that, at similar temperatures, these systems undergo more rapid fusion, as the energetic barriers for atomic diffusion and coalescence are likely lower. Further MD simulations were performed at 500 K (Supplementary Fig. 27) and 550 K (Supplementary Fig. 28), with two Au144(SCH3)60 clusters initially positioned 3.08 Å apart to assess the critical temperature necessary for coalescence. However, no complete merging was observed under these conditions. Since no coalescence was observed during these simulations, we reduced the initial cluster–cluster distance to 1.6 Å and ran simulations at 500 K and 550 K. Although reducing the cluster separation to 500 K, as shown in Supplementary Fig. 29 did not lead to coalescence, fragmentation was observed. However, when the temperature was increased to 550 K with the reduced distance, coalescence was achieved, as illustrated in Fig. 8a–i.

Supplementary Table 1 provides a summary of the fragmentation events observed during the simulations of two Au144(SR)60 clusters at different temperatures. At 400 K, with a cluster separation of 3.08 Å and a simulation duration of 87.60 ns, limited fragmentation was observed, resulting in the detachment of only two distinct fragments: one Au7(SR)7 and one Au4(SR)4 complex. In contrast, the simulation at 550 K, with a cluster separation of 1.6 Å over 109.33 ns, exhibited significantly more fragmentation. This simulation produced a diverse array of fragments, including single fragments of Au8(SR)8, Au6(SR)6, Au3(SR)4, and Au(SR)2. Furthermore, we observed recurring detachments of certain fragments, Au5(SR)5 separated from the main cluster, which occurred three times on separate occasions, while Au4(SR)4 fragments appeared four times throughout the simulation.

Figure 8a–i displays a series of snapshots capturing the coalescence mechanism during the simulation. Figure 8a shows the two clusters at the beginning of the simulation, positioned 1.6 Å apart. As the simulation progresses, Fig. 8b at 2.39 ns shows atomic fluctuations, with the outer shells beginning to deform, while the core Au atoms remain largely intact. Furthermore, fragmentation occurs, resulting in the formation of Au4(SR)4 and Au3(SR)4. Incorporation of core Au atoms in the fragment is observed in the Au3(SR)4 structure. Throughout the rest of the simulation, fragments that are far from the remaining clusters are not depicted in the snapshots; however, a complete list of all fragments is provided in Supplementary Table 1. By Fig. 8c at 14.20 ns, the clusters show increased interaction, with atoms from the outer shells beginning to form a bridge. This bridge not only includes Au atoms from the ligand staples but also incorporates core Au atoms (as indicated by the difference in their sphere sizes), which have migrated to this region and contribute to the bridge formation. In Fig. 8d at 21.52 ns, more fragmentation is observed, along with the first instance of core Au atoms migration from the cluster in purple to the second cluster. This migration is pointed out in Fig. 8d using blue dashed lines. Figure 8e–h illustrate the progressive stages of cluster merging. In Fig. 8e at 24.13 ns, the two clusters move closer together, followed by further integration in Fig. 8f at 24.92 ns. An atom string, consisting of both purple and red Au atoms along with yellow and green S atoms from each cluster, becomes visible. By the final snapshot in Fig. 8i at 109.33 ns, a short atom string, incorporating atoms from both clusters, can still be observed. The merged cluster in Fig. 8i adopts an nearly spherical shape with a stoichiometry of Au239(SR)69, consisting of 197 core Au atoms and 42 Au atoms within the Au42(SR)69 found at the various staple motifs. The optimized Au239(SR)69 cluster displays an approximately ellipsoidal gold core with a more compact gold core and Au-S interface region. Our neighbor analysis, shown in Supplementary Fig. 30 confirmed the absence of any sulfur atoms (hence thiols) trapped inside the gold core during the coalescence process.

It is very interesting to note that the molecular composition of Au239(SR)69 falls closely within the estimated composition of Au210−230 (p-mercaptobenzoic acid (pMBA))70−80, identified in a 2019 study41 reporting a synthesis of a plasmonic, water-soluble gold cluster. Although our merged cluster is a “product” of a completely different chemical process as compared to room-temperature, pH-controlled synthesis of a water-soluble gold cluster, the similar stoichiometry strongly suggests that the ACE potential is able to maintain a very realistic ligand density at the cluster surface after coalescence, by ejecting surplus material as gold-thiolate fragments out of the system.

To study further the structure of the Au239(SR)69 cluster, we started from the last configuration from the 550 K MD run, and stepwise cooled it down to 300 K with 50 K steps (13.33 ns for each step), and finally optimized the cluster by using the ACE potential. The analysis of the optimized structure is shown in Fig. 9. We find an inner metal core (50 gold atoms) to have a twinned FCC structure, with the surface region having multiple gold-thiolate units Au n(SR)n+1 (n=1…6) and one additional Au9(SR)9 moiety (Fig. 9b, c). We modeled the powder X-ray diffraction (PXRD) pattern of the structure (see “Methods”) and compared it to the measured PXRD data for the cluster Au210−230 (pMBA)70−80, synthesized in ref. 41. As Fig. 9a shows, the computed PXRD pattern agrees quite well with the measured data for Au210−230 (pMBA)70−80, which was also determined to have the twinned FCC metal core as the most likely structure41. Thus, the ACE potential is able not only to produce a merged larger cluster from two Au144(SR)60 clusters with a realistic chemical composition but also with a realistic metal core symmetry.

Methods
ACE model
The ACE model15 was trained on over 30,000 DFT calculations, including various packings of gold core (icosahedral, decahedral, and face-centered cubic (FCC) structures) and sizes up to Au144. The dataset included MD trajectories and active learning sampled clusters, with 31,942 structures used for training and 11,702 for validation. DFT calculations were performed, ensuring consistency with prior studies. The trained ACE potential accurately predicted energies and atomic forces, achieving a root mean square error of 11.0 meV per atom for the training set and 2.27 meV per atom for the testing set. MD simulations up to 0.12 μs revealed key cluster transformations, including chiral isomerization in Au25(SR)18, where chiral isomers Au25C1 and Au25C2 were identified, leading to a high-energy intermediate Au25D with an open facet. The study also captured intercluster reactions and coalescence mechanisms, demonstrating that two Au25(SR)18 clusters at 500 K fused through ligand exchange and rearrangement, forming transient intermediates of varying sizes Au50−n(SR)36−n. The simulations revealed the formation of (Au–S)n rings (n=4, 5, 6, 8, 12, 13), which facilitated size equilibration and metal exchange. Additionally, high-temperature simulations at 450 K and 500 K showed ligand-shell rearrangements, leading to the fusion of Au25(SR)18 into larger clusters such as Au38(SR)24, aligning with experimental observations.

MD simulations
The MD simulations were conducted using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)43, an open-source MD code that enables modeling of atomic and coarse-grained systems using various forcefields. For this study, we employed the Au-S-C-H ACE potential15, a machine-learned interatomic potential. Recently, it was parametrized for thiolate-protected gold clusters using an extensive DFT database for energies and atomic forces over a range of cluster sizes and compositions from Au18(SR)14 to Au144(SR)60.

All MD simulations were performed using the performant atomic cluster expansion (PACE) implementation within LAMMPS (ML-PACE)44, using the NVT ensemble with a Langevin thermostat45 for temperature control. The MD timestep was 1 fs. The Langevin thermostat mimics the experimental thermalization process by introducing a stochastic force and a frictional force. The stochastic force emulates realistic thermal fluctuations, and the frictional force provides a feedback to keep the average kinetic energy of the system at a value that corresponds to the defined temperature via the classical equipartition theorem. The damping coefficient of the thermostat was 1.0 ps−1. For all cases, whether simulating a single cluster or studying the coalescence of two clusters, thermalization was achieved within 9–20 ps from the start of the simulation run. To minimize the energy of the initial structure, conjugate gradient minimization was performed with strict energy and force tolerances of 10−4 and 10−6, respectively.

RMSD analysis
The RMSD for gold atoms in layers was calculated from Eq. (1) at each timestep where the structure data was saved, with reference to the initial time t0. Atoms were assigned a given layer according to the initial configuration of each replica run.
RMSD(t) = sqrt(1/NAu * sum_{i=1}^{NAu} (ri(t) - ri(t0))^2) (1)

XRD analysis
Theoretical XRD patterns were calculated as described in ref. 20. Modeling of powder XRD curves includes the thermal damping by a Debye–Waller factor exp(−Bs2/2) related to thermal vibrations where s=2sinθ/λ, with λ = 1.54 Å and B = 0.04 nm2. In addition, atomic numbers were used to describe the form factors for different atoms. Experimental powder XRD curve of the protected cluster with estimated composition of Au210−230 (SR)70−80 was taken from ref. 41.

Technical details of MD performance
The computational performance was assessed using a single NVIDIA Volta V100 GPU provided by the Finnish national supercomputing center CSC, achieving an average simulation speed of 83 ns per day for a single Au144(SCH3)60 cluster and 51 ns per day for two clusters in the same computational unit cell. Our simulations employed LAMMPS with the Type Label Framework46, using the GPU-accelerated KOKKOS implementation of PACE to optimize computational efficiency. All structural and simulation results were visualized using VMD (1.9.4a57)47 software.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript \"Thermal Dynamics and Coalescence of Au144(SR)60 clusters from a Machine-Learned Potential\" is a\npurely theoretical molecular dynamics study of such clusters. First, the authors investigate the structural variability of the\nclusters as a function of temperature, and then explore how this variability may affect the potential coalescence of two\nclusters. The paper is mostly well written and the simulations are performed in a mostly appropriate fashion. I would\nrecommend some methodological clarifications and further work, but I don't think these are very important. However, my\nmain concern with respect to publication at Nature Communications is the level of novelty, or rather the lack thereof. I found\nthe paper a technical paper that does not present any new physics or even unexpected result that would give its readers\nnew intuition on the topic (except some incremental knowledge). As such, I cannot recommend it for publication in\nNature Communications.\nHoping that my review will help the authors find a right outlet for their good work, I would also like to add a few\nquestions/comments, which will hopefully help improve the quality of the paper:\n1. The abstract does not communicate any particular importance of the work. It should be clearer why this is a high impact\npaper.\n2. I believe the Methodology section could be more detailed. For example, what is the statistical ensemble used and how\ndoes it reflect the experimental system that is simulated? How were the particles thermalised? In the case of coalescence,\nwere they equilibrated in advance, and, if yes, for how long? Why were the two particular distances between the particles\nchosen? Since there was a thermostat used, why would the authors expect that a closer distance between clusters affect\ntheir degree of coalescence? In my opinion, the authors seem to focus too much on the implementation of the ML potential\n(which is fine, since this constitutes a methodological advance or novelty), but not enough on the basics of MD.\n3. Since velocities are appointed randomly to the atoms in an MD simulation, equilibration is important. Also, statistics. Are\nthe RMSDs of various temperatures reproducible, or are they a result of randomness? Without any statistics we cannot be\nsure whether the results are physically meaningful. For example, it is not easy to believe that the different features between\nthe 325 and 330 K simulations are down to the 5 K difference. We do have 3 cases for 450 K; why not for the other\ntemperatures? I do not think that 3 cases are enough, by the way, especially considering that they gave so different results. The\nsimulation setup is basic and the authors could easily run a great number of simulations without too much effort.\n4. I am not 100% convinced with the analysis of the RMSD results. RMSD take averages for all atoms of each layer;\ntherefore, if a protrusion is formed (or, especially when a part of the cluster is detached) that totally upsets the averaged\nRMSD values. I do not see the \"migration between layers\" the authors discuss about. For example, in lines 190-93: at 350 K,\nL2 crosses 0.8 A and L3 approaches 1.5 A, but I don't see any line mixing in Fig. 4. In my opinion, all features in Fig. 4 are\nrelated to protrusions, not atomic rearrangement.\nI hope that the authors will take the time to work on these issues for the benefit of their already good work so far, and that my\nsuggestions will help them publish a very good paper in another journal.", "reviewer_2": "Metal clusters (NCs) have attracted extensive interest in recent years. The authors selected one of the most studied, near\ncritical sized Au nanoclusters- Au144(SR)60 for thermal dynamics investigation at 300 K - 550 K by applying a recently\ndeveloped machine-learned atomistic potential (ACE). They found that the temperature-induced disordering of the cluster\ntakes place via layer-by-layer mechanism and the outermost gold-thiolate layer, consisting of 30 RS-Au-SR units,\nspontaneously forms open (polymer-like) or closed (ring-like) gold-thiolate moieties. These fragments may detach and again\njoin the cluster in a dynamic fashion. They also investigated interactions and a reaction between two Au144(SR)60 clusters:\nat elevated temperatures, clusters can exchange gold atoms and thiolates by first establishing a cluster-cluster bridge from a\nlong gold-thiolate polymer; when at close enough contact, this interaction can lead to a complete coalescence of the cluster\nto form a new roughly spherical cluster. Overall, these findings and results are very interesting, novel and inspiring, which\nare mirrored by some previous experiments, therefore I can recommend the publication of this nice work after addressing the\nfollowing issues:\n1. Why are the durations of 120 ns, 87.27 ns, and 89.85 ns considered in this work? For realistic experiments, they last hours\nor even days. Does it mean that the simulation can not really reflect the realistic situation?\n2. What is the MD simulation limitations(disadvantages) for metal nanocluster investigations?\n3. Can the authors give a precision and efficiency (time-consuming) comparisons between AIMD method and ML-potential in\nterm of Au144(SR)60? Which atomic characteristic does ACE model denote? Why did the authors use this model? How\ndoes it accelerate the calculation speed.?\n4. The RMSD results (Fig.4) can be further clarified or interpreted. For example, it is unknown what is the electronic energy\nchange of each sharp change in Figure 4? Can the energy change of each peak indicate the feasibility of the according\nreaction? etc.\n5. One of the authors'opinions is that the restructuring of gold nanoclusters starts from the change of the outmost gold-\nthiolate layer, however, a structure transformation from Au127 to Au126 was very recently reported, in which the innermost\ngold of Au127 was removed with the left structures including the outmost layer remained after the transformation. Basing on\nthe fact, my question is whether there is also the possibility that the restructuring of gold nanoclusters starts from the\ninnermost gold atom?", "reviewer_3": "In this work, the authors utilize the atomic cluster expansion (ACE) machine-learning potential to investigate the thermal\ndynamics and coalescence of Au144(SR)60 clusters. While these clusters have been extensively studied within the cluster\ncommunity, research focusing on their thermal dynamics and coalescence is relatively scarce. This study employs machine\nlearning to extend the time scales that have not previously been explored for Au144, unveiling several intriguing findings\nthat pave the way for a deeper understanding of cluster-cluster interactions and coalescence at the atomistic level. I\nrecommend that the manuscript be published in Nature Communications after minor revisions.\n1. The simulations utilize varying lengths of molecular dynamics (MD). It would be helpful to explain how the decision was\nmade regarding when to terminate the simulations.\n2. The authors conducted three independent simulations at 450 K. What was the original motivation for this approach, and\nwhy do the simulation times differ?\n3. The caption for Figure 6 needs to be corrected."}, "author_response": "General remarks from the authors: We thank all Referees for their careful reading and\ninsightful comments and questions. Referees 2 and 3 recommend publication in Nature\nCommunication after a minor revision while Referee 1 is critical about the novelty and impact of\nthe work, while assessing positively the overall quality. Below, we give a point-by-point\nresponse to all comments by the Referees, but here we first summarize the main revisions:\n1. The abstract has been rewritten to address the novelty, importance and impact of this\nwork\n2. Figure 3 has been redrawn to make it visually more clear\n3. Figure 4 has been redrawn based on the new improved statistical sampling of MD\ntrajectories at all temperatures\n4. Discussion of RMSD data (Figure 4) and thermal behaviour has been extensively revised.\n5. Table 1 has been revised to include data for fragmentation patterns from the improved\nstatistics\n6. Methods section has been extended\nFor the convenience of the Referees, we have provided an auxiliary pdf manuscript file where all\nthe major changes in the text are highlighted by blue font (file MS-Au144-marked.pdf).\nReviewer #1 (Remarks to the Author):\nThe manuscript \"Thermal Dynamics and Coalescence of Au144(SR)60 clusters from a Machine-\nLearned Potential\" is a purely theoretical molecular dynamics study of such clusters. First, the\nauthors investigate the structural variability of the clusters as a function of temperature, and then\nexplore how this variability may affect the potential coalescence of two clusters. The paper is\nmostly well written and the simulations are performed in a mostly appropriate fashion. I would\nrecommend some methodological clarifications and further work, but I don't think these are very\nimportant. However, my main concern with respect to publication at Nature Communications is\nthe level of novelty, or rather the lack thereof. I found the paper a technical paper that does not\npresent any new physics or even unexpected result that would give its readers new intuition on\nthe topic (except some incremental knowledge). As such, I cannot recommend it for publication\nin Nature Communications.\nAuthor response: We thank the Referee for their positive overall assessment of the work.\nHowever, we cordially disagree on the lack of novelty and impact, and have worked to make an\nextensive revision to highlight those point as well as to address the technical questions raised by\nall the referees. We note that Referees 2 and 3 regard the work as important, novel, and suitable\nto be published in Nature Communications.\nBriefly, the novelty and impact of this work is based on the following facts:\n- Extension of the simulation timescale to 0.1 microsecond range is unprecedented for a\ncluster of the size Au144. This timescale is 4-5 orders of magnitude longer than what\ncould be achieved by MD simulations using DFT derived forces. Still, our ACE force\nfield reproduces the energetics of the cluster well as compared to DFT (our fig. 2).\n- Due to this extended timescale we are able to observe several novel atomistic\nmechanisms at temperatures 300 – 450 K for a single cluster\n- For the first time in our field, we are able to simulate cluster-cluster reaction, illustrating\nmechanisms by which two clusters can interact exchanging atoms, and how at higher\ntemperature one sees complete coalescence of the clusters into one, larger cluster\n- This “reaction product” has a composition very close to a gold-thiolate cluster that has\nbeen earlier reported and analysed from experiments. Remarkably, when we optimize the\nlarger cluster, we observe that the atomic structure of the metal core is very similar to the\nexperimentally reported one, based on comparison of calculated and measure PXRD data.\nThis shows that our computational modeling of the cluster-cluster reaction is likely to be\nvery realistic.\n- Our work will open doors to further learn about atomistic mechanisms at higher\ntemperatures that are important to understand clusters as catalysts, cluster-cluster\nreactions, etc.\nHoping that my review will help the authors find a right outlet for their good work, I would also\nlike to add a few questions/comments, which will hopefully help improve the quality of the\npaper:\n1. The abstract does not communicate any particular importance of the work. It should be clearer\nwhy this is a high impact paper.\nAuthor response: We have now rewritten the abstract to highlight the importance, novelty and\nimpact of this work (as mentioned above).\n2. I believe the Methodology section could be more detailed. For example, what is the statistical\nensemble used and how does it reflect the experimental system that is simulated? How were the\nparticles thermalised? In the case of coalescence, were they equilibrated in advance, and, if yes,\nfor how long? Why were the two particular distances between the particles chosen? Since there\nwas a thermostat used, why would the authors expect that a closer distance between clusters\naffect their degree of coalescence? In my opinion, the authors seem to focus too much on the\nimplementation of the ML potential (which is fine, since this constitutes a methodological\nadvance or novelty), but not enough on the basics of MD.\nAuthor response: We have extended the discussion in the method section by providing more\ndetails on the MD simulations. Regarding the questions about the coalescence, both clusters were\nwell equilibrated before they started to interact. The initial distance between the clusters only\ndetermines the time scale by how fast the interaction may happen, otherwise it does not have\nimportance here. Temperature of the coalescence simulations is a more important “parameter” –\nat higher temperatures the clusters have a higher average kinetic energy and reactions over\nhigher energy barrier may take place more easily and in a faster timescale that at low\ntemperatures. This is why we used 550 K as the highest temperature, higher than the maximum\n3. Since velocities are appointed randomly to the atoms in an MD simulation, equilibration is\nimportant. Also, statistics. Are the RMSDs of various temperatures reproducible, or are they a\nresult of randomness? Without any statistics we cannot be sure whether the results are physically\nmeaningful. For example, it is not easy to believe that the different features between the 325 and\nother temperatures? I do not think that 3 cases are enough, by the way, especially considering\nthat they gave so different results. The simulation setup is basic and the authors could easily run\na great number of simulations without too much effort.\nAuthor response: Here we took a major task and performed extensive additional MD runs to\nwork out more details and improve the statistics. At each temperature in the single-cluster MD\nruns, we now have data collected from four parallel, independent “replica” simulations, each\nspanning the same physical timescale of 120 ns. The new data is visualized in Figure 4 which\nshows at each temperature the average behavior of the RMSD from those four replica\nsimulations. We note that the extra simulations took a total GPU time of more than two weeks at\none of the Finnish national supercomputers so the numerical effort was rather large taking into\naccount also the actual wallclock/waiting time in the queue system. The ACE force field for this\nsystem is 4-5 orders of magnitude faster than calculating the interactions “on the fly” from DFT,\nbut still it is much slower than simple classical pairwise force fields (such as Lennard-Jones,\nMorse, …. ) routinely used in atomistic simulations. One can always discuss “what is a long\nenough” MD simulation, since the energy landscape for a system like a cluster of 144 Au atoms\nand 60 thiolate ligand is extremely complex – the number of degrees of freedom is 1332 which is\nalso the dimensionality of the energy phase space. It has become convenient in the MD\ncommunity to perform parallel, independent “replica” simulations for shorter physical times,\ninstead of one extremely long simulation, since the replica simulations explore different areas of\nthe energy phase space due to different initial velocity distribution and anharmonic nature of the\nforce field. In fact, early fundamental studies of the MD technique showed that two simulations\nquickly differ in their phase-space trajectories even when started from the same atomic\nconfiguration but using different initial velocity distributions, as discussed, e.g., in the classic\nbook of Allen&Tildesly (1987). Hence, replica simulations produce a better statistical sampling\nin a shorter waiting (wall-clock) time as compared to one extremely long simulation. In the\nsimulations for this revision, we chose to perform four replicas of 120 ns at temperatures of 300\nK, 325 K, 350 K, 375 K, 400 K, 450 K. This temperature range is relevant for experiments where\nthermal effects on the clusters’ atomic structure is expected play a major role for the cluster’s\nfunctionality, a typical example being heterogeneous catalysis where clusters are deposited on\noxide surface and the system is usually heated up to these temperatures to “activate” the catalyst\nbefore cooling it down to the operational temperature (see, e.g., our ref. 36). Figure 4 shows now\nclearly how the thermal disorder proceeds in a layer-by-layer fashion at range of 300 – 350 K. At\n400 -> K one sees more continuous, “diffusive” behavior over all layers, augmented by the big\njumps in RMDS caused by the partially or fully detached gold-thiolate fragments that start to\nappear around 375 K.\n4. I am not 100% convinced with the analysis of the RMSD results. RMSD take averages for all\natoms of each layer; therefore, if a protrusion is formed (or, especially when a part of the cluster\nis detached) that totally upsets the averaged RMSD values. I do not see the \"migration between\nlayers\" the authors discuss about. For example, in lines 190-93: at 350 K, L2 crosses 0.8 A and\nL3 approaches 1.5 A, but I don't see any line mixing in Fig. 4. In my opinion, all features in Fig.\n4 are related to protrusions, not atomic rearrangement.\nAuthor response: See our response above. We have also replotted Figure 4 with the improved\ndata and have extensively revised the discussion of interpretation of RMSD data. For the specific\npoint raised by the Referee, we respond by attaching the structure figures below, showing clearly\nmigration of gold atoms between the layers in one of the runs at 350 K.\nI hope that the authors will take the time to work on these issues for the benefit of their already\ngood work so far, and that my suggestions will help them publish a very good paper in another\njournal.\nAuthor response: We thank the Referee for their critical but constructive comments.\nReviewer #2 (Remarks to the Author):\nMetal clusters (NCs) have attracted extensive interest in recent years. The authors selected one of\nthe most studied, near critical sized Au nanoclusters- Au144(SR)60 for thermal dynamics\ninvestigation at 300 K - 550 K by applying a recently developed machine-learned atomistic\npotential (ACE). They found that the temperature-induced disordering of the cluster takes place\nvia layer-by-layer mechanism and the outermost gold-thiolate layer, consisting of 30 RS-Au-SR\nunits, spontaneously forms open (polymer-like) or closed (ring-like) gold-thiolate moieties.\nThese fragments may detach and again join the cluster in a dynamic fashion. They also\ninvestigated interactions and a reaction between two Au144(SR)60 clusters: at elevated\ntemperatures, clusters can exchange gold atoms and thiolates by first establishing a cluster-\ncluster bridge from a long gold-thiolate polymer; when at close enough contact, this interaction\ncan lead to a complete coalescence of the cluster to form a new roughly spherical cluster.\nOverall, these findings and results are very interesting, novel and inspiring, which\nare mirrored by some previous experiments, therefore I can recommend the publication of this nice work after\naddressing the following issues:\nAuthor response: We thank the Referee for their positive assessment of our work. We do agree\nthat our simulations with the machine-learned ACE potential reveal several interesting novel\nprocesses in the atomistic mechanism by how a single cluster disorders at high temperatures and\nby how two such clusters can react to form one larger merged particle. The potential impact of\nsuch results is discussed in our response to Referee 1, and we also rewrote the abstract to\nhighlight the novelty and impact of this work.\n1. Why are the durations of 120 ns, 87.27 ns, and 89.85 ns considered in this work? For realistic\nexperiments, they last hours or even days. Does it mean that the simulation can not really reflect\nthe realistic situation?\nAuthor response: As we response to Referee 1, we have now significantly improved the\nstatistical sampling at all temperatures by performing additional MD simulations. It is true that\none can never reach even close to the actual timescale of lab experiments – processes like\ncluster-cluster reactions may indeed take hours to days. At best, we can explore the energy\nlandscape of the clusters up to a few microseconds. This is one reason why we have also\nperformed the cluster-cluster reaction simulation at a rather high temperature, up to 550 K, to\nspeed-up the atomistic dynamics and to facilitate the systems to overcome critical energy barriers\nfor reactions to take place. We also wish to mention that our results should not (yet) be taken\nliterally as describing cluster-cluster reactions that take place in a solvent, since we are not\nmodeling solvent effects at all. Incorporating those would be one natural extension of our work\nin the future.\n2. What is the MD simulation limitations(disadvantages) for metal nanocluster investigations?\nAuthor response: In general terms, MD simulations depend on the quality of the force field\nused to advance the atomic dynamics. We have verified that the ACE force field used in this\nwork is of good quality as compared to DFT results (see Figure 2 in the manuscript and point 3\nbelow). Another limitation is that even with the most efficient force fields and supercomputers,\none can simulate MD trajectories of this kind of systems perhaps only up to a few microseconds\n(see our point 1 above).\n3. Can the authors give a precision and efficiency (time-consuming) comparisons between\nAIMD method and ML-potential in term of Au144(SR)60? Which atomic characteristic does\nACE model denote? Why did the authors use this model? How does it accelerate the calculation\nspeed.?\nAuthor response: We base our work on an earlier report (our ref. 15) that describes the ACE\nmodel and its fitting to DFT data (structures, forces, energies) of thiolate-protected gold clusters.\nWe also give more details of the model in the Supplementary Information file. The ACE force\nfield is roughly 4-5 orders of magnitude faster than calculating forces “on the fly” from DFT at\neach atomic configuration (we could think of achieving just a few picosecond physical\nsimulation time from DFT-MD for this system in a supercomputer, instead of about 100 ns with\nACE). The accuracy of energies as predicted from ACE vs. DFT energies for the same\nconfigurations was shown in Figure 2 and discussed in that context already in the original\nmanuscript, showing a rather good linear correlation (R2 = 0.94).\n4. The RMSD results (Fig.4) can be further clarified or interpreted. For example, it is unknown\nwhat is the electronic energy change of each sharp change in Figure 4? Can the energy change of\neach peak indicate the feasibility of the according reaction? etc.\nAuthor response: We have extensively revised the discussion on Figure 4. The abrupt changes\nin the RMSD are connected to qualitative changes in the atomic structure, such as formation of\npolymer like gold-thiolate units or rings that are partially or totally detached from the cluster\nsurface.\n5. One of the authors'opinions is that the restructuring of gold nanoclusters starts from the\nchange of the outmost gold-thiolate layer, however, a structure transformation from Au127 to\nAu126 was very recently reported, in which the innermost gold of Au127 was removed with the\nleft structures including the outmost layer remained after the transformation. Basing on the fact,\nmy question is whether there is also the possibility that the restructuring of gold nanoclusters\nstarts from the innermost gold atom?\nAuthor response: Restructuring of gold clusters (or any metal cluster) can in principle start also\nfrom inner parts of the metal core, but it is generally observed (both simulations and\nexperiments) that disordering or structural changes of clusters and nanoparticles start on\nsurfaces. This is simply because atoms or molecular moieties on a nanoparticle surface are less\ncoordinated and have a smaller binding energy than atoms inside the particles, so thermal effects\nare more likely to break chemical bonds first at surfaces.\nReviewer #3 (Remarks to the Author):\nIn this work, the authors utilize the atomic cluster expansion (ACE) machine-learning potential\nto investigate the thermal dynamics and coalescence of Au144(SR)60 clusters. While these\nclusters have been extensively studied within the cluster community, research focusing on their\nthermal dynamics and coalescence is relatively scarce. This study employs machine learning to\nextend the time scales that have not previously been explored for Au144, unveiling several\nintriguing findings that pave the way for a deeper understanding of cluster-cluster interactions\nand coalescence at the atomistic level. I recommend that the manuscript be published in Nature\nCommunications after minor revisions.\nAuthor response: We thank the Referee for their positive assessment of our work.\n1. The simulations utilize varying lengths of molecular dynamics (MD). It would be helpful to\nexplain how the decision was made regarding when to terminate the simulations.\nAuthor response: Please see our response to comment 1 of Referee 2.\n2. The authors conducted three independent simulations at 450 K. What was the original\nmotivation for this approach, and why do the simulation times differ?\nAuthor response: Running parallel MD simulations for the same system but starting from\ndifferent initial conditions (initial velocities of atoms) yields an enhanced statistical sampling, as\nthe inherent anharmonic effects of MD force field at higher temperatures lead to chaotic\ndynamics and drive each simulation to explore different parts of the phase space. In principle, the\nsame could be achieved by simply running one MD simulation long enough times, but in our\ncase where one simulation already takes a few days in a supercomputer, one can get a much\nmore efficient data collection by independent parallel MD simulations (called “replicas” in the\nMD community) that can be launched at the same time. In the additional work for the revision,\nwe also synchronized the lengths (physical simulation time) of all replicas and run 4 replicas for\nall temperatures discussed in the paper.\n3. The caption for Figure 6 needs to be corrected.\nAuthor response: We thank the Referee for noting the misprint which has now been corrected."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I would like to thank the authors for heavily revising their original manuscript according to the suggestions by all the\nreviewers. However, while I appreciate their effort and understand their position, in all honesty, I still do not consider the\nmanuscript fit to publish in Nature Communications. Let me explain:\nFirst, unless I am still missing something important, I find the results rather expectable. In principle, we have two metallic\nnanoparticles whose ligand shell protects them from coalescing (which would happen spontaneously if unprotected), and\nwhen the shell is disrupted by spontaneous fragmentation, the metallic cores can get in touch and the two clusters coalesce.\nI am not an expert on ligand-protected nanoparticles, but I don't see what is novel here (for example, isn't this, more or less,\nsimilar to Bae et al. Nanoletters, 2020, 20, 8704?).\nAs I said, I am not an expert on ligand-protected nanoparticles, so let us assume that this result is, indeed, groundbreaking. I\nam still not sure about how convincing it is. For example, the authors obliged by performing additional simulations. Then,\nthey present in figure 4 the average RMSD evolution of 4 simulations/temperature. Bearing in mind that the individual RMSD\ngraphs from SI are so different to each other, I am not sure there is a point in averaging them. For example, the two spikes\naround 60 and 80 ns at 375 K come from simulation II, which has a very flat profile otherwise; some protruding chains are\ncreated, which soon reunite with the cluster. The slope in the average profile of figure 4d after 100 ns comes from simulation\nIII, where such spikes were not evident. I don't see the point of averaging.\nInterestingly, the SASA probing is not performed for all simulation runs.\nFurther, with respect to our previous discussion about diffusion (see Response Letter), I still cannot consider these\nprotrusions \"diffusion\" of Au atoms to an external shell. The whole structure is totally disrupted, exploding one might say; that\nis not how I understand diffusion. But, again, perhaps this is just a matter of interpretation.\nCan we be sure that this fragmentation is not an artifact of the inter-atomic potential or of poor relaxation/equilibration? I've\nhad similar effects in similar simulations, and it always turned out I was doing something wrong. This is clearly something\nvery sensitively related to the potential. Has this instability at high temperatures been confirmed by, say, DFT-MD?\nAgain, with respect to the statistics: the authors claim that when the two coalescing nanoparticles were brought to a higher\nproximity, they could coalesce (at least more probably than at longer distances). While I don’t find this completely\nimplausible, shouldn’t this probability depend more on the exact configuration? For example, if two nanoparticles that had a\nhigh tendency to fragment were brought together, wouldn’t they have a higher probability to coalesce fully? This is yet\nanother claim that needs to be statistically supported further. Can the authors be sure that the same two nanoparticles that\ncoalesced in their simulation would still coalesce if brought together at the same distance but with a different relative\norientation?\nI understand that my comments may be frustrating, after the authors having performed a lot of work, and especially\nconsidering that the other two reviewers gave positive assessment. However, I find it hard to accept the manuscript, on\ngrounds of doubts about the conclusions made from the results. I trust the editor can evaluate the situation, and if they find I\nam\nI am being unreasonably doubtful I would be happy to hear the paper was accepted.", "reviewer_2": "The authors addressed my conerns (comments), but it is suggested that the authors added the according discussions in the\nrevised manuscipt to increase the readability.", "reviewer_3": "The authors have addressed all my comments, and I recommend that the manuscript be published as an article in Nature\nCommunications."}, "author_response": "General remarks from the authors: Referees 2 and 3 now recommend acceptance. Referee 1\nhas further technical questions but after careful consideration we have decided not to revise the\nscientific content further, but we provide a point-by-point response below. However, Referee 1\npoints out a very interesting prior work in the literature that we now added as the new ref. 42 and\nmention at the end of Discussion. Other content of the MS remains unchanged.\nReviewer #1 (Remarks to the Author):\nI would like to thank the authors for heavily revising their original manuscript according to the\nsuggestions by all the reviewers. However, while I appreciate their effort and understand their\nposition, in all honesty, I still do not consider the manuscript fit to publish in Nature\nCommunications. Let me explain:\nFirst, unless I am still missing something important, I find the results rather expectable. In\nprinciple, we have two metallic nanoparticles whose ligand shell protects them from coalescing\n(which would happen spontaneously if unprotected), and when the shell is disrupted by\nspontaneous fragmentation, the metallic cores can get in touch and the two clusters coalesce.\nI am not an expert on ligand-protected nanoparticles, but I don't see what is novel here (for\nexample, isn't this, more or less, similar to Bae et al. Nanoletters, 2020, 20, 8704?).\nAuthor response: We thank the Referee for pointing out the nice experimental work of Bae et\nal. that directly supports our theoretical findings. We now cite this paper at the end of\nDiscussion. That work shows, via TEM imaging, how 2-3 nm gold nanoparticles can merge\ntogether to make one larger particle. The theoretical part of that work deals only with classical\nforce field simulations of planar gold surfaces with the ligands used in the experiments,\ndemonstrating the ligand interdigitizing process that “locks” the particles first together when they\nstart to interact. In our work we proved for the first time a full atomistic picture of how\ncoalescence may take place (but with different chemical ligands and interactions as compared to\nBae’s work), starting from two clusters of experimentally known size and composition and\nproducing a new cluster that also has a known experimental size and composition. This\nfinding is indeed remarkable.\nAs I said, I am not an expert on ligand-protected nanoparticles, so let us assume that this result\nis, indeed, groundbreaking. I am still not sure about how convincing it is. For example, the\nauthors obliged by performing additional simulations. Then, they present in figure 4 the average\nRMSD evolution of 4 simulations/temperature. Bearing in mind that the individual RMSD\ngraphs from SI are so different to each other, I am not sure there is a point in averaging them. For\nexample, the two spikes around 60 and 80 ns at 375 K come from simulation II, which has a very\nflat profile otherwise; some protruding chains are created, which soon reunite with the cluster.\nThe slope in the average profile of figure 4d after 100 ns comes from simulation III, where such\nspikes were not evident. I don't see the point of averaging.\nAuthor response: Collecting statistics in nanosystems that are on the verge of large structural\ntransitions is always a bit open question. The example case raised by the Referee belongs to T\nregion where significant changes in the structure start to take place, and we observe that in part\nof the “replica” runs. All runs are still valid paths in the configurational phase space and there is\nno fundamental reason why one could not average the data.\nInterestingly, the SASA probing is not performed for all simulation runs.\nAuthor response: We do not expect to get any additional useful information by doing more of\nthe SASA analysis.\nFurther, with respect to our previous discussion about diffusion (see Response Letter), I still\ncannot consider these protrusions \"diffusion\" of Au atoms to an external shell. The whole\nstructure is totally disrupted, exploding one might say; that is not how I understand diffusion.\nBut, again, perhaps this is just a matter of interpretation.\nAuthor response: Atomic “diffusion” in small particles and at high temperatures is indeed a bit\na question of semantics, but we remind the Referee that diffusion takes place also in amorphous\nmaterials (disordered systems without long scale order) and we do not see any problem with\nusing this term here. Atomic diffusion effects certainly take place when atoms change layers in\nthe clusters at high temperatures.\nCan we be sure that this fragmentation is not an artifact of the inter-atomic potential or of poor\nrelaxation/equilibration? I've had similar effects in similar simulations, and it always turned out I\nwas doing something wrong. This is clearly something very sensitively related to the potential.\nHas this instability at high temperatures been confirmed by, say, DFT-MD?\nAuthor response: The point in our work is that with the developed ACE ML force field we can\nperform these dynamical simulations to hundreds of nanosecond timescales. Doing similar\nsimulations with forces calculated from DFT, we could reach perhaps max 10 ps timescale. So,\ndirect verification of the high temperature dynamics is clearly not possible.\nAgain, with respect to the statistics: the authors claim that when the two coalescing nanoparticles\nwere brought to a higher proximity, they could coalesce (at least more probably than at longer\ndistances). While I don’t find this completely implausible, shouldn’t this probability depend\nmore on the exact configuration? For example, if two nanoparticles that had a high tendency to\nfragment were brought together, wouldn’t they have a higher probability to coalesce fully? This\nis yet another claim that needs to be statistically supported further. Can the authors be sure that\nthe same two nanoparticles that coalesced in their simulation would still coalesce if brought\ntogether at the same distance but with a different relative orientation?\nAuthor response: Starting the coalescence simulations from various nanoparticle distances\nwould lead to coalescence eventually but with varying physical timescales.\nI understand that my comments may be frustrating, after the authors having performed a lot of\nwork, and especially considering that the other two reviewers gave positive assessment.\nHowever, I find it hard to accept the manuscript, on grounds of doubts about the conclusions\nmade from the results. I trust the editor can evaluate the situation, and if they find I am being\nunreasonably doubtful I would be happy to hear the paper was accepted.\nAuthor response: Referees 2 and 3 recommend acceptance and we leave the case now to the\nhands of the Editor."}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have answered all my previous comments in a somewhat satisfactory way. I fully agree with their decision not to\nretouch their manuscript, as this wouldn't have done it any good. Therefore, I accept the manuscript for publication."}, "author_response": ""}]

Decision letter