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Dynamic Ensembles of Phosphine-Stabilized Gold Nanoclusters

This study employs machine-learned molecular dynamics and Markov state models to demonstrate that phosphine-stabilized gold nanoclusters exist as dynamic ensembles at finite temperatures, where experimentally observed crystal structures often represent minor metastable states rather than the dominant thermodynamic configurations.

Original authors: Caitlin A. McCandler, Disha Sanwal, Jutta Rogal

Published 2026-08-21

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Caitlin A. McCandler, Disha Sanwal, Jutta Rogal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of nanoscience, researchers often build tiny machines out of gold, wrapping them in a protective shell of organic molecules to keep them stable and functional. These are not the bulk gold bars found in jewelry or electronics, but clusters so small they contain only a handful of atoms, yet large enough to exhibit unique chemical properties useful for sensing or catalysis. For decades, the standard way to understand these tiny structures has been to freeze them in a crystal and take a picture using X-rays. This method provides a sharp, static image of where every atom sits, much like a photograph captures a single moment in time. However, a photograph cannot show how a living thing moves, breathes, or changes shape. In the real world, whether in a solution or a gas, these clusters are warm and jiggling, constantly shifting their shapes. The question that has long lingered is whether the frozen picture we take in the lab truly represents the shape the cluster actually prefers when it is free to move and react.

A team of researchers has now tackled this question by looking beyond the static photograph to see the full movie of how these gold clusters behave. Instead of relying on a single snapshot, they used powerful computer simulations to watch how phosphine-stabilized gold nanoclusters move and change shape at room temperature. By combining advanced computer models that predict how atoms interact with a statistical framework designed to track long-term behavior, they mapped out the energy landscapes of these tiny structures. Their findings reveal a surprising truth: the specific shapes scientists have identified and published for years are often not the most common or stable forms the clusters take when they are free. In many cases, the famous crystal structures are actually rare, fleeting configurations, while the clusters spend most of their time in different, more dynamic shapes that were previously invisible to standard observation.

The researchers focused on a series of gold clusters wrapped in phosphine ligands, which are molecules that bind to the gold surface. They simulated these clusters in a gas phase at 300 Kelvin, a temperature comparable to a warm day, to see how they would naturally settle. Using a machine-learned potential, a type of artificial intelligence trained on quantum physics calculations, they ran simulations that captured the motion of atoms over time scales reaching up to 1.5 microseconds. This allowed them to observe rare events that happen too quickly or too infrequently for traditional methods to catch. They then organized this massive amount of movement data into a map of distinct shapes, or isomers, and tracked how the clusters switched between them.

The results challenged the conventional view that the crystal structure is the definitive shape of the cluster. For several cluster sizes, including those with seven or eight gold atoms, the structure found in the crystal database turned out to be a minor player, representing less than 12 percent of the time the cluster spends in its natural state. In some instances, the crystal structure was so rare it accounted for less than one percent of the population. Instead, the clusters favored different geometries that were more stable at room temperature. Even more striking was the case of the nine-gold cluster, where four different crystal structures reported in the literature were found to be merely different snapshots of the same single, rapidly shifting shape. These variations were not separate, stable forms but rather quick fluctuations within a single dynamic family, connected by thermal energy that allowed them to flip back and forth in mere picoseconds.

The study also uncovered how the number of protective molecules attached to the gold core changes the cluster's behavior. When the gold cluster had no protective ligands, it preferred a flat, two-dimensional shape. As the researchers added more ligands, the cluster was forced to fold into a three-dimensional ball. Interestingly, clusters with a medium amount of ligands showed the most variety, bouncing between many different shapes, while those fully covered by ligands settled into a few specific, compact forms. This added protection also made the clusters change shape faster, lowering the energy barriers that usually keep them stuck in one form. The researchers found that the more ligands were added, the quicker the cluster could rearrange itself, moving from one shape to another in a fraction of a second.

Perhaps the most significant implication of this work concerns how these clusters might be used in chemistry, particularly in catalysis, where they help speed up reactions. To work, a catalyst needs to expose its gold surface to other molecules. The simulations showed that the most stable, common shapes of these clusters often hide the gold surface deep inside, making it hard for other molecules to reach it. The shapes that expose the most gold surface area were often the rare, high-energy ones. However, because the clusters are so dynamic, they constantly visit these rare, exposed shapes. This means that even if the most stable form is hidden, the cluster still spends enough time in an open, accessible form to be useful for reactions. The researchers calculated that for a cluster with eleven gold atoms, the dominant shape exposed only about one square angstrom of surface, while rare shapes exposed up to ten times that amount.

This work suggests that scientists need to stop thinking of these nanoclusters as rigid, unchanging objects defined by a single crystal structure. Instead, they should be viewed as a dynamic ensemble, a collection of constantly shifting shapes that exist in a balance of probabilities. The crystal structure is just one possible moment in a long, fluid history. By understanding the full range of shapes a cluster can take and how quickly it moves between them, researchers can better predict how these tiny machines will behave in real-world applications. The study establishes a new framework for looking at nanoclusters, one that prioritizes movement and probability over static images, offering a more accurate picture of how these tiny gold structures truly function in the world.

Technical Summary: Dynamic Ensembles of Phosphine-Stabilized Gold Nanoclusters

Problem Statement
Atomically precise phosphine-stabilized gold nanoclusters are traditionally characterized by single-crystal X-ray diffraction, which provides static, zero-temperature structural snapshots. However, the extent to which these crystallographic structures represent the finite-temperature behavior of nanoclusters in solution or the gas phase remains unclear. Experimental observations indicate that distinct crystal structures can interconvert or decay in solution, raising questions about the persistence of atomically precise geometries under operating conditions. Furthermore, the functional properties of these clusters, particularly in catalysis and sensing, are tightly coupled to geometry and structural rigidity. Current static descriptions fail to capture the dynamic behavior, equilibrium populations, and isomerization kinetics required to understand structure–property relationships in these systems.

Methodology
To address these limitations, the authors established a computational framework combining machine-learned interatomic potentials (MLIPs) with Markov state models (MSMs) to explore the free-energy landscapes and dynamics of phosphine-stabilized gold nanoclusters ( $Au_n(PR_3)_m$ ) in the gas phase at 300 K.

Machine-Learned Interatomic Potential (MLIP): An Atomic Cluster Expansion (ACE) potential was trained on approximately 42,000 density functional theory (DFT+D3) calculations involving $Au_n(PH_3)_m$ and $Au_n(P(CH_3)_3)_m$ clusters ( $n \le 12, m \le n$ ). Trimethylphosphine ( $P(CH_3)_3$ ) was used as a computationally tractable model for bulky phosphine ligands to avoid the unrealistic ligand dissociation observed with simpler $PH_3$ models. The potential achieves a mean absolute error of 2.14 meV/atom in energies and 43.0 meV/Å in forces.
Molecular Dynamics (MD) Simulations: Gas-phase MD simulations were performed using LAMMPS. To ensure adequate sampling of the configurational landscape, replica exchange simulations were used to select 32 representative initial structures, followed by independent NVT production runs totaling 288–1536 ns per system. Trajectories were reweighted to correct for non-equilibrium initialization.
Markov State Models (MSMs): The resulting trajectories were analyzed using MSMs. Structural descriptors based on ACE functions were computed for each frame, projected into a low-dimensional space using Time-lagged Independent Component Analysis (TICA), and clustered into microstates. These were coarse-grained into metastable macrostates (isomers) using Perron-Cluster Cluster Analysis (PCCA+). This approach allows for the quantification of equilibrium populations, identification of metastable states, and calculation of isomerization kinetics.

Key Results
The study analyzed a series of experimentally reported clusters ( $Au_4L_4$ through $Au_{11}L_{10}$ ) and varying ligand coverages on $Au_8$ .

Discrepancy Between Crystal and Gas-Phase Structures: The analysis reveals that experimentally determined crystal structures frequently correspond to minor metastable states or transient configurations rather than the dominant finite-temperature structures. For example, the crystal structure of $Au_7L_6$ corresponds to only 11.8% of the equilibrium ensemble, while the dominant isomer is structurally distinct. Similarly, crystal structures for $Au_8L_6$ and $Au_8L_8$ do not coincide with any metastable basins in the simulated gas-phase landscapes. Conversely, crystal structures for $Au_9L_8$ and $Au_{11}L_{10}$ do correspond to dominant equilibrium isomers.
Dynamic Ensembles and Polymorphism: Multiple crystallographically distinct polymorphs of $Au_9(PR_3)_8$ were found to reside within a single free-energy basin, interconverting rapidly via thermal fluctuations rather than representing separate metastable isomers. This suggests that crystallographic differences in these cases reflect snapshots of a single dynamic gas-phase isomer.
Impact of Ligand Coverage: Ligand coverage systematically alters thermodynamics and kinetics.
- Thermodynamics: Bare $Au_8$ clusters prefer planar geometries. Adding ligands drives a transition to three-dimensional (3D) gold cores. Intermediate ligand coverages (e.g., $Au_8L_6$ ) exhibit maximum structural diversity with shallow free-energy landscapes and many competing minima. Fully ligated clusters ( $Au_8L_8$ ) return to deep, well-defined basins dominated by compact 3D geometries.
- Kinetics: Isomerization kinetics are significantly accelerated by ligand coverage. Sparsely ligated clusters exhibit slow implied timescales ( $\sim 10^4$ ps), whereas heavily ligated clusters show faster interconversion ( $10^2$ – $10^3$ ps).
Trade-off Between Stability and Accessibility: A trade-off exists between structural stability and catalytic accessibility. Highly protected clusters (e.g., $Au_{11}L_{10}$ ) are thermodynamically stable but expose minimal gold surface area (Solvent-Accessible Surface Area, SASA $\sim 1$ Å²). However, minor isomers in the equilibrium ensemble can expose significantly more surface area (up to $\sim 10$ Å²). These higher-energy, accessible geometries are dynamically accessible (interconversion timescale $\sim 156$ ps) rather than kinetically trapped, suggesting they may disproportionately contribute to catalytic activity despite low equilibrium populations.

Significance and Claims
The paper claims that ligand-protected nanoclusters must be viewed as dynamic ensembles of interconverting isomers rather than single static structures. The authors assert that finite-temperature behavior cannot be fully captured by crystallographic structures alone. By providing a general framework to resolve free-energy landscapes and isomerization networks directly from molecular dynamics, the work demonstrates that:

Experimentally observed crystal geometries often represent minor components of the gas-phase equilibrium ensemble.
Ligand coverage governs not only equilibrium structures but also the kinetics of structural rearrangement and the accessibility of reactive sites.
Understanding and controlling the properties of atomically precise nanoclusters requires moving beyond static descriptions toward an ensemble-based understanding that accounts for entropic stabilization and dynamic accessibility.

The authors note that while the framework is applicable to other ligand-protected systems, the current study is limited to gas-phase calculations; explicit treatment of solvation and ligand exchange remains beyond the scope of the present model.

Technical Summary: Dynamic Ensembles of Phosphine-Stabilized Gold Nanoclusters

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