Structure, thermodynamics, and rearrangement mechanisms in gold clusters-insights from the energy landscapes framework

Charting Nanocluster Structures via Convolutional Neural Networks

A general method to obtain a representation of the structural landscape of nanoparticles in terms of a limited number of variables is proposed. The method is applied to a large data set of parallel tempering molecular dynamics simulations of gold clusters of 90 and 147 atoms, silver clusters of 147 atoms, and copper clusters of 147 atoms, covering a plethora of structures and temperatures. The method leverages convolutional neural networks to learn the radial distribution functions of the nanoclusters and distills a low-dimensional chart of the structural landscape. This strategy is found to give rise to a physically meaningful and differentiable mapping of the atom positions to a low-dimensional manifold in which the main structural motifs are clearly discriminated and meaningfully ordered. Furthermore, unsupervised clustering on the low-dimensional data proved effective at further splitting the motifs into structural subfamilies characterized by very fine and physically relevant differences such as the presence of specific punctual or planar defects or of atoms with particular coordination features. Owing to these peculiarities, the chart also enabled tracking of the complex structural evolution in a reactive trajectory. In addition to visualization and analysis of complex structural landscapes, the presented approach offers a general, low-dimensional set of differentiable variables that has the potential to be used for exploration and enhanced sampling purposes.

Minimum Free-Energy Shapes of Ag Nanocrystals: Vacuum vs Solution

We use two variants of replica-exchange molecular dynamics (MD) simulations, parallel tempering MD and partial replica exchange MD, to probe the minimum free-energy shapes of Ag nanocrystals containing 100-200 atoms in a vacuum, ethylene glycol (EG) solvent, and EG solvent with a PVP polymer containing 100 repeat units. Our simulations reveal a shape intermediate between a Dh and an Ih, a Dh-Ih, that has distinct structural signatures and magic sizes. We find several prominent features associated with entropy: pure FCC nanocrystals are less common than FCC crystals containing stacking faults, and crystals with the minimum potential energy are not always preferred over the range of relevant temperatures. The shapes of the nanocrystals in solution are influenced by the chemical identities of the solution-phase molecules. Comparing Ag nanocrystal shapes in EG to those in an EG+PVP solution, we find more icosahedra in EG and more decahedra in EG+PVP across all of the nanocrystal sizes probed in this study. At certain critical sizes, nanocrystal shapes can change dramatically with the addition and removal of a single atom or with a change in temperature at a fixed size. The information in our study could be useful in efforts to devise processing routes to achieve selective nanocrystal shapes.

Comparison of Taboo Search Methods for Atomic Cluster Global Optimization with a Basin-Hopping Algorithm

The basin-hopping algorithm (BHA) allows for the efficient exploration of atomic cluster potential energy surfaces by random perturbations in configuration space, followed by energy minimizations. Here, the taboo search method is incorporated to prevent the search from revisiting recently visited regions of the search space. Two taboo search modes are implemented, one mode resets the search to random coordinates upon encountering the taboo region, while the other simply rejects any proposed move into the taboo region. These two modes are tested and compared on a variety of potential energy surfaces─several clusters where atomic interactions are described by the Lennard-Jones potential, and Au55 where a semi-empirical tight binding potential is used to describe atomic interactions. Some differences in performance between the two taboo search modes were noted for LJ38 and Au55, with the mode that rejects all hops into the taboo region performing better, offering a means to improve the efficiency of the BHA for multifunnel systems. However, both taboo search modes failed to significantly improve performance on multifunnel systems where more than two funnels were present in the system.

Structural transformations in Cu, Ag, and Au metal nanoclusters

Finite-temperature structures of Cu, Ag, and Au metal nanoclusters are calculated in the entire temperature range from 0 K to melting using a computational methodology that we proposed recently [M. Settem et al., Nanoscale 14, 939 (2022)]. In this method, Harmonic Superposition Approximation (HSA) and Parallel Tempering Molecular Dynamics (PTMD) are combined in a complementary manner. HSA is accurate at low temperatures and fails at higher temperatures. PTMD, on the other hand, effectively samples the high temperature region and melts. This method is used to study the size- and system-dependent competition between various structural motifs of Cu, Ag, and Au nanoclusters in the size range 1-2 nm. Results show that there are mainly three types of structural changes in metal nanoclusters, depending on whether a solid-solid transformation occurs. In the first type, the global minimum is the dominant motif in the entire temperature range. In contrast, when a solid-solid transformation occurs, the global minimum transforms either completely to a different motif or partially, resulting in the co-existence of multiple motifs. Finally, nanocluster structures are analyzed to highlight the system-specific differences across the three metals.

Theoretical and computational methodologies for understanding coordination self-assembly complexes

This perspective highlights three theoretical and computational methods to capture the coordination self-assembly processes at the molecular level: quantum chemical modeling, molecular dynamics, and reaction network analysis. These methods cover the different scales from the metal-ligand bond to a more global aspect, and approaches that are best suited to understand the coordination self-assembly from different perspectives are introduced. Theoretical and numerical researches based on these methods are not merely ways of interpreting the experimental studies but complementary to them.

Theory of Anisotropic Metal Nanostructures

A significant challenge in the development of functional materials is understanding the growth and transformations of anisotropic colloidal metal nanocrystals. Theory and simulations can aid in the development and understanding of anisotropic nanocrystal syntheses. The focus of this review is on how results from first-principles calculations and classical techniques, such as Monte Carlo and molecular dynamics simulations, have been integrated into multiscale theoretical predictions useful in understanding shape-selective nanocrystal syntheses. Also, examples are discussed in which machine learning has been useful in this field. There are many areas at the frontier in condensed matter theory and simulation that are or could be beneficial in this area and these prospects for future progress are discussed.

Thermally Activated Microstructural Evolution of PtIrCu Alloyed Nanorings: Insights from Molecular Dynamics Simulations

Nanoalloys have attracted extensive interest from the research and industrial community due to their unique properties. In this work, the thermally activated microstructural evolution and resultant collapse of PtIrCu nanorings were investigated using molecular dynamics simulations. Three PtIrCu nanorings with a fixed outer radius and varied inner radii were addressed to investigate the size effects on their thermal and shape stabilities. The shape factor was introduced to monitor their shape changes, and a common neighbor analysis was employed to characterize the local structures of atoms. The results reveal that both the thermal and shape stabilities of these nanorings can be remarkably improved by decreasing the inner radius. Furthermore, they all experienced the evolutionary process from ring to pie and spherelike morphologies, finally resulting in structural collapse. The stacking faults were observed in these rings during the heating process. Our work sheds light on the fundamental understanding of alloyed nanorings subjected to heating, hence offering a theoretical foundation for their syntheses and applications.

Relative Populations and IR Spectra of Cu38 Cluster at Finite Temperature Based on DFT and Statistical Thermodynamics Calculations

The relative populations of Cu₃₈ isomers depend to a great extent on the temperature. Density functional theory and nanothermodynamics can be combined to compute the geometrical optimization of isomers and their spectroscopic properties in an approximate manner. In this article, we investigate entropy-driven isomer distributions of Cu₃₈ clusters and the effect of temperature on their IR spectra. An extensive, systematic global search is performed on the potential and free energy surfaces of Cu₃₈ using a two-stage strategy to identify the lowest-energy structure and its low-energy neighbors. The effects of temperature on the populations and IR spectra are considered via Boltzmann factors. The computed IR spectrum of each isomer is multiplied by its corresponding Boltzmann weight at finite temperature. Then, they are summed together to produce a final temperature-dependent, Boltzmann-weighted spectrum. Our results show that the disordered structure dominates at high temperatures and the overall Boltzmann-weighted spectrum is composed of a mixture of spectra from several individual isomers.

Artificial neural network potential for Au20clusters based on the first-principles

The search of ground-state structures (GSSs) of gold (Au) clusters is a formidable challenge due to the complexity of potential energy surface (PES). In this work, we have built a high-dimensional artificial neural network (ANN) potential to describe the PES of Au₂₀clusters. The ANN potential is trained through learning the GSS search process of Au₂₀by the combination of density functional theory (DFT) method and genetic algorithm. The root mean square errors of energy and force are 7.72 meV atom^-1and 217.02 meV Å^-1, respectively. As a result, it can find the lowest-energy structure (LES) of Au₂₀clusters that is consistent with previous results. Furthermore, the scalability test shows that it can predict the energy of smaller size Au_16-19clusters with errors less than 22.85 meV atom^-1, and for larger size Au_21-25clusters, the errors are below 36.94 meV atom^-1. Extra attention should be paid to its accuracy for Au_21-25clusters. Applying the ANN to search the GSSs of Au_16-25, we discover two new structures of Au₁₆and Au₂₁that are not reported before and several candidate LESs of Au_16-18. In summary, this work proves that an ANN potential trained for specific size clusters could reproduce the GSS search process by DFT and be applied in the GSS search of smaller size clusters nearby. Therefore, we claim that building ANN potential based on DFT data is one of the most promising ways to effectively accelerate the GSS pre-screening of clusters.

Exploring AuRh nanoalloys: a computational perspective on the formation and physical properties

We studied the formation of AuRh nanoalloys (between 20–150 atoms) in the gas phase by means of Molecular Dynamics (MD) calculations, exploring three possible formation processes: one‐by‐one growth, coalescence, and nanodroplets annealing. As a general trend, we recover a predominance of Rh@Au core‐shell ordering over other chemical configurations. We identify new structural motifs with enhanced thermal stabilities. The physical features of those selected systems were studied at the Density Functional Theory (DFT) level, revealing profound correlations between the nanoalloys morphology and properties. Surprisingly, the arrangement of the inner Rh core seems to play a dominant role on nanoclusters’ physical features like the HOMO‐LUMO gap and magnetic moment. Strong charge separations are recovered within the nanoalloys suggesting the existence of charge‐transfer transitions.

Tempering of Au nanoclusters: capturing the temperature-dependent competition among structural motifs

A computational approach to determine the equilibrium structures of nanoclusters in the whole temperature range from 0 K to melting is developed. Our approach relies on Parallel Tempering Molecular Dynamics (PTMD) simulations complemented by Harmonic Superposition Approximation (HSA) calculations and global optimization searches, thus combining the accuracy of global optimization and HSA in describing the low-energy part of configuration space, together with the PTMD thorough sampling of high-energy configurations. This combined methodology is shown to be instrumental towards revealing the temperature-dependent structural motifs in Au nanoclusters of sizes 90, 147, and 201 atoms. The reported phenomenology is particularly rich, displaying a size- and temperature-dependent competition between the global energy minimum and other structural motifs. In the case of Au₉₀ and Au₁₄₇, the global minimum is also the dominant structure at finite temperatures. In contrast, the Au₂₀₁ cluster undergoes a solid-solid transformation at low temperature (<200 K). Results indicate that PTMD and HSA very well agree at intermediate temperatures, between 300 and 400 K. For higher temperatures, PTMD gives an accurate description of equilibrium, while HSA fails in describing the melting range. On the other hand, HSA is more efficient in catching low-temperature structural transitions. Finally, we describe the elusive structures close to the melting region which can present complex and defective geometries, that are otherwise difficult to characterize through experimental imaging.

Understanding the strain-dependent structure of Cu nanocrystals in Ag-Cu nanoalloys

The structure of octahedral Ag-Cu nanoalloys is investigated by means of basin hopping Monte Carlo (BHMC) searches involving the optimization of shape and chemical ordering. Due to the significant size mismatch between Ag and Cu, the misfit strain plays a key role in determining the structure of Ag-Cu nanoalloys. At all the compositions, segregated chemical ordering is observed. However, the shape of the Cu nanocrystal and the associated defects are significantly different. At lower amounts of Cu (as little as 2 atom %), defects close to the surface are observed leading to a highly non-compact shape of the Cu nanocrystal which is non-trivial. The number of Cu-Cu bonds is relatively lower in the non-compact shape which is contrary to the preference of bulk Ag-Cu alloys to maximize the homo-atomic bonds. Due to the non-compact shape, {100} Ag-Cu interfaces are observed which are not expected. As the amount of Cu increases, the Cu nanocrystal undergoes a shape transition from non-compact to a compact octahedron. The associated defect structure is also modified. The structural changes due to the strain effects have been explained by calculating the atomic pressure maps and the bond length distributions. The trends relating to the structure have also been verified at larger sizes.

Theoretical Prediction of Structures, Vibrational Circular Dichroism, and Infrared Spectra of Chiral Be4B8 Cluster at Different Temperatures

Lowest-energy structures, the distribution of isomers, and their molecular properties depend significantly on geometry and temperature. Total energy computations using DFT methodology are typically carried out at a temperature of zero K; thereby, entropic contributions to the total energy are neglected, even though functional materials work at finite temperatures. In the present study, the probability of the occurrence of one particular Be4B8 isomer at temperature T is estimated by employing Gibbs free energy computed within the framework of quantum statistical mechanics and nanothermodynamics. To identify a list of all possible low-energy chiral and achiral structures, an exhaustive and efficient exploration of the potential/free energy surfaces is carried out using a multi-level multistep global genetic algorithm search coupled with DFT. In addition, we discuss the energetic ordering of structures computed at the DFT level against single-point energy calculations at the CCSD(T) level of theory. The total VCD/IR spectra as a function of temperature are computed using each isomer's probability of occurrence in a Boltzmann-weighted superposition of each isomer's spectrum. Additionally, we present chemical bonding analysis using the adaptive natural density partitioning method in the chiral putative global minimum. The transition state structures and the enantiomer-enantiomer and enantiomer-achiral activation energies as a function of temperature evidence that a change from an endergonic to an exergonic type of reaction occurs at a temperature of 739 K.

Reversible disorder-order transitions in atomic crystal nucleation

Nucleation in atomic crystallization remains poorly understood, despite advances in classical nucleation theory. The nucleation process has been described to involve a nonclassical mechanism that includes a spontaneous transition from disordered to crystalline states, but a detailed understanding of dynamics requires further investigation. In situ electron microscopy of heterogeneous nucleation of individual gold nanocrystals with millisecond temporal resolution shows that the early stage of atomic crystallization proceeds through dynamic structural fluctuations between disordered and crystalline states, rather than through a single irreversible transition. Our experimental and theoretical analyses support the idea that structural fluctuations originate from size-dependent thermodynamic stability of the two states in atomic clusters. These findings, based on dynamics in a real atomic system, reshape and improve our understanding of nucleation mechanisms in atomic crystallization.

Real-time insight into nanostructure evolution during the rapid formation of ultra-thin gold layers on polymers

Ultra-thin metal layers on polymer thin films attract tremendous research interest for advanced flexible optoelectronic applications, including organic photovoltaics, light emitting diodes and sensors. To realize the large-scale production of such metal-polymer hybrid materials, high rate sputter deposition is of particular interest. Here, we witness the birth of a metal-polymer hybrid material by quantifying in situ with unprecedented time-resolution of 0.5 ms the temporal evolution of interfacial morphology during the rapid formation of ultra-thin gold layers on thin polystyrene films. We monitor average non-equilibrium cluster geometries, transient interface morphologies and the effective near-surface gold diffusion. At 1 s sputter deposition, the polymer matrix has already been enriched with 1% gold and an intermixing layer has formed with a depth of over 3.5 nm. Furthermore, we experimentally observe unexpected changes in aspect ratios of ultra-small gold clusters growing in the vicinity of polymer chains. For the first time, this approach enables four-dimensional insights at atomic scales during the gold growth under non-equilibrium conditions.

Modeling microsolvation clusters with electronic-structure calculations guided by analytical potentials and predictive machine learning techniques

We propose a new methodology to study, at the density functional theory (DFT) level, the clusters resulting from the microsolvation of alkali-metal ions with rare-gas atoms. The workflow begins with a global optimization search to generate a pool of low-energy minimum structures for different cluster sizes. This is achieved by employing an analytical potential energy surface (PES) and an evolutionary algorithm (EA). The next main stage of the methodology is devoted to establish an adequate DFT approach to treat the microsolvation system, through a systematic benchmark study involving several combinations of functionals and basis sets, in order to characterize the global minimum structures of the smaller clusters. In the next stage, we apply machine learning (ML) classification algorithms to predict how the low-energy minima of the analytical PES map to the DFT ones. An early and accurate detection of likely DFT local minima is extremely important to guide the choice of the most promising low-energy minima of large clusters to be re-optimized at the DFT level of theory. In this work, the methodology was applied to the Li+Krn (n = 2-14 and 16) microsolvation clusters for which the most competitive DFT approach was found to be the B3LYP-D3/aug-pcseg-1. Additionally, the ML classifier was able to accurately predict most of the solutions to be re-optimized at the DFT level of theory, thereby greatly enhancing the efficiency of the process and allowing its applicability to larger clusters.

Exploration of Free Energy Surface and Thermal Effects on Relative Population and Infrared Spectrum of the Be6B11- Flux-Ional Cluster

The starting point to understanding cluster properties is the putative global minimum and all the nearby local energy minima; however, locating them is computationally expensive and difficult. The relative populations and spectroscopic properties that are a function of temperature can be approximately computed by employing statistical thermodynamics. Here, we investigate entropy-driven isomers distribution on Be₆B₁₁⁻ clusters and the effect of temperature on their infrared spectroscopy and relative populations. We identify the vibration modes possessed by the cluster that significantly contribute to the zero-point energy. A couple of steps are considered for computing the temperature-dependent relative population: First, using a genetic algorithm coupled to density functional theory, we performed an extensive and systematic exploration of the potential/free energy surface of Be₆B₁₁⁻ clusters to locate the putative global minimum and elucidate the low-energy structures. Second, the relative populations’ temperature effects are determined by considering the thermodynamic properties and Boltzmann factors. The temperature-dependent relative populations show that the entropies and temperature are essential for determining the global minimum. We compute the temperature-dependent total infrared spectra employing the Boltzmann factor weighted sums of each isomer’s infrared spectrum and find that at finite temperature, the total infrared spectrum is composed of an admixture of infrared spectra that corresponds to the spectra of the lowest-energy structure and its isomers located at higher energies. The methodology and results describe the thermal effects in the relative population and the infrared spectra.

Towards kinetic control of coordination self-assembly: a case study of a Pd3L6 double-walled triangle to predict the outcomes by a reaction network model

Numerical analysis of self-assembly process (NASAP) was performed for a [Pd3L6]6+ double-walled triangle (DWT) complex. With a chemical reaction network and a parameter set of the reaction rate constants obtained from a numerical search in an eighteen-dimensional parameter space to obtain a good fit to the data from the experimental counterpart (quantitative analysis of self-assembly process, QASAP), a refined calculation resulted in a detailed time evolution of each molecular species. Analysis based on those clues revealed dominant self-assembly pathways and a balance between inter- and intramolecular reactions, and enabled prediction of the reaction outcomes depending on the initial stoichiometric ratio under kinetic control.

Density-functional tight-binding: basic concepts and applications to molecules and clusters

The scope of this article is to present an overview of the Density Functional based Tight Binding (DFTB) method and its applications. The paper introduces the basics of DFTB and its standard formulation up to second order. It also addresses methodological developments such as third order expansion, inclusion of non-covalent interactions, schemes to solve the self-interaction error, implementation of long-range short-range separation, treatment of excited states via the time-dependent DFTB scheme, inclusion of DFTB in hybrid high-level/low level schemes (DFT/DFTB or DFTB/MM), fragment decomposition of large systems, large scale potential energy landscape exploration with molecular dynamics in ground or excited states, non-adiabatic dynamics. A number of applications are reviewed, focusing on -(i)- the variety of systems that have been studied such as small molecules, large molecules and biomolecules, bare orfunctionalized clusters, supported or embedded systems, and -(ii)- properties and processes, such as vibrational spectroscopy, collisions, fragmentation, thermodynamics or non-adiabatic dynamics. Finally outlines and perspectives are given.

Evidencing the relationship between isomer spectra and melting: the 20- and 55-atom silver and gold cluster cases

The present work highlights the links between melting properties and structural excitation spectra of small gold and silver clusters. The heat capacity curves are computed for Ag₂₀, Au₂₀, Ag₅₅, Au₅₅ and their ions, using a parallel-tempering molecular dynamics scheme to explore the density functional based tight binding (DFTB) potential energy surfaces and the multiple histogram method. It is found that clusters having very symmetric lowest energy structures (Au₂₀, Ag₅₅ and their ions) present sharp or relatively sharp solid-to-liquid transitions and large melting temperatures, important structural excitation energies and a discrete isomer spectrum. Opposite trends are observed for less ordered clusters (Ag₂₀, Au₅₅ and their ions). Regarding the structural evolution with temperature, very symmetric clusters exhibit minor evolution up to the starting melting temperature. The present study also highlights that, in contrast with the case of Au₂₀, a single electron excess or deficiency is not determinant regarding the melting characteristics, even quantitatively, for clusters containing 55 atoms, for gold as for silver.

Correlating Oxygen Reduction Reaction Activity and Structural Rearrangements in MgO-Supported Platinum Nanoparticles

We develop a multi‐scale approach towards the design of metallic nanoparticles with applications as catalysts in electrochemical reactions. The here discussed method exploits the relationship between nanoparticle architecture and electrochemical activity and is applied to study the catalytic properties of MgO(100)‐supported Pt nanosystems undergoing solid‐solid and solid‐liquid transitions. We observe that a major increment in the activity is associated to the reconstruction of the interface layers, supporting the need for a full geometrical characterisation of such structures also when in‐operando.

Structural properties of sub-nanometer metallic clusters

At the nanoscale, the investigation of structural features becomes fundamental as we can establish relationships between cluster geometries and their physicochemical properties. The peculiarity lies in the variety of shapes often unusual and far from any geometrical and crystallographic intuition clusters can assume. In this respect, we should treat and consider nanoparticles as a new form of matter. Nanoparticle structures depend on their size, chemical composition, ordering, as well as external conditions e.g. synthesis method, pressure, temperature, support. On top of that, at finite temperatures nanoparticles can fluctuate among different structures, opening new and exciting horizons for the design of optimal nanoparticles for advanced applications. This article aims to overview geometrical features of transition metal clusters and of their various rearrangements.

Multitribe evolutionary search for stable Cu-Pd-Ag nanoparticles using neural network models

We present an approach based on two bio-inspired algorithms to accelerate the identification of nanoparticle ground states. We show that a symbiotic co-evolution of nanoclusters across a range of sizes improves the search efficiency considerably, while a neural network constructed with a recently introduced stratified training scheme delivers an accurate description of interactions in multielement systems. The method's performance has been examined in extensive searches for stable elemental (30-80 atoms), binary (50, 55, and 80 atoms), and ternary (50, 55, and 80 atoms) Cu-Pd-Ag clusters. The best candidate structures identified with the neural network model have consistently lower energy at the density functional theory level compared with those found with traditional interatomic potentials.

Density-functional tight-binding approach for metal clusters, nanoparticles, surfaces and bulk: application to silver and gold

Density-functional based tight-binding (DFTB) is an efficient quantum mechanical method that can describe a variety of systems, going from organic and inorganic compounds to metallic and hybrid materials. The present topical review addresses the ability and performance of DFTB to investigate energetic, structural, spectroscopic and dynamical properties of gold and silver materials. After a brief overview of the theoretical basis of DFTB, its parametrization and its transferability, we report its past and recent applications to gold and silver systems, including small clusters, nanoparticles, bulk and surfaces, bare and interacting with various organic and inorganic compounds. The range of applications covered by those studies goes from plasmonics and molecular electronics, to energy conversion and surface chemistry. Finally, perspectives of DFTB in the field of gold and silver surfaces and NPs are outlined.

Au147 nanoparticles: Ordered or amorphous?

Structural aspects of the Au₁₄₇ cluster have been investigated through a density functional based tight binding global optimization involving a parallel tempering molecular dynamics scheme with quenching followed by geometries relaxation at the Density Functional Theory (DFT) level. The focus is put on the competition between relaxed ordered regular geometries and disordered (or amorphous) structures. The present work shows that Au₁₄₇ amorphous geometries are relevant low energy candidates and are likely to contribute in finite temperature dynamics and thermodynamics. The structure of the amorphous-like isomers is discussed from the anisotropy parameters, the atomic coordinations, the radial and pair distribution functions, the IR spectra, and the vibrational DOS. With respect to the regular structures, the amorphous geometries are shown to be characterized by a larger number of surface atoms, a less dense volume with reduced coordination number per atom, a propensity to increase the dimension of flat facets at the surface, and a stronger anisotropy. Moreover, all amorphous clusters have similar IR spectra, almost continuous with active frequencies over the whole spectral range, while symmetric clusters are characterized by a few lines with large intensities.