Solid−Solid Structural Transformations in Lennard-Jones Clusters: Accurate Simulations versus the Harmonic Superposition Approximation

100 Years of the Lennard-Jones Potential

It is now 100 years since Lennard-Jones published his first paper introducing the now famous potential that bears his name. It is therefore timely to reflect on the many achievements, as well as the limitations, of this potential in the theory of atomic and molecular interactions, where applications range from descriptions of intermolecular forces to molecules, clusters, and condensed matter.

Analysis and interpretation of first passage time distributions featuring rare events

In this contribution we consider theory and associated computational tools to treat the kinetics associated with competing pathways on multifunnel energy landscapes. Multifunnel landscapes are associated with molecular switches and multifunctional materials, and are expected to exhibit multiple relaxation time scales and associated thermodynamic signatures in the heat capacity. Our focus here is on the first passage time distribution, which is encoded in a kinetic transition network containing all the locally stable states and the pathways between them. This network can be renormalised to reduce the dimensionality, while exactly conserving the mean first passage time and approximately conserving the full distribution. The structure of the reduced network can be visualised using disconnectivity graphs. We show how features in the first passage time distribution can be associated with specific kinetic traps, and how the appearance of competing relaxation time scales depends on the starting conditions. The theory is tested for two model landscapes and applied to an atomic cluster and a disordered peptide. Our most important contribution is probably the reconstruction of the full distribution for long time scales, where numerical problems prevent direct calculations. Here we combine accurate treatment of the mean first passage time with the reliable part of the distribution corresponding to faster time scales. Hence we now have a fundamental understanding of both thermodynamic and kinetic signatures of multifunnel landscapes.

Combining experiment and energy landscapes to explore anaerobic heme breakdown in multifunctional hemoproteins

To survive, many pathogens extract heme from their host organism and break down the porphyrin scaffold to sequester the Fe2+ ion via a heme oxygenase. Recent studies have revealed that certain pathogens can anaerobically degrade heme. Our own research has shown that one such pathway proceeds via NADH-dependent heme degradation, which has been identified in a family of hemoproteins from a range of bacteria. HemS, from Yersinia enterocolitica, is the main focus of this work, along with HmuS (Yersinia pestis), ChuS (Escherichia coli) and ShuS (Shigella dysenteriae). We combine experiments, Energy Landscape Theory, and a bioinformatic investigation to place these homologues within a wider phylogenetic context. A subset of these hemoproteins are known to bind certain DNA promoter regions, suggesting not only that they can catalytically degrade heme, but that they are also involved in transcriptional modulation responding to heme flux. Many of the bacterial species responsible for these hemoproteins (including those that produce HemS, ChuS and ShuS) are known to specifically target oxygen-depleted regions of the gastrointestinal tract. A deeper understanding of anaerobic heme breakdown processes exploited by these pathogens could therefore prove useful in the development of future strategies for disease prevention.

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.

Experimental absence of the non-perovskite ground state phases of MaPbI3 explained by a Funnel Hopping Monte Carlo study based on a neural network potential

Methylammonium lead iodide is a material known for its exceptional opto-electronic properties that make it a promising candidate for many high performance applications, such as light emitting diodes or solar cells. A recent computational structure search revealed two previously unknown non-perovskite polymorphs, that are lower in energy than the experimentally observed perovskite phases. To investigate the elusiveness of the non-perovskite phases in experimental studies, we extended our Funnel Hopping Monte Carlo (FHMC) method to periodic systems and performed extensive MC simulations driven by a machine learned potential. FHMC simulations that also include these newly discovered non-perovskite phases show that above temperatures of 200 K the perovskite phases are thermodynamically preferred. A comparison with the quasi-harmonic approximation highlights the importance of anharmonic effects captured by FHMC.

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.

Rare events and first passage time statistics from the energy landscape

We analyze the probability distribution of rare first passage times corresponding to transitions between product and reactant states in a kinetic transition network. The mean first passage times and the corresponding rate constants are analyzed in detail for two model landscapes and the double funnel landscape corresponding to an atomic cluster. Evaluation schemes based on eigendecomposition and kinetic path sampling, which both allow access to the first passage time distribution, are benchmarked against mean first passage times calculated using graph transformation. Numerical precision issues severely limit the useful temperature range for eigendecomposition, but kinetic path sampling is capable of extending the first passage time analysis to lower temperatures, where the kinetics of interest constitute rare events. We then investigate the influence of free energy based state regrouping schemes for the underlying network. Alternative formulations of the effective transition rates for a given regrouping are compared in detail to determine their numerical stability and capability to reproduce the true kinetics, including recent coarse-graining approaches that preserve occupancy cross correlation functions. We find that appropriate regrouping of states under the simplest local equilibrium approximation can provide reduced transition networks with useful accuracy at somewhat lower temperatures. Finally, a method is provided to systematically interpolate between the local equilibrium approximation and exact intergroup dynamics. Spectral analysis is applied to each grouping of states, employing a moment-based mode selection criterion to produce a reduced state space, which does not require any spectral gap to exist, but reduces to gap-based coarse graining as a special case. Implementations of the developed methods are freely available online.

Nested Basin-Sampling

We report an embarrassingly parallel method for the evaluation of thermodynamic properties over an energy landscape exhibiting broken ergodicity, nested is the likelihood of the observed data D givenbasin-sampling (NBS). We also introduce the No Galilean U-Turn Sampler (NoGUTS), a new sampling scheme based on the No U-Turn Sampler (NUTS) introduced by Hoffman and Gelman (2014) that works with the Galilean Monte Carlo scheme introduced by Betancourt (2012) to aid the efficient generation of new live points. NoGUTS can be thought of as a form of reflective slice sampling with an automatic stopping criterion. We apply this approach to a benchmark atomic cluster of 31 Lennard-Jones atoms, which exhibits a low temperature solid-solid heat capacity peak. The calculated heat capacity is compared with results generated by parallel tempering (PT), basin-sampling parallel tempering (BSPT), and standard nested sampling (NS) simulations. NBS reproduces the full heat capacity curve predicted by PT and BSPT, while the NS calculation with similar computational cost fails to resolve the low-temperature solid-solid phase transition.

Temperature and isomeric effects in nanoclusters

The canonical thermodynamics of clusters is determined quantum mechanically in the general case of multiple minima of the potential energy surface (PES) using the superposition approximation. As an illustration of the consequences of our approach, we study in detail the thermodynamic properties of CuN clusters with N from 2 to 150 as a function of cluster size, temperature, and number of isomers. Thereby, for instance, solid-solid transition temperatures for several cluster sizes are determined. We show that the cluster vibrations have a strong impact on the stability of the clusters and that this can be observed not only at medium and high temperatures but also at low temperatures and even at T = 0 K. Thus, including zero-point corrections can change the relative energetic ordering of different isomers. This isomeric effect results in an oscillatory dependence of the heat capacity on cluster size at moderate and high temperatures. Moreover, for the identification of magic clusters at non-vanishing temperature, the Helmholtz free energy is analyzed as a function of cluster size and temperature within a one-, two-, and three-minima model of the PES. Thereby, we demonstrate that the concept of magic clusters is strongly temperature dependent so that in several cases clusters change from being magic or non-magic at T = 0 K to be the opposite at non-vanishing temperature. We emphasize that all these effects are not specific for copper clusters alone but can also be observed in other metal or semiconductor nanoclusters.

Exploring Energy Landscapes

Recent advances in the potential energy landscapes approach are highlighted, including both theoretical and computational contributions. Treating the high dimensionality of molecular and condensed matter systems of contemporary interest is important for understanding how emergent properties are encoded in the landscape and for calculating these properties while faithfully representing barriers between different morphologies. The pathways characterized in full dimensionality, which are used to construct kinetic transition networks, may prove useful in guiding such calculations. The energy landscape perspective has also produced new procedures for structure prediction and analysis of thermodynamic properties. Basin-hopping global optimization, with alternative acceptance criteria and generalizations to multiple metric spaces, has been used to treat systems ranging from biomolecules to nanoalloy clusters and condensed matter. This review also illustrates how all this methodology, developed in the context of chemical physics, can be transferred to landscapes defined by cost functions associated with machine learning.

Atomic clusters with addressable complexity

A general formulation for constructing addressable atomic clusters is introduced, based on one or more reference structures. By modifying the well depths in a given interatomic potential in favour of nearest-neighbour interactions that are defined in the reference(s), the potential energy landscape can be biased to make a particular permutational isomer the global minimum. The magnitude of the bias changes the resulting potential energy landscape systematically, providing a framework to produce clusters that should self-organise efficiently into the target structure. These features are illustrated for small systems, where all the relevant local minima and transition states can be identified, and for the low-energy regions of the landscape for larger clusters. For a 55-particle cluster, it is possible to design a target structure from a transition state of the original potential and to retain this structure in a doubly addressable landscape. Disconnectivity graphs based on local minima that have no direct connections to a lower minimum provide a helpful way to visualise the larger databases. These minima correspond to the termini of monotonic sequences, which always proceed downhill in terms of potential energy, and we identify them as a class of biminimum. Multiple copies of the target cluster are treated by adding a repulsive term between particles with the same address to maintain distinguishable targets upon aggregation. By tuning the magnitude of this term, it is possible to create assemblies of the target cluster corresponding to a variety of structures, including rings and chains.

Retracted Article: A highest stable cluster Au58 (C 1) re-optimized via a density-functional tight-binding (DFTB) approach

The vibrational spectrum ω i of a re-optimized neutral gold cluster Au58 has been calculated using a numerical finite-difference approach and the density-functional tight-binding (DFTB) method. We have exactly predicted the vibrational frequency ranging from 3.88 through to 304.49 cm-1 which depends on the size and the arrangement of the atoms in the nanoparticle morphology of the cluster at ΔE = 0. Our investigation has revealed that the vibrational spectrum is strongly influenced by size and structure. It is well known that gold atomic clusters can have planar or hollow cage-like structures due to their relativistic effect. However, in our study, by first principles calculations on a Au58 cluster we have proposed that gold clusters of medium size can form a shell-like structure (skeleton/helmet), this is demonstrated by the remarkable robustness of a double shell structure with a hollow inner shell of about ten atoms. Finally, the structure symmetry (C 1) is confirmed through the cluster size, vibrational spectroscopy, and by studying the effect of temperature on a neutral gold cluster for the first time.

Energy landscapes for machine learning

Machine learning techniques are being increasingly used as flexible non-linear fitting and prediction tools in the physical sciences. Fitting functions that exhibit multiple solutions as local minima can be analysed in terms of the corresponding machine learning landscape. Methods to explore and visualise molecular potential energy landscapes can be applied to these machine learning landscapes to gain new insight into the solution space involved in training and the nature of the corresponding predictions. In particular, we can define quantities analogous to molecular structure, thermodynamics, and kinetics, and relate these emergent properties to the structure of the underlying landscape. This Perspective aims to describe these analogies with examples from recent applications, and suggest avenues for new interdisciplinary research.

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

We consider finite-size and temperature effects on the structure of model AuN clusters (30 ≤ N ≤ 147) bound by the Gupta potential. Equilibrium behaviour is examined in the harmonic superposition approximation, and the size-dependent melting temperature is also bracketed using molecular dynamics simulations. We identify structural transitions between distinctly different morphologies, characterised by various defect features. Reentrant behaviour and trends with respect to cluster size and temperature are discussed in detail. For N = 55, 85, and 147 we visualise the topography of the underlying potential energy landscape using disconnectivity graphs, colour-coded by the cluster morphology; and we use discrete path sampling to characterise the rearrangement mechanisms between competing structures separated by high energy barriers (up to 1 eV). The fastest transition pathways generally involve metastable states with multiple fivefold disclinations and/or a high degree of amorphisation, indicative of melting. For N = 55 we find that reoptimising low-lying minima using density functional theory (DFT) alters their energetic ordering and produces a new putative global minimum at the DFT level; however, the equilibrium structure predicted by the Gupta potential at room temperature is consistent with previous experiments.

Energy Landscape and Pathways for Transitions between Watson-Crick and Hoogsteen Base Pairing in DNA

The recent discovery that Hoogsteen (HG) base pairs are widespread in DNA across diverse sequences and positional contexts could have important implications for understanding DNA replication and DNA-protein recognition. While evidence is emerging that the Hoogsteen conformation could be a thermodynamically accessible conformation of the DNA duplex and provide a means to expand its functionality, relatively little is known about the molecular mechanism underlying the Watson-Crick (WC) to HG transition. In this Perspective, we describe pathways and kinetics for this transition at an atomic level of detail, using the energy landscape perspective. We show that competition between the duplex conformations results in a double funnel landscape, which explains some recent experimental observations. The interconversion pathways feature a number of intermediates, with a variable number of WC and HG base pairs. The relatively slow kinetics, with possible deviations from two-state behavior, suggest that this conformational switch is likely to be a challenging target for both simulation and experiment.

Decoding heat capacity features from the energy landscape

A general scheme is derived to connect transitions in configuration space with features in the heat capacity. A formulation in terms of occupation probabilities for local minima that define the potential energy landscape provides a quantitative description of how contributions arise from competition between different states. The theory does not rely on a structural interpretation for the local minima, so it is equally applicable to molecular energy landscapes and the landscapes defined by abstract functions. Applications are presented for low-temperature solid-solid transitions in atomic clusters, which involve just a few local minima with different morphologies, and for cluster melting, which is driven by the landscape entropy associated with the more numerous high-energy minima. Analyzing these features in terms of the balance between states with increasing and decreasing occupation probabilities provides a direct interpretation of the underlying transitions. This approach enables us to identify a qualitatively different transition that is caused by a single local minimum associated with an exceptionally large catchment volume in configuration space for a machine learning landscape.

Impurity effects on solid-solid transitions in atomic clusters

We use the harmonic superposition approach to examine how a single atom substitution affects low-temperature anomalies in the vibrational heat capacity (C_V) of model nanoclusters. Each anomaly is linked to competing solidlike "phases", where crossover of the corresponding free energies defines a solid-solid transition temperature (T_s). For selected Lennard-Jones clusters we show that T_s and the corresponding C_V peak can be tuned over a wide range by varying the relative atomic size and binding strength of the impurity, but excessive atom-size mismatch can destroy a transition and may produce another. In some tunable cases we find up to two additional C_V peaks emerging below T_s, signalling one- or two-step delocalisation of the impurity within the ground-state geometry. Results for Ni₇₄X and Au₅₄X clusters (X = Au, Ag, Al, Cu, Ni, Pd, Pt, Pb), modelled by the many-body Gupta potential, further corroborate the possibility of tuning, engineering, and suppressing finite-system analogues of a solid-solid transition in nanoalloys.

Smart darting diffusion Monte Carlo: Applications to lithium ion-Stockmayer clusters

In a recent investigation [K. Roberts et al., J. Chem. Phys. 136, 074104 (2012)], we have shown that, for a sufficiently complex potential, the Diffusion Monte Carlo (DMC) random walk can become quasiergodic, and we have introduced smart darting-like moves to improve the sampling. In this article, we systematically characterize the bias that smart darting moves introduce in the estimate of the ground state energy of a bosonic system. We then test a simple approach to eliminate completely such bias from the results. The approach is applied for the determination of the ground state of lithium ion-n-dipoles clusters in the n = 8-20 range. For these, the smart darting diffusion Monte Carlo simulations find the same ground state energy and mixed-distribution as the traditional approach for n < 14. In larger systems we find that while the ground state energies agree quantitatively with or without smart darting moves, the mixed-distributions can be significantly different. Some evidence is offered to conclude that introducing smart darting-like moves in traditional DMC simulations may produce a more reliable ground state mixed-distribution.

Grand and Semigrand Canonical Basin-Hopping

We introduce grand and semigrand canonical global optimization approaches using basin-hopping with an acceptance criterion based on the local contribution of each potential energy minimum to the (semi)grand potential. The method is tested using local harmonic vibrational densities of states for atomic clusters as a function of temperature and chemical potential. The predicted global minima switch from dissociated states to clusters for larger values of the chemical potential and lower temperatures, in agreement with the predictions of a model fitted to heat capacity data for selected clusters. Semigrand canonical optimization allows us to identify particularly stable compositions in multicomponent nanoalloys as a function of increasing temperature, whereas the grand canonical potential can produce a useful survey of favorable structures as a byproduct of the global optimization search.

Intrinsically disordered energy landscapes

Analysis of an intrinsically disordered protein (IDP) reveals an underlying multifunnel structure for the energy landscape. We suggest that such 'intrinsically disordered' landscapes, with a number of very different competing low-energy structures, are likely to characterise IDPs, and provide a useful way to address their properties. In particular, IDPs are present in many cellular protein interaction networks, and several questions arise regarding how they bind to partners. Are conformations resembling the bound structure selected for binding, or does further folding occur on binding the partner in a induced-fit fashion? We focus on the p53 upregulated modulator of apoptosis (PUMA) protein, which adopts an α-helical conformation when bound to its partner, and is involved in the activation of apoptosis. Recent experimental evidence shows that folding is not necessary for binding, and supports an induced-fit mechanism. Using a variety of computational approaches we deduce the molecular mechanism behind the instability of the PUMA peptide as a helix in isolation. We find significant barriers between partially folded states and the helix. Our results show that the favoured conformations are molten-globule like, stabilised by charged and hydrophobic contacts, with structures resembling the bound state relatively unpopulated in equilibrium.

Exploring the potential energy landscape of the Thomson problem via Newton homotopies

Locating the stationary points of a real-valued multivariate potential energy function is an important problem in many areas of science. This task generally amounts to solving simultaneous nonlinear systems of equations. While there are several numerical methods that can find many or all stationary points, they each exhibit characteristic problems. Moreover, traditional methods tend to perform poorly near degenerate stationary points with additional zero Hessian eigenvalues. We propose an efficient and robust implementation of the Newton homotopy method, which is capable of quickly sampling a large number of stationary points of a wide range of indices, as well as degenerate stationary points. We demonstrate our approach by applying it to the Thomson problem. We also briefly discuss a possible connection between the present work and Smale's 7th problem.

Superposition-Enhanced Estimation of Optimal Temperature Spacings for Parallel Tempering Simulations

Effective parallel tempering simulations rely crucially on a properly chosen sequence of temperatures. While it is desirable to achieve a uniform exchange acceptance rate across neighboring replicas, finding a set of temperatures that achieves this end is often a difficult task, in particular for systems undergoing phase transitions. Here we present a method for determination of optimal replica spacings, which is based upon knowledge of local minima in the potential energy landscape. Working within the harmonic superposition approximation, we derive an analytic expression for the parallel tempering acceptance rate as a function of the replica temperatures. For a particular system and a given database of minima, we show how this expression can be used to determine optimal temperatures that achieve a desired uniform acceptance rate. We test our strategy for two atomic clusters that exhibit broken ergodicity, demonstrating that our method achieves uniform acceptance as well as significant efficiency gains.

A conformational factorisation approach for estimating the binding free energies of macromolecules

We present a conformational factorization approach. The theory is based on a superposition partition function, decomposed as a sum over contributions from local minima. The factorisation greatly reduces the number of minima that need to be considered, by employing the same local configurations for groups that are sufficiently distant from the binding site. The theory formalises the conditions required to analyse how our definition of the binding site region affects the free energy difference between the apo and holo states. We employ basin-hopping parallel tempering to sample minima that contribute significantly to the partition function, and calculate the binding free energies within the harmonic normal mode approximation. A further significant gain in efficiency is achieved using a recently developed local rigid body framework in both the sampling and the normal mode analysis, which reduces the number of degrees of freedom. We benchmark this approach for human aldose reductase (PDB code 2INE). When varying the size of the rigid region, the free energy difference converges for factorisation of groups at a distance of 14 Å from the binding site, which corresponds to 80% of the protein being locally rigidified.

An inversion-relaxation approach for sampling stationary points of spin model Hamiltonians

Sampling the stationary points of a complicated potential energy landscape is a challenging problem. Here, we introduce a sampling method based on relaxation from stationary points of the highest index of the Hessian matrix. We illustrate how this approach can find all the stationary points for potentials or Hamiltonians bounded from above, which includes a large class of important spin models, and we show that it is far more efficient than previous methods. For potentials unbounded from above, the relaxation part of the method is still efficient in finding minima and transition states, which are usually the primary focus of attention for atomistic systems.

Melting and Freezing of Metal Clusters

Recent developments allow heat capacities to be measured for size-selected clusters isolated in the gas phase. For clusters with tens to hundreds of atoms, the heat capacities determined as a function of temperature usually have a single peak attributed to a melting transition. The melting temperatures and latent heats show large size-dependent fluctuations. In some cases, the melting temperatures change by hundreds of degrees with the addition of a single atom. Theory has played a critical role in understanding the origin of the size-dependent fluctuations, and in understanding the properties of the liquid-like and solid-like states. In some cases, the heat capacities have extra features (an additional peak or a dip) that reveal a more complex behavior than simple melting. In this article we provide a description of the methods used to measure the heat capacities and provide an overview of the experimental and theoretical results obtained for sodium and aluminum clusters.

Elastic Lennard-Jones polymers meet clusters: differences and similarities

We investigate solid-solid and solid-liquid transitions of elastic flexible off-lattice polymers with Lennard-Jones monomer-monomer interaction and anharmonic springs by means of sophisticated variants of multicanonical Monte Carlo methods. We find that the low-temperature behavior depends strongly and nonmonotonically on the system size and exhibits broad similarities to unbound atomic clusters. Particular emphasis is dedicated to the classification of icosahedral and nonicosahedral low-energy polymer morphologies.

Metal clusters with hidden ground states: Melting and structural transitions in Al115(+), Al116(+), and Al117(+)

Heat capacities measured as a function of temperature for Al(115)(+), Al(116)(+), and Al(117)(+) show two well-resolved peaks, at around 450 and 600 K. After being annealed to 523 K (a temperature between the two peaks) or to 773 K (well above both peaks), the high temperature peak remains unchanged but the low temperature peak disappears. After considering the possible explanations, the low temperature peak is attributed to a structural transition and the high temperature peak to the melting of the higher enthalpy structure generated by the structural transition. The annealing results show that the liquid clusters freeze exclusively into the higher enthalpy structure and that the lower enthalpy structure is not accessible from the higher enthalpy one on the timescale of the experiments. We suggest that the low enthalpy structure observed before annealing results from epitaxy, where the smaller clusters act as a nucleus and follow a growth pattern that provides access to the low enthalpy structure. The solid-to-solid transition that leads to the low temperature peak in the heat capacity does not occur under equilibrium but requires a superheated solid.

Quantum disordering versus melting in Lennard-Jones clusters

The ground states of Lennard-Jones clusters for sizes up to n=147 are estimated as a function of the de Boer quantum delocalization length Lambda , and the n-Lambda "phase diagram" is constructed. The increase in Lambda favors more disordered and diffuse structures over more symmetric and compact ones, eventually making the liquidlike motif most energetically favorable. The analogy between the quantum- and thermally-induced structural transitions is explored.