Lennard-Jones Clusters Research Articles

Cluster structure prediction via revised particle-swarm optimization algorithm

The global minimization of cluster structure at the atomic level is emerging as a state-of-the-art approach to accelerate the functionality-driven discovery of cluster-based materials. In this work, we have developed a method for global optimization of Lennard-Jones (LJ), elemental metal and bimetallic clusters through revised particle swarm optimization (RPSO) algorithm within random learning procedure, competition operation and confusion mechanism. The approach requires only size and chemical composition for a given cluster to predict stable or metastable structures at given external conditions. Random learning procedure significantly improves the performance of RPSO such as converge rate and optimization efficiency. Moreover, competition operation can ensure the superiority of outstanding individuals and accept the bad solution with a certain probability. Confusion mechanism allows the clusters to fly in a wider space and makes large-scale optimization possible. Results of optimization based on test functions show that the convergence of RPSO is much faster and more reliable than several other algorithms. In addition, RPSO can also perform well on optimization of Lennard-Jones clusters, Pt and Pt–Pd clusters with various sizes and compositions. The high success rate of RPSO demonstrates the reliability of this methodology and provides crucial insights for understanding the rich and complex structures of clusters.

The Structure and Crystallization Process of Amorphous Iron Nanoparticles

This paper studies the crystallization process and structure of amorphous iron nanoparticles by molecular dynamics method. The study shows that amorphous iron nanoparticles could not be crystallized at 300 K and 500 K. Iron nanoparticle, annealed at 900 K over a long time, was crystallized into a BCC crystal structure. The structure of crystallized iron nanoparticle at 900 K was analyzed through the pair radial distribution function and the number of crystal atoms upon various regions in nanoparticles. The simulation revealed that the first nuclei was formed most frequently in the area near the surface of the nanoparticle. Then the crystal cluster grew toward the centre of the nanoparticle. The completely crystallized nanoparticle had two components: the core with a BCC crystal structure and surface with an amorphous structure. As for the amorphous nanoparticle at 300 or 500 K, crystal-clusters were too small to grow large enough to crystallize the nanoparticle.
 
 Keywords
 Iron nanoparticle, crystallize, annealing, crystal atom, crystal cluster.
 References
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Hybrid advanced player selection strategy based population search for global optimization

Motivation: Finding the optimum solution of real world complex optimization problem needs efficient solver system. Meta-heuristic like swarm, evolutionary or physics based methods essentially form an expert system which may strive for the search of optimum solution. However, these methods have their own merits and demerits. To supersede the performance of existing systems in context of real time and advanced problems and to propose new intelligent method having an optimal balance between exploration and exploitation phase have become a center of focus of the expert system design. Therefore, being inspired from such a scenario, a hybrid advanced player selection strategy based population search (HAPS-PS) technique is proposed and evaluated in this paper in context of modern applications.Methods: Sport is a multi-functional activity where different learning methods like cooperative, competitive and self-learning variants along with interactive environmental features help to improve the performance of the players. The properties of playing activity and the thematic learning process of sports, along with some features like, chaotic map based initialization, pool based topological structure selection, integration of blended laplacian operator etc. contributes in the design of HAPS-PS. Also a novel mechanism of team reformation through elimination of comparatively poor players and recruitment of new talents is employed. It provides diversification and regulates the search agents in a promising direction.Results: We have comprehensively evaluated the performance of the HAPS-PS by applying it on standard benchmark problems and on various shifted rotated, hybrid and composite problems of CEC2013 and CEC2014. Also some of the competitive problems of CEC 2017 are tested using the proposed method and they are also compared with other state-of-the-art algorithm to validate the efficacy of the system. Further, tests are carried out on some real-world problems from CEC2011 which includes applications related to parameter optimization of frequency-modulated (FM) sound wave, minimization of molecular potential energy of Lennard-Jones cluster, controlling the chemical reactors, and evaluation of the inter atomic tersoff potentials for covalent systems of silicon. The proposed method is also applied on spread spectrum radar polyphase code design problem as a part of real world application. The comparative results validate the effectiveness and efficiency of HAPS-PS.

Lorentz Excitable Lattice Gas Automata (LELGA) for optimization of Lennard-Jones atomic cluster size up to N≤383

Abstract From the perspective of cellular-automaton and by exploiting the interactions between travelling and localized waves the Lorentz Excitable Lattice Gas Automata (LELGA) is proposed and applied to the optimization of Lennard-Jones (LJ) atomic clusters. Our approach exploits the fact that a reaction-diffusion phenomenon with self-localized excitations cells which behave like quasi-particles, could be potentially used to implement dynamical computation. As an example of collision-based computation in reaction-diffusion systems based on cellular automata we provided our experimental results for unbiased optimization of Lennard-Jones (LJ) atomic clusters. We also demonstrated that the proposed method has successfully located the known global minima of LJ clusters for configuration space size of N ≤ 383.

"Teamwork Makes the Dream Work": Tribal Competition Evolutionary Search as a Surrogate for Free-Energy-Based Structural Predictions.

Crystal structure prediction has been a grand challenge in material science owing to the large configurational space that one must explore. Evolutionary (genetic) algorithms coupled with first principles calculations are commonly used in crystal structure prediction to sample the ground and metastable states of materials based on configurational energies. However, crystal structure predictions at finite temperature ( T), pressure ( P), and composition ( X) require a free-energy-based search that is often computationally expensive and tedious. Here, we introduce a new machine-learning workflow for structure prediction that is based on a concept inspired by the evolution of human tribes in primitive society. Our tribal genetic algorithm (GA) combines configurational sampling with evolutionary optimization to accurately predict entropically stabilized phases at finite ( T, P, X), at a computational cost that is an order of magnitude smaller than that required for a free-energy-based search. In a departure from standard GA techniques, the populations of individuals are divided into multiple tribes based on a bond-order fingerprint, and genetic operations are modified to ensure that cluster configurations are sampled adequately to capture entropic contributions. Team competition introduced into the evolutionary process allows winning teams (representing a better set of individuals) to expand their sizes; this translates into a more expanded search of the phase space allowing us to explore solutions near possible global minimum. Each team explores a specific section of the structural phase space and avoids bias on solutions arising from the use of individual populations in a purely energy-based search. We demonstrate the efficacy of our approach by performing the structural prediction of a representative two-dimensional two-body system as well as Lennard-Jones clusters over a range of temperatures up to its melting point. Our approach outperforms the standard GA approaches and enables structural search under "real nonambient conditions" on both bulk systems and finite-sized clusters.

Single-root networks for describing the potential energy surface of Lennard-Jones clusters

Potential energy surface (PES) holds the key in understanding a number of atomic clusters or molecular phenomena. However, due to the high dimension and incredible complexity of PES, only indirect methods can be used to characterize a PES of a given system in general. In this paper, a branched dynamic lattice searching method was developed to travel the PES, which was described in detail by a single-root network (SRN). The advantage of SRN is that it reflects the topological relation between different conformations and highlights the size of each structure energy trap. On the basis of SRN, to demonstrate how to transform one conformation to another, the transition path that connects two local minima in the PES was constructed. Herein, we take Lennard-Jones (LJ) clusters at the sizes of 38, 55, and 75 as examples. It is found that the PES of these three clusters have many local funnels and each local funnel represents one morphology. If a morphology is located more frequently, it will lie in a larger local funnel. Besides, certain steps of the transition path were generated successfully, such as changing from icosahedral to truncated octahedral of the LJ38-cluster. Though we do not exhibit all the parts of the PES or all transition paths, this method indeed works well in the local area and can be used more widely.

Deviation from equilibrium conditions in molecular dynamic simulations of homogeneous nucleation.

We present a comparison between Monte Carlo (MC) results for homogeneous vapour-liquid nucleation of Lennard-Jones clusters and previously published values from molecular dynamics (MD) simulations. Both the MC and MD methods sample real cluster configuration distributions. In the MD simulations, the extent of the temperature fluctuation is usually controlled with an artificial thermostat rather than with more realistic carrier gas. In this study, not only a primarily velocity scaling thermostat is considered, but also Nosé-Hoover, Berendsen, and stochastic Langevin thermostat methods are covered. The nucleation rates based on a kinetic scheme and the canonical MC calculation serve as a point of reference since they by definition describe an equilibrated system. The studied temperature range is from T = 0.3 to 0.65 ϵ/k. The kinetic scheme reproduces well the isothermal nucleation rates obtained by Wedekind et al. [J. Chem. Phys. 127, 064501 (2007)] using MD simulations with carrier gas. The nucleation rates obtained by artificially thermostatted MD simulations are consistently lower than the reference nucleation rates based on MC calculations. The discrepancy increases up to several orders of magnitude when the density of the nucleating vapour decreases. At low temperatures, the difference to the MC-based reference nucleation rates in some cases exceeds the maximal nonisothermal effect predicted by classical theory of Feder et al. [Adv. Phys. 15, 111 (1966)].

Classical Optimal Control for Energy Minimization Based On Diffeomorphic Modulation under Observable-Response-Preserving Homotopy.

We introduce the so-called "Classical Optimal Control Optimization" (COCO) method for global energy minimization based on the implementation of the diffeomorphic modulation under observable-response-preserving homotopy (DMORPH) gradient algorithm. A probe particle with time-dependent mass m( t;β) and dipole μ( r, t;β) is evolved classically on the potential energy surface V( r) coupled to an electric field E( t;β), as described by the time-dependent density of states represented on a grid, or otherwise as a linear combination of Gaussians generated by the k-means clustering algorithm. Control parameters β defining m( t;β), μ( r, t;β), and E( t;β) are optimized by following the gradients of the energy with respect to β, adapting them to steer the particle toward the global minimum energy configuration. We find that the resulting COCO algorithm is capable of resolving near-degenerate states separated by large energy barriers and successfully locates the global minima of golf potentials on flat and rugged surfaces, previously explored for testing quantum annealing methodologies and the quantum optimal control optimization (QuOCO) method. Preliminary results show successful energy minimization of multidimensional Lennard-Jones clusters. Beyond the analysis of energy minimization in the specific model systems investigated, we anticipate COCO should be valuable for solving minimization problems in general, including optimization of parameters in applications to machine learning and molecular structure determination.

Recognizing Local and Global Structural Motifs at the Atomic Scale

Most of the current understanding of structure-property relations at the molecular and the supramolecular scales can be formulated in terms of the stability of and the interactions between a limited number of recurring structural motifs (e.g., H-bonds, coordination polyhedra, and protein secondary structure). Here we demonstrate an algorithm to automatically recognize such patterns, based on the identification of local maxima in the probability distributions observed in atomistic computer simulations, which is robust to the dimensionality and the sparsity of the reference atomistic data. We first discuss its main features, demonstrating some on artificial data sets, and then show how it can be applied to identify coordination environments in Lennard-Jones clusters and to recognize secondary-structure patterns in the simulation of an oligopeptide. To assess the applicability of this algorithm for motifs that involve several interdependent degrees of freedom, we also employ it to identify groups of conformers of the cluster and the polypeptide, considered in their entirety. The motifs identified by analyzing atomistic simulations can be used to interpret and rationalize the stability and behavior of the system at hand, and also as a tool to accelerate sampling, in association with biased molecular dynamics schemes.

Methods to locate saddle points in complex landscapes.

We present a class of simple algorithms that allows us to find the reaction path in systems with a complex potential energy landscape. The approach does not need any knowledge on the product state and does not require the calculation of any second derivatives. The underlying idea is to use two nearby points in the configuration space to locate the path of the slowest ascent. By introducing a weak noise term, the algorithm is able to find even low-lying saddle points that are not directly reachable by means of the slowest ascent path. Since the algorithm only makes use of the value of the potential and its gradient, the computational effort to find saddle points is linear in the number of degrees of freedom if the potential is short-ranged. We test the performance of the algorithm for three potential energy landscapes. For the Müller-Brown surface, we find that the algorithm always finds the correct saddle point. For the modified Müller-Brown surface, which has a saddle point that is not reachable by means of the slowest ascent path, the algorithm is still able to find this saddle point with high probability. For the case of a three-dimensional Lennard-Jones cluster, the algorithm is able to find the lowest energy barrier with high probability, showing that the method is also efficient in landscapes with many dimensions.

A particle swarm optimization algorithm with random learning mechanism and Levy flight for optimization of atomic clusters

Swarm intelligence optimization algorithms are mainstream algorithms for solving complex optimization problems. Among these algorithms, the particle swarm optimization (PSO) algorithm has the advantages of fast computation speed and few parameters. However, PSO is prone to premature convergence. To solve this problem, we develop a new PSO algorithm (RPSOLF) by combining the characteristics of random learning mechanism and Levy flight. The RPSOLF algorithm increases the diversity of the population by learning from random particles and random walks in Levy flight. On the one hand, we carry out a large number of numerical experiments on benchmark test functions, and compare these results with the PSO algorithm with Levy flight (PSOLF) algorithm and other PSO variants in previous reports. The results show that the optimal solution can be found faster and more efficiently by the RPSOLF algorithm. On the other hand, the RPSOLF algorithm can also be applied to optimize the Lennard-Jones clusters, and the results indicate that the algorithm obtains the optimal structure (2–60 atoms) with an extraordinary high efficiency. In summary, RPSOLF algorithm proposed in our paper is proved to be an extremely effective tool for global optimization.

Rayleigh limit and fragmentation of multiply charged Lennard-Jones clusters: Can charged clusters provide clues to investigate the stability of electrospray droplets?

The fragmentation of multiply charged clusters composed of N≤1000 Lennard-Jones particles augmented with electrostatic interactions is explored by classical Monte Carlo and molecular dynamics simulations with the stated goal of establishing possible analogies with electrospray droplets. Clusters with few charge carriers are shown to be only subject to particle ejection and their Rayleigh limit can be estimated by quantifying the loss of charged particles. On the contrary, uniformly charged clusters can both evaporate particles and undergo fission, making them better candidates to model electrospray droplets. Critical charges delimiting regions of instability of these clusters are defined from the calculation of lower order multipole moments and asymmetry parameters based on the knowledge of moments of inertia. The first discontinuity of quadrupole moments and asymmetry parameters is related to cluster elongation before twofold fission and the corresponding charge is deemed to be a good estimate of the Rayleigh limit. Octopole moments are negligible about this charge, their discontinuities arising at higher charges when threefold fissions are allowed. The size dependence of these critical charges is qualitatively predicted from Rayleigh's formula and the expression of surface energy advocated in liquiddrop models. Deviations below 15% are commonly achieved when comparing Rayleigh limits extracted from experimental data with theoretical predictions based on Monte Carlo simulations or liquiddrop models for a set of eleven atomic and molecular liquid clusters. Although manifold fission of uniformly charged clusters is unlikely close to the Rayleigh limit, successive asymmetric fissions are found to occur in conjunction with other fragmentation mechanisms, including the expansion of ring-shaped structures, at charges more than twice as large as the Rayleigh limit.

Global optimization through a stochastic perturbation of the Polak–Ribière conjugate gradient method

We develop a new modified Polak–Ribière conjugate gradient method by considering a random perturbation. Our approach is suitable for solving a large class of optimization problems on a rectangle of Rn or unconstrained problems. Theoretical results ensure that the proposed method converges to a global minimizer. Numerical experiments are achieved on some typical test problems, particularly the engineering problem of Lennard-Jones clusters. A comparison with well known methods is carried out to show the performance of our algorithm.

Conquering the hard cases of Lennard-Jones clusters with simple recipes

Lennard-Jones clusters are the best-known benchmark for global cluster structure optimization. For a few cluster sizes, the landscape is deceptive, featuring several funnels, with the global minimum not being in the widest one. More than a decade ago, several non-deterministic global search algorithms were presented that could solve these cases, mostly using additional tools to ensure structural diversity. Recently, however, many publications have advertised new search algorithms, claiming efficiency but being unable to solve these harder benchmark cases. Here, we demonstrate that evolutionary algorithms can solve these hard cases efficiently, if enhanced with one of several very different diversity measures (niching) which were set up in an ad-hoc way, without extensive deliberation, testing or tuning. Hence, these hard benchmark cases should definitely be considered solvable. Additionally, these niching concepts offer insights into the different Lennard-Jones structural types, and into the way niching works in evolutionary algorithms.

Formation of regular polyicosahedral structure during growth of large Lennard-Jones clusters from their global minimum

Simulated growth of four global-minimum Lennard-Jones clusters of sizes N=561, 823, 850 and 923, representing multishell icosahedra and decahedron, always leads to formation of regular polyicosahedral clusters. Observation of cluster structure evolution revealed that new atoms form anti-Mackay islands spreading over the cluster surface by making strong island-island junctions at cluster edges. Analysis of potential energies of atoms composing different local structures shows that energy-driven preference for decahedral arrangement of several atoms initiating the junction of pentagonal symmetry on the cluster surface is responsible for kinetic effect in the cluster growth.

Structure diagram of binary Lennard-Jones clusters.

We analyze the structure diagram for binary clusters of Lennard-Jones particles by means of a global optimization approach for a large range of cluster sizes, compositions, and interaction energies and present a publicly accessible database of 180 000 minimal energy structures (http://softmattertheory.lu/clusters.html). We identify a variety of structures such as core-shell clusters, Janus clusters, and clusters in which the minority species is located at the vertices of icosahedra. Such clusters can be synthesized from nanoparticles in agglomeration experiments and used as building blocks in colloidal molecules or crystals. We discuss the factors that determine the formation of clusters with specific structures.

Self-consistent phonons: An accurate and practical method to account for anharmonic effects in equilibrium properties of general classical or quantum many-body systems

The self-consistent phonons (SCP) method is a practical approach for computing structural and dynamical properties of a general quantum or classical many-body system while incorporating anharmonic effects. However, a convincing demonstration of the accuracy of SCP and its advantages over the standard harmonic approximation is still lacking. Here we apply SCP to classical Lennard-Jones (LJ) clusters and compare with numerically exact results. The close agreement between the two reveals that SCP accurately describes structural properties of the classical LJ clusters from zero-temperature (where the method is exact) up to the temperatures at which the chosen cluster conformation becomes unstable. Given the similarities between thermal and quantum fluctuations, both physically and within the SCP ansatz, the accuracy of classical SCP over a range of temperatures suggests that quantum SCP is also accurate over a range of quantum de Boer parameter Λ=ℏ/(σmε), which describes the degree of quantum character of the system.

Anisotropy effect of multi-center Lennard-Jones molecular clusters

Models of randomly packed hard molecules of Multi-Center Lennard-Jones (MCLJ) clusters exhibit some features of anisotropy effects. To investigate the anisotropy effect on the structures of molecular clusters, four highly symmetrical molecules are modeled: 4-atom tetrahedron (TLJ), 6-atom octahedron (OLJ), 8-atom cube (CLJ), and 12-atom icosahedron (ILJ). The intermolecular interactions are described using the MCLJ model. Using the funnel hopping algorithm, we located the putative global minimum structures of these four MCLJ molecular clusters up to cluster size N=80. The structural patterns of these MCLJ clusters are much different to that of Lennard-Jones clusters, and some new rules and structures are found. The structures of TLJ clusters are most irregular due to the highest anisotropy effect of the tetrahedral molecule. For OLJ clusters, a new structural pattern, rhombic dodecahedron, is viewed. The structures of CLJ clusters are also interesting, which look like oblique face-centered cubic (fcc) structures. For ILJ clusters, icosahedral motifs are favored at cluster size N⩽19, and fcc motifs are favored more at N>19.

Improving the Local Search Ability of Spider Monkey Optimization Algorithm Using Quadratic Approximation for Unconstrained Optimization

Spider monkey optimization (SMO) algorithm, which simulates the food searching behavior of a swarm of spider monkeys, is a new addition to the class of swarm intelligent techniques for solving unconstrained optimization problems. The purpose of this article is to study the performance of SMO after incorporating quadratic approximation (QA) operator in it. The proposed version is named as QA‐based spider monkey optimization (QASMO). An experimental study has been carried out to check the validity and applicability of QASMO. For validation purpose, the performance of QASMO is tested over a benchmark set of 46 scalable and nonscalable problems, and results are compared with the original SMO algorithm. In order to test the applicability of the proposed algorithm in solving real‐life optimization problems, one of the most challenging optimization problems, namely, Lennard–Jones (LJ) problem is considered. LJ clusters containing atoms from three to ten have been taken into consideration, and results are presented. To the best of our knowledge, this is the first attempt to apply SMO and its proposed variant on a real‐life problem. The results demonstrate that incorporation of QA in SMO has positive effects on its performance in terms of reliability, efficiency, and accuracy.

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 (CV) of model nanoclusters. Each anomaly is linked to competing solidlike "phases", where crossover of the corresponding free energies defines a solid-solid transition temperature (Ts). For selected Lennard-Jones clusters we show that Ts and the corresponding CV 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 CV peaks emerging below Ts, 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.