Lennard-Jones Atomic Clusters Research Articles

Impurity effects on phase change in Lennard-Jones atomic clusters

Impurity effects on phase change in Lennard-Jones atomic clusters

Optimization and Symmetry Property of Lennard-Jones Atomic Clusters Using Cooperative Quantum Particle Swarm Algorithm with Lévy Flights

Optimization and Symmetry Property of Lennard-Jones Atomic Clusters Using Cooperative Quantum Particle Swarm Algorithm with Lévy Flights

Ternary and quaternary Lennard-Jones atomic clusters: The effects of atomic sizes on the compositions, geometries, and relative stability

Global-minimum geometries of ternary and quaternary Lennard-Jones clusters have been calculated with constraints on atomic compositions of the clusters. In the present study, the constraints were removed to obtain optimal compositions. The size ratios of the largest-sized atom to the smallest-sized one ranged from 1.1 to 1.6 whereas the depths of the interatomic potentials were constant. The heuristic method combined with the geometrical perturbations and atom-type conversion was used to search for the global minima of the clusters with up to 50 atoms. The smallest-sized and largest-sized atoms usually occupy cores and outer shells, respectively, and the atoms with intermediate sizes are often lacking. The size ratio has pronounced effects on the compositions, structures, and relative stability of the clusters.

Binary Lennard-Jones atomic clusters: Structural features induced by large-sized atoms

Global-minimum geometries of binary Lennard-Jones clusters (BLJN where N is the number of atoms) are previously elucidated when the size ratio of the large B atom to the small A atom is 1.05–1.3. In the present study, BLJ clusters for larger size ratios (1.4–2.0) are investigated to clarify the structural features. The heuristic method combined with geometrical perturbations and atom-type conversion is developed to search for the global minima of the BLJ clusters. In the lowest-energy geometries of the BLJ clusters (N⩽50), A and B atoms form cores and outer shells, respectively. The existence of fairly large sized atoms induces complicated structural growth sequence patterns (including icosahedron, triangular orthobicupola, cuboctahedron, etc.). New global minima of BLJ59 for s=1.15, BLJ68 for s=1.15 and BLJ70 for s=1.2 are also reported.

Anisotropy effect on global minimum structures of clusters: Two-center Lennard-Jones model

Using a two-center Lennard-Jones (2CLJ) model, the simplest anisotropic case, we investigated how anisotropy affects global minimum structures of clusters and obtained some interesting results. The anisotropy parameter, R, is defined as the ratio of the bond length of 2CLJ dimer to the LJ equilibrium pair separation, where a larger R value means higher anisotropy. For low R values, the structures resemble those of the Lennard-Jones atomic clusters. However, as the pairwise interaction becomes more anisotropic, the "magic numbers" change, and several novel cluster patterns emerge as particularly stable structures, and the global minima change from icosahedral, to polyicosahedral and to novel irregular structures. Moreover, increasing the anisotropy effectively softens the 2CLJ potential. Given the general importance of the LJ cluster as a simple model cluster, 2CLJ model can provide a straightforward and useful analysis of the effect of molecular shape on the structures of clusters.

Novel Method for Geometry Optimization of Molecular Clusters:  Application to Benzene Clusters

A heuristic and unbiased method for searching optimal geometries of clusters of nonspherical molecules was constructed from the algorithm recently proposed for Lennard-Jones atomic clusters. In the method, global minima are searched by using three operators, interior, surface, and orientation operators. The first operator gives a perturbation on a cluster configuration by moving molecules near the center of mass of a cluster, and the second one modifies a cluster configuration by moving molecules to the most stable positions on the surface of a cluster. The moved molecules are selected by employing a contribution of the molecules to the potential energy of a cluster. The third operator randomly changes the orientations of all molecules. The proposed method was applied to benzene clusters. It was possible to find new global minima for (C6H6)11, (C6H6)14, and (C6H6)15. Global minima for (C6H6)16 to (C6H6)30 are first reported in this article.

High-symmetry global minimum geometries for small mixed Ar/Xe Lennard-Jones clusters

We report several new global minima for mixed Ar/Xe Lennard-Jones atomic clusters. The geometries deviate significantly from those predicted by the icosahedral packing which dominates the structure of small homogeneous Lennard-Jones clusters. The structural motif common to all the reported structures is the ArXe 4 square pyramid.

Fast Global Optimization of Difficult Lennard-Jones Clusters

The minimization of the potential energy function of Lennard-Jones atomic clusters has attracted much theoretical as well as computational research in recent years. One reason for this is the practical importance of discovering low energy configurations of clusters of atoms, in view of applications and extensions to molecular conformation researchs another reason of the success of Lennard Jones minimization in the global optimization literature is the fact that this is an extremely easy-to-state problem, yet it poses enormous difficulties for any unbiased global optimization algorithm. In this paper we propose a computational strategy which allowed us to rediscover most putative global optima known in the literature for clusters of up to 80 atoms and for other larger clusters, including the most difficult cluster conformations. The main feature of the proposed approach is the definition of a special purpose local optimization procedure aimed at enlarging the region of attraction of the best atomic configurations. This effect is attained by performing first an optimization of a modified potential function and using the resulting local optimum as a starting point for local optimization of the Lennard Jones potential. Extensive numerical experimentation is presented and discussed, from which it can be immediately inferred that the approach presented in this paper is extremely efficient when applied to the most challenging cluster conformations. Some attempts have also been carried out on larger clusters, which resulted in the discovery of the difficult optimum for the 102 atom cluster and for the very recently discovered new putative optimum for the 98 atom cluster.

The optimal geometry of Lennard-Jones clusters: 148–309

This paper deals with the global optimization problem of determining the n-atom cluster configuration that yields the minimum Lennard-Jones potential energy. To approach this problem we propose a genetic algorithm combined with a stochastic search procedure on icosahedral lattices. Although the potentials obtained with our method for n=148,…,309 are in fact only upper bounds for the global minima, we believe that most of these upper bounds are tight. We provide a geometrical description of the optimal configurations found, whose structures are either icosahedral or Marks decahedral in character. Also, we were able to discover a novel morphology – called FD here – for Lennard-Jones atomic clusters.

An Efficient Deformation-Based Global Optimization Method (Self-Consistent Basin-to-Deformed-Basin Mapping (SCBDBM)). Application to Lennard-Jones Atomic Clusters

A recently proposed method to surmount the multiple-minima problem in protein folding is applied here to global optimization of Lennard-Jones atomic clusters. The method, self-consistent basin-to-deformed-basin mapping (SCBDBM), locates a group of large basins containing low-energy minima (hereafter referred to as superbasins) in the original energy surface by coupling the superbasins in the original surface to basins in a highly deformed energy surface (which contains a significantly reduced number of minima, compared to the original rugged energy surface). Various kinds of deformation based on the distance scaling method (DSM) have been tested. The method was able to locate all the lowest-energy structures of Lennard-Jones atomic clusters with a size of up to 100 atoms, except for clusters of 75−77 atoms. In these cases, the method found the previously known second-to-the-lowest energy structures.

Global optimization using bad derivatives: Derivative-free method for molecular energy minimization

A general method designed to isolate the global minimum of a multidimensional objective function with multiple minima is presented. The algorithm exploits an integral “coarse-graining” transformation of the objective function, U, into a smoothed function with few minima. When the coarse-graining is defined over a cubic neighborhood of length scale ϵ, the exact gradient of the smoothed function, 𝒰ϵ, is a simple three-point finite difference of U. When ϵ is very large, the gradient of 𝒰ϵ appears to be a “bad derivative” of U. Because the gradient of 𝒰ϵ is a simple function of U, minimization on the smoothed surface requires no explicit calculation or differentiation of 𝒰ϵ. The minimization method is “derivative-free” and may be applied to optimization problems involving functions that are not smooth or differentiable. Generalization to functions in high-dimensional space is straightforward. In the context of molecular conformational optimization, the method may be used to minimize the potential energy or, preferably, to maximize the Boltzmann probability function. The algorithm is applied to conformational optimization of a model potential, Lennard–Jones atomic clusters, and a tetrapeptide. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1445–1455, 1998

Smoothing techniques of global optimization: Distance scaling method in searches for most stable Lennard-Jones atomic clusters

Spatial averaging of the potential energy function facilitates the search for the most stable configuration of a molecular system. Recently some global optimization methods of this kind have been designed in the literature that rely on physical phenomena such as diffusion, wave function evolution in quantum mechanics, Smoluchowski dynamics, evolution in temperature of canonical ensembles, etc. In the present article we highlight the fact that all these methods, when applied to the Gaussian distributions of an ensemble, represent special cases of a set of differential equations involving the spatially averaged potential energy. Their structure suggests that the nature's strategy to cope with the global optimization is robust and differs only in the details in particular applications. The strategy consists of going downhill of the averaged potential energy, removing the barriers, and hunting for low energy regions by a selective increasing of the spatial averaging. In this study we explore the deformation of the potential rather than its averaging. The deformation comes from scaling of atomic distances and reduces the barriers even more effectively than the Gaussian averaging. The position and widths of the Gaussian distribution evolve similarly to the Gaussian density annealing (GDA), but we allow elliptical instead of spherical Gaussians as well as branching of the single trajectory of the system into multiple ones. When the temperature reaches 0 K, one has a number of independent Gaussian distributions, each corresponding to a structure and (usually low) energy of the system. The multiple elliptic-Gaussian distance scaling method has been applied to clusters of argon atoms (N=5,…,31), a system serving usually as a benchmark domain. The method found the global minima for all but three clusters (of very low energy). The procedure is 20 or more times less expensive than the GDA one. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 2040–2049, 1997