A dynamic lattice searching method with constructed core for optimization of large Lennard-Jones clusters

A variation of the previous dynamic lattice searching (DLS) method, named as DLS with constructed core (DLSc), was proposed for structural optimization of Lennard-Jones (LJ) clusters. In the new method, the starting random structure is generated with an icosahedron or a decahedron as a core. For a cluster with n shells, the atoms in the inner n - 2 shells are set as a fixed core, and the remaining atoms in the outer 2 shells are optimized by DLS. With applications of DLSc to optimization of LJ100-200 and LJ660-670, it was found that all the putative global minima can be obtained by using the DLSc method, and the method was proved to be high efficient compared with the previous DLS, because the searching space is reduced by the use of the fixed core. However, although DLSc is still an unbiased approach for smaller LJ clusters, it turned out to be biased for large ones. Further works are still needed to make it to be a more general method for cluster optimization problem.     

A dynamic lattice searching method for fast optimization of Lennard-Jones clusters

A highly efficient unbiased global optimization method called dynamic lattice searching (DLS) was proposed. The method starts with a randomly generated local minimum, and finds better solution by a circulation of construction and searching of the dynamic lattice (DL) until the better solution approaches the best solution. The DL is constructed adaptively based on the starting local minimum by searching the possible location sites for an added atom, and the DL searching is implemented by iteratively moving the atom located at the occupied lattice site with the highest energy to the vacant lattice site with the lowest energy. Because the DL can greatly reduce the searching space and the number of the time-consuming local minimization procedures, the proposed DLS method runs at a very high efficiency, especially for the clusters of larger size. The performance of the DLS is investigated in the optimization of Lennard-Jones (LJ) clusters up to 309 atoms, and the structure of the LJ(500) is also predicted. Furthermore, the idea of dynamic lattice can be easily adopted in the optimization of other molecular or atomic clusters. It may be a promising approach to be universally used for structural optimizations in the chemistry field.     

Clever and efficient method for searching optimal geometries of lennard-jones clusters

An unbiased algorithm for determining global minima of Lennard-Jones (LJ) clusters is proposed in the present study. In the algorithm, a global minimum is searched by using two operators: one modifies a cluster configuration by moving atoms to the most stable positions on the surface of a cluster and the other gives a perturbation on a cluster configuration by moving atoms near the center of mass of a cluster. The moved atoms are selected by employing contribution of the atoms to the potential energy of a cluster. It was possible to find new global minima for LJ506, LJ521, LJ536, LJ537, LJ538, and LJ541 together with putative global minima of LJ clusters of 10-561 atoms reported in the literature. This indicates that the present method is clever and efficient for cluster geometry optimization.     

A conformational analysis method for understanding the energy landscapes of clusters.

A newly developed unbiased structural optimization method, named dynamic lattice searching (DLS), is proposed as an approach for conformational analysis of atomic/molecular clusters and used in understanding the energy landscape of large clusters. The structures of clusters are described in terms of the number of basic tetrahedron (BT) units they contain. We found that the hit numbers of different structural motifs in DLS runs is proportional to the number of BTs. A parameter T(max) is defined to limit the maximal number of atoms moved in a structural transition. Results show that T(max) is a key parameter for modulating the efficiency of the DLS method and has a great influence on the hit number of different motifs in DLS runs. Finally, the effect of potential range on the conformational distribution of the (Morse)(98) cluster is also discussed with different potential-range parameters.     

Lennard-Jones团簇最低能量构型的预测

针对Lennard-Jones(LJ)团簇的结构优化问题,在前人工作的基础上,提出了一个新的无偏优化算法,即DLS-TPIO(dynamic lattice searching method with two-phase local searchand interior operation)算法.对LJ2-650,LJ660,LJ665-680这666个实例进行了优化计算.为其中每个实例所找到的构型其势能均达到了剑桥团簇数据库中公布的最好记录.对LJ533与LJ536这两个算例,所达到的势能则优于先前的最好记录.在DLS-TPIO算法中,采用了内部操作,两阶段局部搜索方法以及动态格点搜索方法.在优化的前一阶段,内部操作将若干能量较高的表面原子移入团簇的内部,从而降低团簇的能量,并使其构型逐渐地变为有序.与此同时,两阶段局部搜索方法指导搜索进入更有希望的构型区域.这种做法显著地提高了算法的成功率.在优化的后一阶段,借用动态格点搜索方法对团簇表面原子的位置作进一步优化,以再一次降低团簇的能量.另外,为识别二十面体构型的中心原子,本文给出了一个简单的新方法.相比于文献中一些著名的无偏优化算法,DLS-TPIO算法具...     

Global optimization of Lennard-Jones clusters by a parallel fast annealing evolutionary algorithm

A parallel fast annealing evolutionary algorithm (PFAEA) was presented and applied to optimize Lennard-Jones (LJ) clusters. All the lowest known minima up to LJ(116) with both icosahedral and nonicosahedral structure, including the truncated octahedron of LJ(38), central fcc tetrahedron of LJ(98), the Marks' decahedron of LJ(75)(-)(77), and LJ(102)(-)(104), were located successfully by the unbiased algorithm. PFAEA is a parallel version of fast annealing evolutionary algorithm (FAEA) that combines the aspect of population in genetic algorithm and annealing algorithm with a very fast annealing schedule. A master-slave paradigm is used to parallelize FAEA to improve the efficiency. The performance of PFAEA is studied, and the scaling of execution time with the cluster size is approximately cubic, which is important for larger scale energy minimization systems.     

Evolution of the properties of Al(n)N(n) clusters with size

A global optimization of stoichiometric (AlN)(n) clusters (n = 1-25, 30, 35, ..., 95, 100) has been performed using the basin-hopping (BH) method and describing the interactions with simple and yet realistic interatomic potentials. The results for the smaller isomers agree with those of previous electronic structure calculations, thus validating the present scheme. The lowest-energy isomers found can be classified in three different categories according to their structural motifs: (i) small clusters (n = 2-5), with planar ring structures and 2-fold coordination, (ii) medium clusters (n = 6-40), where a competition between stacked rings and globular-like empty cages exists, and (iii) large clusters (n > 40), large enough to mix different elements of the previous stage. All the atoms in small and medium-sized clusters are in the surface, while large clusters start to display interior atoms. Large clusters display a competition between tetrahedral and octahedral-like features: the former lead to a lower energy interior in the cluster, while the latter allow for surface terminations with a lower energy. All of the properties studied present different regimes according to the above classification. It is of particular interest that the local properties of the interior atoms do converge to the bulk limit. The isomers with n = 6 and 12 are specially stable with respect to the gain or loss of AlN molecules.     

Dynamic lattice searching methods for optimization of clusters

Global optimization of clusters is a subject of intense interest in computational chemistry. Especially for large clusters, locating the global minima is a challenging problem. Two strategies are generally used for the problem, i.e., the stochastic optimization and the static modeling strategy. The former is known as unbiased global optimization method, while the latter is more efficient but biased. This review describes the development of a dynamic lattice searching (DLS) approach. In DLS, the lattices are constructed dynamically and optimization is achieved by searching these lattices. Therefore, DLS possesses the characteristics of both the stochastic and static methods. With the aim of improving the efficiency of DLS for optimization of large clusters, several variants of the method have been developed. The results show that DLS methods may be promising tools for fast modeling of large clusters. With this review, greater interests are expected for global optimization of atomic or molecular clusters.     

Structural transitions and melting in LJ(74-78) Lennard-Jones clusters from adaptive exchange Monte Carlo simulations

Phase changes in Lennard-Jones (LJ) clusters containing between 74 and 78 atoms are investigated by means of exchange Monte Carlo simulations in the canonical ensemble. The replica temperatures are self-adapted to facilitate the convergence. Although the 74- and 78-atom clusters have icosahedral global minima, the clusters with 75-77 atoms have decahedral ground-state structures and they undergo a structural transition to icosahedral minima before melting. The structural transitions are characterized by quenching and by looking at the Q4 and Q6 orientational bond order parameters. The transition temperatures are estimated to be 0.114, 0.065, and 0.074 reduced units for LJ75, LJ76, and LJ77, respectively. These values, their ordering and the associated latent heats are compared with other estimates based on the harmonic superposition approach.     

Structural transition from icosahedra to decahedra of large Lennard-Jones clusters

The lowest icosahedral and decahedral energies of LJ1001-1610 clusters are obtained using a greedy search method (GSM) based on lattice construction. By comparing the lowest energies of icosahedral and decahedral clusters with the same atoms, the structural transition of LJ clusters is studied. Results show that the critical size from icosahedra to decahedra is located at N = 1034. When the cluster size is larger than 1034, the optimal structures are decahedra except the LJ1367-1422 clusters near the magic number, 1402, of icosahedra. However, the energies of icosahedra near the next magic number, 2044, are higher than that of decahedra, which implies that decahedra will be the optimal structure when the cluster size is larger than 1422, even for those clusters near the magic numbers of icosahedra.     

PDECO: Parallel differential evolution for clusters optimization

The optimization of the atomic and molecular clusters with a large number of atoms is a very challenging topic. This article proposes a parallel differential evolution (DE) optimization scheme for large‐scale clusters. It combines a modified DE algorithm with improved genetic operators and a parallel strategy with a migration operator to address the problems of numerous local optima and large computational demanding. Results of Lennard–Jones (LJ) clusters and Gupta‐potential Co clusters show the performance of the algorithm surpasses those in previous researches in terms of successful rate, convergent speed, and global searching ability. The overall performance for large or challenging LJ clusters is enhanced significantly. The average number of local minimizations per hit of the global minima for Co clusters is only about 3–4% of that in previous methods. Some global optima for Co are also updated. We then apply the algorithm to optimize the Pt clusters with Gupta potential from the size 3 to 130 and analyze their electronic properties by density functional theory calculation. The clusters with 13, 38, 54, 75, 108, and 125 atoms are extremely stable and can be taken as the magic numbers for Pt systems. It is interesting that the more stable structures, especially magic‐number ones, tend to have a larger energy gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital. It is also found that the clusters are gradually close to the metal bulk from the size N > 80 and Pt₃₈ is expected to be more active than Pt₇₅ in catalytic reaction. © 2013 Wiley Periodicals, Inc.     

Structures and relative stability of medium- and large-sized silicon clusters. VI. Fullerene cage motifs for low-lying clusters Si(39), Si(40), Si(50), Si(60), Si(70), and Si(80)

We performed a constrained search, combined with density-functional theory optimization, of low-energy geometric structures of silicon clusters Si(39), Si(40), Si(50), Si(60), Si(70), and Si(80). We used fullerene cages as structural motifs to construct initial configurations of endohedral fullerene structures. For Si(39), we examined six endohedral fullerene structures using all six homolog C(34) fullerene isomers as cage motifs. We found that the Si(39) constructed based on the C(34)(C(s):2) cage motif results in a new leading candidate for the lowest-energy structure whose energy is appreciably lower than that of the previously reported leading candidate obtained based on unbiased searches (combined with tight-binding optimization). The C(34)(C(s):2) cage motif also leads to a new candidate for the lowest-energy structure of Si(40) whose energy is notably lower than that of the previously reported leading candidate with outer cage homolog to the C(34)(C(1):1). Low-lying structures of larger silicon clusters Si(50) and Si(60) are also obtained on the basis of preconstructed endohedral fullerene structures. For Si(50), Si(60), and Si(80), the obtained low-energy structures are all notably lower in energy than the lowest-energy silicon structures obtained based on an unbiased search with the empirical Stillinger-Weber potential of silicon. Additionally, we found that the binding energy per atom (or cohesive energy) increases typically >10 meV with addition of every ten Si atoms. This result may be used as an empirical criterion (or the minimal requirement) to identify low-lying silicon clusters with size larger than Si(50).     

Impact of different potentials on the structures and energies of clusters

Cluster studies have attracted much interest in the past decades because of their extraordinary properties. To describe the interaction between atoms or molecules and predict the energies and structures, potential functions are developed. However, different potentials generally produce different structures and energies for a cluster. To study the effect of potentials on the structure of a cluster, He clusters in the size range of 13-140 are investigated by Lennard-Jones (LJ), Pirani, and Hartree-Fock-dispersion individual damping (HFD-ID) potential with dynamic lattice searching (DLS) method. Potential function curves, cluster structures, bonds, and energies of the global minima are compared. The results show that cluster energies decrease with the values of the potential functions, the differences between structures depend upon the disagreements of the potentials, and the preferable motif of a cluster changes from icosahedron to decahedron with the increase of the derivative of the short-range part of the potentials.     

Geometry optimization and conformational analysis of (C60)N clusters using a dynamic lattice-searching method.

A newly developed unbiased global optimization method, named dynamic lattice searching (DLS), is used to locate putative global minima for all (C6O)N clusters with Girifalco potential up to N=150. A simple greedy strategy is adopted for the basic frame, so DLS has a very high convergence speed and may converge at various configurations. As most structures are packed by basic tetrahedra, some sequences are defined by both configurations and the size of the basic tetrahedra. A sequence-based conformational analysis is carried out with the defined sequences by counting the hit number over 10,000 independent DLS runs for all the cases up to N = 5. It was found that the hit rate of a sequence is related to the size of the basic tetrahedra. U.e of this method proved that the Leary tetrahedral sequence is dominant in a certain range of cluster sizes, although the sequence has no potential energy advantage. The calculation results are also consistent with those of annealing experiments at high temperature, both in magic numbers and height of the peaks in the mass spectrum.     

Parallel random tunneling algorithm for structural optimization of Lennard-Jones clusters up to N=330

A random tunneling algorithm (RTA) is derived from the terminal repeller unconstrained subenergy tunneling (TRUST) algorithm, and the parallelization of the RTA is implemented with an island parallel paradigm. Combined with the techniques of angular moving, the parallel random tunneling algorithm (PRTA) is applied to the optimization of Lennard-Jones (LJ) atomic clusters, and all the global minima of LJ clusters with size up to 200 are successfully located. For the optimization of larger cluster, a PRTA with an improved seeding technique is developed and successfully applied to the optimization of LJ151-LJ309. Furthermore, the optimized structures of LJ309-330 with the PRTA, which have not been studied before, are also provided.     

液态Ca7Mg3合金快速凝固过程中团簇结构的形成特性

采用分子动力学方法对液态Ca7Mg3合金凝固过程中团簇结构的形成特性进行了模拟研究. 采用双体分布函数、Honeycutt-Andersen(HA)键型指数法、原子团类型指数法(CTIM)以及遗传跟踪等方法对凝固过程中团簇结构的形成演变特性进行了分析. 结果表明: 在以冷速为1×1012 K·s-1 的快速凝固条件下, 系统形成以1551、1541、1431键型为主的非晶态结构; 二十面体基本原子团(12 0 12 0)在快速凝固过程中对非晶态结构的形成起决定性作用; 在合金凝固过程中, 团簇的稳定性不仅与构成团簇的基本原子团类型有关, 还与中心原子类型以及中心原子之间的连接方式有关. 由于(12 0 12 0)基本原子团能量较低并且在低温具有较好的遗传特性, 基本原子团之间很容易连接在一起组成更大的团簇. 所形成的团簇结构显著不同于那些由气相沉积、离子溅射等方法所获得的团簇结构.     

Predicting structural models for silicon clusters

This article introduces an efficient method to generate structural models for medium-sized silicon clusters. Geometrical information obtained from previous investigations of small clusters is initially sorted and then introduced into our predictor algorithm in order to generate structural models for large clusters. The method predicts geometries whose binding energies are close (95%) to the corresponding value for the ground-state with very low computational cost. These predictions can be used as a very good initial guess for any global optimization algorithm. As a test case, information from clusters up to 14 atoms was used to predict good models for silicon clusters up to 20 atoms. We believe that the new algorithm may enhance the performance of most optimization methods whenever some previous information is available.     

Particle-swarm structure prediction on clusters

We have developed an efficient method for cluster structure prediction based on the generalization of particle swarm optimization (PSO). A local version of PSO algorithm was implemented to utilize a fine exploration of potential energy surface for a given non-periodic system. We have specifically devised a technique of so-called bond characterization matrix (BCM) to allow the proper measure on the structural similarity. The BCM technique was then employed to eliminate similar structures and define the desirable local search spaces. We find that the introduction of point group symmetries into generation of cluster structures enables structural diversity and apparently avoids the generation of liquid-like (or disordered) clusters for large systems, thus considerably improving the structural search efficiency. We have incorporated Metropolis criterion into our method to further enhance the structural evolution towards low-energy regimes of potential energy surfaces. Our method has been extensively benchmarked on Lennard-Jones clusters with different sizes up to 150 atoms and applied into prediction of new structures of medium-sized Li(n) (n = 20, 40, 58) clusters. High search efficiency was achieved, demonstrating the reliability of the current methodology and its promise as a major method on cluster structure prediction.     

具有多体效应的胶体聚团的特征

考察了多体效应（用Ｓｔｕｔｔｏｎ－Ｃｈｅｎ势，ＳＣ）对胶体聚团的影响并与双体（ＬＪ）势下的结果作了比较。研究表明，ＳＣ和ＬＪ势下簇团的性质有其相似的一面，如：随着剪应力的增加，系统里颗粒的平均势能增加，而每个簇团的颗粒数减少；在较强的剪应力场里，簇团沿剪应力方向（Ｘ轴）被明显拉长且其主轴偏离Ｘ轴等。但它们间的差异也是明显的，在剪应力下ＳＣ系统内颗粒排列得更合理，从而使得平均位能比ＬＪ系统低约１－３