Dynamic Lattice Searching Research Articles

Application of Optimization Algorithms in Clusters.

The structural characterization of clusters or nanoparticles is essential to rationalize their size and composition-dependent properties. As experiments alone could not provide complete picture of cluster structures, so independent theoretical investigations are needed to find out a detail description of the geometric arrangement and corresponding properties of the clusters. The potential energy surfaces (PES) are explored to find several minima with an ultimate goal of locating the global minima (GM) for the clusters. Optimization algorithms, such as genetic algorithm (GA), basin hopping method and its variants, self-consistent basin-to-deformed-basin mapping, heuristic algorithm combined with the surface and interior operators (HA-SIO), fast annealing evolutionary algorithm (FAEA), random tunneling algorithm (RTA), and dynamic lattice searching (DLS) have been developed to solve the geometrical isomers in pure elemental clusters. Various model or empirical potentials (EPs) as Lennard–Jones (LJ), Born–Mayer, Gupta, Sutton–Chen, and Murrell–Mottram potentials are used to describe the bonding in different type of clusters. Due to existence of a large number of homotops in nanoalloys, genetic algorithm, basin-hopping algorithm, modified adaptive immune optimization algorithm (AIOA), evolutionary algorithm (EA), kick method and Knowledge Led Master Code (KLMC) are also used. In this review the optimization algorithms, computational techniques and accuracy of results obtained by using these mechanisms for different types of clusters will be discussed.

A modified dynamic lattice searching method for structural optimization of metal oxide clusters

Abstract The stable structures of metal oxide clusters were obtained using a modified dynamic lattice searching (DLS) method with a classic rigid-ion model. Due to the important effect of the electrostatic charge force on the structure of metal oxide clusters, the clusters with various formal charges of metal (q), i.e., (M+2O)n, (M3+2.67O4)n, (M2+3O3)n, (M+4O2)n, (M2+5O5)n and (M+6O3)n, were investigated. Results showed that (MO)n clusters preferred to form highly coordinated structures, and (M3O4)n clusters generally adopt cage-based structures. When q value was larger, the lowest energy clusters tended to form some compact structures with one or two dangling M O bonds, even more. Finally, the relationship between the number of M O bond and the q values were discussed.

Stable structures and potential energy surface of the metallic clusters: Ni, Cu, Ag, Au, Pd, and Pt

Metallic clusters have been widely studied due to their special electrical, optical, and catalytic properties. The many-body Gupta potential is applied to describe the interatomic interaction of Ni, Cu, Ag, Au, Pd, and Pt clusters, and their global minimal structures within 100 atoms are optimized using dynamic lattice searching (DLS) method. The configurational distribution of global minima is analyzed, and the geometrical difference among these clusters is demonstrated. Results show that the dominant motif of Ni and Cu clusters is the icosahedron, and in Ag and Au clusters the number of decahedra is slightly larger than that of the icosahedra. However, more face-centered cubic (fcc), stacking fault fcc, and amorphous structures are formed in Au clusters than in Ag clusters. Furthermore, the main motif of Pd and Pt clusters is the decahedron. In particular, Ni98 adopts a Leary tetrahedral motif, and Pt54 is a central vacant icosahedron. The difference related to the potential parameters of these metallic clusters is further investigated by energy analysis. Moreover, the potential energy surfaces (PES) of 38-atom metallic clusters is characterized in terms of conformational analysis. It was found that the sequence of the number of local minima on the PES from large to low is Ni, Cu, Ag, Pt, Pd, and Au.

A modified adaptive immune optimization algorithm for geometrical optimization of Pd-Pt clusters

Bimetallic Pd-Pt clusters have attracted wide interest because of their special catalytic, optical, electronic, and magnetic properties. However, the geometrical optimization of Pd-Pt cluster has been a difficult task due to the homotopic problem, i.e., in some binary clusters, these clusters are identical in configuration, but different in relative arrangement of two types of atoms. For a fixed geometrical configuration the iterated local search（ILS) method is adopted to search the optimal homotop. By the combination of the merit of heuristic optimization algorithm and the idea of dynamic lattice searching（DLS), an adaptive immune optimization algorithm（AIOA) is modified, and the modified AIOA is called AIOA-BDLS-ILS method. To evaluate the efficiency of the improved method, the optimization of binary Lennard-Jones clusters up to 100 atoms is performed. The Results show that the CPU time for one hit of the global minima is less than 5000 s for all clusters and it is less than 1000 s for most clusters. Compared with previously reported BDLS-ILS method, the proposed method is very efficient. The method is thus proved to be efficient. It can be deduced that the method should be a universal algorithm for the fast optimization of binary or bimetallic clusters. Furthermore, the Gupta potential is used to describe the interatomic interactions in Pd-Pt clusters, which is based on the second moment approximation to tight binding theory, and the corresponding potential parameters are fitted to the experimental values of cohesive energy, lattice constant, and elastic constants for the face centered cubic crystal structure at 0 K. The structural optimizations of Pd-Pt clusters with 34, 50 and 79 atoms are performed by the AIOA-BDLS-ILS method. Results show that for optimizing the 34-atom Pd-Pt clusters, 12 new structures with lower energies are found. In 34-atom bimetallic Pd-Pt clusters, the motifs can be categorized into five classes, i.e., 12 decahedral structures, 3 decahedral structures with close packing anti-layers, 7 incomplete Mackay icosahedral structures, 6 poly-icosahedral structures, and 5 structures composed of two 19-atom double icosahedra. In 50- and 79-atom Pd-Pt clusters, the structural characteristics and the atomic distributions are analyzed. The results indicate that the decahedral and decahedral structures with close-packed configurations are dominant, and twin face centered cubic and partial icosahedral structures are also found. Moreover, the order parameter is adopted to analyze the distributions of different types of atoms in Pd-Pt clusters, which are calculated by the average distance of Pd or Pt atoms from the center of a cluster. The results show that there exists the segregation phenomenon of Pd and Pt atoms in Pd-Pt clusters, i.e., Pd atoms tend to occupy the surface sites, and Pt atoms prefer to occupy the inner core sites. This is explained by the lower surface energy of Pd（125-131 meV-2) than that of Pt（155-159 meV-2).

Fast optimization of binary clusters using a novel dynamic lattice searching method.

Global optimization of binary clusters has been a difficult task despite of much effort and many efficient methods. Directing toward two types of elements (i.e., homotop problem) in binary clusters, two classes of virtual dynamic lattices are constructed and a modified dynamic lattice searching (DLS) method, i.e., binary DLS (BDLS) method, is developed. However, it was found that the BDLS can only be utilized for the optimization of binary clusters with small sizes because homotop problem is hard to be solved without atomic exchange operation. Therefore, the iterated local search (ILS) method is adopted to solve homotop problem and an efficient method based on the BDLS method and ILS, named as BDLS-ILS, is presented for global optimization of binary clusters. In order to assess the efficiency of the proposed method, binary Lennard-Jones clusters with up to 100 atoms are investigated. Results show that the method is proved to be efficient. Furthermore, the BDLS-ILS method is also adopted to study the geometrical structures of (AuPd)79 clusters with DFT-fit parameters of Gupta potential.

Geometrical structures and energetics of gold clusters from Au13 to Au300

The growth pattern of gold clusters containing up to 300 atoms was studied. The interatomic interaction is modeled by Gupta potential with density-functional-theory-fitted parameters. The stable structures of Au13–300 clusters are obtained by dynamic lattice searching (DLS) and DLS with constructed cores (DLSc) methods. In the optimized structures of Au13–88 clusters, most of clusters adopt the decahedral motifs, except for twelve face-centered cubic (FCC) and stacking fault (SF) FCC motifs. With respect to the wide appearance of the three kinds of motifs, corresponding inner cores are constructed in the DLSc method to optimize Au90–300 clusters with several tens number of atoms. Results show that the major motifs are FCC and SF FCC, except for decahedra at Au110, Au160, Au180, Au190, and Au230. Furthermore, the ratio of SF FCC structures is obviously increasing, it can, thus, be deduced that SF FCC motif is more stable in large size gold clusters.

An adaptive immune optimization algorithm with dynamic lattice searching operation for fast optimization of atomic clusters

Geometrical optimization of atomic clusters is performed by a development of adaptive immune optimization algorithm (AIOA) with dynamic lattice searching (DLS) operation (AIOA-DLS method). By a cycle of construction and searching of the dynamic lattice (DL), DLS algorithm rapidly makes the clusters more regular and greatly reduces the potential energy. DLS can thus be used as an operation acting on the new individuals after mutation operation in AIOA to improve the performance of the AIOA. The AIOA-DLS method combines the merit of evolutionary algorithm and idea of dynamic lattice. The performance of the proposed method is investigated in the optimization of Lennard-Jones clusters within 250 atoms and silver clusters described by many-body Gupta potential within 150 atoms. Results reported in the literature are reproduced, and the motif of Ag61 cluster is found to be stacking-fault face-centered cubic, whose energy is lower than that of previously obtained icosahedron.

Geometrical structures of gold clusters on Gupta and Sutton-Chen potentials

The putative stable structures of Au clusters are investigated up to 200 atoms. The interaction of Au atoms is described by Gupta potential and Sutton-Chen potential with experimental-fitted parameters and Sutton-Chen potential with density-functional-theory (DFT)-fitted parameters. The stable structures of Au clusters are optimized by dynamic lattice searching (DLS) and DLS with constructed cores (DLSc) methods. The topological geometries, growth patterns, and energies of Au3–38 clusters for both potentials are studied, and their difference could be found from the second finite difference of the energy. Furthermore, for both potentials of Au100–200 clusters, the dominant motif is decahedron, except for face-centered-cubic (fcc) structures at Au150 and Au160 and stacking fault (SF) fcc motif at Au130 and Au140 of Sutton-Chen potential.

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 of Atomic Clusters Using a Heuristic Method with Dynamic Lattice Searching

In this paper, a global optimization method is presented to determine the global-minimum structures of atomic clusters, where several already existing techniques are combined, such as the dynamic lattice searching method and two-phase local minimization method. The present method is applied to some selected large-sized Lennard-Jones (LJ) clusters and silver clusters described by the Gupta potential in the size range N = 13-140 and 300. Comparison with the results reported in the literature shows that the method is highly efficient and a lot of new global minima missed in previous papers are found for the silver clusters. The method may be a promising tool for the theoretical determination of ground-state structure of atomic clusters. Additionally, the stabilities of silver clusters are also analyzed and it is found that in the size range N = 13-140 there exist 12 particularly stable clusters.

Structural optimization of silver clusters from [formula omitted] to [formula omitted] using a Modified Dynamic Lattice Searching method with Constructed core

The structures of silver clusters from Ag 141 to Ag 310 are systematically optimized with a modified version of the dynamic lattice searching method with constructed core (DLSc). The interaction between silver atoms is modeled by the Gupta potential. The optimized results indicate that the present method reproduces the known global minima for 22 clusters and improves the known global minima for 19 clusters. For the other clusters under investigation, putative global minima are first reported in this Letter. It is found that the definition of the energy of a single atom is a key factor affecting the efficiency of the dynamic lattice searching method.

Growth Pattern of Truncated Octahedra in AlN (N ≤ 310) Clusters

The growth sequence of aluminum clusters containing up to 310 atoms was studied. The interaction of aluminum atoms is modeled by the NP-B potential fitted by highly accurate electronic structure datum for aluminum clusters and nanoparticles. The putative global minimum structures of Al(2-310) clusters are obtained by dynamic lattice searching (DLS) and DLS with constructed cores (DLSc) method. Lower energy structures of Al(63) and Al(64) were found in comparison with the previously reported Al(2-65) clusters. In the optimized structures of Al(63-310), all clusters are identified as truncated octahedra (TO) except for five decahedral structures at Al(64), Al(72), Al(74), Al(76), and Al(101), four stacking fault face-centered cubic structures at Al(91), Al(99), Al(129), and Al(135), and one icosahedron at Al(147). Therefore, the structural transition from small clusters to bulk metal may occur around Al(65). At the same time, the results show that aluminum clusters adopt TO growth pattern, and the growth is found to be based on six complete TO at Al(38), Al(79), Al(116), Al(140), Al(201), and Al(260).

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.

A dynamic lattice searching method with interior operation for unbiased optimization of large Lennard‐Jones clusters

For improving the efficiency of dynamic lattice searching (DLS) method for unbiased optimization of large Lennard-Jones (LJ) clusters, a variant of the interior operation (IO) proposed by Takeuchi was combined with DLS. The method is named as DLS-IO. In the method, the IO moves outer atoms with higher energy toward the coordinates center, i.e., (0, 0, 0), of a cluster and a local minimization (LM) follows each IO. This makes the interior atoms more compact and the outer atoms more uniformly distributed with lower potential energy. Therefore, the starting structure for DLS operations is closer to the global optimum compared with the randomly generated structures. On the other hand, a method to identify the central atom is proposed for the central vacancy problem. Optimizations of LJ(500), LJ(561), LJ(660), LJ(665), and LJ(670) were investigated with the DLS-IO, and the structural transition during the optimization was analyzed. It was found that the method is efficient and unbiased for optimization of large LJ clusters, and it may be a promising approach to be universally used for structural optimizations.

Structural Variation of Silver Clusters from Ag13 to Ag160

The structures of silver clusters from Ag(121) to Ag(160) were optimized with a modified dynamic lattice searching (DLS) method, named as DLS with constructed core (DLSc). The interaction among silver atoms is modeled by the Gupta potential. Structural characteristic of silver clusters with the growth of cluster size is investigated with the newly optimized structures and our previous results from Ag(13) to Ag(120). A set of amorphous structures was obtained in the size range of 13-48, together with several ordered structures. The putative stable motif is an icosahedron from Ag(49) to Ag(61) and then changes to a decahedron in the size range of 62-160. Some of the results are consistent with experiments. Furthermore, it was also found that, for clusters with decahedral motif, the stable structure is a result of the competition among the different Marks decahedral motifs. On the other hand, different from the Lennard-Jones cluster, there are some silver clusters with the face-centered cubic (fcc) motif in the size range of 13-160. But the fcc motif can only be obtained for some specific sizes.

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 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.

Structural optimization of silver clusters from Ag 61 to Ag 120 by dynamic lattice searching method

Structures of silver clusters from Ag 61 to Ag 120 were optimized with an unbiased global optimization algorithm named dynamic lattice searching (DLS). The interaction among silver atoms is modeled by the Gupta potential. New global minima of Ag 79 and Ag 80 were found. The results show that there are two magic number clusters Ag 75 and Ag 101 from Ag 61 to Ag 120. Most of the clusters in the studied sizes have decahedral motifs, however, there are 9 clusters with non-decahedral pattern between the two neighboring magic numbers. These results might help us understand the growth rules of medium sized silver clusters.

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.

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.