Size‐guided multi‐seed heuristic method for geometry optimization of clusters: Application to benzene clusters

Since searching for the global minimum on the potential energy surface of a cluster is very difficult, many geometry optimization methods have been proposed, in which initial geometries are randomly generated and subsequently improved with different algorithms. In this study, a size‐guided multi‐seed heuristic method is developed and applied to benzene clusters. It produces initial configurations of the cluster with n molecules from the lowest‐energy configurations of the cluster with n − 1 molecules (seeds). The initial geometries are further optimized with the geometrical perturbations previously used for molecular clusters. These steps are repeated until the size n satisfies a predefined one. The method locates putative global minima of benzene clusters with up to 65 molecules. The performance of the method is discussed using the computational cost, rates to locate the global minima, and energies of initial geometries. © 2018 Wiley Periodicals, Inc.     

Nickel and copper doped palladium clusters from a first-principles perspective

A complete study on the evolution of structures and the variation of the energy properties of MPd_n−1 (M = Ni and Cu; n = 2-13) clusters is presented. The study was performed employing auxiliary density functional theory. The obtained results were compared with the results of Pd_n clusters studied with the same methodology. For each cluster size, several structures were studied to determine the lowest energy structures. The initial structures for the geometry optimization were taken along ab initio Born-Oppenheimer molecular dynamics trajectories. Different potentials energy surfaces were studied. All cluster structures were fully optimized without any symmetry restriction. Stable structures, frequencies, spin multiplicities, averaged bond lengths, spin density plots, different energy properties, dipole and magnetic moments as well as charge transfers are reported. This investigation indicates that the palladium clusters doped with a Ni atom are the most stable and potentially the most chemical active ones.     

Stochastic search of the quantum conformational space of small lithium and bimetallic lithium-sodium clusters

In this paper we report the results obtained by an implementation and application of the simulated annealing optimization procedure to the exploration of the conformational space of small neutral and charged lithium clusters (Li(n)(q), n = 5, 6, 7; q = 0, +/-1) and of the bimetallic lithium/sodium clusters (Li5Na) in their lowest spin states. Our methodology eliminates the structure guessing procedure in the process of generating cluster configurations. We evaluate the quantum energy, typically with the Hartree-Fock Hamiltonian, of randomly generated points in the conformational space and use a modified Metropolis test in the annealing algorithm to generate candidate structures for atomic clusters. The structures are further optimized by analytical methods (gradient following) at the M?ller-Plesset second order perturbation theory level (MP2), in conjunction with basis sets including polarization functions with and without diffuse functions. High accuracy ab initio energies at the coupled clusters level, with single, double, and triple substitutions from the Hartree-Fock determinant (CCSD(T)), on the MP2 geometries were calculated and used to establish the relative stability of the isomers within each potential energy surface. Various cluster properties were computed and compared to existing values in order to validate our methods. Our results show excellent agreement with previous experimental and theoretical reports. Even at these small sizes, evidence for 10 new structures never reported before for the lithium clusters and four new structures for the bimetallic clusters is presented.     

Global search for minimum energy (H2O)n clusters, n = 3-5

The Gaussian-3 (G3) model chemistry method has been used to calculate the relative deltaG(o) values for all possible conformers of neutral clusters of water, (H2O)n, where n = 3-5. A complete 12-fold conformational search around each hydrogen bond produced 144, 1728, and 20,736 initial starting structures of the water trimer, tetramer, and pentamer. These structures were optimized with PM3, followed by HF/6-31G* optimization, and then with the G3 model chemistry. Only two trimers are present on the G3 potential energy hypersurface. We identified 5 tetramers and 10 pentamers on the potential energy and free-energy hypersurfaces at 298 K. None of these 17 structures were linear; all linear starting models folded into cyclic or three-dimensional structures. The cyclic pentamer is the most stable isomer at 298 K. On the basis of this and previous studies, we expect the cyclic tetramers and pentamers to be the most significant cyclic water clusters in the atmosphere.     

Size-dependent properties of Zn(m)S(n) clusters: a density-functional tight-binding study

We present the results of our theoretical calculations on structural and electronic properties of ligand-free Zn(n)S(n) [with n ranging from 4 to 104 (0.8-2.0-nm diameter)] clusters as a function of size of the clusters. We have optimized the structure whereby our initial structures are spherical parts of either zinc-blende or wurtzite structure. We have also considered some hollow bubblelike structures. The calculations are performed by using a parametrized linear combination of atomic orbitals-density-functional theory-local-density approximation-tight-binding method. We have focused on the variation of radial distribution function, Mulliken populations, electronic energy levels, band gap, and stability as a function of size for both zinc-blende and wurtzite-derived ZnS clusters. We have also reported the results of some nonstoichiometric Zn(m)S(n) (with m+n=47, 99, 177) clusters of zinc-blende modification.     

An ab initio study of water clusters in gas phase and bulk aqueous media: (H2O)n,n=1–12

Geometries of several clusters of water molecules including single minimum energy structures of n‐mers (n=1–5), several hexamers and two structures of each of heptamer to decamer derived from hexamer cage and hexamer prism were optimized. One structural form of each of 11‐mer and 12‐mer were also studied. The geometry optimization calculations were performed at the RHF/6‐311G* level for all the cases and at the MP2/6‐311++G** level for some selected cases. The optimized cluster geometries were used to calculate total energies of the clusters in gas phase employing the B3LYP density functional method and the 6‐311G* basis set. Frequency analysis was carried out in all the cases to ensure that the optimized geometries corresponded to total energy minima. Zero‐point and thermal free energy corrections were applied for comparison of energies of certain hexamers. The optimized cluster geometries were used to solvate the clusters in bulk water using the polarized continuum model (PCM) of the self‐consistent reaction field (SCRF) theory, the 6‐311G* basis set, and the B3LYP density functional method. For the cases for which MP2/6‐311++G** geometry optimization was performed, solvation calculations in water were also carried out using the B3LYP density functional method, the 6‐311++G** basis set, and the PCM model of SCRF theory, besides the corresponding gas‐phase calculations. It is found that the cage form of water hexamer cluster is most stable in gas phase among the different hexamers, which is in agreement with the earlier theoretical and experimental results. Further, use of a newly defined relative population index (RPI) in terms of successive total energy differences per water molecule for different cluster sizes suggests that stabilities of trimers, hexamers, and nonamers in gas phase and those of hexamers and nonamers in bulk water would be favored while those of pentamer and decamer in both the phases would be relatively disfavored. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 90–104, 2001     

On the simultaneous optimization of pressure and layout for gas permeation membrane systems

We present a methodology for the simultaneous optimization of pressure and network configurations for gas separation membrane permeators. The methodology targets and refines pressure clusters for efficient operation of membrane networks and follows a three-stage strategy. The first stage produces a pressure target curve (PTC) that allows the identification of Pareto optimal pressure cluster combinations. This is followed by a second stage, where the different optimal pressure ratios are used in an optimal search for process structures to identify the performance of the individual clusters. The third stage processes the information generated in the first two stages in a generalized process superstructure model. Throughout the methodology, a modified process synthesis model for membrane network optimization and design is employed which can be optimized robustly using the simulated annealing algorithm. Three illustrative examples are presented to demonstrate the proposed methodology for simultaneous pressure and layout optimization.     

Parametric effects of the potential energy function on the geometrical features of ternary lennard-jones clusters

The impact of parameters in potential function for describing atomic or molecular clusters is complex due to the complicated potential energy surface. Ternary Lennard-Jones (TLJ) A(l)B(m)C(n) clusters with two-body potential are investigated to study the effect of parameters. In the potential, the size parameter (σ(AA)) of A atoms is fixed, and corresponding parameters of B and C atoms (relative to A atoms), i.e., σ(BB)/σ(AA) and σ(CC)/σ(AA) > 1.00, are used to control the atomic interaction among A, B, and C atoms in TLJ clusters. The minimum energy configurations of A(l)B(m)C(n) clusters with different species are optimized by adaptive immune optimization algorithm. Ternary cluster structures, bonds, and energies of the putative minima are studied. The results show that two different structures based on double-icosahedra are found in 30-atom TLJ clusters. Furthermore, with increasing potential size parameters of B and C atoms, A atoms tend to be more compact for the increasing numbers of A-A bonds, but the short-range attractive part in TLJ clusters becomes weaker. To lower the potential energy, B and C atoms grow around the A atoms in pursuit of a compact configuration. The results are also approved in A(l)B(m)C(n) (l + m + n = 9-55) clusters and A(l)B(m)C(n) (l = 13, m + n = 42) clusters.     

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.     

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).     

Structure of hydrogenated silicon clusters. Small clusters

The ground state structures of silicon hydride clusters Si_nH_m containing up to 12 silicon atoms are obtained by numerical modeling. The cluster geometry is optimized for a wide set of initial structures using the MINDO/3 approximation for Monte Carlo simulation of interatomic interactions. The energy of the cluster depending on the content of hydrogen is studied, and it is shown that the Si-H and Si-Si bond energies depend little on the cluster size.     

杂合型全局优化法优化水分子团簇结构

基于遗传算法、快速模拟退火及共轭梯度方法提出了一种快速的杂合型全局优化方法(fast hybrid global optimization algorithm, FHGOA),并将这一方法应用于TIP3P和TIPS2模型水分子团簇(H2O)n结构的优化.在进行TIP3P模型水分子团簇结构的优化过程中,发现了能量比文献值更低的团簇结构,且执行效率有较大提高.把该方法应用到优化TIPS2模型的水分子团簇,发现最优结构和采用TTM2-F模型优化的水分子团簇结构在n < 17时完全相同,为全表面结构;而在n=17、19、22时为单中心水分子笼状结构;在n=25、27时为双中心水分子笼状结构.说明随着团簇中水分子个数的增加,采用TIPS2和TTM2-F势能函数优化的团簇最优结构有相同的变化趋势.     

Binding energy of large icosahedral and cuboctahedral Lennard-Jones clusters

It is widely believed that the lowest energy configurations for small rare gas clusters have icosahedral symmetry. This contrasts with the bulk crystal structures which have cuboctahedral fcc symmetry. It is of interest to understand the transition between this finite and bulk behavior. To model this transition in rare gas clusters we have undertaken optimization studies within the Lennard-Jones pair potential model. Using a combination of Monte Carlo and Partan Search optimization methods, the lowest energy relaxed structures of Lennard-Jones clusters having icosahedral and cuboctahedral symmetry were found. Studies were performed for complete shell clusters ranging in size from one shell having 13 atoms to 14 shells having 10,179 atoms. It was found that the icosahedral structures are lower in energy than the cuboctahedral structures for cluster sizes having 13 shells or fewer. Additional studies were performed using the more accurate Aziz-Chen [HFD-C] pair potential parameterized for argon. The conclusions appear to be relatively insensitive to the form of the potential.     

Electronic and structural shell closure in AgCu and AuCu nanoclusters

The structures of AgCu clusters containing 40 atoms are investigated. The most promising structural families (fcc clusters, capped decahedra, and two types of capped polyicosahedra) are singled out by means of global optimization techniques within an atom-atom potential model. Then, representative clusters of each family are relaxed by means of density-functional methods. It is shown that, for a large majority of compositions, a complex interplay of geometric and electronic shell-closure effects stabilizes a specific polyicosahedral family, whose clusters are much lower in energy and present large HOMO-LUMO gaps. Within this family, geometric and quantum effects concur to favor magic structures associated with core-shell chemical ordering and high symmetry, so that these clusters are very promising from the point of view of their optical properties. Our results also suggest a natural growth pathway of AgCu clusters through high-stability polyicosahedral structures. Results for AuCu clusters of the same size are reported for comparison, showing that the interplay of the different effects is highly material specific.     

Structure,spectroscopy and electronic properties of neutral lattice-like (MgO) n clusters: a study based on a blending of DFT with stochastic algorithms inspired by natural processes

In this article, we explore the feasibility of using stochastic optimization techniques, which are inspired by natural processes, namely simulated annealing (SA) and genetic algorithm (GA) in association with DFT, to find out the global minimum structures of (MgO) _n clusters with n being in the range of 2–15. To check whether the structures are indeed the correct ones, we proceed to do several property calculations like IR-spectroscopic modes, vertical excitation energy, cluster-formation energy, vertical ionization potential, the HOMO–LUMO gap as well as polarizability and hyperpolarizability—both static and dynamic. We emphasize on the point that an initial determination of structure using SA/GA leads to very quick relaxation to structures which are very close to the structures predicted from quantum chemical calculations done from the outsets like DFT. The general pattern of these systems to form beautiful three-dimensional lattice networks is also evident from our study.     

硫原子团簇负离子的螺旋结构

在使用B3LYP密度泛函进行几何构型优化和振动频率计算得到的硫原子团簇负离子的结构中,分子的总能量最低的S9- 到 S13-的同分异构体呈螺旋状构型。另外也计算了螺旋状的Sn- (n = 14~20)的结构。大多数的的硫负离子是链状结构,这与相应中性硫原子团簇的环状构型完全不同。     

Monte Carlo temperature basin paving with effective fragment potential: an efficient and fast method for finding low-energy structures of water clusters (H2O)20 and (H2O)25

Determining low-energy structures of large water clusters is a challenge for any optimization algorithm. In this work, we have developed a new Monte Carlo (MC)-based method, temperature basin paving (TBP), which is related to the well-known basin hopping method. In the TBP method, the Boltzmann weight factor used in MC methods is dynamically modified based on the history of the simulation. The states that are visited more are given a lower probability by increasing their temperatures and vice versa. This allows faster escapes from the states frequently visited in the simulation. We have used the TBP method to find a large number of low-energy minima of water clusters of size 20 and 25. We have found structures energetically same to the global minimum structures known for these two clusters. We have compared the efficiency of this method to the basin-hopping method and found that it can locate the minima faster. Statistical efficiency of the new method has been investigated by running a large number of trajectories. The new method can locate low-energy structures of both the clusters faster than some of the reported algorithms for water clusters and can switch between high energy and low-energy structures multiple times in a simulation illustrating its efficiency. The large number of minima obtained from the simulations is used to get both general and specific features of the minima. The distribution of minima for these two clusters based on the similarity of their oxygen frames shows that the (H(2)O)(20) can have different variety of structures, but for (H(2)O)(25), low-energy structures are mostly cagelike. Several (H(2)O)(25) structures are found with similar energy but with different cage architectures. Noncage structures of (H(2)O)(25) are also found but they are 6-7 kcal/mol higher in energy from the global minimum. The TBP method is likely to play an important role for exploring the complex energy landscape of large molecules.     

Geometry optimizations of benzene clusters using a modified genetic algorithm

A modified genetic algorithm with real-number coding, non-uniform mutation and arithmetical crossover operators was described in this paper. A local minimization was used to improve the final solution obtained by the genetic algorithm. Using the exp-6-1 interatomic energy function, the modified genetic algorithm with local minimization (MGALM) was applied to the geometry optimization problem of small benzene clusters (C6H6)N(N = 2-7) to obtain the global minimum energy structures. MGALM is simple but the structures optimized are comparable to the published results obtained by parallel genetic algorithms.     

PdSi_n(n=1～15)团簇电子结构与光谱性质的理论研究

在卡里普索(CALYPSO)结构预测的基础上,采用密度泛函理论(DFT)B3LYP方法,优化得到PdSi_n(n=1~15)团簇的基态结构,对其电子性质、红外光谱和拉曼光谱进行了讨论.结果表明,PdSi_n(n=1~15)团簇的基态构型随n值的增大由平面结构向立体结构演化;当n≤4时,PdSi_n团簇的红外与拉曼活性在450~500 cm-1范围内表现较好,当n≥5时,PdSi_n团簇的红外与拉曼活性在50~500 cm~(-1)范围内表现较好.     

B2Cn+(n=1～9)团簇的结构及其稳定性

采用密度泛函理论(DFT)的B3LYP泛函, 在6-311G*水平上对B2Cn+(n=1～9)团簇的几何构型和电子结构进行了优化和振动频率计算. 结果表明, 在B2Cn+(n=1～9)团簇的基态构型中, B2C2+、B2C3+为具有D∞h对称性的线形结构, B2C7+为具有Cs对称性的立体环状结构, 其余均为平面构型; 其成键顺序为C—C成键优于B—C 成键, B—C成键优于B—B成键. 进一步得到了B2Cn+(n=1～9)团簇的总能量(ET)、零点能(EZ)、摩尔热容(Cp)、标准熵(S0)以及原子化能(ΔEn+). 其结果显示, 随着n的递增, ET、EZ、Cp、S0和ΔEn+数值均呈现增大趋势, 其中EZ数值呈现近似等梯度的增加趋势. 通过对B2Cn+(n=1～9)团簇基态结构的垂直电子亲合势的研究发现, n为奇数的B2Cn+团簇比n为偶数的稳定.