Electronic shells and shells of atoms in metallic clusters

Intensity anomalies (magic numbers) have been observed in the mass spectra of sodium clusters containing up to 22 000 atoms. For small clusters (Na _n ,n≤1500) the anomalies appear to be due to the filling of electronic shells (groups of subshells having the same energy). The shells can be characterized rather well by a pseudoquantum-number, indicating the possible existence of a symmetry higher than spherical. The mass spectra of larger clusters (1500≤n≤22 000) are well explained by the completion of icosahedral or cuboctahedral shells of atoms. The fact that the two types of shells (electron and atom) occur in distinct and non-overlapping size intervals might indicate the existence of a “liquid” to “solid” transition in going from small to large clusters.     

Theoretical Study of Structural Symmetry in Ternary Clusters

The geometrical symmetry presents an intriguing theoretical problem in many kinds of clusters. The diversity of geometrical structures is associated with cluster sizes, different model functions and potential parameters, and ternary clusters are investigated to study the relationship between geometrical symmetry and homotopic symmetry. Ternary Lennard-Jones model potential is studied with different parameters, and the putative global minimum structures of A₁₃B₃₀C₁₂ clusters are optimized using an adaptive immune optimization algorithm. The results show that there mainly exist five geometrical symmetry structures, i.e., Mackay icosahedral, fivefold partial Mackay icosahedral, sixfold pancake, partial double Mackay icosahedral, and amorphous structures. Furthermore, the number of bonds is used to distinguish the geometrical symmetry. The importance of geometrical symmetry and homotopic symmetry determined by potential parameters is discussed. It was found that in the optimization it is more important to generate geometrical symmetry than to optimize homotopic symmetry.     

Mackay,Anti-Mackay,Double-Mackay,Pseudo-Mackay,and Related Icosahedral Shell Clusters

Mackay introduced two important crystallographic concepts in a short paper published 40 years ago. One is the icosahedral shell structure (iss) consisting of concentric icosahedra displaying fivefold rotational symmetry. The number of atoms contained within these icosahedral shells and subshells agrees well with the magic numbers in rare gas clusters, (C₆₀) _N molecules, and some metal clusters determined by mass spectroscopy or simulated on energy considerations. The cluster of 55 atoms within the second icosahedral shell occurs frequently and has been called Mackay icosahedron, or simply MI, which occurs not only in various clusters, but also in intermetallic compounds and quasicrystals. The second concept is the hierarchic icosahedral structures caused by the presence of a stacking fault in the fcc packing of the successive triangular faces in the iss. For instance, a fault occurs after the ABC layers resulting an ABCB packing. This is, in fact, a hierarchic icosahedral structure of a core icosahedron connected to 12 outer icosahedra by vertex sharing, or an icosahedron of icosahedra (double MI. Contrary to Mackay's iss, a faulted hierarchic icosahedral shell is, in fact, a twinlike face capping of the underlying triangles; it is, therefore, called an anti-Mackay cluster. The hierarchic icosahedral structure in an Al-Mn-Pd icosahedral quasicrystal has a core of body-centered cube rather than an icosahedron and, therefore, is called a pseudo-Mackay cluster. The hierarchic icosahedral structures have been studied separately in the past in the fields of clusters, nanoparticles, intermetallic compounds, and quasicrystals, but the underlying geometry should be the same. In the following a unified geometrical analysis is presented.     

5d过渡金属原子中心镶嵌Ag团簇M@Ag12(M＝Hf～Hg) Ih和Oh构型的密度泛函理论研究

采用相对论密度泛函理论方法对Ih和Oh构型M@Ag12 (M＝Hf～Hg)的几何和电子结构进行了系统的研究. 研究表明, 原子半径之和与团簇的电子结构共同决定了M—Ag键长的大小. M@Ag12的成键能来自中心原子的嵌入能和Ag12笼子的形变能. 最高占据轨道为成键轨道的团簇比反键轨道的团簇的稳定性强. 我们发现在此系列中, Ih构型不一定总比Oh构型稳定. Hf@Ag12, Ir@Ag12, Au@Ag12和Hg@Ag12的Oh构型比Ih构型稳定.     

Composition-induced structural transitions in mixed Lennard-Jones clusters: global reparametrization and optimization

As extended benchmarks to global cluster structure optimization methods, we provide a first systematic point of entry into the world of strongly mixed rare gas clusters. A new set of generalized Lennard-Jones pair potentials is generated for this purpose, by fitting them to high-end ab initio reference data. Employing these potentials in our genetic algorithm-based global structure optimization framework, we examined various systems from binary to quinary mixtures of atom types. A central result from this study is that the famous fcc structure for 38 atoms can survive for certain binary mixtures but appears to be prone to collapsing into the dominating icosahedral structure, which we observed upon introduction of one single atom of a ternary type.     

Transition between icosahedral and cuboctahedral nanoclusters of lead

We have used ab initio methods to study the possible transition between icosahedral (ico) and cuboctahedral (fcc) structures in lead nanoclusters of sizes up to 309 atoms. Spontaneous fcc-to-ico transition in Pb(13) was observed in the ab initio molecular dynamics (MD) simulations at various temperatures. The transition path can be described predominantly by an angular variable s, which can, generally be applied to the similar transitions in clusters of larger sizes and was observed to follow the Mackay model. We have calculated the two-dimensional energy surface that describes the transition in Pb(13) and found a barrierless fcc-to-ico transition path, which is consistent with the observed spontaneous transition in the ab initio MD simulations. The atomic displacements in the transition were identified as one of the vibrational eigenmodes of these two Pb(13) clusters. For clusters of larger sizes (Pb(n), where n = 55, 147, and 309), the possible transitions following similar paths were determined not to be barrierless and the sizes of the barriers were determined by the ab initio elastic band method.     

Theory of the structural fluctuation of Au55 clusters

Stabilities and structural fluctuations of both neutral and charged Au₅₅ clusters are examined and discussed in relation to recent experimental observations of small gold particles with an electron microscope. Transition probabilities between the icosahedral and cuboctahedral structures are calculated according to the transition state theory using a model potential consisting of attractive many-body, repulsive pairwise and Coulomb parts. It is shown that for a neutral cluster the cuboctahedral structure has too short life time to be observed around room temperature and that, on the other hand, for more than 6-fold multiply charged clusters, both structures have life times of the order of 0.1 s around room temperature and, therefore, the transition between them can be observed.     

锡原子簇的结构和相对稳定性

基于从固体锡确立的多体展开势能函数, 采用座标直接优化方法预测了锡原子簇分子(Sn_2-Sn_(300))的结构和相对稳定性, 并用蒙特卡罗方法验证了有关小的原子簇(Sn_2-Sn_(15))的所有结果. 优化结果表明: (1)小的锡原子簇分子(Sn_2-Sn_(15))倾向于密堆积结构, 与锡晶体结构无关; (2)中等大小的簇分子, 如Sn_(47), Sn_(71), Sn_(87)及Sn_(147)等, 则呈α-Sn晶体的畸变结构, 其外围各层原子到中心原子的距离受到压缩, 且某些层被劈裂为两层或多层; (3)随着原子簇尺寸的进一步增大, 结构畸变逐渐减弱, 簇分子的单原子平均结合能缓慢增大, 其外椎值大约在Sn_(740)处趋近于α-Sn的结合能.     

多体展开势能函数在碳族元素原子簇研究中的应用III. 锗原子族的结构和相对稳定性

基于从晶体锗确立的多体展开势能函数, 本文通过坐标完全优化, 发现小的锗原子簇分子(Ge~2~Ge~14)倾向于形成密堆积结构, 表面原子分布以蝶形四元环(D~2d)为主; 常见立方晶体“微观晶体碎片”的分层优化结果表明, 在Ge~15~Ge~100范围内, 多数壳层的原子到分子中心的距离均受到压缩, 且以畸变的简单立方、面心立方及体心立方较为稳定; 在这些畸变密堆积结构中, 表面原子向内压缩最为严重, 使整个分子趋于球形化。较为开放的金刚石类层状原子族只有当所含原子数达数百以上时才可能相对更为稳定。     

Stability of relaxed Lennard-Jones models made of 500 to 6000 atoms

We present a study of the stability of clusters models made of a numberN of atoms in the range 500 to 6000 atoms, freely interacting through the Lennard-Jones potential. The potential energy per atom, calculated for relaxed models, shows that stable models belong to an icosahedral sequence whenN<1600 and to a decahedral sequence beyond. A coexistence size range of both structures is discussed in connection with experimental results on argon clusters in free jet expansions.     

The mathematical modelling of cluster geometry

Calculations of minimum energy configurations for aggregates of up to forty atoms, commonly referred to as clusters, are presented. In contrast to previous studies, random initial configurations have been optimised to find the lowest energy structure for a given number of atoms. Three different two-body, bireciprocal potential functions were used in these calculations and in the case of the Lennard-Jones potential, previously calculated results have been confirmed. New structures obtained using softer potentials are also presented. Minimum energy structures of small clusters containing two different types of atoms have also been calculated, and the relationship between the geometry of a cluster and the relative sizes of its constituent atoms examined.     

Continuation calculations of boron- (aluminum-, titanium-, and nickel-) doped La13 clusters

In this work, we have calculated boron-, aluminum-, titanium-, and nickel-doped La13 clusters by DMOL method based on the density-functional theory. Two doping modes are employed: surface and center doping. The boron, aluminum, and nickel atoms prefer to occupy the surface sites while the titanium atom prefers to occupy the center site. The doped La13 clusters with these four kinds of atoms have lower binding energy than pure La13 clusters. The icosahedral isomers are of lower binding energy than cubotahedral and decahedral isomers for La12B(-1), La12Al(-1), and La12Ni, while doping makes the cubotahedral La12Ti stable with a binding energy a little lower than icosahedral La12Ti. There are electronic shell effects in icosahedral La12B(-1) and La12Al(-1). The icosahedral La12B(-1) is promising to be formed during the doped process experimentally. Furthermore, we have also discussed the distorted structures of center doping by bond lengths, density of states, and charge transfers.     

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 transitions and melting of copper clusters

Molecular dynamics is used to study the melting and structural transitions of small copper clusters. The melting temperature is found to be proportional to the average coordination number. Small icosahedral clusters melt at slightly higher temperatures than the cubic structures. Small cuboctahedral clusters are not stable but transform via a nondiffusive transition to icosahedral structure.     

Thermodynamic properties of a shell molecule-gas particles system

A Lennard-Jones potential is used to derive formulaefor the potential of interaction between shell macromolecules and gas atoms. Calculations are carried as far as thespherical symmetry approximation. The potential of interaction determines the stability of a shell topological compound. When the distance R between a shell molecule and a gas atom is equal the radius 1 of the outer shell, the potential is the highest and determines a barrier, which must be lower than an energy of the trapped/by the shell molecule/atom to be released.     

Evolution of the structural stability of large Cu,Ni, Pd,and Ag clusters with size: An analysis within the embedded atom method

A study of the structural stability of clusters made up of a single component has been carried out within the Embedded Atom Method. Perfect icosahedral and cuboctahedral Cu, Ni, Pd, and Ag clusters with up to 5083 atoms have been compared. The icosahedron is found to be the stable structure for small clusters, and a change of structure from icosahedral to cuboctahedral is found as the cluster size increases. A contraction of the interatomic distances results when the cluster size decreases.     

Taming the rugged landscape: production, reordering, and stabilization of selected cluster inherent structures in the X13-nYn system

We present studies of the potential energy landscape of selected binary Lennard-Jones 13 atom clusters. The effect of adding selected impurity atoms to a homogeneous cluster is explored. We analyze the energy landscapes of the studied systems using disconnectivity graphs. The required inherent structures and transition states for the construction of disconnectivity graphs are found by combination of conjugate gradient and eigenvector-following methods. We show that it is possible to controllably induce new structures as well as reorder and stabilize existing structures that are characteristic of higher-lying minima. Moreover, it is shown that the selected structures can have experimentally relevant lifetimes.     

Surveying a potential energy surface by eigenvector-following

We have developed a method to search potential energy surfaces which avoids some of the difficulties associated with trapping in local minima. Steps are directly taken between minima using eigenvector-following. Exploration of this space by low temperature Metropolis Monte Carlo is a useful global optimisation tool. This method successfully finds the lowest energy icosahedral minima of Lennard- Jones clusters from random starting configurations, but cannot find the global minimum in a reasonable time for difficult cases such as the 38-atom Lennard-Jones cluster where the face-centred-cubic truncated octahedron is lowest in energy. However, by performing searches at higher temperatures, we have found a pathway between the truncated octahedron and the lowest energy icosahedral minima. Such a pathway may be illustrative of some of the structural transformations that are observed for supported metal clusters by electron microscopy.     

Geometrical and electronic structure of small copper nanoclusters: XANES and DFT analysis

The geometrical and electronic structure of small copper nanoclusters was studied by density functional theory (DFT) and analysis of X-ray absorption spectra. It was shown that the icosahedral geometry of small copper nanoclusters of 13 atoms was energetically more stable than cuboctahedral geometry. The binding energies in these structures were compared; the theoretical XANES spectra were compared with experiment. The paper gives the results of ab initio calculations of the electronic structure of copper clusters differing in size.     

A density functional molecular dynamics study of the bonding and stability of Mg n clusters (n=2−13)

A study of the structure and the bonding nature of Mg clusters having 2 to 13 atoms has been made using the density functional molecular dynamics method within the local density approximation. The calculated lowest energy structures can be described in terms of a tetrahedron and a trigonal prism. Mg₄ and Mg₁₀ are magic clusters and Mg₁₃ is neither an icosahedron nor a cuboctahedron. The bonding nature varies from atom to atom in a cluster and the transition from weakly bonded dimer to bulk like metallic behaviour is oscillatory and slow.