Optimization of bimetallic Cu–Au and Ag–Au clusters by using a modified adaptive immune optimization algorithm

A modified adaptive immune optimization algorithm (AIOA) is designed for optimization of Cu–Au and Ag–Au bimetallic clusters with Gupta potential. Compared with homoatom clusters, there are homotopic isomers in bimetallic cluster, so atom exchange operation is presented in the modified AIOA. The efficiency of the algorithm is tested by optimization of Cu_nAu_38‐n (0 ≤ n ≤ 38). Results show that all the structures with the putative global minimal energies are successfully located. In the optimization of Ag_nAu_55‐n (0 ≤ n ≤ 55) bimetallic clusters, all the structures with the reported minimal energies are obtained, and 36 structures with even lower potential energies are found. On the other hand, with the optimized structures of Cu_nAu_55‐n, it is shown that all 55‐atom Cu–Au bimetallic clusters are Mackay icosahedra except for Au₅₅, which is a face‐centered cubic (fcc)‐like structure; Cu₅₅, Cu₁₂Au₄₃, and Cu₁Au₅₄ have two‐shell Mackay icosahedral geometries with I_h point group symmetry. © 2009 Wiley Periodicals, Inc. J Comput Chem 2009     

Geometric and topological analysis of icosahedral structures of samson Mg2Zn11 (cP39) Phases, K6Na15Tl18H (cP40), and Tm3In7Co9 (cP46): Nanocluster precursors, self-assembly mechanism, and superstructure ordering

Geometric and topological analysis and 3D reconstruction of self-assembly of icosahedral structures of Samson Mg₂Zn₁₁ clusters (space group Pm[`3]Pm\bar 3, cP39, 10 compounds) and the K<sub>6</sub>Na<sub>15</sub>Tl<sub>18</sub>H and Tm<sub>3</sub>In<sub>7</sub>Co<sub>9.29</sub> structures were performed by computer methods (the TOPOS program package). The complete decomposition of the 3D graph of the crystal structures into cluster substructures showed the existence of the crystal-forming nanocluster precursor A comprising 45 atoms (A-45). The S-6 cluster spacers were identified in Mg<sub>2</sub>Zn<sub>11</sub>, and the S-7 cluster spacers were found in K<sub>6</sub>Na<sub>15</sub>Tl<sub>18</sub>H. In Tm<sub>3</sub>In<sub>7</sub>Co<sub>9.29</sub>, the S-6 and S-7 cluster spacers with the centers statistically occupying the same position were determined. The A-45, S-6 (octahedron), and S-7 (centered octahedron) clusters have symmetry [`3]m\bar 3m. The A-45 nanocluster contains an inner Zn(Zn)₁₂ template icosahedron and an external quasi-spherical shell composed of 32 atoms (deltahedron D32). A-45 is equivalent to the Bergman cluster used as the approximant of the local structure of quasicrystals. For deltahedron D32, the existence of a hierarchical structure was identified as a result of self-assembly involving two types of cyclic clusters: K-7 with an atom in the center of the sixth ring and three-atom cyclic clusters K-3. The atoms of the K-3 and K-7 clusters occupy all possible positions over the 12 vertices and 20 faces of an icosahedron and thereby form an edge net of bonds made of triangles. For the K₆(Na₁₄MTl₁₈) structures (M = Mg, An, Cd, Hg), the cluster nature of superstructure ordering of three chemically different atoms (14Na, M, and 18Tl) over 33 positions of the Zn atoms in the unit cell of the basis Mg₂Zn₁₁(Mg₆Zn₃₃) structure was considered.     

Melting of icosahedral gold nanoclusters from molecular dynamics simulations

Molecular dynamics simulations show that gold clusters with about 600-3000 atoms crystallize into a Mackay icosahedron upon cooling from the liquid. A detailed surface analysis shows that the facets on the surface of the Mackay icosahedral gold clusters soften but do not premelt below the bulk melting temperature. This softening is found to be due to the increasing mobility of vertex and edge atoms with temperature, which leads to inter-layer and intra-layer diffusion, and a shrinkage of the average facet size, so that the average shape of the cluster is nearly spherical at melting.     

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.     

Nickel cluster structure determined from the adsorption of molecular nitrogen: Ni49-Ni71

The geometrical structures of nickel clusters in the size range from 49 to 71 atoms are studied by the chemical probe method. Saturation coverages of molecular nitrogen are determined for each cluster and from this data specific structures are proposed (except for Ni₆₆ and Ni₆₇). The results indicate that icosahedral packing is the dominant structural configuration throughout this size range, in agreement with earlier results based on water and ammonia adsorption. In addition, it seems that for clusters larger than Ni₅₄ the excessive strain in the surface of the 55-atom regular icosahedron often leads to rear-rangements of the surface atoms to relieve that strain. Ni₅₅, in particular, is found to have two isomers, the regular icosahedron and a structure in which a single apex atom is displaced to the center of an opposite face. Ni₇₁ occurs as a 55-atom regular icosahedron with a 16-atom cap. The results suggest that the atoms in the cap adopt an ABA configuration relative to the underlying icosahedron rather than an icosahedral arrangement. For some clusters the saturation with nitrogen causes a small degree of surface reconstruction that leads to the adsorption of additional nitrogen molecules.Work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, under Contract No. W-31-109-Eng-38     

Structural and electronic properties of Al12X+ (X=C, Si, Ge, Sn, and Pb) clusters

Using the first-principles method with the generalized gradient approximation, the authors have studied the structural and electronic properties of Al(12)X(+) (X=C, Si, Ge, Sn, and Pb) clusters in detail. The ground state of Al(12)C(+) is a low symmetry C(s) structure instead of an icosahedron. However, the Si, Ge, Sn, and Pb atom doped cationic clusters favor icosahedral structures. The ground states for Al(12)Si(+) and Al(12)Ge(+) are icosahedra, while the C(5nu) structures optimized from an icosahedron with a vertex capped by a tetravalent atom have the highest binding energy for Al(12)Sn(+) and Al(12)Pb(+) clusters. The I(h) structure and the C(5nu) structure are almost degenerate for Al(12)Ge(+), whose binding energy difference is only 0.03 eV. The electronic properties are altered much by removing an electron from the neutral cluster. The binding strength of a valence electron is enhanced, while the binding energy of the cluster is reduced much. Due to the open electronic shell, the band gaps between the highest occupied molecular orbital and the lowest unoccupied molecular orbital are approximately 0.3 eV for the studied cationic clusters.     

Zn7Cu6: a magic cluster of brass?

Density functional theory was applied to a series of 13-atom Zn–Cu alloy clusters. We did a thorough search for the low-energy isomers by global optimization, plus explicit optimization of all homotops of the icosahedron. Structures of copper rich clusters tend to be compact, often icosahedra, whereas zinc rich clusters ( ${{\rm Zn}_8{\rm Cu}_{5}^{+}}$ , Zn₉Cu₄, Zn₁₁Cu₂) have compact copper cores surrounded by an incomplete shell of solvating Zn atoms. The icosahedral structure, low total energy, and large hardness of Zn₇Cu₆ indicate that it has special stability among Zn _x Cu _y clusters. However, Zn₇Cu₆ has many low lying isomers and a small cohesive energy compared to brass, which suggest that it is not stable in a broader sense.     

Theoretical Analysis of the Mackay Icosahedral Cluster Pd55(PiPr3)12(μ3-CO)20: An Open-Shell 20-Electron Superatom

The electronic structure of the spherical Mackay icosahedral nanosized cluster Pd₅₅(PiPr₃)₁₂(μ₃-CO)₂₀ is analyzed by using DFT calculations. Results reveal that it can be considered as a regular superatom with a “magic” electron count of 20, characterized by a 1S² 1P⁶ 1D¹⁰ 2S² jellium configuration. Its open shell nature is associated with partial occupation of non-jellium, 4d-type, levels located on the interior of the Pd₅₅ kernel. This shows that the superatom model can be used to rationalize the bonding and stability of spherical ligated group 10 clusters, despite their apparent 0-electron count.     

Isolation and Total Structure Determination of an All‐Alkynyl‐Protected Gold Nanocluster Au144

Total structure determination of a ligand‐protected gold nanocluster, Au₁₄₄, has been successfully carried out. The composition of title nanocluster is Au₁₄₄(C≡CAr)₆₀ ( 1 ; Ar=2‐FC₆H₄‐). The cluster 1 exhibits a quasi‐spherical Russian doll‐like architecture, comprising a Au₅₄ two‐shelled Mackay icosahedron (Au₁₂@Au₄₂), which is further enclosed by a Au₆₀ anti‐Mackay icosahedral shell. The Au₁₁₄ kernel is enwrapped by thirty linear ArC≡C‐Au‐C≡CAr staple motifs. The absorption spectrum of 1 shows two bands at 560 and 620 nm. This spectrum is distinctly different from that of thiolated Au₁₄₄, which was predicted to have an almost identical metal kernel and very similar ligands arrangement in 1 . These facts indicate the molecule‐like behavior of 1 and significant involvement of ligands in the electronic structure of 1 . The cluster 1 is hitherto the largest coinage metal nanocluster with atomically precise molecular structure in the alkynyl family. The work not only addresses the concern of structural information of Au₁₄₄, which had been long‐pursued, but also provides an interesting example showing ligand effects on the optical properties of ligand protected metal nanoclusters.     

Insight into the Geometric and Electronic Structures of Gold/Silver Superatomic Clusters Based on Icosahedron M13 Units and Their Alloys

Metal superatomic nanoclusters, with electronic structures similar to those of one certain atom, are an important type of metal clusters. Interestingly, metal clusters with metal cores composed of either icosahedral M₁₃ or icosahedral assemblies always have a greater potential to become superatomic clusters. Furthermore, superatomic clusters with similar electronic compositions could possess various geometric structures, owing to differences in the shells; this provides a deeper understanding of the metal superatomic cluster and the assembly for nanomaterials. Therefore, this review focuses on the geometric and electronic structures of gold/silver superatomic clusters based on icosahedron M₁₃ units and their alloys, which will facilitate the development of various applications of superatomic clusters.     

Icosahedron oligomerization and condensation in intermetallic compounds. Bonding and electronic requirements

Icosahedron-based clustering has been found to be very common in intermetallics, particularly for group 13 and early p-block icosogen elements. Linking of the icosahedral building blocks depends on the valence electron concentrations. Vertex-, edge-, or face-sharing icosahedra occur as the structure compensates for electron deficiency. Some examples of icosahedron-based clusters have been selected for an analysis of the relationships between the structural features (icosahedron oligomerization, atomic defects, etc.) and the bonding and electronic requirements. The extended Hückel method has been used with either a molecular approach or an electronic band structure calculation to rationalize bonding in the intermetallic framework.     

Structural transformations in silver nanoparticles

A molecular dynamics simulation was performed for silver clusters of 147, 309, and 561 atoms with the initial cuboctahedral habit in the temperature range 0–1000 K with an embedded atom potential for silver. Structural transitions of the silver clusters to complex twins (icosahedral habit) with coherent (111)/(111) boundaries over all edges of icosahedra were found, which started at temperatures of 50 K, 350 K, and 700 K, respectively. To analyze the structural transformations in nanoparticles, an algorithm is proposed based on a simplicial Delaunay decomposition (Delaunay triangulation). It was found that after the transition of silver nanoparticles to complex twins, the atomic motion becomes vibrational; the atoms vibrate around the sites that correspond to the vertices of the regular polyhedra. In the case of the 147-atom silver nanoparticle, the polyhedra are arranged in the following sequence, starting from the center of mass: icosahedron (12 atoms), icosododecahedron (30 atoms), icosahedron (12 atoms), dodecahedron (20 atoms), truncated icosahedron (60 atoms, isostructural with fullerene C₆₀), icosahedron (12 atoms), and one atom at the center of mass.     

Thermally-induced surface reconstructions of Mackay icosahedra

Simulations of 561- and 923-atom Mackay icosahedra reveal that at energies below the melting point vacancies are generated in the surface layer of the icosahedra at the vertices and edges, and that the resulting adatoms diffuse across the surface. However, in contrast to the behaviour of smaller Mackay icosahedra, there is also a surface reconstruction before melting, in which the atoms of the surface layer occupy the hexagonal close-packed sites with respect to the twenty face-centred-cubic tetrahedra which make up the Mackay icosahedron.     

Heteroatomic Centering of Icosahedral Clusters. Crystal and Electronic Structure of the K(6)(NaCd)(2)Tl(12)Cd Compound Containing the Not-So-Naked Tl(12)Cd(12-) Polyanion

The title compound was synthesized in a niobium container by fusion of the elements followed by slow cooling. In the first stage, the stoichiometric proportion KNaCd(3)Tl(7) yielded a heterogeneous product containing a few single crystals of the compound K(6)(Na(2.36(9))Cd(1.64(9)))Tl(12)Cd, the structure of which was established by a single crystal X-ray diffraction technique (cubic, Im&thremacr;, a = 11.352(2) ?, Z = 2, R(F) = 3.24%, Rw(F) = 4.60%). Occurrence of a stoichiometry range for the compound was indicated after a new reaction starting from the composition K(6)Na(2)Cd(3)Tl(12) gave a quite homogeneous and well-crystallized product (refined composition K(6)(Na(1.93(7))Cd(2.07(7)))Tl(12)Cd, Im&thremacr;, a = 11.321(2) ?, Z = 2, R(F) = 3.98%, Rw(F) = 4.99%). The structure of K(6)(NaCd)(2)Tl(12)Cd is distinguishable from that reported for Na(4)K(6)Tl(13) by replacement of the icosahedron centering thallium and of half the sodium cations by cadmium. Statistical occupation disorder occurs on the 8(c) position of the outer Cd/Na atom. The structure contains the 50-electron closed shell centered Tl(12)Cd(12-) icosahedral cluster with &thremacr;m symmetry (T(h)). Extended Hückel molecular orbital and band calculations were carried out to analyze the centering effect on the anion stability and look at the electron transfer, especially from cadmium lying within the first coordination shell of the icosahedral cluster. Electron localization within the Cd-centered icosahedron is not as evident as in the Tl-centered thallium icosahedral clusters described elsewhere; actually, cadmium is found to bridge icosahedra within a more three-dimensional network than sodium by forming bonds that are mainly covalent. The compound is a semiconducting Zintl phase with closed shell bonding.     

SOME NOVEL Au-Ag SUPRACLUSTERS: THE SYNTHESES AND CHAR- ACTERIZATION OF 25-ATOM,37-ATOM,AND 38-ATOM CLUSTERS

<正> The reduction of mixtures of mononuclear Au(I)and Ag(I) phosphine halide complexes with sodium,boronhydride in different solvents gave rise to two types of 25-atom clusters,and 37-atom and 38-atom clusters. These clusters were formed by vertex-sharing of Au-centered icosahedral cluster units (Au7Ag6). The nuclearity of these clusters is given by (13n-e) , where n is the number of the cluster units and e is the edges of the polyhedron formed by centers of the icosahedral cluster units . The structures of these novel 25-atom,37-atom and 38-atom clusters can be described as two icosahedra sharing one vertex (2×13-1 = 25)or three icosahedra sharing three vertices in a triangle(3×13-3 = 36)plus capping atom(s).     

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.     

First principles study of the carbon-(silicon-) doped La13 clusters

The structural stability and physical properties have been studied for carbon-(silicon-) doped La(13) clusters using DMOL method based on density-functional theory. Doped La(13) clusters prefer to be icosahedron. Substitutional doping with a carbon or silicon impurity makes some clusters closed electronic shell, especially in icosahedral isomers. Substitutional doping of icosahedral La(13) clusters is found to be favorable at surface sites of clusters, especially for Si-doped La(13) cluster, which is very likely to be formed during the doping process. In addition, the structural distortions due to the doping are discussed.     

Theory of cluster self-organization for crystal-forming systems: A model of transition from disodered to ordered systems

The current state of ideas concerning the self-organization of crystal-forming systems where long-range order spontaneously appears in the arrangement of nanolevel structural units of any nature (micro- and macromolecules or atomic clusters) that initially existed in a dynamic state as a chaotic mixture is considered. Three partially overlapping stages of self-organization of a system accepted in physical models of “order-disorder” kinetic transitions are matched to those used in supramolecular chemistry. An algorithmically constructed model of transition from disordered to hierarchically ordered systems is considered. The geometrical and topological modeling of density fluctuations of n-atomic species (clusters) An in a crystal-forming medium is carried out. A specific set of An clusters with block-diagonal connectivity matrices is recognized. These types of clusters (S ₃⁰), having “sectional” or “hierarchic” partition, are defined as precursors of crystal structures that are capable of evolving most rapidly to give rise to a long-range order in structures. For an S ₃⁰ ring cluster shaped as a triangle, geometrical and topological modeling is carried out for all of the eight topologically and symmetrically possible types of S ₃¹ primary chains built of S ₃⁰ using theory of one-dimensional symmetry groups. Thirty three structural variants of morphologically and topologically different types of S ₃² micronets described by two-dimensional groups of symmetry are considered. Algorithms are presented for combinatorial and topological analysis to search for precursor clusters and restore a three-dimensional net of covalent and noncovalent bonds in a crystal structure by the matrix (cluster) self-assembly mechanism. The model advanced is universal. Examples of self-assembly of a series of cluster-assembled structures of AB₂ alloys of the unique Friauf-Laves family (which counts in 1400 of binary and ternary compounds) are given: for MgCu₂ (cF24) (with its superstructures of ZrCu ₅ and MgSnCu₄ types), MgZn₂ (hP12), and MgNi₂ (hP24) (from AB₂ or A₂B + B₃ three-atom clusters); for ZrZn₂₂ icosahedral structures (from a suprapolyhedral cluster built of a ZrZn₁₆ Friauf polyhedron and two ZnZn₁₂ icosahedra); NaCd₂ (from one A cluster with 61 atoms and two B clusters with 63 atoms); and for a bimolecular compound C₇₈H₃₀ (which is formed of fullerene C₆₀ and a C₂₈H₃₀ molecule). The scenario of formation of self-curving nets with icosahedral symmetry is considered: to form a B₁₂ icosahedron from two isomers with n = 3, a C₂₀ dodecahedron from two isomers with n = 5, and C₆₀ fullerene from pentagonal clusters with n = 5.     

Pseudo‐Fivefold Diffraction Symmetries in Tetrahedral Packing

We review the way in which atomic tetrahedra composed of metallic elements pack naturally into fused icosahedra. Orthorhombic, hexagonal, and cubic intermetallic crystals based on this packing are all shown to be united in having pseudo‐fivefold rotational diffraction symmetry. A unified geometric model involving the 600‐cell is presented: the model accounts for the observed pseudo‐fivefold symmetries among the different Bravais lattice types. The model accounts for vertex‐, edge‐, polygon‐, and cell‐centered fused‐icosahedral clusters. Vertex‐centered and edge‐centered types correspond to the well‐known pseudo‐fivefold symmetries in I_h and D_5h quasicrystalline approximants. The concept of a tetrahedrally‐packed reciprocal space cluster is introduced, the vectors between sites in this cluster corresponding to the principal diffraction peaks of fused‐icosahedrally‐packed crystals. This reciprocal‐space cluster is a direct result of the pseudosymmetry and, just as the real‐space clusters, can be rationalized by the 600‐cell. The reciprocal space cluster provides insights for the Jones model of metal stability. For tetrahedrally‐packed crystals, Jones zone faces prove to be pseudosymmetric with one another. Lower and upper electron per atom bounds calculated for this pseudosymmetry‐based Jones model are shown to accord with the observed electron counts for a variety of Group 10–12 tetrahedrally‐packed structures, among which are the four known Cu/Cd intermetallic compounds: CdCu₂, Cd₃Cu₄, Cu₅Cd₈, and Cu₃Cd₁₀. The rationale behind the Jones lower and upper bounds is reviewed. The crystal structure of Zn₁₁Au₁₅Cd₂₃, an example of a 1:1 MacKay cubic quasicrystalline approximant based solely on Groups 10–12 elements is presented. This compound crystallizes in Im$\bar 3$ (space group no. 204) with a=13.842(2) Å. The structure was solved with R₁=3.53 %, I>2σ;=5.33 %, all data with 1282/0/38 data/restraints/parameters.     

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.