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.     

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     

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.     

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.     

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.     

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.     

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     

Symmetry lowering by cage doping in spherical superatoms: Evaluation of electronic and optical properties of 18-electron W@Au12Pt n (n = 0-4) superatomic clusters from relativistic DFT calculations

Attempts to expand the versatility of well defined clusters are a relevant issue in the design of building blocks for functional nanostructures. Here, we investigate the plausible formation of related structures from the emblematic highly symmetrical 18-e [W@Au₁₂] cluster. The calculated [W@Au₁₂Pt_n] series, with n = 0, 1, 2, 3, and 4, show cohesion energies, HOMO-LUMO gap, adiabatic electron affinities (AEAs) and adiabatic ionization potentials (AIPs), indicating a relative stability to the parent cluster [W@Au₁₂] experimentally characterized, where clusters with n = 1 and n = 4 are suggested as the most stable with respect to oxidation. The resulting symmetry lowering away from the high icosahedral symmetry upon adding Pt atoms induces a sizable splitting of the frontiers shells, which in turn effectively modify the properties of the calculated clusters, as observed from calculated optical properties. The estimated absorption spectra show an interesting broadening effect of the absorption peaks, which appears as a useful approach for further design of broad black absorbers, which are able to absorb light in a wider range, with potential capabilities to enhance the efficiency of thin film solar cells and photocatalysis processes, among other applications.     

Quantum chemical study of the structure of molybdenum pentafluoride

The geometrical and electronic structure of various configurations of molybdenum pentafluoride are studied in the nonrelativistic approximation of the DV-X_α method. It is shown in a cluster approximation that the presence of the MoF₅ monomer with a distorted trigonal bipyramid (C_2v symmetry) and cyclic trimer [MoF₅]₃(D_3h symmetry) configurations is most probable for the liquid and gaseous phases. High probability of the existence of cyclic tetramers [MoF_{5 4} of lower symmetry (D_2h) is confirmed for the crystalline state. The geometrical parameters calculated for the most stable clusters and the data on their stability agree well with the experimental data.     

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.     

Quantum chemical study of the geometrical and electronic structure of niobium pentafluoride

The geometrical and electronic structure of various configurations of niobium pentafluoride is studied in the nonrelativistic approximation of the DV-X_α method. It is shown in a cluster approximation that the presence of the NbF₅ monomer in various geometrical configurations and of the cyclic [NbF₅]₃ trimer (D_3h symmetry) is most probable for the liquid and gaseous phases. Formation of infinite [NbF₅]_n chains is unlikely. For the most stable clusters, the calculated geometrical parameters are in satisfactory agreement with the available experimental data.     

On the solid- and liquidlike nature of quantum clusters in their ground state

The ground state of pristine clusters of (paraH(2))(N) and (orthoD(2))(N) of size ranging from N=11 to 55 is examined by means of the variational path integral method. The chemical potential is calculated for two different interaction models and it is shown that the location of magic numbers indeed depends on the chosen interaction potential. Density profiles are calculated and reveal the difference between the two isotopes with regards to shell structure. The magnitude of relative pair distance and position fluctuations is used to asses the rigidity of these finite-size quantum systems. A comparison of generic and specific distance fluctuations as a function of cluster size is proposed as a probe of the appearance of rigidity in the clusters. It is found that smaller (paraH(2))(N) clusters are fluidlike and start to display increased rigidity for clusters of size N> or =26, whereas (orthoD(2))(N) clusters of N=13 and N> or =19 are rigid. Small clusters exhibit structures loosely based on an anti-Mackay icosahedral motif. An anti-Mackay to Mackay transition at N=41-42 is suggested.     

The structures of small clusters of C60 molecules

Using a pairwise additive atom-atom intermolecular potential to describe the interaction between C₆₀ molecules, we calculated the lowest-energy structures of (C60)N clusters up to N = 15 and compared the results with predictions derived using Girifalco’s spherical potential. The cluster binding energies calculated on the basis of the former potential are in all cases significantly higher than those obtained from the latter. Moreover, the atom-atom potential predicts that small fullerene clusters have structures based on icosahedral packing, a finding which, for N = 14 and 15, contrasts with the results obtained using Girifalco’s approximation.     

Rehybridization and π-orbital alignment: the key to the existence of spheroidal carbon clusters

The rehybridization and bonding in large carbon spheroids (C_n) has been analyzed with POAV/3D HMO theory; larger clusters are shown to be favored, and the (lower) critical value of n for stable clusters is discussed. In the generic series of truncated icosahedral spheroids of I_h symmetry, C₂₄₀ is found to be significantly more stable than its lower homologue C₆₀.     

Crystal Structure of ϵ‐Ag7+xMg26–x – A Binary Alloy Phase of the Mackay Cluster Type

The binary alloy phase ϵ‐Ag_7+xMg_26–x with x ≈ 1 and small amounts of the β′‐AgMg phase crystallize by annealing of Ag–Mg alloys with starting compositions between 24–28 At‐% Ag at 390 to 420 °C. A model structure for the ϵ‐phase consisting of a fcc packing of Mackay clusters was derived from the known structure of the ϵ′‐Ag₁₇Mg₅₄ phase. Crystals of the ϵ‐phase were obtained by direct melting of the elements and annealing. The examination of a single crystal yielded a face‐centered cubic unit cell, space group Fm3 with a = 1761.2(5) pm. The refinement was started with the parameters of the model: wR2(all) = 0.0925 for 1093 symmetrically independent reflections. A refinement of the occupancy parameters indicated a partial replacement of silver for magnesium at two metal atom sites, resulting in the final composition ϵ‐Ag_7+xMg_26–x with x = 0.96(2). There are 264 atoms in the unit cell and the calculated density is 3.568 gcm^–3. The topology of the model was confirmed. Mackay icosahedra are located at the lattice points of a face‐centered cubic lattice. Differences between model and refined structure and their effects on the powder patterns are discussed. The new binary structure type of ϵ‐Ag_7+xMg_26–x can be described in terms of the I3‐cluster concept.     

Structural optimization of Cu-Ag-Au trimetallic clusters by adaptive immune optimization algorithm

The putative global minimum structures of Cu-Ag-Au trimetallic clusters with 19 and 55 atoms are obtained by adaptive immune optimization algorithm (AIOA) with the Gupta potential. For the 19-atom trimetallic clusters, the results indicate that all of them have double-icosahedral motifs. For the optimized structures of Cu(13)Ag(n)Au(42-n) (n = 1-41), the clusters can be categorized into 19 Mackay icosahedral structures, 1 6-fold pancake structure, and 21 ring-like structures linked by three face-sharing double-icosahedra. Furthermore, the segregation phenomena of the Cu, Ag, and Au atoms in the Cu-Ag-Au trimetallic clusters are studied to provide useful information for geometric character. Results show that Cu and Ag atoms prefer to locate in the inner-shell and on the surface, respectively, whereas Au atoms mainly locate in the middle-shell and tend to solve into Cu and Ag atoms.     

Symmetry and ferromagnetic clusters

Isomers of pure Fe₁₃ and icosahedral Fe₁₂X clusters are studied using the all-electron linear-combination-of-Gaussian-type-orbital (LCGTO) local-density-functional (LDF) methods that allow the spin and geometry of the cluster to be determined self-consistently. The Fe₁₃ ground state is icosahedral. The icosahedral cluster also has the greatest magnetic moment because of increased symmetry-required orbital degeneracy for electrons of different spins. The central atom of the icosahedral iron cluster has been varied to optimize the spin of the cluster keeping the oribital contribution to the magnetic moment quenched. Varying the central atom under this constraint can alter the magnetic moment by more than 20%. Similar studies have begun on 55-atom icosahedral iron clusters.     

Adamant-1,3,5-triyl super dodecahedranes

By quantum chemical DFT M06-2X/6-31G(d,p) method, the equilibrium parameters of rigid and stable hydrocarbon clusters of icosahedral symmetry with a dodecahedron structure whose sites are occupied by 20 adamant-1,3,5-triyl moieties linked with each other either directly or through bridges containing two or four carbon atoms are determined. The radius of the smallest studied quasispherical molecules is 1.05 nm and that of the largest one is 1.76 nm. The radius of the inner cavity in them varies from 0.37 nm (C₂₀₀H₂₆₀) to 1.06 nm (C₃₂₀H₂₆₀). Perfluorination increases the outer and decreases the inner radius of super dodecahedrane.     

Ionization potentials and geometrical structure of pure and oxygenated barium clusters

In this article, we present results obtained on structures of small bare and oxygenated barium clusters produced by the laser vaporization technique. Contrary to the very weak Ba—Ba binding (0.202 eV), the Ba—O binding is rather strong (5.8 eV). As a consequence, extremely small amounts of oxygen in the cluster source are sufficient to induce the nucleation of oxygenated barium clusters. Surprisingly, at least for substoechiometric systems, Ba_nO_m clusters exhibit the same icosahedral structure as bare clusters do. Mass spectra seem to indicate that replacing a barium atom with a strongly polar BaO molecule does not affect the geometrical structure of the cluster. This aspect is more easily understandable if we assume that the oxygen atom takes place inside the structure of the Ba_n cluster without significant distortions. Ab-initio calculations have been undertaken in order to give a satisfactory account of this hypothesis.     

Ab initio studies of small AlmFen clusters

The equilibrium geometrical structures of small Al_mFe_n clusters have been determined through ab initio calculation of the cluster total energy at the UB3LYP/Lanl2dz level. For dimers of iron and aluminum, the dissociation energies, the equilibrium atomic distances, and the vibrational frequencies were calculated. The agreement between calculations and experiments is reasonable. The ground stable geometrical structures of Fe₄, FeAl₃, Fe₃Al and Fe₂Al₂ clusters favor three-dimension configurations, but Al₄ tetramers are planar structures. The Al-rich tetramers are more stable than the other two composition tetramers. This is different from that of bulk alloys.