# Icosahedral symmetry in clusters

01 Jan 1997-Progress in Crystal Growth and Characterization of Materials (Pergamon)-Vol. 34, pp 95-131

TL;DR: In this paper, it is shown that the electronic structure plays an important role in the atomic structure and related properties of other magic clusters, such as rare gases, metals, covalently bonded systems and water.

Abstract: First principles calculations and simulations based on interatomic potentials together with experimental studies of abundance spectrum suggest icosahedral structures to be common for some magic clusters of diverse systems such as rare gases, metals, covalently bonded systems and water. Close packing models obtained from pair potentials are shown to be good representations of the structure of rare gas clusters. However, the electronic structure is found to play the important role in the atomic structure and related properties of other clusters. Results of recent studies of fullerenes, their derivatives as well as some large icosahedral metal clusters containing several thousand atoms are also presented. Further results on doped icosahedral clusters are discussed which hold promise for the development of new materials and for understanding the occurrence of icosahedral order in several aluminum alloys.

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TL;DR: In this article, the authors extend the Lennard-Jones potential to obtain analytical expressions for the lattice parameters, cohesive energy, and bulk modulus using the solid state parameters of cubic lattices and hcp.

Abstract: The many-body expansion ${V}_{\mathrm{int}}={\ensuremath{\sum}}_{ilj}{V}^{(2)}({r}_{ij})+{\ensuremath{\sum}}_{iljlk}{V}^{(3)}({r}_{ij},{r}_{ik},{r}_{jk})+\ensuremath{\cdots},$ in terms of interaction potentials between rare-gas atoms converges fast at distances $rg{r}_{\mathit{HS}}$, with ${r}_{\mathit{HS}}$ being the hard-sphere radius at the start of the repulsive wall of the interaction potential. Hence, for the solid state where the minimum distance is always above ${r}_{\mathit{HS}}$, a reasonable accuracy is already obtained for the lattice parameters and cohesive energies of the rare-gas elements using precise two-body terms. All tested two-body potentials show a preference of the hcp over the fcc structure. We demonstrate that this is always the case for the Lennard-Jones potential. We extend the Lennard-Jones potential to obtain analytical expressions for the lattice parameters, cohesive energy, and bulk modulus using the solid-state parameters of Lennard-Jones and Ingham [Proc. R. Soc. London, Ser. A 107, 636 (1925)], which we evaluate up to computer precision for the cubic lattices and hcp. The inclusion of three-body terms does not change the preference of hcp over fcc, and zero-point vibrational effects are responsible for the transition from hcp to fcc as shown recently by Rosciszewski et al. [Phys. Rev. B 62, 5482 (2000)]. More precisely, we show that it is the coupling between the harmonic modes which leads to the preference of fcc over hcp, as the simple Einstein approximation of moving an atom in the static field of all other atoms fails to describe this difference accurately. Anharmonicity corrections to the crystal stability are found to be small for argon and krypton. We show that at pressures higher than $15\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ three-body effects become very important for argon and good agreement is reached with experimental high-pressure density measurements up to $30\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, where higher than three-body effects become important. At high pressures we find that fcc is preferred over the hcp structure. Zero-point vibrational effects for the solid can be successfully estimated from an extrapolation of the cluster zero-point vibrational energies with increasing cluster size $N$. For He, the harmonic zero-point vibrational energy is predicted to be always above the potential energy contribution for all cluster sizes up to the solid state at structures obtained from the two-body force. Here anharmonicity effects are very large which is typical for a quantum solid.

106 citations

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TL;DR: In this article, the lowest energy geometric structures and their corresponding magnetic moments of CoN (N = 2 − 13 ) clusters have been studied using first-principles method based on the density functional theory.

Abstract: The lowest-energy geometric structures and their corresponding magnetic moments of CoN ( N = 2 – 13 ) clusters have been studied using first-principles method based on the density functional theory. In the calculation, the Jahn–Teller effect plays an important role because there are many isomers near the ground state for small cobalt clusters. And our results find that the magnetism is more sensitive to the symmetry relative to interatomic spacing and cobalt clusters grow in an icosahedral pattern. The results of the formation energy and the second derivative of binding energy show oscillatory behavior for small cobalt clusters and that 6-, 10- and 12-atom clusters are so-called magic clusters. Further, for the small cobalt clusters, the HOMO–LUMO gap and the mean magnetic moments show strong odd–even alternation as cluster size.

101 citations

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TL;DR: The structures, binding energies, and magnetic moments of Fe N (N = 2 − 13, 15, 19 ) clusters have been obtained by all-electron density functional theory.

Abstract: The structures, binding energies, and magnetic moments of Fe N ( N = 2 – 13 , 15 , 19 ) clusters have been obtained by all-electron density functional theory. The Jahn–Teller effect plays an important role in determining the ground state of certain geometric structures. For Fe3 and Fe4, new ground states are found. The results indicate that the magnetic moment per atom shows only small variations with cluster size and remains in the vicinity of 3.0 μ B / atom over this size range. With increasing atom number the mean binding energy monotonically decreases and the second derivative of the binding energy indicates that N = 6 and N = 10 are magic numbers for neutral FeN clusters.

75 citations

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TL;DR: This comprehensive review presents results of many such developments in this fast-growing field including endohedrally doped Al, Ga, and In clusters, and performs ab initio calculations to present updated results of the most stable atomic structures and fundamental electronic properties of the endohedral doped cage clusters.

Abstract: The discovery of carbon fullerene cages and their solids opened a new avenue to build materials from stable cage clusters as “artificial atoms” or “superatoms” instead of atoms. However, cage clust...

54 citations

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TL;DR: The DMol cluster method based on density-functional theory has been employed to study the structural stability and electronic structure of La(n) (n=2-14) clusters and the results show strong odd-even alternation and that 7- and 13-atom clusters are magic.

Abstract: The DMol cluster method based on density-functional theory has been employed to study the structural stability and electronic structure of Lan (n=2–14) clusters. The ground states have been found out for lanthanum clusters. The Jahn-Teller effect plays an important role in this process because there are many isomers near the ground state. The magnetism is not sensitive to interatomic spacing when the change of interatomic spacing is in a small range. Lanthanum clusters grow in an icosahedral pattern. The results of the mean binding energy, of the second derivative of binding energy, and of the formation energy show strong odd–even alternation and that 7- and 13-atom clusters are magic. Further, the HOMO-LUMO gap, the mean nearest bond lengths, and the mean magnetic moments suggest that the convergence to bulk is slow and it shows an oscillatory behavior for small lanthanum clusters.

49 citations

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TL;DR: In this paper, the Hartree and Hartree-Fock equations are applied to a uniform electron gas, where the exchange and correlation portions of the chemical potential of the gas are used as additional effective potentials.

Abstract: From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed. These methods are exact for systems of slowly varying or high density. For the ground state, they lead to self-consistent equations analogous to the Hartree and Hartree-Fock equations, respectively. In these equations the exchange and correlation portions of the chemical potential of a uniform electron gas appear as additional effective potentials. (The exchange portion of our effective potential differs from that due to Slater by a factor of $\frac{2}{3}$.) Electronic systems at finite temperatures and in magnetic fields are also treated by similar methods. An appendix deals with a further correction for systems with short-wavelength density oscillations.

42,177 citations

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TL;DR: In this article, the authors proposed a truncated icosahedron, a polygon with 60 vertices and 32 faces, 12 of which are pentagonal and 20 hexagonal.

Abstract: During experiments aimed at understanding the mechanisms by which long-chain carbon molecules are formed in interstellar space and circumstellar shells1, graphite has been vaporized by laser irradiation, producing a remarkably stable cluster consisting of 60 carbon atoms. Concerning the question of what kind of 60-carbon atom structure might give rise to a superstable species, we suggest a truncated icosahedron, a polygon with 60 vertices and 32 faces, 12 of which are pentagonal and 20 hexagonal. This object is commonly encountered as the football shown in Fig. 1. The C60 molecule which results when a carbon atom is placed at each vertex of this structure has all valences satisfied by two single bonds and one double bond, has many resonance structures, and appears to be aromatic. Before 1985, it was generally accepted that elemental carbon exists in two forms, or allotropes: diamond and graphite. Then, Kroto et al. identified the signature of a new, stable form of carbon that consisted of clusters of 60 atoms. They called this third allotrope of carbon 'buckminsterfullerene', and proposed that it consisted of polyhedral molecules in which the atoms were arrayed at the vertices of a truncated icosahedron. In 1990, the synthesis of large quantities of C60 [see Nature 347, 354–358 (1990)] confirmed this hypothesis.

12,456 citations

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TL;DR: In this article, a unified scheme combining molecular dynamics and density-functional theory is presented, which makes possible the simulation of both covalently bonded and metallic systems and permits the application of density functional theory to much larger systems than previously feasible.

Abstract: We present a unified scheme that, by combining molecular dynamics and density-functional theory, profoundly extends the range of both concepts. Our approach extends molecular dynamics beyond the usual pair-potential approximation, thereby making possible the simulation of both covalently bonded and metallic systems. In addition it permits the application of density-functional theory to much larger systems than previously feasible. The new technique is demonstrated by the calculation of some static and dynamic properties of crystalline silicon within a self-consistent pseudopotential framework.

8,457 citations

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TL;DR: In this article, a new form of pure, solid carbon has been synthesized consisting of a somewhat disordered hexagonal close packing of soccer-ball-shaped C60 molecules.

Abstract: A new form of pure, solid carbon has been synthesized consisting of a somewhat disordered hexagonal close packing of soccer-ball-shaped C60 molecules. Infrared spectra and X-ray diffraction studies of the molecular packing confirm that the molecules have the anticipated 'fullerene' structure. Mass spectroscopy shows that the C70 molecule is present at levels of a few per cent. The solid-state and molecular properties of C60 and its possible role in interstellar space can now be studied in detail.

6,429 citations

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TL;DR: In this article, a metallic solid with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations, was observed and its diffraction spots are as sharp as those of crystals but cannot be indexed to any Bravais lattice.

Abstract: We have observed a metallic solid (Al-14-at.%-Mn) with long-range orientational order, but with icosahedral point group symmetry, which is inconsistent with lattice translations. Its diffraction spots are as sharp as those of crystals but cannot be indexed to any Bravais lattice. The solid is metastable and forms from the melt by a first-order transition.

5,343 citations