Structural stability of boron clusters with octahedral and tetrahedral symmetries

The structural stability of cagelike boron clusters with octahedral and tetrahedral symmetries has been investigated by means of first-principles calculations. Twenty-eight cluster models, ranging from B(10) to B(66), were systematically constructed using regular and semiregular polyhedra as prototypes. The binding energies per atom were, on the whole, slightly lower than those of icosahedral clusters B(80) and B(100), which are supposed to be the most stable in the icosahedral group. The larger clusters did not always have higher binding energies. Isothermal molecular dynamics simulations were performed to determine the deformation temperatures at which clusters began to break or change their structures. We found eight clusters that had nonzero deformation temperatures, indicating that they are in metastable states. The octahedral cluster B(18) had the highest deformation temperature among these, similar to that of icosahedral B(80) and B(100). The analysis of the electronic structure of B(18) showed that it attained this high stability owing to Jahn-Teller distortion.     

Structural Principles of Polyhedral Allotropes of Phosphorus

We derive the structural principles of polyhedral allotropes of phosphorus, introducing three distinct families of black phosphorus nanostructures. The predicted tetrahedral, octahedral, and icosahedral phosphorus cages can also be considered as phosphorus fullerenes. Phosphorus cages up to P₈₈₈ are systematically investigated by quantum chemical methods, and their thermodynamic stabilities are compared with the experimentally known allotropic forms of phosphorus. The tetrahedral cages are thermodynamically favored over the octahedral and icosahedral structures, although large octahedral structures become nearly as stable as the tetrahedral ones. The stability trends of the studied polyhedral families can be rationalized on the basis of their structural characteristics. The phosphorus polyhedra can be further stabilized by fitting smaller structures inside larger ones, resulting in multilayered, bulk‐like cages. The synthesis of the predicted black phosphorus nanostructures is suggested to be viable from the thermodynamic point of view, and several approaches for their experimental preparation can be envisaged.     

Orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra

The necessity of the development of the orbital structure of vibrations of nanoparticles, clusters, and coordination polyhedra is dictated by synthesis of clusters, supermolecules, and other structures of nanoscale dispersion for which translational symmetry is absent and the crystal system is inapplicable. The composition of complicated molecules, polynuclear complexes, and clusters is described, in addition to the chemical formula, by the composition equations derived from analysis of the symmetry properties of molecular structures. This analysis enables the derivation of analytical formulas for the types of molecular orbitals of structures with arbitrary groups of symmetry. Here, we use the representation of nanoscale structures described by the orbital system as a set of concentric nested spherical orbits of atoms, orbits of faces of different order, and orbits of edges. The orbits are grouped into shells shaped as polyhedra with vertices, edges, or faces accommodating atoms with different types of packing. In such a way, the sets of molecular orbitals of all high-, intermediate-, and low-symmetry groups have been determined depending on the number of atoms in the axial, planar, and primitive orbits.     

Molecular structures of magnesium dichloride sheets and nanoballs

The structures and relative stabilities of (MgCl(2))(n)() sheetlike clusters and nanoballs were studied by quantum chemical methods. The sheets as discrete molecules were studied up to Mg(100)Cl(200). Their stabilities increase systematically as a function of the size of the sheet. Periodic ab initio calculations were performed for (001) monolayer sheets of alpha- and beta-MgCl(2), beta-sheet being slightly favored. Nanoballs were constructed from Archimedean polyhedra, producing tetrahedral, octahedral, and icosahedral symmetries, and were studied up to Mg(60)Cl(120). Nanoballs prefer to take the shape of truncated cuboctahedron (Mg(48)Cl(96)). Comparisons to sheetlike clusters and periodic calculations suggest that magnesium dichloride nanoballs are stable.     

Spiral codes and Goldberg representations of icosahedral fullerenes and octahedral analogues

An icosahedral fullerene may be considered as a tessellation of the sphere specified by an ordered pair of integers, or as a tightly wound spiral of faces. Explicit analytical relations for interconverting the two representations are given, enabling the canonical spiral code to be constructed for an icosahedral fullerene of any size. Analogous relations hold for the octahedral square + hexagon polyhedra that have been mentioned as possible candidates for boron-nitride "fullerenes".     

Molecular structures of alumina nanoballs and nanotubes: a theoretical study

Molecular structures of alumina nanoballs and nanotubes have been determined. Tetrahedral, octahedral, and icosahedral alumina nanostructures were derived from Platonic solids and Archimedean polyhedra and were optimized by quantum chemical methods. I(h)-symmetric balls, resembling their isovalence electronic analogues, fullerenes, are preferred. The nanoballs consist of adjacent Al(5)O(5) and Al(6)O(6) rings, similar to C(5)- and C(6)-rings of fullerenes. The structural characteristics of alumina nanoballs are dominated by pi-electron donation from oxygen to aluminum. Alumina nanotubes can be derived from icosahedral nanoballs. The tubes alternate between D(5d)- and D(5h)-symmetries and are capped by halves of the icosahedral balls.     

A computational study of the lowest singlet and triplet states of neutral and dianionic 1,2-substituted icosahedral and octahedral o-carboranes

This work introduces a calibrated B3LYP/6-31G(d) study on the electronic structure of singlet and triplet neutral species of 1,2-substituted icosahedral 1,2-R(2)-1,2-C(2)B(10)H(10) and octahedral 1,2-R(2)-1,2-C(2)B(4)H(4) molecules with R = {H, OH, SH, NH(2), PH(2), CH(3), SiH(3)} and their respective dianions formed by proton removal on each R group. A variety of small adiabatic singlet-triplet gaps DeltaE(ST) are obtained from these systems ranging from 2.93 eV (R = NH(2)) 相似文献   

Architecture of Platonic and Archimedean polyhedral links

A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic and Archimedean solids by using our method of the ‘three-cross-curve and dou- ble-twist-line covering’. There are five classes of polyhedral links that can be explored: the tetrahedral and truncated tetrahedral links; the hexahedral and truncated hexahedral links; the dodecahedral and truncated dodecahedral links; the truncated octahedral and icosahedral links. Our results show that the tetrahedral and truncated tetrahedral links have T symmetry; the hexahedral and truncated hexahedral links, as well as the truncated octahedral links, O symmetry; the dodecahedral and truncated dodeca- hedral links, as well as the truncated icosahedral links, I symmetry, respectively. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA and protein catenanes.     

Nesting of fullerenes and Frank-Kasper polyhedra

The duality relationship between fullerenes and Frank-Kasper polyhedra suggests that these two families of polyhedra appear nested in solid state structures. Magic numbers, described by simple mathematical relationships, identify four families of fullerenes with tetrahedral and icosahedral symmetries.     

Microscopic Symmetry Imposed by Rotational Symmetry Boundary Conditions in Molecular Dynamics Simulation

A large number of viral capsids, as well as other macromolecular assemblies, have icosahedral structure or structures with other rotational symmetries. This symmetry can be exploited during molecular dynamics (MD) to model in effect the full viral capsid using only a subset of primary atoms plus copies of image atoms generated from rotational symmetry boundary conditions (RSBC). A pure rotational symmetry operation results in both primary and image atoms at short range, and within nonbonded interaction distance of each other, so that nonbonded interactions can not be specified by the minimum image convention and explicit treatment of image atoms is required. As such an unavoidable consequence of RSBC is that the enumeration of nonbonded interactions in regions surrounding certain rotational axes must include both a primary atom and its copied image atom, thereby imposing microscopic symmetry for some forces. We examined the possibility of artifacts arising from this imposed microscopic symmetry of RSBC using two simulation systems: a water shell and human rhinovirus 14 (HRV14) capsid with explicit water. The primary unit was a pentamer of the icosahedron, which has the advantage of direct comparison of icosahedrally equivalent spatial regions, for example regions near a 2-fold symmetry axis with imposed symmetry and a 2-fold axis without imposed symmetry. Analysis of structural and dynamic properties of water molecules and protein atoms found similar behavior near symmetry axes with imposed symmetry and where the minimum image convention fails compared with that in other regions in the simulation system, even though an excluded volume effect was detected for water molecules near the axes with imposed symmetry. These results validate the use of RSBC for icosahedral viral capsids or other rotationally symmetric systems.     

Copper Keplerates: High‐Symmetry Magnetic Molecules

Keplerates are molecules that contain metal polyhedra that describe both Platonic and Archimedean solids; new copper keplerates are reported, with physical studies indicating that even where very high molecular symmetry is found, the low‐temperature physics does not necessarily reflect this symmetry.     

Predicting the shape and structure of face-centered cubic gold nanocrystals smaller than 3 nm.

Although a number of computational studies have examined the relative stability of icosahedral and decahedral gold clusters from 1 to 3 nm in size, few studies have focussed on the variety of face-centered cubic (fcc) nanoparticles in this size regime. In most cases small fcc gold particles are assumed to adopt the truncated octahedral shape, but in light of the fact that the shape and structure of gold nanoparticles is known to vary, the relative stability of fcc polyhedra may change with size. Presented here are results of first-principles calculations investigating the preferred shape of gold particles less than 3 nm in size. Our results indicate that the equilibrium shape of fcc gold nanoparticles less than 1 nm is the cuboctahedron, but this shape rapidly becomes energetically unstable with respect to the truncated octahedron, octahedron and truncated cube shapes as the size increases.     

On methods to determine the surface areas of molecules

The known methods to determine the surface areas of molecules are discussed and a new method is proposed based on the molecule representation in the form of a molecular Voronoi-Dirichlet polyhedron. A linear correlation is found between the surface areas of molecules calculated using the molecular Voronoi-Dirichlet polyhedra and determined by the already known methods. The efficiency of the use of molecular Voronoi-Dirichlet polyhedra for the analysis of intermolecular interactions in crystal structures is demonstrated by concrete examples.     

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.     

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.     

A method for modeling icosahedral virions: Rotational symmetry boundary conditions

We present two techniques for implementing a new method of simulating an entire virion. Earlier computer simulations of a capsid protein revealed large edge effects due to the use of free standing boundaries. Because of the size of a given protomer, conventional three-dimensional periodic boundary conditions would be extremely wasteful. This would require an extremely large number of solvent molecules, and therefore would be computationally feasible for only a fragment of the entire virion. The new method employs non-space-filling computational cells in molecular modeling and molecular dynamics with the boundary conditions based on the icosahedral group. The method is general and could be used for any molecular system with a point group symmetry. With this method, the dynamical and spatial intra and interprotomer correlations can be studied at atomic levels. The technique is applicable to any virion with icosahedral symmetry. A sample calculation involving a geometry optimization of the human rhinovirus coat proteins is given to demonstrate the technique.     

From Octahedral to Icosahedral Metal–Organic Polyhedra Assembled from Two Types of Polyoxovanadate Clusters

Design and synthesis of metal–organic polyhedra (MOPs) with targeted geometries from predetermined secondary building units (SBUs) is a long‐standing challenge in chemistry and material science. Theoretical prediction shows that there are 6 possible polyhedra from the 3‐coordinated, 4‐coordinated octahedron ((3,4)‐c octahedron) to (3,5)‐c icosahedron with minimal transitivity (simplest possible). Except for one missing polyhedron ( mtr ) due to the unfavorable angles, we report five MOPs based on these structures, including an octahedral (3,4)‐c VMOP‐ 21 ( rdo ), an icosahedral (3,5)‐c VMOP‐ 25 ( trc ), and three intermediate derived trinodal (3,4,5)‐c VMOP‐ 22 – 24 ( ghm , hmg , xum ). Remarkably, all these MOPs obey the minimal transitivity principle and are consistent with geometrical predictions.     

The Chirality of Icosahedral Fullerenes: a Comparison of the Tripling (leapfrog), Quadrupling (chamfering), and Septupling (capra) Transformations

Fullerene polyhedra of icosahedral symmetry have the midpoints of their 12 pentagonal faces at the vertices of a macroicosahedron and can be characterized by the patterns of their hexagonal faces on the (triangular) macrofaces of this macroicosahedron. The numbers of the vertices in fullerene polyhedra of icosahedral symmetry satisfy the Goldberg equation v=20(h ²+hk+k ²), where h and k are two integers and 0 <h≥ k≥ 0 and define a two-dimensional Goldberg vector G = (h, k). The known tripling (leapfrog), quadrupling (chamfering), and septupling (capra) transformations correspond to the Goldberg vectors (1, 1), (2, 0), and (2, 1), respectively. The tripling and quadrupling transformations applied to the regular dodecahedron generate achiral fullerene polyhedra with the full I _h point group. However, the septupling transformation destroys the reflection operations of the underlying icosahedron to generate chiral fullerene polyhedra having only the I icosahedral rotational point group. Generalization of the quadrupling transformation leads to the fundamental homologous series of achiral fullerene polyhedra having 20 n ² vertices and Goldberg vectors (n, 0). A related homologous series of likewise achiral fullerene polyhedra having 60 n ² vertices and Goldberg vectors (n, n) is obtained by applying the tripling transformation to regular dodecahedral C₂₀ to give truncated icosahedral C₆₀ followed by the generalized operations (as in the case of quadrupling) for obtaining homologous series of fullerenes. Generalization of the septupling (capra) transformation leads to a homologous series of chiral C_20m fullerenes with the I point group and Goldberg vectors G=(h, 1) where m=h ²+h+1.     

Molecular Hume-Rothery compounds [M(ZnR)n] and [M(ZnR)a(GaR)b] (a + 2b = n ≥ 8): relations of coordination polyhedra and electronic structure

The icosahedral complex [Mo(ZnMe)(9)(ZnCp*)(3)] is discussed as the prototype for a whole family of high-coordinate, metal-rich compounds [M(ZnR)(n)] and [M(ZnR)(a)(GaR)(b)] (a + 2b = n ≥ 8; for the same metal M). In contrast to other highly coordinate complexes of classic, monodentate (nonchelating) nonmetal atom ligator ligands, for the (weakly) bonding metal atom ligators ZnR and GaR, attractive ligand-ligand interactions play an important role. The structures of the compounds were evaluated by the method of continuous-shape measures, and the bonding situation of models (R = H) was analyzed on the density functional level of theory. The structures and coordination polyhedra of [M(M'R)(n)] (M' = Zn, Ga) turned out to be independent of the central metal or the nature of the metals M' in the ligand shell, and the resulting molecular orbital schemes vary only slightly as a result of the different symmetries, however resulting in the same coordination polyhedra (structures) for all complexes. This result may be viewed as a molecular representation for the situation in extended solid-state intermetallic phases of the Hume-Rothery type.     

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.