Icosahedral and ring-shaped allotropes of phosphorus

The existence of two new allotropic forms of phosphorus, icosahedral cages and ring-shaped chains, is predicted. The cages and rings are nanostructural modifications of the black and the red phosphorus, respectively. The icosahedral and ring-shaped allotropes are compared with the experimentally known allotropic forms of phosphorus by quantum chemical methods. Both the cages and the rings are thermodynamically favored over the white phosphorus, the rings being comparable to the Hittorf's violet phosphorus and to the recently discovered fibrous red phosphorus. The stabilities of the icosahedral cages increase as a function of their size, having structural resemblance with the rhombohedral black phosphorus. The high thermodynamic stability of the phosphorus nanostructures suggests their experimental synthesis to be viable.     

Structural and electronic trends among group 15 polyhedral fullerenes

We have investigated the structural and electronic characteristics of tetrahedral, octahedral, and icosahedral fullerenes composed of group 15 elements phosphorus, arsenic, antimony, and bismuth. Systematic quantum chemical studies at the DFT and MP2 levels of theory were performed to obtain periodic trends for the structural principles, stabilities, and electronic properties of the elemental nanostructures. Calibration calculations for polyhedral clusters with up to 20 atoms showed the applied theoretical approaches to be in good agreement with high-level CCSD(T)/cc-pVTZ results. By studying fullerenes up to P₈₈₈, As₅₄₀, Sb₆₂₀, and Bi₆₂₀, we found their structures and stabilities to converge smoothly toward their experimental bulk counterparts. The diameters of the largest studied cages were 4.8, 3.7, 4.8, and 5.1?nm for the P, As, Sb, and Bi fullerenes, respectively. Comparisons with the experimentally known allotropes of the studied elements suggest the predicted polyhedral cages to be thermodynamically stable. All studied group 15 polyhedral fullerenes were found to be semiconducting, and density of states analysis illustrated clear periodic trends in their electronic structure. Relativistic effects become increasingly important when moving from P to Bi and taking the spin?Corbit effects into account by using a two-component procedure had a significant positive effect on the relative stability of bismuth clusters.     

Icosahedral and ring-shaped allotropes of arsenic.

We predict the existence of two novel families of arsenic nanostructures: icosahedral cages and ring-shaped chains. Quantum chemical calculations on the cages, rings, and the experimentally known allotropes of arsenic suggest the nanostructures to be thermodynamically stable. The icosahedral cages are modifications of the gray allotrope of arsenic, while the ring-shaped chains are structurally related to the red allotrope of phosphorus. Comparisons between the analogous allotropes of arsenic and phosphorus show distinct differences. While phosphorus favors the ring-shaped chains over the icosahedral cages, large cages become favorable for arsenic. From the thermodynamical point of view, experimental preparation of the proposed families of arsenic nanostructures is expected to be viable.     

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.     

Two‐, One‐, and Zero‐Dimensional Elemental Nanostructures Based on Ge9‐Clusters

We investigated the structural principles of novel germanium modifications derived by oxidative coupling of Zintl‐type [Ge₉]^4?clusters in various ways. The structures, stabilities, and electronic properties of the predicted {²_∞[Ge₉]_n} sheet, {¹_∞[Ge₉]_n} nanotubes, and fullerene‐like {Ge₉}_n cages were studied by using quantum chemical methods. The polyhedral {Ge₉}_n cages are energetically comparable with bulk‐like nanostructures of the same size, in good agreement with previous experimental findings. Three‐dimensional structures derived from the structures of lower dimensionality are expected to shed light on the structural characteristics of the existing mesoporous Ge materials that possess promising optoelectronic properties. Furthermore, 3D networks derived from the polyhedral {Ge₉}_n cages lead to structures that are closely related to the well‐known LTA zeolite framework, suggesting further possibilities for deriving novel mesoporous modifications of germanium. Raman and IR spectra and simulated X‐ray diffraction patterns of the predicted materials are given to facilitate comparisons with experimental results. The studied novel germanium modifications are semiconducting, and several structure types possess noticeably larger band gaps than bulk α‐Ge.     

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.     

The architecture of Platonic polyhedral links

A new methodology for understanding the construction of polyhedral links has been developed on the basis of the Platonic solids by using our method of the ‘n-branched curves and m-twisted double-lines covering’. There are five classes of platonic polyhedral links we can construct: the tetrahedral links; the hexahedral links; the octahedral links; the dodecahedral links; the icosahedral links. The tetrahedral links, hexahedral links, and dodecahedral links are, respectively, assembled by using the method of the ‘3-branched curves and m-twisted double-lines covering’, whereas the octahedral links and dodecahedral links are, respectively, made by using the method of the ‘4-branched curves’ and ‘5-branched curves’, as well as ‘m-twisted double-lines covering’. Moreover, the analysis relating topological properties and link invariants is of considerable importance. Link invariants are powerful tools to classify and measure the complexity of polyhedral catenanes. This study provides further insight into the molecular design, as well as theoretical characterization, of the DNA polyhedral catenanes.     

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 relationships in close-packed AB2O4 oxides involving spinel,olivine, and hexagonal LiFeSnO4 structures

Close-packed structures with formula AB₂O₄ were studied in terms of polyhedra arrangement. The junction between octahedral layers (kagome) and mixed (octahedral and tetrahedral) layers were analyzed; this association allows one to determine two types of double layers whose packing leads to three closely related structural types: spinel, double-hexagonal LiFeSnO₄, and a hypothetical hexagonal structure. Olivine structure shows a polyhedral arrangement closely related to those of these three structures and can be described alike in terms of mixed layers and double layers. AB₂O₄ close-packed oxides can exhibit polymorphism; from this analysis a mechanism, involving geometrical operations applied to the double layers, is proposed for the different transitions, really observed as (DH) LiFeSnO₄ spinel and olivine-spinel or theoretical as (DH) LiFeSnO₄—hypothetical hexagonal and olivine—hypothetical hexagonal.     

Some aspects of the symmetry and topology of possible carbon allotrope structures

Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite and diamond known since antiquity. Such fullerenes contain polyhedral carbon cages in which all vertices have degree 3 and all faces are either pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are completely surrounded by hexagons so that no two pentagonal faces share an edge. The smallest fullerene structures satisfying the IPR are the known truncated icosahedral C₆₀ of I _h symmetry and ellipsoidal C₇₀ of D _5h symmetry. The multiple IPR isomers of families of larger fullerenes such as C₇₆, C₇₈, C₈₂ and C₈₄ can be classified into families related by the so-called pyracylene transformation based on the motion of two carbon atoms in a pyracylene unit containing two linked pentagons separated by two hexagons. Larger fullerenes with 3ν vertices can be generated from smaller fullerenes with ν vertices through a so‐called leapfrog transformation consisting of omnicapping followed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n ² carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the corresponding carbon allotropes constructed from carbon networks of negative curvature are the polymeric schwarzites. The negative curvature in schwarzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on minimal surfaces with negative Gaussian curvature, particularly the so-called P and D surfaces with local cubic symmetry. The smallest unit cell of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C₆₀ fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C₁₆₈ schwarzite unit cell has local O _h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C₆₀ fullerene considered as the isomorphous PSL(2,5) group. The schwarzites, which are still unknown experimentally, are predicted to be unusually low density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

Theoretical studies of aluminoxane chains, rings, cages, and nanostructures

Alumina nanostructures and three families of aluminoxanes, linear, cyclic, and cagelike structures, have structures that resemble their isovalent electronic hydrocarbon analogues. Specific examples of each family are the counterparts of fullerene, allene, benzene, and cubane, respectively. The aluminoxanes and alumina nanostructures are related to each other; the latter can be regarded as a hydrogen- or alkyl-free form of aluminoxane. By exploiting this relationship, the relative stabilities of the three families of aluminoxanes, alumina nanostructures, and alumina crystal lattices have been estimated. According to ab initio calculations, aluminoxane cages, which take the form of a truncated octahedron and related polyhedra, are favored. The stability of the preferred cage, T-symmetric Al28O28H28, is practically equal to that of the alpha-alumina crystal lattice.     

Shape- and Size-Tunable Synthesis of Covalent Organic Cages through Rh-Catalyzed Regioselective [2+2+2] Cycloaddition

Covalent organic cages have potential applications in molecular inclusion/recognition and porous organic crystals. Bridging arene units with sp³ atoms enables facile construction of rigid isolated internal vacancies, and various prismatic arene cages have been synthesized by kinetically controlled covalent bond formation. However, the synthesis of a tetrahedral one, which requires twice as much bond formation as prismatic ones, has been limited to a thermodynamically controlled dynamic S_NAr reaction, and this reversible covalent bond formation made the resulting cage product chemically unstable. Here we report the Rh-catalyzed high-yielding and highly 1,3,5-selective room temperature [2+2+2] cycloaddition of push-pull alkynes and its application to the synthesis of chemically stable aryl ether cages of various shapes and sizes, including prismatic and tetrahedral forms. These aryl ether cages are highly crystalline and intertwine with each other to form regular packing structures. Some aryl ether cages encapsulated isolated water molecules in their hydrophobic cavity by hydrogen bonding with the multiple ester moieties.     

Trapping White Phosphorus within a Purely Organic Molecular Container Produced by Imine Condensation

Three tetrahedral organic cages have been obtained by condensing a triamino linker with a set of three ostensibly analogous triformyl precursors. Despite the large number of imine bonds formed, the corresponding cages were obtained in exceptionally high yields. Both theory and experimental results demonstrate that intramolecular CH⋅⋅⋅π interactions within all of the cage frameworks play an important role in abetting the condensations and contributing to the near‐quantitative synthetic yields. The three cages of this study exhibit high thermodynamic and kinetic stability. A variety of small neutral guest molecules with complementary sizes and geometries may be used as templates in the cage forming reactions. Among the guests that may be used in this way is white phosphorus (P₄), whose inherent reactivity towards oxygen is almost fully attenuated when bound within one of the cages.     

Cluster Models of the VOx/TiO2 Supported Catalyst

Molecular structures of the active vanadium phase of the VO _x /TiO₂ supported catalyst are calculated in the framework of the cluster approximation of density functional theory (DFT). It is shown that vanadium can be stabilized on the anatase (001) surface both in the tetrahedral and octahedral coordinations with the formation of monoxo- and dioxovanadyl structures. The energy of the dioxovanadyl structure binding to the support surface is 600–800 kJ/mol. The formation of dioxovanadyl structures from monoxovanadyl ones with the formation of water molecules is energetically favorable. The effect of support on the electronic state and acidic properties of the supported vanadium phase is discussed.     

One-pot fabrication of single-crystalline octahedral Pd-Pt nanocrystals with enhanced electrocatalytic activity for methanol oxidation

This study reports the synthesis of octahedral Pd-Pt bimetallic alloy nanocrystals through a facile, one-pot, templateless, and seedless hydrothermal method in the presence of glucose and hexadecyl trimethyl ammonium bromide. The morphologies, compositions, and structures of the Pd-Pt nanocrystals were fully characterized by various physical techniques, thereby demonstrating their highly alloying octahedral nanostructures. The formation or growth mechanism of the Pd-Pt bimetallic alloy nanocrystals was explored and is discussed here based on the experimental observations. In addition, the synthesized Pd-Pt nanocrystals were applied to the methanol oxidation reaction (MOR) in alkaline media, which proved that the as-prepared catalysts exhibit enhanced electrocatalytic activity for MOR. Pd₁Pt₃ exhibited the best stability and durability, and its mass activity was 3.4 and 5.2 times greater than those of Pt black and Pd black catalysts, respectively. The facile synthetic process and excellent catalytic performance of the as-prepared catalysts demonstrate that they have the potential to be used in direct methanol fuel cell techniques.     

Self‐Assembled Cage‐Like Protein Structures

Proteins and protein‐based assemblies represent the most structurally and functionally diverse molecules found in nature. Protein cages, viruses and bacterial microcompartments are highly organized structures that are composed primarily of protein building blocks and play important roles in molecular ion storage, nucleic acid packaging and catalysis. The outer and inner surface of protein cages can be modified, either chemically or genetically, and the internal cavity can be used to template, store and arrange molecular cargo within a defined space. Owing to their structural, morphological, chemical and thermal diversity, protein cages have been investigated extensively for applications in nanotechnology, nanomedicine and materials science. Here we provide a concise overview of the most common icosahedral viral and nonviral assemblies, their role in nature, and why they are highly attractive scaffolds for the encapsulation of functional materials.     

Effect of temperature on the extraction of octahedral and tetrahedral cobalt(II) by 8-hydroxyquinoline and/or dibenzo-18-crown-6 or dibenzylamine

The extraction behavior of octahedral and tetrahedral cobalt(II) complexes from aqueous nitrate medium was studied in the system 8-hydroxyquinoline (HOX) and dibenzo-18-crown-6 (Db 18C6) or dibenzylamine (DBA) in chloroform at different temperatures to evaluate the thermodynamic functions as well as the equilibrium constants of each reaction. The stoichiometry of the extracted organic phase species were established to be Co(OX)₂·Db18C6 for the octahedral cobalt and Co(OX)₂·DBA for the tetrahedral cobalt.     

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.     

An [FeIII34] Molecular Metal Oxide

The dissolution of anhydrous iron bromide in a mixture of pyridine and acetonitrile, in the presence of an organic amine, results in the formation of an [Fe₃₄] metal oxide molecule, structurally characterised by alternate layers of tetrahedral and octahedral Fe^III ions connected by oxide and hydroxide ions. The outer shell of the complex is capped by a combination of pyridine molecules and bromide ions. Magnetic data, measured at temperatures as low as 0.4 K and fields up to 35 T, reveal competing antiferromagnetic exchange interactions; DFT calculations showing that the magnitudes of the coupling constants are highly dependent on both the Fe‐O‐Fe angles and Fe?O distances. The simplicity of the synthetic methodology, and the structural similarity between [Fe₃₄], bulk iron oxides, previous Fe^III–oxo cages, and polyoxometalates (POMs), hints that much larger molecular Fe^III oxides can be made.     

A correlation between the structure and some physical properties of binary molybdates (Tungstates) of uni- and bivalent metals

General principles of arrangement are discussed and analyzed for 13 known structural types (families) of binary molybdates and tungstates of uni-and bivalent metals, which were classified according to the degree of condensation of M²⁺O_n polyhedral and MoO₄ tetrahedral constructions. The structural features of some phases correlate with their ferroelectric, ion-conducting, and luminescent (laser) properties. We suggest crystallochemical criteria and mechanisms of distortion (ferroelectric) phase transitions in palmierites, langbeinites, and Rb₄Mn(MoO₄)₃ and show the paths of ionic transport in phases with NaCo₂₃₁(MoO₄)₃, alluaudite, and Na₂Mg₅(MoO₄)₆ structures. On the basis of the analysis performed, we outlined the compounds that are the most promising for the development of inorganic materials with efficient physical properties. Institute of Inorganic Chemistry, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 6, pp. 145–157, November–December, 1994. Translated by T. Yudanova