The enumeration of electron-rich and electron-poor polyhedral clusters

We present as dual processes the capping of closed triangulated polyhedra with apical atoms and the making of holes in such polyhedra either by puncture of the surface or by excision of atoms and their edges. These processes are shown to generate stable chemical species containing respectively less or more than 2n + 2 skeletal electrons. The former species are designated as electron-poor whereas the latter are called electron-rich. Pólya's enumeration method is used to enumerate the distinct ways of capping and excising the closed, triangulated polyhedra to yield systems containing from four to twelve vertices. For the enumeration of cappings the appropriate cycle index is that of the dual of the polyhedron being capped, whilst for the enumeration of the excisions the cycle index is that of the polyhedron being excised.     

Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

The dimensionality and topology of chemical bonding manifolds in metal clusters and related compounds

The chemical bonding manifolds in metal cluster skeletons (as well as in skeletons of clusters of other elements such as boron or carbon) may be classified according to their dimensionalities and their chemical homeomorphism to various geometric structures. The skeletal bonding manifolds of discrete metal cluster polyhedra may be either one-dimensional edge-localized or three-dimensional globally delocalized, although two-dimensional face-localized skeletal bonding manifolds are possible in a few cases. Electron precise globally delocalized metal cluster polyhedra withv vertices have 2v + 2 skeletal electrons and form deltahedra with no tetrahedral chambers having total skeletal bonding manifolds chemically homeomorphic to a closed ball. Electron-rich metal cluster polyhedra withv vertices have more than 2v + 2 skeletal electrons and form polyhedra with one or more non-triangular faces, whereas electron-poor metal cluster polyhedra withv vertices have less than 2v + 2 skeletal electrons and form deltahedra with one or more tetrahedral chambers. Fusion of metal cluster octahedra by sharing (triangular) faces forms three-dimensional analogues of polycyclic aromatic hydrocarbons such as naphthalene, anthracene, and perinaphthenide. Fusion of metal cluster octahedra by sharing edges can be extended infinitely into one and two dimensions forming chains (e.g. Gd₂Cl₃) and sheets (e.g. ZrCl), respectively. Infinite extension of such fusion of metal cluster octahedra into all three dimensions leads to bulk metal structures. Unusual anionic platinum carbonyl clusters can be contructed from stacks of Pt₃ triangles or Pt₅ pentagons. The resulting platinum polyhedra appear to exhibit edge-localized bonding, supplemented by unusual types of delocalized bonding at the top and the bottom of the stacks. Superconducting ternary molybdenum chalcogenides and lanthanide rhodium borides consist of infinite lattices of electronically linked edge-localized Mo₆ octahedra or Rh₄ tetrahedra, leading naturally to the idea of porous delocalization in superconducting materials.     

Structural paradigms in macropolyhedral boranes

Macropolyhedral borane clusters are concave polyhedra constituting fused convex simple polyhedra. They are formally obtained by condensation of simple polyhedral boranes under elimination of between one and four BH(3) or isoelectronic units. The number of eliminated vertexes from simple polyhedra equals the number of shared vertexes in macropolyhedral boranes. For each of the eight classes with general formulae ranging from B(n)H(n-4) to B(n)H(n+10), more than one structure type is possible, differing in the number of shared vertexes and in the types of the two combined cluster fragments. However, only one type of "potential structures" is represented by experimentally known examples and is found to be favored by theoretical calculations. A sophisticated system exists among the favored macropolyhedral borane structures. For each class of macropolyhedral boranes, the number of skeletal electron pairs is directly related to the general formula, the number of shared vertexes and the type of fused cluster fragments. In order to predict the distribution of vertexes among the fused fragments, we propose the concept of preferred fragments. Preferred fragments are those usually present in the thermodynamically most stable structure of a given class of macropolyhedral boranes and are also frequently observed in the experimentally known structures. This allows us to completely predict the cluster framework of the thermodynamically most stable macropolyhedral borane isomers.     

Topological aspects of chemically significant polyhedra

This paper considers both static and dynamic properties of chemically significant polyhedra. Static properties of polyhedra consider relationships between the numbers and degrees/sizes of polyhedral vertices, edges, and faces; polyhedral symmetries; and numbers of topologically distinct polyhedra of various types. Dynamic properties of polyhedra involve studies of polyhedral isomerizations from both macroscopic and microscopic points of view. Macroscopic aspects of polyhedral isomerization can be described by graphs called topological representations in which the vertices correspond to different permutational isomers and the edges to single degenerate polyhedral isomerization steps. Such topological representations are presented for isomerizations of polyhedra having five, six, and eight vertices. Microscopic aspects of polyhedral isomerizations arise from consideration of the details of polyhedral topology, such as the topological aspects of diamond-square-diamond processes. In this connection, Gale diagrams are useful for describing isomerizations of five- and six-vertex polyhedra, including the Berry pseudorotation of a trigonal bipyramid through a square pyramid intermediate and the Bailar or Ray and Dutt twists of an octahedron through a trigonal prism intermediate.     

Note on ordering and complexity of Platonic and Archimedean polyhedra based on solid angles

The missing values for the solid angles of the two snub semiregular polyhedra have been calculated, and integrated into the whole series of Platonic and Archimedean polyhedra. This is the only criterion which so far gives an unambiguous answer (without any degeneracy leading to posets) on how to order these polyhedra according to their increasing complexity.     

Nonconvex polyhedra by repeated truncation of semiregular polyhedra

Some of the semiregular (Archimedean) polyhedra (1–13 in Table 1) afford on truncation polyhedra that contain vertices where the sum of planar degrees for the faces which meet at those vertices is equal to (for 17, 18, and 23 in Table 3) or higher than 360° (21, 22, 24–26 in Table 3). Therefore such polyhedra are nonconvex.     

Codes in platonic, Archimedean, Catalan, and related polyhedra: a model for maximum addition patterns in chemical cages

The notion of d-code is extended to general polyhedra by defining maximum sets of vertices with pairwise separation > or =d. Codes are enumerated and classified by symmetry for all regular and semiregular polyhedra and their duals. Partial results are also given for the series of medials of Archimedean polyhedra. In chemistry, d-codes give a model for maximal addition to or substitution in polyhedral frameworks by bulky groups. Some illustrative applications from the chemistry of fullerenes and boranes are described.     

The rich stereochemistry of eight-vertex polyhedra: a continuous shape measures study

A stereochemical study of polyhedral eight-vertex structures is presented, based on continuous shape measures (CShM). Reference polyhedra, shape maps, and minimal-distortion interconversion paths are presented for eight-vertex polyhedral and polygonal structures within the CShM framework. The application of these stereochemical tools is analyzed for several families of experimental structures: 1) coordination polyhedra of molecular transition-metal coordination compounds, classified by electron configuration and ligands; 2) edge-bonded polyhedra, including cubane structures, realgar, and metal clusters; 3) octanuclear transition-metal supramolecular architectures; and 4) coordination polyhedra in extended structures in inorganic solids. Structural classification is shown to be greatly facilitated by these tools, and the detection of less common structures, such as the gyrobifastigium, is straightforward.     

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.     

On topological form in structures

This paper begins with a review of the Euler relation for the polyhedra and presents the corresponding Schläfli relation in n, the polygonality, and p, the connectivity of the polyhedra. The use of ordered pairs as given by (n, p), the Schläfli symbols, to organize the mapping of the polyhedra and its extension into the two-dimensional (2D) and three-dimensional (3D) networks is described. The topological form index, represented by l, is introduced and is defined as the ratio of the polygonality, n, to the connectivity, p, in a structure, it is given by l = n/p. Next a discussion is given of establishing a conventional metric of length in order to compare topological properties of the polyhedra and networks in 2D and 3D. A fundamental structural metric is assumed for the polyhedra. The metric for the polyhedra is, in turn, used to establish a metric for tilings in the Euclidean plane. The metrics for the polyhedra and 2D plane are used to establish a metric for networks in 3D. Once the metrics have been established, a conjecture is introduced, based upon the metrics assumed, that the area of the elementary polygonal circuit in the polyhedra and 2D and 3D networks is proportional to a function of the topological form index, l, for these structures. Data of the form indexes and the corresponding elementary polygonal circuit areas, for a selection of polyhedra and 2D and 3D networks is tabulated, and the results of a least squares regression analysis of the data plotted in a Cartesian space are reported. From the regression analysis it is seen that a quadratic in l, the form index, successfully correlates with the corresponding elementary polygonal circuit area data of the polyhedra and 2D and 3D networks. A brief discussion of the evident rigorousness of the Schläfli indexes (n, p) over all the polyhedra and 2D and 3D networks, based upon the correlation of the topological form index with elementary polygonal circuit area in these structures, and the suggestion that an Euler–Schläfli relation for the 2D and 3D networks, is possible, in terms of the Schläfli indexes, concludes the paper.     

The principle of maximum filling and characteristics of sublattices of hydrogen atoms

The most important characteristics of the Voronoi-Dirichlet polyhedra of 4005928 crystallographically different atoms in sublattices only containing hydrogen atoms were determined in the structures of 151044 inorganic, organoelement, and organic compound crystals. In such sublattices, the Voronoi-Dirichlet polyhedra of H atoms most often had 15 faces, and these faces were most often pentagonal. A comparative analysis of H- and C-sublattices was performed to find that the most important geometric and topological characteristics of H-sublattices were caused by the absence of short-range and local long-range order in the mutual arrangement of hydrogen atoms in the structure of crystals.     

Chemical applications of topology and group theory

This paper characterizesforbidden polyhedra, which are polyhedra with fewer than 9 vertices which cannot be formed using only the 9s,p, andd atomic orbitals. In this connection polyhedra are of particular interest if their symmetry groups are direct product groups of the typeR × C′ _s in whichR is a group containing only proper rotations andC′ _s is eitherC _s orC _i in which the non-identity element is an inversion center or a reflection plane which is called theprimary plane of the groupR ×C′ _s . Using this terminology polyhedra of the following types are shown always to be forbidden polyhedra: (1) Polyhedra having 8 vertices, such direct product symmetry point groups, and either an inversion center or aprimary plane fixing either 0 or 6 vertices; (2) Polyhedra having a 6-fold or higherC _n rotation axis. However, these conditions are not necessary for a polyhedron to be forbidden since in addition to one 7-vertex polyhedron and ten 8-vertex polyhedra satisfying one or both of the above conditions there are two forbiddenC _3v 8-vertex polyhedra which satisfy neither of the above conditions. For part 15 of this series see reference 1.     

Structure of stable and metastable water. Analysis of Voronoi polyhedra of molecular dynamics models

Conclusions The basic results that we have obtained in regard to the structure of water, by an analysis of Voronoi polyhedra and simplified Voronoi polyhedra, are as follows: the network of hydrogen bonds in water is not similar to an ideal random tetrahedral network under any conditions; in the actual network, there are major deviations from the tetrahedral directions of the bonds, such that molecules that are distant with respect to the bonds may be located closer than the first molecules. When the density is reduced from 1.0 to 0.8 g/cm³, the tetrahedral coordination of the hydrogen bond network is improved: the angles between bonds approach tetrahedral. This can be interpreted as a consequence of a decrease in the internal pressure that had forced the network to be deformed under normal conditions. However, the capability for rectifying the random quasitetrahedral network by removing the pressure has a limit corresponding to a density of 0.8 g/cm³. When the density is further reduced, the network becomes unstable, forming discontinuities and cavities that are large on a molecular scale.The results that we have set forth in this article give a rather clear demonstration of the possibilities of the Voronoi polyhedra method in analyzing the construction of the hydrogen bond networks that are the basis of water structure. An important feature of the method is that it does not depend on an exact definition of the hydrogen bond. This makes it possible to examine the properties of the network critically in terms of greater or lesser tetrahedricity without constructing the network itself. More detailed information, more convenient for interpretation, is obtained when the change is made from conventional to simplified Voronoi polyhedra in which no account is taken of most of the neighbors that are second along the bonds.Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy of Sciences. Dortmund University, Germany. Translated from Zhurnal Strukturnoi Khimii, Vol. 33, No. 2, pp. 79–87, March–April, 1992.     

Reticular chemistry of metal-organic polyhedra

Metal-organic polyhedra (MOPs), are discrete metal-organic molecular assemblies. They are useful as host molecules that can provide tailorable internal volume in terms of metrics, functionality, and active metal sites. As a result, these materials are potentially useful for a variety of applications, such as highly selective guest inclusion and gas storage, and as nanoscale reaction vessels. This review identifies the nine most important polyhedra, and describes the design principles for the five polyhedra most likely to result from the assembly of secondary building units, and provides examples of these shapes that are known as metal-organic crystals.     

Polytopal form and isomerism

A thesis is developed that accurate structural data for molecules in the solid state can be utilized to derive direct information about the geometric parameters for solution reaction mechanisms, This is specifically illustrated for intramolecular rearrangements but the basic approach should be applicable to bimolecular reactions. A dihedral angle criterion is employed to quantitatively assess shape parameters for polyhedra found in coordination compounds and cluster molecules. These data are expressed in reaction coordinate form whereby a real structure is related to two idealized polyhedra (a reaction path) or to three or more idealized polyhedra (a reaction cycle or chain). It is demonstrated through an analysis of structural data for five coordinate complexes that the Berry type of rearrangement is the lowest energy physical pathway for rearrangements in ML₅ molecules or ions. Solvents may alter the relative energies of ground and excited state forms but should not significantly alter the physical character of the rearrangement process unless the solvent strongly interacts with the molecules. This feature is discussed with respect to polytopal polymorphism in clusters, e.g., B₈H₈^2?.     

Into the Next Dimension: Nanometer-Sized, Oligonuclear Coordination Compounds with C(3)-Symmetric Ligands

Nanometer-sized cavities are present in oligonuclear coordination compounds formed in molecular self-assembly processes from C(3)-symmetric ligands and appropriate metal complex fragments. The structures obtained can be described as basic polyhedra such as tetrahedron, hexahedron, or octahedron (see picture).