Supraicosahedral polyhedra in carboranes and metallacarboranes: The role of local vertex environments in determining polyhedral topology and the anomaly of 13-vertex closo polyhedra

Metal-free carboranes having 13 vertices are anomalous since their closo polyhedra having the expected 28 skeletal electrons are not the usual deltahedra with exclusively triangular faces but instead polyhedra with one or two trapezoidal faces obtained by removal of one or more edges from the corresponding 13-vertex deltahedron. Removal of such edges converts degree 6 boron vertices in the 13-vertex deltahedron into more favorable degree 5 boron vertices while lowering the degree of nearby carbon vertices. Thus the anomaly of the 13-vertex carborane closo polyhedron can be rationalized by the preference of boron for degree 5 vertices. The 12-vertex tetracarbon carborane (CH₃)₄C₄B₈H₈ with a nido electron count of 28 skeletal electrons but with two quadrilateral faces has a solid state structure derived from a 13-vertex “closo” polyhedron with one quadrilateral face by removal of a degree 4 vertex to give the second quadrilateral face. However, the corresponding tetraethyl derivative (C₂H₅)₄C₄B₈H₈ has a different solid state structure derived from removal of a degree 6 vertex from an unusual 13-vertex deltahedron with three degree 6 vertices to give an open hexagonal face rather than two quadrilateral faces. In contrast to the 13-vertex closo polyhedra, the 14-vertex closo polyhedron is a true deltahedron, namely the D_6d bicapped hexagonal antiprism, which is found in a carborane derivative as well as in several dimetallacarboranes with the metal atoms always at the degree 6 vertices. However, the 15-vertex closo polyhedron, so far found only in the metallaborane 1,2-μ-(CH₂)₃C₂B₁₂H₁₂Ru(η⁶-p-cymene), is a non-deltahedron with one quadrilateral face.     

Systematics of bonding in non-icosahedral carbon clusters

General formulas are presented for the vertex numbers, , of pentagon+hexagon polyhedra of icosahedral, tetrahedral or dihedral symmetries. Criteria for uniqueness of representation, isomer counts and grouping of pentagons are established. All polyhedra with 256 vertices or less and belonging to T, D ₅, D₆or their supergroups are listed. With the addition of C₃ to the dihedral and higher groups, at least one pentagon+hexagon cluster is found for all even 20 except for = 22 which is unrealisable in any symmetry, and = 46 (for which a C₃ polyhedron exists). Carbon clusters with closed electronic shells are shown to be generated by a geometrical leapfrog procedure: for all = 60+6k (where k is zero or greater than one) at least one closed shell structure is predicted. In dihedral symmetry closed shells also exist for some other values of . Separation of the 12 pentagonal faces is not sufficient to ensure a closed electronic shell but appears to be a necessary condition in dihedral or tetrahedral symmetry.     

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 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.     

Liquid crystalline behaviour of mixtures of structurally dissimilar mesogens in binary systems

We have studied four binary systems comprising four ester components, viz. 4-nitrophenyl-4′-n-alkoxybenzoates (wheren-alkoxy isn-butoxy, C₄,n-hexyloxy, C₆,n-octyloxy, C₈ andn-decyloxy, C₁₀) and one azo component, 4-n-decyloxy phenylazo-4′-isoamyloxy benzene. A variety of mesomorphic properties are observed in these mixtures. The properties of these systems are discussed on the basis of phase diagrams.     

Cubane derivatives

NewN,N′-bis- andN,N,N′, N′-tetrakis-hydroxyalkyl-substituted 1,4-cubanedicarboxamides were synthesized. Nitration of these compounds yielded the corresponding nitrates. The reaction of 1,4-cubanedicarboxylic acid dichloride with ethylene glycol mononitrate and glycerol dinitrate gave ester 1,4-[R¹R²CHOC(O)]₂C₈H₆, where R¹=H and R²=CH₂ONO₂; and R¹=R²=CH₂ONO₂, respectively. The cardiopharmacological activity of some of the synthesized compounds was determined. This allowed us to find for the first time cubane derivatives that exhibit this kind of biological activity. The antiischemic activity of one of these compounds,N,N′-bis(2-nitroxythyl)-1,4-cubanedicarboxdiamide, is higher than that of the well-known Nicorandil. For Part 2, see Ref. 1. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1169–1172, June, 1998.     

The Undecakisicosahedral Group and a 3-regular Carbon Network of Genus 26

Three projective special linear groups PSL(2,p), those with p = 5, 7 and 11, can be seen as p-multiples of tetrahedral, octahedral and icosahedral rotational point groups, respectively. The first two have already found applications in carbon chemistry and physics, as PSL(2,5) ≡ I is the rotation group of the fullerene C₆₀ and dodecahedrane C₂₀H₂₀, and PSL(2,7) is the rotation group of the 56-vertex all-heptagon Klein map, an idealisation of the hypothetical genus-3 “plumber’s nightmare” allotrope of carbon. Here, we present an analysis of PSL(2,11) as the rotation group of a 220-vertex, all 11-gon, 3-regular map, which provides the basis for a more exotic hypothetical sp ² framework of genus 26. The group structure and character table of PSL(2,11) are developed in chemical notation and a three dimensional (3D) geometrical realisation of the 220-vertex map is derived in terms of a punctured polyhedron model where each of 12 pentagons of the truncated icosahedron is connected by a tunnel to an interior void and the 20 hexagons are connected tetrahedrally in sets of 4.      

The exponent in the general Randić index

We determine conditions for the parameters n and δ, for which the general Randić index R _δ is not an acceptable index of branching of n-vertex trees, i.e., for which the n-vertex star and the n-vertex path have not extremal R _δ-values among all n-vertex trees. Analogous results are established also in the case of n-vertex chemical trees. Numerous other results for the general Randić index of trees and chemical trees are obtained.      

Quenching of Li (2P) by H2: potential energy surfaces, conical intersection seam, and diabatic bases

The quenching of Li (1s ²2p; ²P) to Li (1s ²2s; ²S) by H₂ is considered using coupled-cluster and multireference configuration-interaction techniques. C _{2 v} (²A₁, ²B₂) and C _{∞ v} (²Π,²Σ⁺) sections of the 1²A^′ and 2²A^′ potential energy surfaces are determined. The C _{2 v} portion of the 1²A^′−2²A^′ seam of conical intersection is studied. Perhaps the most significant finding is a surprising trifurcation of this seam into a portion with only C _s symmetry and the aforementioned C _{2 v} portion. The adiabatic-to-diabatic state transformation is considered in the vicinity of the seam of conical intersection using both perturbation theory and the dipole moment operator. The ²B₂ section of the 2²A^′ potential energy surface exhibits an exciplex in the general vicinity of the seam of conical intersection. The ²Π section of the 2²A^′ potential energy surface possesses a global minimum lying 1.86kcal/mol below the Li (²P)+H₂ asymptote. A van der Waals-like minimum with C _{∞ v} symmetry was found on the 1²A^′ potential energy surface. Received: 14 August 1998 / Accepted: 20 August 1998 / Published online: 11 November 1998     

Large-Scale Chirality Measures and General Symmetry Deficiency Measures for Functional Group Polyhedra of Proteins

Functional Group Polyhedra provide a simplified representation of the most essential spatial features of macromolecules, especially, of globular proteins. Since the functional group polyhedron model focuses on large scale features, the chirality and other symmetry deficiency measures of these molecules, when adapted to these polyhedra, should also be based on the characterization of large scale shape features. Two new approaches for the evaluation of such symmetry deficiency and chirality measures are presented.     

Variation of interatomic distances in oxides

The average interatomic distances D in oxygen polyhedra MO _n of isostructural oxides were proposed to be estimated using the equation D = Kɛ(R _M + R _O) or D = Ax ² + Bx + C, where x = ɛ(R _M + R _O), ɛ is the ionicity of the M-O bond, R _M is the ionic radius of the cation M with account for the coordination in the polyhedron, and R _O is the ionic radius of oxygen. Calculations were made for MO oxides having the rock-salt structure; Ln₂O₃ oxides, where Ln = Ce-Yb; and the MO₂ oxides having the rutile and fluorite structures.     

Crystal structure of (5-methylcyclopentadienyl)-(1,5-cyclooctadiene) iridium(I)

XRD is used to determine the structure of Cp′Ir(cod) at a temperature of 150(2) K. Crystallographic data for C₁₄H₁₉Ir are: a = 10.8272(5) ?, b = 9.7746(4) ?, c = 10.9180(5) ?, β = 97.3310(10)°, monoclinic symmetry, space group P2₁/n, V = 1146.02(9) ?³, Z = 4, d _calc= 2.199 g/cm³, R = 0.0246. The structure is molecular, built of neutral molecules. The metal atom coordinates carbon atoms of two cyclic ligands: 5-methylcyclopentadienyl-ion (Cp′) and 1,5-cyclooctadiene (cod). Five Ir-C_Cp′ distances lie in the range of 2.21–2.28 ?; four Ir-C_cod distances differ insignificantly, and their average value is 2.114(13) ?. The C₁₁C₁₂C₁₃C₁₄C₁₅ and C₁C₂C₅C₆ planes of ligand fragments are almost parallel, and the angle between normals is 1.9°. In the crystal, molecules are bonded only by van der Waals interactions; in the structure, the eight shortest Ir...Ir distances are in the range of 5.608–7.257 ?. Original Russian Text Copyright ? 2009 by K. V. Zherikova, N. B. Morozova, and I. A. Baidina __________ Translated from Zhurnal Strukturnoi Khimii, Vol. 50, No. 3, pp. 591–594, May–June, 2009.     

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.     

Synthesis and crystal structures of cyclopentadienyl dimolybdenum carbonyl complexes

Reactions of the fulvenes C₅H₄C(R ₁ R ₂) [(R ₁ = CH₂CH₃, R ₂ = CH₃ (1); R ₁ = R ₂ = C₂H₅ (2); R ₁, R ₂ = (CH₂)₄ (3), R ₁,R ₂ = (CH₂)₅ (4)] with Mo(CO)₆ in refluxing xylene gave the corresponding cyclopentadienyl dimolybdenum carbonyl complexes [(η⁵-C₅H₄CR₁′R₂′Mo(CO)₃]₂ [(R ₁′ = CH₂CH₃, R ₂′ = CH₃ (5); R ₁′ = R ₂′ = C₂H₅ (6); R ₁′, R ₂′ = CH(CH₂)₃ (7); R ₁′, R ₂′ = CH(CH₂)₄ (8)], which were characterized by elemental analysis, IR and ¹H NMR spectra. The molecular structures were determined by single-crystal X-ray diffraction. The results indicated the exocyclic double bond of the ligands 1 and 2 changed into a single bond and the exocyclic double bond of the ligands 3 and 4 underwent a double-bond isomerization process.     

Flattening of rhodium vertices in mixed rhodium-nickel carbonyl clusters: relationships to borane and zintl ion structures

The flattened deltahedra and related polyhedra found in hypoelectronic bare group 13 metal cluster anions are also found in some anionic mixed rhodium-nickel carbonyl clusters. In all cases the rhodium vertices rather than the nickel vertices are involved in the flattening process so that the rhodium vertices contribute four internal orbitals and the nickel vertices three internal orbitals to the skeletal bonding of the cluster. Thus, the 11-vertex cluster Rh(5)Ni(6)(CO)(21)(3-) has a D(3h) triflattened pentacapped trigonal prismatic structure similar to that found in the In(11)(7-) anion of the intermetallic K(8)In(11). Similarly the polyhedra in the 11-vertex cluster RhNi(10)(CO)(19)(3-) and the 9-vertex cluster Rh(3)Ni(6)(CO)(17)(3-) are both derived from a 10-vertex isocloso polyhedron by capping (for RhNi(10)(CO)(19)(3-)) or vertex removal (for Rh(3)Ni(6)(CO)(17)(3-)) followed by flattening all of the rhodium vertices. A D(3h) icosahedron with flattened rhodium vertices is found in the 12-vertex cluster Rh(3)Ni(9)(CO)(22)(3-).     

Comparative study on interaction of bovine serum albumin with dissymmetric and symmetric gemini surfactant by spectral method

The interaction between bovine serum albumin (BSA) with N, N′-bis(dimethylalkyl) ethylammonium dibromide (C₁₂C₂C_m, m = 8, 12) was investigated by spectral methods. It can be seen that C₁₂C₂C₈ and C₁₂C₂C₁₂ mainly interact with tryptophan residues of BSA from synchronous fluorescence spectra. Fluorescence, far-UV, and near-UV circular dichrosim spectra of BSA are changed by addition of dissymmetric and symmetric gemini surfactant. For surfactant solution, the polarity of the microenvironment surrounding pyrene is lower while the fluorescence lifetime of it is longer and the microviscosity is higher in the presence of BSA than those in the absence of BSA. But compared with C₁₂C₂C₁₂, C₁₂C₂C₈ has lower binding ability with BSA due to the shorter hydrophobic tail and lower symmetry.     

Algorithm for variational calculations of molecular symmetry in solving anharmonic vibrational problems

A universal program for variational calculations of molecular symmetry in solving anharmonic vibrational problems, realized by the author, is described. The program uses the group-theoretical method. Symmetrized basis wave functions are constructed with the aid of the generalized KJebsch-Gordan series suggested by the author. The method of constructing symmetrized basis wave functions and the program for adequate calculations of molecular symmetry were verified for many molecules of different symmetry groups: O_h, O, T_d, T_h, T, D_∞h, C_t8v, D_nd, D_nh, D_n, C_nv, C_nh, S_2n, C_n, C_i, C_s, and C₁ where 2 ≤n ≤6. It was confirmed that the program provides correct results and high-speed operation. Translated fromZhurnal Strukturnoi Khimii, Vol. 38, No. 6, pp. 1146–1153, November–December, 1997.     

Synthesis of mixed-ligand managanese(II) complexes with alkyl xanthate ions and nitrogen-containing heterocycles. Crystal and molecular structure of the [Mn(S2COC3H7−i )2(2,2′-Bipy)] complex

We have synthesized high-spin mixed-ligand Mn²⁺ complexes Mn(S₂COR)₂L where R=i−C₃H₇, i−C₄H₉; [L=1,10-phenanthroline (Phen), 2,2′-bipyridyl (2,2′-Bipy), 4,4′-bipyridyl (4,4′-Bipy)]. As solids, the compounds are stable to oxidation by atmospheric oxygen. An X-ray structural study of the [Mn(S₂COC₃H₇−i)₂(2,2′-Bipy)] complex was carried out. The structure is composed of discrete monomeric molecules. The corrdination polyhedron of the Mn atom is a distorted [4S+2N] octahedron. The molecules are bonded by van der Waals interactions. Institute of Inorganic Chemistry, Siberian Branch, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 35, No. 2, pp. 106–111, March–April, 1994. Translated by T. Yudanova     

Synthesis and crystal structures of two structurally related kryptoracemates

Kryptoracemates are racemic compounds (pairs of enantiomers) that crystallize in Sohnke space groups (space groups that contain neither inversion centres nor mirror or glide planes nor rotoinversion axes). Thus, the two symmetry‐independent molecules cannot be transformed into one another by any symmetry element present in the crystal structure. Usually, the conformation of the two enantiomers is rather similar if not identical. Sometimes, the two enantiomers are related by a pseudosymmetry element, which is often a pseudocentre of inversion, because inversion symmetry is thought to be favourable for crystal packing. We obtained crystals of two kryptoracemates of two very similar compounds differing in just one residue, namely rac‐N‐[(1S ,2R ,3S )‐2‐methyl‐3‐(5‐methylfuran‐2‐yl)‐1‐phenyl‐3‐(pivalamido)propyl]benzamide, C₂₇H₃₂N₂O₃, (I), and rac‐N‐[(1S ,2S ,3R )‐2‐methyl‐3‐(5‐methylfuran‐2‐yl)‐1‐phenyl‐3‐(propionamido)propyl]benzamide dichloromethane hemisolvate, C₂₅H₂₈N₂O₃·0.5CH₂Cl₂, (II). The crystals of both compounds contain both enantiomers of these chiral molecules. However, since the space groups [P 2₁2₁2₁ for (I) and P 1 for (II)] contain neither inversion centres nor mirror or glide planes nor rotoinversion axes, there are both enantiomers in the asymmetric unit, which is a rather uncommon phenomenon. In addition, it is remarkable that (II) contains two pairs of enantiomers in the asymmetric unit. In the crystal, molecules are connected by intermolecular N—H…O hydrogen bonds to form chains or layered structures.     

Chemical applications of topology and group theory. 26. The sextuple diamond-square-diamond rearrangement of the icosahedron through a cuboctahedron intermediate

The details of the symmetry factoring of the graphs corresponding to the icosahedron and the cuboctahedron are presented. Such symmetry factoring procedures use the sequence of two-foldC ₂ and threefoldC ₃ elementsC ₂ xC ₂ x CZ x C3 to give disconnected graphs having eigenvalue spectra similar to those of the original polyhedra but with components having only one and two vertices. In addition, the same symmetry factoring sequence is used to determine the eigenvalue spectrum of an intermediate in the sextuple diamond-square process for conversion of the icosahedron to the cuboctahedron.This paper is dedicated to Professor Frank Harary in recognition of his pioneering work in areas of graph theory closely related to chemical problems. For part 25 of this series, see ref. [1].