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.     

Symmetry properties of chemical graphs. IV. Rearrangement of tetragonal-pyramidal complexes

Using a canonical numbering of vertices for the graph corresponding to a particular rearrangement of tetragonal–pyramidal complexes, all 120 permutations defining the symmetry for the rearrangement are derived. An examination of the permutations points to the symmetric group S₅, which has previously been found for isomerizations of trigonal–bipyramidal complexes and in the rearrangement of homotetrahedryl cations.     

Chemical applications of topology and group theory. XXII : Lowest degree chirality polynomials for regular polyhedra [1]

The lowest degree chirality polynomials for the regular octahedron, cube, and regular icosahedron are discussed. All three of these regular polyhedra are chirally degenerate since they have more than one lowest degree chiral ligand parition by the Ruch-Schönhofer scheme. The two lowest degree chirality polynomials for the octahedron have degree 6 and can be formed from three degree 3 generating polynomialsf,g, andh through the relationshipsf(g +h) andf(g –h), wheref,g, andh measure the effects of the three separating reflection planes (h), the four threefold rotation axes, and the three fourfold rotation axes, respectively. The permutation groups of the vertices of the cube and icosahedron contain only even permutations, which leads to a natural pairing of their chiral ligand partitions according to equivalence of the corresponding Young diagrams upon reflection through their diagonals. The two lowest degree chirality polynomials for the cube have degree 4 and can be formed from two degree 4 generating polynomialsf andg through the relationships –2g andf –2g, wheref andg measure the effects of theS ₆ improper rotation andC ₄, proper rotation axes. respectively. The four lowest degree chiral ligand partitions for the icosahedron have degree 4 and lead naturally to a single degree 4 chirality polynomial with 120 terms of the general type (x –y)² (z –w)². This chirality polynomial for the icosahedron cannot be broken down into simpler generating polynomials, in contrast to the lowest degree chirality polynomials for the octahedron and cube. This appears to relate to the origin of the icosahedral group from the simple alternating groupA ₅. The full icosahedral chirality polynomial can be simplified to give a chirality polynomial for the chiral boron-monosubstituted ortho and meta carboranes of the general formula B₂C₁₀H₁₁X.     

Burnside Rings: Application to Molecules of Icosahedral Symmetry

The Burnside ring, B(G), of a group G is the set of isomorphism classes of orbits of G together with the operations of addition and product. The addition is defined as the disjoint union, and the product as the Cartesian product. This paper describes basic facts about this algebraic structure and develops some applications in chemistry, as the labelling of atoms in molecules of high symmetry and the construction of symmetry-adapted functions. For illustrating such applications, the concept of Burnside ring is applied to the icosahedral symmetry. Sets of points which are isomorphic to the orbits of the I group are described and the multiplication table of B({I}) is obtained from the table of marks. This multiplication table allows us to obtain an elegant labelling of the atoms of the buckminsterfullerene which is consistent with the icosahedral symmetry. Also, we obtain complete sets of symmetry-adapted functions for the buckminsterfullerene which span the Boyle and Parker's icosahedral representations.     

Complexes of Co(II) and Zn(II) with N-(Thio)phosphorylthioureas AdNHC(S)NHP(O)(OiPr)2 and MeNHC(S)NHP(S)(OiPr)2

The reaction of the potassium salt of the N-(thio)phosphorylated thioureas AdNHC(S)NHP(O)(OiPr)₂ (HL^I , Ad = Adamantyl) and MeNHC(S)NHP(S)(OiPr)₂ (HL^II ) with Co(II) and Zn(II) in aqueous EtOH leads to [ML^I,II ₂] chelate complexes. They were investigated by UV-vis, ¹H and ³¹P NMR spectroscopy, and microanalysis. The molecular structures of [ML^I ₂] were elucidated by single crystal X-ray diffraction analysis. The metal centers in both complexes are found to be in a distorted-tetrahedral O₂S₂ environment formed by the C=S sulfur atoms and the P=O oxygen atoms of two deprotonated L^I ligands. The photoluminescence properties of [ZnL^II ₂] are also reported.     

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.     

The mechanisms of substitution reactions in octahedral complexes from the point of view of ligand field theory

Two possible mechanisms for substitution reactions in octahedral complexes, ML₆, are discussed in terms of molecular orbital theory. Jorgensen's model with angular parameters is used to calculate the change in activation energy on forming complexes of the type ML₅ (D₃h symmetry), ML₅ (C_4v symmetry), and ML₇ D_5h symmetry). Analysis of the quantities obtained (Table 4) shows that high spin ML₆ octahedral complexes of metals with d³ or d⁸ electronic configurations, and low spin complexes with d⁶ electronic configurations are particularly stable. An S_N1 mechanism is, apparently, characteristic for complexes of metals with d¹ or d² electronic configurations. The formation of bonds facilitates the course of substitution reactions in octahedral complexes. The results we have obtained explain the available experimental material and permit us to make some predictions.     

More Icosahedral Fulleroids

The discovery of the famous fullerene has raised an interest in the study of other candidates for a modeling of carbon molecules. Motivated by a P. Fowler's question Delgado Friedrichs and Deza defined I(a,b)-fulleroids as cubic convex polyhedra having only a-gonal and b-gonal faces and the symmetry groups isomorphic with the rotation group of the regular icosahedron. In this note we prove that for every n8 there exist infinitely many I(5,n)-fulleroids. This answers positively questions posed recently by Delgado Friedrichs and Deza.     

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.      

Coordination chemistry and biological activity of nickel(II) and copper(II) ion complexes with nitrogen–sulphur donor ligands derived from S-benzyldithiocarbazate (SBDTC)

A bidentate and a quadridentate Schiff base having NS and NNSS donor sequences were prepared by condensing S-benzyldithiocarbazate (NH₂NHCSSCH₂Ph) with 2,3-butanedione (1:1 and 1:2 mole ratio). Ni^II and Cu^II complexes of these ligands were studied and characterised by elemental analyses and various physico-chemical techniques. The nickel complexes, [Ni(NS)₂] and [Ni(SNNS)], were diamagnetic with square-planar and five-coordinate structures, respectively. The copper complex was, however, pentacoordinated. The ligands and the complexes were screened for anticancer activity against T-lymphoblastic leukemic cells (CEM-SS) and colon cancer cells (HT-29). The NS Schiff base was strongly active against leukemic cells with a CD₅₀ value of 2.05 g cm^–3. The nickel and copper complexes were found to be stronger antioxidants than Vitamin E.     

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.     

Speciation in the Vanadium(III)–Carnosine System

Speciation in the aqueous V(III)–carnosine system has been determined from potentiometric and spectroscopic (UV-Vis absorption and CD) data. Application of the Hyperquad program to the experimental potentiometric data indicates that under our experimental conditions (I=0.5 mol⋅L⁻¹ NaClO₄, pH=2 to 6.5, and L/M>5) only ML₂H₄, ML₂H₃, ML₂H₂ and ML₂H form. These potentiometric results prove that stable complexes form and, with use of the spectroscopic methods, the binding sites are identified.     

New computer program to calculate the symmetry of molecules

In this paper we, present some MATLAB and GAP programs and use them to find the automorphism group of the Euclidean graph of the C₈₀ fullerence with connectivity and geometry of I _h symmetry point group. It is proved that this group has order 120 and is isomorphic to I _h ≊Z₂×A₅, where Z₂ is, a cyclic group of order 2 and A₅ is the alternating group on five symbols.     

Symmetry properties of chemical graphs. V. Internal rotation in XY⋅3XY⋅2XY3

Structures XY?₃XY?₂XY₃ of symmetry C_2v (of which propane is an example) are examined and the rearrangement due to the internal rotation of the end groups XY₃ studied. The isomerization graph is constructed, various forms of which are displayed and the symmetry of which has been determined. The order of the group is 72. There are nine prime (irreducible) representations (4A + E + 4G) with the following partitioning of the elements into classes: 1, 4², 6², 9, 12², 18. When the mechanism for rearrangement is generalized to include enantiomers, a duplex graph is produced with the order of the group 144 which is isomorphic to the group S₂(S₃,S₂) (generalized wreath product of the symmetric group S₂ and S₃). The corresponding graph has been constructed and displayed in one of more symmetrical forms. Isomorphism of groups of order 144 is discussed and a procedure is outlined in which correspondence between distinctive combinatorial objects is established by inducing permutations of m elements from available permutations of n elements. The scheme is based on selection of suitable graph invariants in one system and their labeling as m objects which form the basis for representation of the symmetry for the other system.     

Magnetic,spectroscopic and biological properties of copper(II) complexes of the tridentate ligand, α-N-methyl-S-methyl-β-N-(2-pyridyl)methylenedithiocarbazate(NNS) and the X-ray crystal structure of the [Cu(NNS)I2] complex

Copper(II) complexes of general formula, Cu(NNS)X ₂ · nH₂O (NNS = the 2-formylpyridine Schiff base of N-methyl-S-methyldithiocarbazate; X = Cl^–, Br^–, I^–, NCS^–; n = 0, 2) have been synthesized and characterized by elemental analysis and by magnetic and spectroscopic techniques. Based on magnetic and spectroscopic data, a monomeric five-coordinate square-pyramidal structure is assigned to these complexes. The crystal and molecular structure of [Cu(NNS)I₂] has been determined by X-ray diffraction. The complex has a monomeric square-pyramidal structure with the ligand coordinated to the copper(II) ion via the pyridine nitrogen atom, the azomethine nitrogen atom and the thione sulfur atom. The fourth and fifth coordination sites are occupied by the iodide ligands. Antimicrobial tests indicate that Schiff base is inactive against the bacteria, Bacillus subtilis (mutant defective DNA repair), Pseudomonas aeruginosa, methicillin resistant Staphylococcus aureus and Bacillus subtilis (wild type) and weakly active against the fungi, Candida albicans, Candida lypolytica, Saccharomyces cereviseae and Aspergillus ochraceous but its copper(II) complexes, Cu(NNS)X ₂ are strongly active against these organisms. A cytotoxicity study of the compounds against leukemic and cervical cancer cells showed that the Schiff base is inactive, but the complexes, [Cu(NNS)I₂] and [Cu(NNS)(NCS)₂] · 2H₂O exhibit significant activity against cervical cancer cells with CD₅₀ values of 4.8 and 4.2 g, respectively.     

Icosahedral hybrid orbitals

A set of twelve equivalent icosahedral hybrid orbitals pointing from the centre to the corners of a regular icosahedron has been obtained. Such hybrids can be used to explain the geometry of twelve-coordinate complexes of a rare-earth atom. Using group theoretical considerations, it is shown that these hybrids can be constructed by linear combination of one s, three p, five d and three f-orbitals. Bearing in mind that the twelve hybrids have identical shape but are oriented differently in space, their mathematical expressions have been obtained by applying geometrical transformations to the sp³d⁵f³ hybrid pointing along the positive z-axis. In order to obtain elegant mathematical expressions, the x, y and z axes have been chosen to be coincident with three orthogonal binary axes of the icosahedron.     

Synthesis,Oxygenation and Catalytic Epoxidation Performance of Salen and Salophen Transition-Metal Complexes with Aza-Crown or Morpholino Pendants

Co^II and Mn^III complexes with aza-crown or morpholino substituted Salen and Salophen ligands were synthesized starting from benzo-10-aza-15-crown-5 or morpholine. The saturated oxygen uptake of the Co^II complexes CoL¹–CoL⁴ in MeOCH₂CH₂OMe solution was determined at different temperatures. The equilibrium constant (KO₂) and thermodynamic parameters (H ⁰, S ⁰) for oxygenation were calculated. Meanwhile, the corresponding Mn^III complexes, MnL¹Cl–MnL⁴Cl, were employed as models of mimic mono-oxygenase to catalyze PhCH=CH₂ epoxidation at ambient temperature and pressures. The modulation of O₂-binding capabilities and catalytic oxidation performance by these pendant substituents in the complexes were investigated and compared with the parent complexes ML⁵(Msalen) and ML⁶(Msalophen). The results indicate that the dioxygen affinities and catalytic oxidation activities of these complexes have been much more enhanced by aza-crown pendants than by morpholino pendants. Moreover, the O₂-binding capabilities of bis(aza-crown ether) Co^II complexes, CoL¹ and CoL², would also be improved by adding alkali metal (Li⁺, Na⁺ and K⁺) cations to the system. Adding K⁺ shows the most significant enhancement of dioxygen affinity through its forming sandwich-type complexes with two aza-crown ethers of CoL¹ and CoL². Likewise, the bis(aza-crown ether) Mn^III complexes, MnL¹ and MnL², exhibit the best catalytic activity: the conversions of PhCH=CH₂ attain 76.6, 79.5% respectively.     

New fac-tricarbonyl rhenium(I) semicarbazone complexes: synthesis,characterization, and biological evaluation

Expanding our previous work on salicylaldehyde semicarbazone metal complexes as prospective anti-trypanosomal agents, five new fac-Re^I(CO)₃-containing complexes with ligands of this semicarbazone series were synthesized and characterized. An atypical coordination mode of these potentially tridentate ligands through only the carbonylic oxygen and the azomethine nitrogen (the so-called N,O fashion) was demonstrated by IR spectroscopy and supported by theoretical calculations. Three of the compounds showed moderate in vitro anti-Trypanosoma cruzi activity and increased activity with respect to the corresponding free ligands. The brominated ligands, 5-bromo-2-hydroxybenzaldehyde semicarbazone (L2) and 5-bromo-2-hydroxy-3-methoxybenzaldehyde semicarbazone (L5), led to the most active rhenium(I) complexes. These compounds are among the few reported examples of rhenium complexes bearing in vitro activity against T. cruzi.     

Complex Equilibria of Organotin(IV) in Aqueous Solution with Imidazole Derivatives

Summary. Equilibria studies in aqueous solution containing 25% dioxane (V/V) are reported for dimethyltin(IV) and trimethyltin(IV) (M) complexes with some imidazole derivatives (L). Stoichiometry and stability constants for the complexes formed were determined at 25°C and ionic strength 0.1M NaNO₃. The results of the dimethyltin(IV) complexes showed the best fit of the titration curves when complexes ML, ML ₂, ML ₂H_–1, and ML ₂H_–2 were expected beside the hydrolysis products of the dimethyltin(IV) cation, while the calculations of the trimethyltin(IV) complexes reported the presence of only the complexes ML, MLH_–1, and the hydrolysis products of the trimethyltin(IV) cation. The concentration distribution of each species of the complexes in solution was evaluated. The stability of all complexes formed was investigated and discussed in terms of molecular structure of the ligand imidazole and the nature of the alkyltin cation. It is deduced that the stability of the complex formed increases as the basicity of the ligand imidazole is increased. On the other hand, the trimethyltin(IV) cation has a very low ability to form complexes compared to the dimethyltin(IV) cation.Received November 22, 2002; accepted (revised) March 3, 2003 Published online August 18, 2003     

Electrochemical oxidation and reduction of osmium and iridium complexes with fullerene C<Subscript>60</Subscript>

The electrochemical oxidation and reduction of a series of complexes of platinum-group metals (Os, Ir, and Rh) with fullerene C₆₀ with common formula C₆₀[ML _n ] (ML _n is the metal and the metal-bonded ligands) in the tetrahydrofuran solution are studied using the method of cyclic voltammetry. The effect of metal and ligands on the variations in the redox properties of complexes and stability of anions, which are formed in the reduction, is considered. It is shown that all complexes studied are reduced more difficultly than free fullerene, and the reduced forms of metallofullerenes are less stable as compared with free fullerene anions.