A note on pólya enumeration theory

The power group method of de Bruijn and Harary in enumeration under group action of mappings between finite sets is extended to include correlation of group actions on domain and range. By relaxing the restriction of weight functions to be constant over orbits, more specific results concerning the enumeration of orbits by weight are obtained.This paper is dedicated to Professor Dr. Ernst Ruch on the occasion of this 65th birthday     

Structures and Syntheses of Carboranes

Following a discussion of bonding, the structures of the known carboranes of the clovo type and of the incompletely condensed types are described; the recently discovered carboranes with high carbon contents, such as tetracarbahexaborane (“boracarbane”) are also discussed. The preparation of carboranes from polyboranes and the possibility of obtaining organo carboranes from organoboranes are then described. Some rearrangements which take place within the carborane skeleton are also mentioned.     

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.     

Reaction of a ruthenium B-carboranyl hydride complex and BH3(SMe2): Selective formation of a pincer-supported metallaborane LRu(B3H8)

Reaction of a boryl hydride pincer complex (POBOP)Ru(H)(PPh₃) (POBOP?=?1,7-OP(i-Pr)₂-m-2-carboranyl) and BH₃(SMe₂) at 70?°C led to the selective formation of a pincer-supported metallaborane (POBOP)Ru(B₃H₈). Single crystal structure of (POBOP)Ru(B₃H₈) was determined. This complex features coordination of the carborane cluster through adjacent boryl and borane groups that impose significantly different trans-influence on the coordinated B₃H₈ fragment.     

Metallacarborane chemistry of the hypho-[6,7-C2B6H13] anion: Reaction with nickelocene and the formation of three multimetallic nickel-boron clusters

The reaction of the hypho-[6,7-C₂B₆H₁₃]⁻ anion (1) with nickelocene and an excess of ‘proton sponge’ (1,8-bis-(dimethylamino-naphthalene)) in boiling acetonitrile leads to the formation of a pair of isomeric trimetallic nickel-boron clusters, [6,7,8-(CpNi)₃-1-CB₅H₆] (2) and [6,7,8-(CpNi)₃-2-CB₅H₆] (3), in a combined yield of 55%. Isomer (2) had been previously prepared from nido-2-CB₅H₉ but in much lower yield. Isomer (3) is without precedent and has been characterized using multi-nuclear NMR spectroscopy and mass spectrometry. Isomer (3) undergoes conversion to (2) via heating in boiling toluene. In addition to this isomeric pair, an interesting nido dimetallacarborane of constitution [6,6′-(CpNi)₂-7,7′-C₂B₆H₈] (4) has been isolated from the same reaction in 5% yield and characterized by single-crystal X-ray diffraction analysis.     

The relevance of boranes and metallaboranes to the structures of p-block element nanoparticles

Recent d-block element metallaborane chemistry, in which metal identity is varied with constant ancillary ligand, demonstrates how the rising energy of the d orbitals as one moves to earlier metals gives rise to non spherical cluster shapes that permit low formal cluster electron counts. In essence, the separation of frontier orbitals from “nonbonding” orbitals required by the isolobal analogy breaks down and the resulting mixing generates additional high-lying empty orbitals concurrently with shape change. A very similar mechanism explains recent p-block cluster chemistry albeit with variation in extent of external cluster ligation as the variable and separation of external lone pair orbitals from cluster bonding as the problem. Sensible, novel explanations of the shape/electron count relationships can be discovered for large group 13 clusters by recognizing the perturbation in cluster orbital energies when stabilization by ligand interactions is removed. These observations are pertinent to an understanding of large p-block clusters with internal atoms often referred to as nanoparticles.     

Chemistry related to cluster-borane analogues of the cyclopentadienide anion and ferrocene: New developments

A review on cluster-borane analogues of the cyclopentadienide anion (Cp) and ferrocene is presented. Analogues of Cp that have been so far isolated and characterised are the 11-vertex triheteroboranes of general structure [nido-E₃B₈H₈]⁻ (where E = CH or P and their combinations), the molecules of which contain an open pentagonal face. These anions were used as effective ligands for the preparation of “half- and full-sandwich” complexes [CpFeE₃B₈H₈] and [Fe(E₃B₈H₈)₂], respectively - analogues of ferrocene. Developments in this area of cluster-borane chemistry that include recent results in the synthesis and Fe-complexation reactions of 11-vertex tricarbaboranes (tricarbollides), phosphadicarbollides, and diphosphacarbollides are the subject of this work.     

The synthesis and characterisation of some closed polyhedral metallocarbon clusters

Ligands containing unsaturated C₂ and C₄ units have been reacted with triruthenium dodecacarbonyl to produce new organometallic clusters with simple closo-Ru_xC_y polyhedral frameworks which may be regarded as quasi-carboranes. The thermolysis of [Ru₃(CO)₁₂] with 1,4-diphenybutadiene yields the new clusters [Ru₃(CO)₈(μ₃-CPh(CH)₂CPh)] 2 and [Ru₄(CO)₉(μ₄-CPhCCH₂CH₂Ph)] 3, while treatmentof a solution of [Ru₃(CO)₁₂] and diphenylacetylene with trimethylamine N–oxide (Me₃NO) yields [Ru₂(CO)₆(μ-{C₂Ph₂}₂CO)] 4 as the major product and the new cluster [Ru₄(CO)₁₁(μ₄-C₂Ph₂)₂] 5. The solid-state structures of 2, 3 and 5 have been established by single crystal X-ray diffraction analyses and are shown to possess closo-Ru₃C₄ pentagonal bipyramidal, closo-Ru₄C₂ octahedral and closo-Ru₄C₄ dodecahedral skeletons, respectively. The structure and bonding in all three clusters may be rationalised using the Wade–Mingos polyhedral skeletal electron pair approach.     

Stability of polyhedral borane anions and carboranes

Polyhedral borane anions and carboranes that can be constructed formally from the interaction of rings and caps will be stable with six interstitial electrons. Interstitial electron count is obtained by summing the number of π electrons of the ring and the electrons of the caps involved in ring cap binding. Thus B₇H₇ ⁻² (D_5h) has 6 interstitial electrons (none from the B₅H₅ ring, two each from the twobh caps and two negative charge),mndo calculations on isoelectronic pyramidal molecules B₆H₆ ⁻⁴ (C_5v), B₅H₅CH⁻³ (C_5v), B₅H₅ ⁻⁴ (C_4v), B₄H₄CH⁻³ (C_4v), B₄H₄ ⁻⁴ (T _d) and B₃H₃CH⁻³ (C_3v) suggests a criterion based on the out-of-plane bendings of the ring B-H bonds to select the best combination of borocycles and BH or CH caps. Three-membered borocycle prefers CH cap, five-membered borocycle prefers BH cap. The preference of four-membered ring for BH or CH cap is not as pronounced. The extra stability of B₁₂H₁₂ ⁻² arises from the geometry of the icosahedron. The relative stabilities ofnido andcloso carboranes follow from these rules.     

A joint electrochemical/spectroelectrochemical inspection (and re-inspection) of high-nuclearity platinum carbonyl clusters

The accurate study of the electron transfer activity of the tetraanion [Pt₁₉(CO)₂₂]⁴⁻ is presented together with that of the dianion [Pt₃₈(CO)₄₄]²⁻, which was previously studied by spectroelectrochemistry but only partially examined from the electrochemical viewpoint. The main feature of the two clusters is that they undergo a sequence of close-spaced pairs of reversible one-electron processes, which are qualitatively reminiscent of those exhibited by the dianion [Pt₂₄(CO)₃₀]²⁻. In order to focus on such unique aspect of the three structurally characterised platinum clusters, we have investigated (and reinvestigated) their electrochemical and spectroelectrochemical redox properties, also reporting on the electron paramagnetic resonance (EPR) spectrum of the monoanion [Pt₂₄(CO)₃₀]⁻, which represents the first successful study of the paramagnetism of homoleptic platinum–carbonyl clusters.     

Study of formation of aqueous Tl/Ag bimetallic clusters by gamma and electron irradiation

The UV–visible spectral characteristics of the Ag, Tl and bimetallic Tl/Ag clusters stabilized by polyethylene glycol and produced by radiation-induced reduction using both gamma and electron irradiation, have been studied in aqueous medium. A comparison of the two reduction methods, suggested that the high dose rate conditions of electron irradiation resulted in intimately alloyed mixed clusters with almost no aggregation at higher dose. Whereas, under the lower dose rate conditions of gamma irradiation, the bimetallic clusters with core of Ag and shell of Tl were formed due to the reduction of Tl⁺ ions over the initially produced Ag nanoparticles. Only the thallium part of the mixed clusters were oxidized by both methyl viologen (MV²⁺) chloride and oxygen, leaving behind absorption due to Ag nanoparticles.     

The classification of tensor surface harmonic functions for clusters and coordination compounds

A simple method for enumerating theL and functions for polyhedral cluster and coordination molecules, within Stone's tensor surface harmonic methodology, is described. The nature of theL orbitals which are generated depends on the polyhedral topology and in particular the number of layers of vertices and the number of vertices within each layer. The functions are enumerated from theL 's by a number of spherical harmonic multiplication rules.     

Selecting high stability water clusters by examining locations of free OH bonds: Application in the study of clusters

Computational studies on water clusters can be quite challenging, especially when an irregular cage with non-equivalent oxygen sites are considered which may yield a large number of starting geometries that differ in relative positions of non-H-bonding H (NHB H, free OH) atoms. A detailed study on water octamers suggests that the fewest number of NHB H atoms on neighboring oxygen sites yields the most stable neutral isomer followed by those with increasing number of NHB H atoms on adjacent sites. The least stable cluster has all the NHB H atoms around a ring. By considering a regular cage structure and minimum number of NHB H atoms on adjacent sites, one can readily identify a limited number of starting geometries that are optimized to highly stable isomers. This method has been verified in the identification of the most stable isomer of (H₂O)₈ cubic cage and (H₂O)₂₀ dodecahedral cage. The same method has been applied in the study of cluster isomers.     

Structures and properties of the tin-doped carbon clusters

A systemic density functional theory study of the tin-doped carbon clusters has been carried out using B3LYP method with TZP+ basis set. For each species, the electronic states, relative energies and geometries of various isomers are reported. Except for smaller SnC₂ and the largest , the Sn-terminated linear or quasi-linear isomer is the most stable structure for clusters. The electronic ground state is alternate between ³Σ (for n-odd member) and ¹Σ (for the n-even member) for linear SnC_n and invariably ²Π for linear and , except for SnC/SnC⁺/SnC⁻,, and . The incremental binding energy diagrams show that strong even–odd alternations in the cluster stability exist for both neutral SnC_n and anionic , with their n-even members being much more stable than the corresponding odd n − 1 and n + 1 ones, while for cationic , the alternation effect is less pronounced. These parity effects also reflect in the ionization potential and electron affinity curves. By comparing with the fragmentation energies accompanying various channels, the most favorable dissociation channel for each kind of the clusters are given. All these results are very similar to those obtained previously for the clusters.     

Subduction of Q-conjugacy representations and characteristic monomials for combinatorial enumeration

A new method of combinatorial enumeration is presented. The subduction of Q-conjugacy representations gives a characteristic subduction table and a characteristic monomial table. A cycle index is defined on the basis of such monomials and used for combinatorial enumeration of isomers. Received: 10 October 1997 / Accepted: 13 February 1998 / Published online: 17 June 1998     

Comments on McBride's completion of Kroner's proof that hydrogens of benzene are homotopic

This paper shows that McBride's completion of Kroner's proof (J. M. McBride, J. Am. Chem. Soc.102, 4134 (1980)) can be immediately obtained using the methods developed by the author in the context of isomer enumeration and NMR-signal enumeration with a theorem of Pólya. The method is illustrated with both rigid and non-rigid molecules.     

Synthesis and characterization of triosmium and triruthenium pyrene clusters

The reaction between 1-pyrenecarboxaldehyde (C₁₆H₉CHO) and the labile triosmium cluster [Os₃(CO)₁₀(CH₃CN)₂] gives rise to the formation of two new compounds by competitive oxidative addition between the aldehydic group and an arene C-H bond, to afford the acyl complex [Os₃(CO)₁₀(μ-H)(μ-COC₁₆H₉)] (1) and the compound [Os₃(CO)₁₀(μ-H) (C₁₆H₈CHO)] (2), respectively. Thermolysis of [Os₃(CO)₁₀(μ-H)(μ-C₁₆H₉CO)] (1) in n-octane affords two new complexes in good yields, [Os₃(CO)₉(μ-H)₂(μ-COC₁₆H₈)] (3) and the pyryne complex [Os₃(CO)₉(μ-H)₂(μ₃-η¹:η¹:η²-C₁₆H₈)] (4).In contrast, when 1-pyrenecarboxaldehyde reacts with [Ru₃(CO)₁₂] only one product is obtained, [Ru₃(CO)₉(μ-H)₂(μ₃-η¹:η¹:η²-C₁₆H₈)] (5), a nonacarbonyl cluster bearing a pyrene ligand. All compounds were characterized by analytical and spectroscopic data, and crystal structures for 1, 2, 4 and 5 were obtained.     

Mechanical and thermal properties and flame retardancy of phosphorus-containing polyhedral oligomeric silsesquioxane (DOPO-POSS)/polycarbonate composites

A novel polyhedral oligomeric silsesquioxane containing 9,10-dihydro-9-oxa-10-phosphaphenanthrene-10-oxide (DOPO-POSS) has been incorporated into polycarbonate (PC) composites in order to study its effect on mechanical and thermal properties and flame retardancy. The mechanical and thermal properties of the DOPO-POSS/PC composites have been investigated by tensile and flexural testing, DSC, and DMA. Slight enhancements of yield stress, and flexural strength and modulus, and obvious decreases of fracture strength and strain of the DOPO-POSS/PC composites were observed with an increase in DOPO-POSS loading. The glass transition temperatures (T_g) of the composites were reduced with increasing DOPO-POSS loading. The morphology of the PC composites was evaluated by SEM, which indicated that the DOPO-POSS was dispersed with a particle size of 100-250 nm in the PC matrix. The thermal degradation behaviour and flame retardancies of PC composites with different DOPO-POSS loadings were investigated by TGA, LOI, UL-94 standard, and cone calorimetry. The composite had an LOI value of 30.5 and a UL-94 rating V-0 when the content of DOPO-POSS was 4%.     

Molecular and electronic structures of some trimers of polyhedral carbon clusters

Geometries and electronic structures of four crimped linear carbon clusters were modeled by the MNDO/PM3 method. Three of these clusters (C₁₈₀ clusters) are trimers ofI _h -C₆₀ fullene, which differ from each other by the mode of linkage of the monomers. The fourth cluster (C₁₇₂ pseudo-trimer) consists of two C₅₈ fragments of C₆₀ fullerene linked to each other through the C₅₆ cluster. The optimum geometric parameters, hieats of formation, and ionization potentials were calculated for the above-mentioned systems as well as for the corresponding C₁₂₀ and C₁₁₆ dimers. The possibility of extrapolation of the data on dimers and trimers to linear oligomers of the C₆₀ and C₅₆ clusters with a larger number of repeating fragments is discussed. The character of linkages of monomers was analyzed for the two trimers under consideriation, which have the most complex mode of binding of the C₆₀ fullerene molecule and its fragments, using the C₆₀H₂₀ and C₇₂H₂₄ molecules (whose carbon skeletons model the structures of these linkages) as examples. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 1, pp. 7–12, January, 1998.