A complete characterization for <Emphasis Type="Italic">k</Emphasis>-resonant Klein-bottle polyhexes

A hexagonal tessellation K(p, q, t) on Klein bottle, a non-orientable surface with cross-cap number 2, is a finite-sized elemental benzenoid which can be produced from a p × q-parallelogram of hexagonal lattice with usual identifications of sides and with torsion t. Unlike torus, Klein bottle polyhex K(p, q, t) is not transitive except for some degenerated cases. We shall show, however, that K(p, q, t) does not depend on t. Accordingly, criteria for K(p, q, t) to be k-resonant for every positive integer k will be given. Moreover, we shall show that K(3, q, t) of 3-resonance are fully-benzenoid.      

k-resonant toroidal polyhexes

A toroidal polyhex H(p, q, t) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, which can be described by a string (p, q, t) of three integers (p≥ 1, q≥ 1, 0≤ t≤ p−1). A set of mutually disjoint hexagons of H(p, q, t) is called a resonant pattern if H(p, q, t) has a prefect matching M such that all haxgons in are M-alternating. A toroidal polyhex H(p, q, t) is k-resonant if any i (1 ≤ i ≤ k) mutually disjoint hexagons form a resonant pattern. In [16], Shiu, Lam and Zhang characterized 1, 2 and 3-resonant toroidal polyhexes H(p, q, t) for min(p, q)≥ 2. In this paper, we characterize k-resonant toroidal polyhexes H(p, 1, t). Furthermore, we show that a toroidal polyhex H(p, q, t) is k-resonant (k≥ 3) if and only if it is 3-resonant.      

The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

Split series potential energy function

A split series potential energy function is developed herein such that it can be primarily reduced into generalized forms of Lennard-Jones, Morse, Buckingham, Linnett, and anything in between them by altering the type-parameters. Unlike the previous approach whereby the type-parameters act as weighting parameter between the inverse power and the exponential decay terms, the current approach attaches the type-parameters as indices to the denominator of every summation term. A change of the index shifts the series summation to mimic a curve that is representative of an inverse power, an exponential index, or a weighted proportion between the two. Comparison with the Lennard-Jones (12–6) and Exponential-6 (both ξ = 13.772 and ξ = 12) in the case of uncharged non-bonded interactions by using type-parameters (p, q) = (0, 0) and (p, q) = (1, 0) respectively in the model gives very good correlation. Comparison with the Morse potential in the case of bonded interaction by using (p, q) = (1, 1) gives excellent agreement. All results show that the use of only five expansion terms is sufficient for providing good agreement with the classical potentials. Better fit to experimental data and ab initio results are expected with the incorporation of coefficients in every term of the series expansion. The proposed model is useful for modeling the transition state from solid to fluid and vice versa.     

Enantiopure Ferrocene‐Based Planar‐Chiral Iridacycles: Stereospecific Control of Iridium‐Centred Chirality

Reaction of [IrCp*Cl₂]₂ with ferrocenylimines (Fc=NAr, Ar=Ph, p‐MeOC₆H₄) results in ferrocene C?H activation and the diastereoselective synthesis of half‐sandwich iridacycles of relative configuration S_p*,R_Ir*. Extension to (S)‐2‐ferrocenyl‐4‐(1‐methylethyl)oxazoline gave highly diastereoselective control over the new elements of planar chirality and metal‐based pseudo‐tetrahedral chirality, to give both neutral and cationic half‐sandwich iridacycles of absolute configuration S_c,S_p,R_Ir. Substitution reactions proceed with retention of configuration, with the planar chirality controlling the metal‐centred chirality through an iron–iridium interaction in the coordinatively unsaturated cationic intermediate.     

<Emphasis Type="Italic">k</Emphasis>-resonance in Toroidal Polyhexes*

This paper considers the k -resonance of a toroidal polyhex (or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). A toroidal polyhex G is said to be k-resonant if, for 1≤ i ≤ k, any i disjoint hexagons are mutually resonant, that is, G has a Kekulé structure (perfect matching) M such that these hexagons are M-alternating (in and off M). Characterizations for 1, 2 and 3-resonant toroidal polyhexes are given respectively in this paper. *This work is supported by FRG, Hong Kong Baptist University; NSFC and TRAPOYT.     

Asymmetric radical cyclization of (2S,3S)-2,3-butanediyl, (2S,4S)-2,4-pentanediyl,and (2S,5S)-2,5-hexanediyl bis(4-vinylbenzoate)s as a model reaction for asymmetric cyclocopolymerization

The asymmetric cyclocopolymerization of bis(4-vinylbezoate) of a chiral diol with styrene is a promising method for the preparation of optically active polystyrene derivatives because of main-chain chirality. However, the mechanism for chirality induction from the chiral diol to the main chain is still unknown. To clarify the chirality induction mechanism, we carried out the radical cyclizations of (2S,3S)-2,3-butanediyl bis(4-vinylbenzoate), (2S,4S)-2,4-pentanediyl bis(4-vinylbenzoate), and (2S,5S)-2,5-hexanediyl bis(4-vinylbenzoate) with tri-n-butyltin hydride or allyltri-n-butyltin as a chain-transfer reagent as model reactions for asymmetric cyclocopolymerization. The absolute configuration was determined with single-crystal X-ray crystallography and a circular dichroism exciton chirality method. The distribution of the stereoisomer showed (R)-configuration selectivity (21–34% diastereomeric excess) in the intramolecular cyclization and an extremely low extent (<1%) of the (S,S)-cyclized product among the four stereoisomers. Therefore, chirality induction is caused by the selective inhibition of the (S,S)-racemo configuration. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 4671–4681, 2004     

Synthesis of novel ferrocenylphosphine-amidine ligands with central and planar chirality and their diastereomeric effect in Pd-catalyzed asymmetric allylic alkylation

Some novel ferrocenylphosphine-amidine ligands with central and planar chirality were prepared from (R,S_p)-PPFNH₂-R 3 and its diastereomer (S,S_p)-PPFNH₂ 3a. The efficiency and diastereomeric impact of these ferrocenylphosphine-amidine ligands in the palladium-catalyzed asymmetric allylic substitution was examined, and up to 96% e.e. with 98% yield was achieved by the use of ligand (R,S_p)-4a with a methyl group in the amidino moiety. The results also indicated that (R)-central chirality and (S_p)-planar chirality in these ferrocenylphosphine-amidine ligands were matched for the palladium-catalyzed asymmetric allylic alkylation.     

Effect of the structural parameters of a series of fluoromonoacrylates on the surface properties of cured films

A series of fluorinated acrylates [F(CF₂)_q? (CH₂)_p? OCOCH?CH₂, where q = 8 and p = 2 ÷ 11) were synthesized and used as comonomers in the photopolymerization of acrylic systems. These fluoroacrylates were synthesized in a three‐step procedure through the radical addition of perfluoroalkyliodides to an unsaturated alcohol followed by their acrylation. The mixtures, containing up to 0.8% (w/w) of the comonomers, produced transparent films after UV curing; they showed changes in the surface properties as a function of the comonomer type and concentration. With contact‐angle and X‐ray photoelectron spectroscopy analyses, the relationship between the structure of the monomers and the surface properties of the UV‐cured films was investigated: the wettability decreased, depending on the length of the fluorinated group (q value) and the hydrogenated segment of the monomer (p value). © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 4227–4235, 2001     

Sorption of Certain Metal Ammine Complexes on Hydrous β-SnO2

Sorption of ammine complexes of Ni⁺², Cu⁺², Zn⁺² and Cd⁺² on hydrous Tin dioxide has been studied at different pH values. It has been found that qΔ and Kd values attain a maximum with increasing pH, beyond which a fall is observed. The selectivity quotients of the exchanger material towards competing pairs (Cu⁺²-Zn⁺², Cu⁺²-Cd⁺², Cu⁺²-Ni⁺²) of complex ions have been measured as a function of the concentration of the external solution. The selectivity of the exchanger material towards the preferred ion decreased with increase in the concentration of the competing ions in the external solution. The selectivity and the sorption values (qδ) follow the same order.     

The Honeysuckle and the Bindweed Revisited: Synthesis and Stereochemistry of Extended Helical Structures from Coordination Complexes

We report structural studies of a chiral tridentate ligand which forms helical cubanes with cobalt(II) and manganese(II). A quadruple helicate with (P)-chirality is obtained using a (S)-ligand with cobalt(II) but the ligand binds manganese(II) in one of two possible orientations and either (P)- or (M)-quadruple helicates may be observed for a given ligand enantiomer. The helicates may be linked into extended structures by p-nitrobenzoate capping ligands which show stacking interactions with neighbouring complexes. With cobalt(II) we find an extended helical structure with (M)-chirality linking helicates which themselves have (P)-chirality. With manganese(II) we observe a remarkable structure with extended (M)-helices coexisting with extended (P)-helices.     

Stereochemistry of planarchiral compounds,part XII: Absolute chiralities and X-ray crystal structures of 2,2′-Bi(1,6-methano[10]annulenyl)s

Summary The absolute chirality of the dextrorotatory main rotamer of 10,10-dibromo-2,2-bi(1,6-methano[10]annulenyl) (1 b), obtained by low temperature chromatography or by crystallisation of the (+)-enantiomer of1 b, was established by observation of anomalous X-ray diffraction as (+)-(R) _p (R) _a (R) _p -1b. This result was confirmed by reduction of (+)-1 b to a dextrorotatory 2,2-bi(1,6-methano[10]annulenyl) (1 a), which in turn was accessible by coupling of (–)-2-Bromo-1,6-methano[10]annulene (3) of known chirality (S) _p and, moreover, by enantioselective chromatography of1 a on triacetyl cellulose in ethanol. This correlation determines also the chirality of (+)-1 a as (S) _p (S) _p .According to the X-ray analyses the main rotamer of (+)-1 b and the meso-form (2 b) adopt crystal structures in which the bromine atoms come close to the CH₂-bridges. The torsional angles around the 2,2-bonds vary between 33°, 53° and 65° for (+)-1 b, racem.1 b and the mesoform2 b, respectively.Dedicated to Prof. Dr. G. Snatzke (Bochum) with cordial wishes on the occasion of the 60th anniversary of his birthday     

New structural parameters of fullerenes for principal component analysis

The Kekulé structure count and the permanent of the adjacency matrix of fullerenes are related to structural parameters involving the presence of contiguous pentagons p, q, r, q/p and r/p, where p is the number of edges common to two pentagons, q is the number of vertices common to three pentagons and r is the number of pairs of nonadjacent pentagons adjacent to another common pentagon. The cluster analysis of the structural parameters allows classification these parameters. Principal component analysis (PCA) of the structural parameters and the cluster analyses of the fullerenes permit their classification. PCA clearly distinguishes five classes of fullerenes. The cluster analysis of fullerenes is in agreement with PCA classification. Cluster analysis shows greatest similarity for the q–q/p and r–r/p pairs. PCA provides five orthogonal factors F ₁–F ₅. The use of F ₁ gives an error of 28%. The inclusion of F ₂ decreases the error to 2%.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresión Latina (QUITEL 2002)     

An exact expression for the wiener index of a TUC<Subscript>4</Subscript>C<Subscript>8</Subscript>(R) nanotorus

The Wiener index of a graph G is defined as , where V(G) is the set of all vertices of G and for denotes the length of a minimal path between x and y. A C ₄ C ₈ net is a trivalent decoration made by alternating squares C ₄ and octagons C ₈. It can cover either a cylinder or a torus. In this paper, an algorithm for computing the distance matrix of a C ₄ C ₈(R) nanotorus T = T[p,q] is given. Using this matrix, the Wiener index of T is computed.     

The incompleteness of Rényi entropies

Since the advent of the notion of mixing character in statistical mechanics, it has been conjectured over and over again that the Rényi entropies provide a mixing isomorphic family, that is: given two probability distributions p and q, the mixing character of q exceeds that of p, m[q]> m[p], if and only if I (q)I (p) for any positive . This conjecture is disproved by means of counterexamples.     

Effect of the variation of swallow-tailed groups on the mesomorphic and electro-optical properties of chiral compounds derived from (S)-2-(6-hydroxy-2-naphthyl)propionic acid

Three homologous series of chiral swallow-tailed compounds, alkyl (S)-2-{6-[4-(4′-alkoxyphenyl)benzoyloxy]-2-naphthyl}propionates, (S)HNP(p,n,q) derived from (S)-2-(6-hydroxy-2-naphthyl)propionic acid in conjugation with a variety of swallow-tailed groups, attached to the external side of the chiral centre, have been synthesized and their mesomorphic and electro-optical properties studied. Both (S)HNP(p,1,2) and (S)HNP(p,1,3) exhibited an enantiotropic antiferroelectric SmC*_A phase. This implys that the swallow-tailed groups in the molecules favour zigzag pairing of the molecules in the smectic phase. The maximum P _S values of compounds (S)HNP(p,1,2) in the antiferroelectric phase were measured in the range 21–30 nC cm^-2; those of compounds (S)HNP(p,1,3) were in the range 15–23 nC cm^-2, indicating that these chiral compounds possess low polarity. The electro-optical response of the compounds in the antiferroelectric SmC*_A phase displayed thresholdless V-shaped switching.     

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.     

Coordination formulation of tetrahedrally close packed structures: An addendum to the observations of Yarmolyuk and Kripyakevich

Ya. P. Yarmolyuk and P. I. Kripyakevich (Kristallographiya 19, 539 (1974)) showed that all tetrahedrally close packed (t.c.p.) structures have coordination formulae P_pQ_qR_rX_x → (PX₂)_i(Q₂R₂X₃)_j (R₃X)_k, where P, Q, R, and X represent coordination numbers (CN) 16, 15, 14, and 12 polyhedra respectively: p, q, r, and x indicate the numbers of such polyhedra in the unit cells of t.c.p. structures and i, j, and k are positive integers. We propose and demonstrate a limitation to the above formulation: if i ≥ 1 and k ≥ 1, then j ≥ 1 (or if both p> 0 and r> 0, then q> 0). We give reasons for this and discuss the Aufbauprinzip of t.c.p. structures and the results of C. B. Shoemaker and D. P. Shoemaker (Acta Crystallogr. B 42, 3 (1986)).     

A partial recurrence relation for reduced class coefficients of the symmetric group

An expression for the product of a single-cycle class [(1)^{N - P}(p)]_N and an arbitrary class [(1)^l1(2)^l …? (N)^lN]_N of the symmetric group has recently been conjectured. This expression involves a sum over a relatively small number of reduced class sums, depending on p indices. A further conjecture is formulated and demonstrated, according to which reduced class coefficients (RCCS ) involving cycles whose length is expressed by means of a single index can be related to corresponding coefficients in the product of [(1)^{N - P+1}(p - 1)]_N with an arbitrary class sum. Consequently, the problem of evaluating the general class sum product reduces to that of obtaining a relatively small set of fundamental RCCS containing no single-index cycles. The conjectures mentioned can be used to evaluate the product [(1)^{N - p}(p)]_N · [(1)^{N - q}(q)]_N in terms of fundamental RCCS that can all be obtained from the product [(r)]_r · [(r)]_r, where r = min(p, q). For the latter product, we use a result due to Boccara.     

Screening of ligands in the asymmetric metallocenethiolatocopper(I)-catalyzed allylic substitution with Grignard reagents

Screening of metallocenethiolate ligands for copper(I)-catalyzed substitution of allylic acetates with Grignard reagents has been carried out. The previously used ligand, lithium (R,S_p)-2-(1-dimethylaminoethyl)ferrocenylthiolate (4a), possessing both central and planar chirality, was the starting point for the screening. It was found that the diastereomeric ligand lithium (R,R_p)-2-(1-dimethylaminoethyl)ferrocenylthiolate (4b) exhibiting reversed planar chirality gave increased enantioselectivity in the allylic substitution, at least when cinnamyl acetate was used as a substrate. The ruthenocene-based ligand lithium (R,S_p)-2-(1-dimethylaminoethyl)ruthenocenylthiolate (4c) gave an enhanced reaction rate, but lower chiral induction. The use of disulfide bis[(R,S_p)-2-(1-dimethylaminoethyl)ferrocenyl]disulfide (7a) as a ligand precursor worked well but resulted in lower enantioselectivity.