Reactions of the cationic diruthenium carbonyl complex [Ru2(μ-dppm)2(CO)4(μ,η-O2CMe)] with bidentate ligands; intramolecularly assisted stereospecific synthesis via the second-sphere face-to-face π–π stacking interactions

The reactions of the diruthenium carbonyl complexes [Ru₂(μ-dppm)₂(CO)₄(μ,η²-O₂CMe)]X (X=BF₄⁻ (1a) or PF₆⁻ (1b)) with neutral or anionic bidentate ligands (L,L) afford a series of the diruthenium bridging carbonyl complexes [Ru₂(μ-dppm)₂(μ-CO)₂(η²-(L,L))₂]X_n ((L,L)=acetate (O₂CMe), 2,2′-bipyridine (bpy), acetylacetonate (acac), 8-quinolinolate (quin); n=0, 1, 2). Apparently with coordination of the bidentate ligands, the bound acetate ligand of [Ru₂(μ-dppm)₂(CO)₄(μ,η²-O₂CMe)]⁺ either migrates within the same complex or into a different one, or is simply replaced. The reaction of [Ru₂(μ-dppm)₂(CO)₄(μ,η²-O₂CMe)]⁺ (1) with 2,2′-bipyridine produces [Ru₂(μ-dppm)₂(μ-CO)₂(η²-O₂CMe)₂] (2), [Ru₂(μ-dppm)₂(μ-CO)₂(η²-O₂CMe)(η²-bpy)]⁺ (3), and [Ru₂(μ-dppm)₂(μ-CO)₂(η²-bpy)₂]²⁺ (4). Alternatively compound 2 can be prepared from the reaction of 1a with MeCO₂H–Et₃N, while compound 4 can be obtained from the reaction of 3 with bpy. The reaction of 1b with acetylacetone–Et₃N produces [Ru₂(μ-dppm)₂(μ-CO)₂(η²-O₂CMe)(η²-acac)] (5) and [Ru₂(μ-dppm)₂(μ-CO)₂(η²-acac)₂] (6). Compound 2 can also react with acetylacetone–Et₃N to produce 6. Surprisingly [Ru₂(μ-dppm)₂(μ-CO)₂(η²-quin)₂] (7) was obtained stereospecifically as the only one product from the reaction of 1b with 8-quinolinol–Et₃N. The structure of 7 has been established by X-ray crystallography and found to adopt a cis geometry. Further, the stereospecific reaction is probably caused by the second-sphere π–π face-to-face stacking interactions between the phenyl rings of dppm and the electron-deficient six-membered ring moiety of the bound quinolinate (i.e. the N-included six-membered ring) in 7. The presence of such interactions is indeed supported by an observed charge-transfer band in a UV–vis spectrum.     

Versatile 2,5‐Pyridinedicarboxylate in Linking Transition‐Metal Atoms into 1D and 2D Coordination Polymers and Concomitant Polymorphs

By employing one bridging ligand, 2,5‐pyridinedicarboxylate (2,5‐pda^2?), in the presence or absence of another bridging ligand, 4,4′‐bipyridine (4,4′‐bpy), one one‐dimensional (1D) {[Co₂(2,5‐pda)(2,5‐Hpda)₂(4,4′‐bpy)(H₂O)₃]·6H₂O}_∞ ( 1 ) and two two‐dimensional (2D) coordination polymers, {[Cu₃(2,5‐pda)₃(H₂O)₃]·6H₂O}_∞ ( 2 ) and {[Co(2,5‐pda)(H₂O)]·2H₂O}_∞ ( 3 ) were synthesized. Complexes 2 and 3 are characterized as concomitant polymorphs from a one‐pot reaction at ambient temperature. A comparison of the coordination geometries of all neutral and anionic coordination polymers containing {M_x(2,5‐pda)_y(H₂O)_z}_∞ available to date is presented.     

On a Family of Diophantine Difference Equations

The family of Diophantine difference equations considered byJ. H. Conway, H. T. Croft, P. Erds and M. J. T. Guy can be solvedin closed form when the parameter is a prime. The associatedcharacterisation problem is shown to be related to the determinationof certain constants, which also happen to be coefficients inthe asymptotic formula for the associated exponential sum.     

Extremal Hosoya index and Merrifield-Simmons index of hexagonal spiders

For any graph G, let m(G) and i(G) be the numbers of matchings (i.e., the Hosoya index) and the number of independent sets (i.e., the Merrifield-Simmons index) of G, respectively. In this paper, we show that the linear hexagonal spider and zig-zag hexagonal spider attain the extremal values of Hosoya index and Merrifield-Simmons index, respectively.     

Methods for estimating the optimal dividend barrier and the probability of ruin

In applications of collective risk theory, complete information about the individual claim amount distribution is often not known, but reliable estimates of its first few moments may be available. For such a situation, this paper develops methods for estimating the optimal dividend barrier and the probability of ruin. In particular, two De Vylder approximations are explained, and the first and second order diffusion approximations are examined. For several claim amount distributions, the approximate values are compared numerically with the exact values. The De Vylder and diffusion approximations can be adapted to the more general situation where the aggregate claims process is a Lévy process with nonnegative increments.     

Direct comparison of equibiaxial elongational viscosity measurements from lubricated squeezing flow and the MultiAxiales Dehnrheometer

We present the first direct comparisons of rheological data from the lubricated squeezing flow (LSF) technique and the MultiAxiales Dehnrheometer (MAD) instrument developed by Meissner and coworkers (J Rheol 47:989–1010, 2003). Comparisons of transient equibiaxial elongational viscosity are carried out at strain rates well into the nonlinear regime on low-density polyethylene and polystyrene melts. We find data obtained using LSF deviate from the MAD data when the Hencky strain reaches a value of approximately 1, which we interpret as a failure of the LSF technique. The strain at which the LSF technique fails is relatively insensitive to experimental parameters including strain rate. For Hencky strains larger than 1, LSF data display behavior that could easily be mistaken for the phenomenon of strain hardening.     

Toward chiral sum-frequency spectroscopy

Chiral sum-frequency (SF) spectroscopy that measures both the real and the imaginary components of the SF spectral response was demonstrated for the first time. It was based on interference of the SF signal with a dispersionless SF reference. Solutions of 1,1'-bi-2-naphthol (BN) were used as model systems, and their chiral SF spectra over the first exciton-split transitions were obtained. Chiral spectra are useful for determination of absolute configuration and conformation of chiral molecules.     

The effects of thermal annealing on the relaxation of rubbing-induced birefringence in polystyrene

We have conducted a systematic study on the effects of post rubbing annealing on the relaxation of rubbing-induced birefringence of polystyrene. It is found that annealing at T₀ only affects the relaxation up to T₀ + T_Lag, where T_Lag is proportional to the logarithm of the annealing time t_A. A theoretical model based on the distribution of relaxation times due to the individual birefringence elements is proposed. To remove its contribution to the net birefringence each element must overcome an energy barrier E = (317 + 1.17ξ)×10³ J/mol, and therefore must have a characteristic relaxation time τ which depends on temperature T and a barrier height which ranges from 340.4 kJ/mol to 445.7 kJ/mol. The relaxation of birefringence is expressed by the equation NB(T, t) = N(ξ)e^-t/τ(T,ξ)dξ, in which both the relaxation time τ(T,ξ) and the distribution function N(ξ) can be extracted from experimental data. The predictions of the model agree well with all the experimental results presented in this work. The differences and similarities of the relaxation of birefringence with respect to the physical aging of quenched PS are discussed. In particular, similarities in terms of the general temperature lag phenomena are noted.