Platinum Carbonyl Clusters Chemistry: Four Decades of Challenging Nanoscience

The state of art of the chemical, spectroscopic and structural characterization of platinum carbonyl clusters is reviewed. We begin by enlightening the fundamental contribution given to this chemistry of two great scientists: Paolo Chini and Larry Dahl, two without equal maestros of science and life. We then focus the main body of this review on the challenge represented by studying molecular ions already belonging to the nano regime by size almost 50 years ago, and the challenges their chemistry continues to present also nowadays. In detail, the possible reasons which enable the [Pt_3n (CO)_6n ]^2? oligomers to grow up to n = 10, and why the oligomers with n > 5 may self-assemble in infinite molecular conductor wires are suggested. The interplay between the CO/Pt_surface ratios and electronic effects in driving the platinum cluster from pseudo one- to tri-dimensional globular structures, often representing chunks of the fcc metal lattice or interpenetrated pentagonal prisms and icosahedra, is examined by means of two significant examples. The nanocapacitor behavior of most high-nuclearity carbonyl clusters is briefly recalled and is confirmed by most recent results. The size-induced transition of their metal kernels from insulator to semiconductor and the expected consequences of their eventual transition to a metallic state are also discussed. Finally, we conclude by commenting the present lack of Pt cluster interstitially lodging a main group element and not yet quantitatively-assessed aggregation phenomena in solution, perhaps peculiar of ionic salts of carbonyl clusters, which could make undetectable the NMR signal of any spin-active nuclei beyond a cluster nuclearity of ca. 20–25.     

Metallocene ion pairs: A direct insight into the reaction equilibria and polymerization by 13C NMR spectroscopy

The reaction equilibria of Cp₂Ti¹³CH₃Cl and Cp₂Ti(CH₃)₂ with AlMe₃ (TMA) and/or methylaluminoxane (MAO) have been investigated by ¹³C NMR. Several adducts have been identified. A study of the ¹³C 90% enriched ethylene polymerization in an NMR tube in the presence of the above catalytic systems, in the most experimentally significant conditions, and a comparison of the NMR data with the catalytic activity have been made as well. It has been shown that: i) some species are side products, inactive for addition ethylene polymerization; ii) active cation-like species such as Cp₂TiMe⁺Cl·[AlMeO]_n- and Cp₂TiMe⁺Me·[AlMeO]_n- are formed in titanocene-MAO systems. Concerning the role of AlMe₃, contained in MAO solutions, it has been shown that: a) AlMe₃ is mainly bound to MAO; b) if some “free” AlMe₃ exists in solution it is not the actual cocatalyst in the metallocene-MAO based catalytic systems; c) the amount of AlMe₃ influences either active or inactive species.     

A Note on the analytic solutions of the Camassa–Holm equation

In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to $H^s(\mathbb{R})$ with $s>3/2$, $6u_0{6}_{L^1}<\infty$ and $u_0?u_{0xx}$ does not change sign, we prove that the solution stays analytic globally in time. To cite this article: M.C. Lombardo et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

The Weight Distribution of C 5(1, n)

In [2] the codes C _q (r,n) over were introduced. These linear codes have parameters , can be viewed as analogues of the binary Reed-Muller codes and share several features in common with them. In [2], the weight distribution of C ₃(1,n) is completely determined.In this paper we compute the weight distribution of C ₅(1,n). To this end we transform a sum of a product of two binomial coefficients into an expression involving only exponentials an Lucas numbers. We prove an effective result on the set of Lucas numbers which are powers of two to arrive to the complete determination of the weight distribution of C ₅(1,n). The final result is stated as Theorem 2.