A limit theorem for additive functions defined on the symmetric group

We are concerned with the value distribution problem of additive functions defined on the symmetric group endowed with the uniform (Haar) measure. For such a function, normalized by a slowly varying sequence, we establish necessary and sufficient conditions for the weak convergence of its distribution to a limit law.     

Tris(eta4-naphthalene)- and tris(1-4-eta4-anthracene)tantalate(1-): first homoleptic arene complexes of anionic tantalum

Reduction of tantalum pentachloride by 6 equiv of sodium naphthalene in 1,2-dimethoxyethane provided, after recrystallization from tetrahydrofuran, 50-55% yields of yellow, pyrophoric [Na(THF)][Ta(C10H8)3]. The product was shown by 1H and 13C NMR spectra and an X-ray study (on the corresponding [Na(crypt 2.2.2)]salt) to be tris(eta4-naphthalene)tantalate(1-), 1, the first homoleptic naphthalene complex of a third row (5d) transition metal. Salts of 1 react under mild conditions with excess CO (1 atm pressure, -60 degrees to +20 degrees C) and 3 equiv of anthracene, C14H10 (20 degrees C), to give 99 and 52% yields of yellow [Ta(CO)6]- and orange [Ta(C14H10)3]-, (2), respectively. The latter is the first homoleptic anthracene complex of a group 5 element and only the third one known, the others being Cr(eta6-C14H10)2 and [Co(eta4-C14H10)2]-. NMR spectra and X-ray structural characterization, as the [Na(crypt 2.2.2)] salt, established 2 to be [Ta(1-4-eta4-C14H10)3]- and is very similar to 1 in solution and in the solid state. Salts of 2 also undergo facile ligand substitution reactions. For example, it reacts with 1,3,5,7-cyclooctatetraene, COT, at 20 degrees in THF to give high yields of the previously known [Ta(COT)3]-, which was structurally characterized as the Na(crypt 2.2.2)salt. One particularly important feature concerning 1 and 2 is that they are the first available synthons for "naked" atomic Ta- and promise to be useful reagents for the general exploration of low-valent tantalum chemistry. Also, 1 and 2 represent the first homoleptic arene tantalum complexes to have been prepared by conventional syntheses. The only previously known substance of this class is the neutral bis(benzene)tantalum(0), which was accessed by the co-condensation of atomic tantalum and benzene vapor in a sophisticated (electron-gun furnace equipped) metal atom reactor.     

Cu(II) Ion Complexation by Excess of β-cyclodextrin in Aqueous Alkaline Solutions

Polarographic investigations of Cu(II) complexation in aqueous alkaline solutions containing an excess of β-cyclodextrin (β-CD) show that the complex formation begins at pH > 11, the concentration of free (uncomplexed) Cu(II) ions being in the range from ca. 10^-12 to ca. 10^-19 M, depending on β-CD concentration and pH. The formation of copper(II) 1:1 hydroxy-complex with β-cyclodextrin anion (CD^2-) was observed at pH 11–14. The logarithm of the stability constant of CuCD(OH)^2- ₂ complex is 19.7 ± 0.2 (20 °C, ionic strength 1.0), the values of the molar extinction coefficient and of the diffusion coefficient of this complex are 50 M^-1 cm^-1 (λ_max = 660 nm) and 1.0 × 10^-6 cm² s^-1, respectively.     

Changes of the Cu electrode real surface area during the process of electroless copper plating

The surface area (nanoscale roughness) of copper coatings deposited from electroless plating solutions containing Quadrol, L(+)- and DL(∓)-tartrate as Cu(II) ion ligands was measured using underpotential deposition thallium monolayer formation. Surface roughness of Cu coatings depends on the plating solution pH and the Cu(II) ligand, and varies over a wide range. In L(+)-tartrate and Quadrol solutions (pH 12.5–13.3) the roughness factor R _f is low and is equal to 2–3 and 4–6, respectively (substrate: electrodeposited Cu; R _f=2.2). Cu coatings of higher surface area are obtained in DL(∓)-tartrate (pH 12.3–12.7) and Quadrol (pH 12.0) solutions: R _f reaches 20–30. The correlation between R _f and Cu deposition rate was found in L(+)-tartrate solution. The Cu surface area changes are discussed in terms of partial electrochemical reactions of the autocatalytic Cu deposition process, and the decisive role of cathodic Cu(II) reduction from adsorbed Cu(II) complex species. Received: 2 November 1999 / Accepted: 22 February 2000     

Peculiarities of Electrochemical Bismuth Film Formation in the Presence of Bromide and Heavy Metal Ions

Bi films were deposited on glassy carbon electrode from solutions with and without KBr. The morphology of both types of the films was characterized by scanning electron microscopy (SEM), and their electrochemical behavior was studied by square wave (SWV) and cyclic voltammetry (CV). Bi films were also co‐deposited with common analyte‐heavy‐metals in the presence of KBr and these films also were characterized by SEM, SWV and CV in order to understand the formation of the mixed metal films. All films studied had a different morphology. Bromide addition made the Bi films more compact and uniform, whereas Pb catalyzed Bi film deposition.     

Asymptotic value distribution of additive functions defined on the symmetric group

We examine the asymptotic value distribution of additive functions defined via the multiplicities of lengths of the cycles comprising a random permutation taken from the symmetric group with equal probability. We establish necessary and sufficient conditions for the weak law of large numbers and for the relative compactness of the sequence of distributions. Considering particular cases, we demonstrate that long cycles play an exceptional role and that, sometimes, in order to obtain a Poisson limit law, their influence must be negligible. The proofs are based on the ideas going back to the seminal papers of I.Z. Ruzsa on the classical additive arithmetic functions.      

Syntheses of symmetric and unsymmetric 2,6‐bis(phosphino)phenols

Compounds bearing the structural motif of 2,6‐bis(phosphino)phenol have been synthesized via two general methods. Double lithium‐halogen exchange occurred in low‐temperature reactions of O‐protected (by methyl‐ or tetrahydropyranyl groups) 2,6‐dibromo‐4‐methylphenol derivatives with BuLi (2 equivalents); quenching the reaction mixtures with chlorophosphines ClPR₂ (R = Ph, ⁱPr) and corresponding O‐deprotection yielded symmetrically substituted 2,6‐bis(phosphino)phenols. Sequential incorporation of  PR₂ functionalities was accomplished via single lithium‐halogen exchange (1 eq. of BuLi) of tetrahydropyranyl‐protected 2,6‐dibromo‐4‐methylphenol followed by ClPR₂ quenches, thus enabling the syntheses of unsymmetric 2,6‐bis(phosphino)phenols. Such compounds were also obtained via sequential ortho‐lithiations of tetrahydropyranyl‐protected 4‐tert‐but ylphenol, followed by ClPR₂ quenches. All of the new compounds have been characterized by spectrometric methods (¹H and ³¹P NMR, and mass spectrometry). In addition, two of the compounds, 1‐(diphenylphosphino)‐3‐(diphenylphosphoryl)‐2‐methoxy‐5‐methylbenzene ( 3a‐ox ) and 1,3‐bis(diphenylphosphino)‐2‐methoxy‐5‐methylbenzene ( 6a ) have also been characterized via single crystal X‐ray diffraction experiments. © 2006 Wiley Periodicals, Inc. Heteroatom Chem 17:656–663, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20251