Dioxygen Activation by Siloxide Complexes of Chromium(II) and Chromium(IV)

The reaction of a tripodal trisilanol with n‐butyllithium and CrCl₂ results in a dinuclear Cr^II complex ( 1 ), which is capable of cleaving O₂ to yield in a unique complex ( 2 ) with an asymmetric diamond core composed of two Cr^IV?O units. Magnetic susceptibility data reveal significant exchange coupling of Cr^II (S=2) in 1 and large zero‐field splitting for Cr^IV (S=1) in 2 owing to strong spin–orbit coupling of the ground state. The Cr^IV?O compound can also be generated using PhIO, and evidence was gathered that although it is the stable product isolated after excessive O₂ treatment, it further activates O₂ to yield an intermediate species that oxidizes THF or Me‐THF. By extensive ¹⁸O labeling studies we were able to show, that in the course of this process ¹⁸O₂ exchanges its label with siloxide O atoms of the ligand via terminal oxido ligands.     

Palladium‐Catalyzed Chlorocarbonylation of Aryl (Pseudo)Halides Through In Situ Generation of Carbon Monoxide

An efficient palladium‐catalyzed chlorocarbonylation of aryl (pseudo)halides that gives access to a wide range of carboxylic acid derivatives has been developed. The use of butyryl chloride as a combined CO and Cl source eludes the need for toxic, gaseous carbon monoxide, thus facilitating the synthesis of high‐value products from readily available aryl (pseudo)halides. The combination of palladium(0), Xantphos, and an amine base is essential to promote this broadly applicable catalytic reaction. Overall, this reaction provides access to a great variety of carbonyl‐containing products through in situ transformation of the generated aroyl chloride. Combined experimental and computational studies support a reaction mechanism involving in situ generation of CO.     

Bounds on absorption times of directionally biased random sequences

A sequence of random variables X₀,X₁, … with values in {0, 1, …, n} representing a general finite-state stochastic process with absorbing state 0 is said to be directionally biased towards 0, if, for all j > 0, ϵ_j: = inf_k>0 {j − E[X_k | X_k−1 = j]} > 0. For such sequences, let t be the expected value of the time to absorption at 0. For a fixed set of biases, the least upper bound for this time can be computed with an algorithm requiring O(n²) steps. Simple upper bounds are described. In particular, t ≤ E[b_x0], where b_i = Σ_j≤i 1/¯ϵ_j and ¯ϵ_j = min_l≥j {ϵ_l}. If all ϵ_j ≤ ϵ_{j + 1} (so ¯ϵ_j = ϵ_j) and ϵ_n < 1, this bound for t is the best possible. For certain finite stochastic processes which we term conditionally independent of X₀ = i, b(i) bounds the expected time given X₀ = i. Similar results are given for lower bounds. The results of this paper were designed to be a useful tool for determining rates of convergence of stochastic optimization algorithms. © 1996 John Wiley & Sons, Inc.     

Density functional description of the early stages of the dioxygenation of [(MeC(CH2PPh2)3)M(catecholate)]+ complexes [M = Co(III), Ir(III)]: toward a rationalization of the catalytic mechanism of ring-opening dioxygenases

Density Functional Theory (DFT) has been applied to characterize the early stages of the reaction of dioxygenation of [(triphos)M(catecholate)]+ complexes [M = Co(III), Ir(III); triphos = MeC(CH2PPh2)3], which have been considered to be models of ring-opening dioxygenases. The structural features of the starting complexes and of the intermediate complexes formed by addition of O2 to the coordinated catecholato ion are well reproduced. The calculations showed that this preliminary stage can be obtained only when the oxygen molecule attacks the molecule on the catecholato site.     

A transformation theorem for one-dimensional F-expansions

If f is a monotone function subject to certain restrictions, then one can associate with any real number x between zero and one a sequence {a_n(x)} of integers such that

$\text{x=f(a}{}_1\text{(x) + f(a}{}_2\text{(x) +f(a}{}_3\text{(x) +…)))}$

. In this paper properties of the function F defined by

$\text{Fx=g(a}{}_1\text{(x) + g(a}{}_2\text{(x) +g(a}{}_3\text{(x) +…)))}$

, where g is any function satisfying the same restrictions as f, are discussed. Principally, F is found to be useful in finding stationary measures on the sequences {a_n(x)}.