Numerical Integration of Peak Functions

We study the logarithmic error of numerical methods for the integration of monotone or unimodal positive functions. We compare adaptive and nonadaptive methods in the worst case setting. It turns out that adaption significantly helps for the class of unimodal functions, but it does not help for the class of monotone functions. We do not assume any smoothness properties of the integrands and obtain numerical methods that are reliable even for discontinuous integrands. Numerical examples show that our method is very competitive in the case of nonsmooth and/or peak functions.     

Enantioselective epoxidation with chiral MN(III)(salen) catalysts: kinetic resolution of aryl-substituted allylic alcohols

A set of aryl-substituted allylic alcohols rac-2 has been epoxidized by chiral Mn(salen*) complexes 1 as the catalyst and iodosyl benzene (PhIO) as the oxygen source. Whereas one enantiomer of the allylic alcohol 2 is preferentially epoxidized to give the threo- or cis-epoxy alcohol 3 (up to 80% ee) as the main product (dr up to >95:5), the other enantiomer of 2 is enriched (up to 53% ee). In the case of 1,1-dimethyl-1,2-dihydronaphthalen-2-ol (2c), the CH oxidation to the enone 4c proceeds enantioselectively and competes with the epoxidation. The absolute configurations of the allylic alcohols 2 and their epoxides 3 have been determined by chemical correlation or CD spectroscopy. The observed diastereo- and enantioselectivities in the epoxidation reactions are rationalized in terms of a beneficial interplay between the hydroxy-directing effect and the attack along the Katsuki trajectory.     

cis-Stilbene and (1 alpha,2 beta,3 alpha)-(2-ethenyl-3-methoxycyclopropyl)benzene as mechanistic probes in the Mn(III)(salen)-catalyzed epoxidation: influence of the oxygen source and the counterion on the diastereoselectivity of the competitive concerted and radical-type oxygen transfer

cis-Stilbene (1) has been epoxidized by a set of diverse oxygen donors [OxD], catalyzed by the Mn(III)(salen)X complexes 3 (X = Cl, PF(6)), to afford a mixture of cis- and trans-epoxides 2. The cis/trans ratios range from 29:71 (extensive isomerization) to 92:8, which depends both on the oxygen source [OxD] and on the counterion X of the catalyst. When (1 alpha,2 beta,3 alpha)-(2-ethenyl-3-methoxycyclopropyl)-benzene (4) is used as substrate, a mechanistic probe which differentiates between radical and cationic intermediates, no cationic ring-opening products are found in this epoxidation reaction; thus, isomerized epoxide product arises from intermediary radicals. The dependence of the diastereoselectivity on the oxygen source is rationalized in terms of a bifurcation step in the catalytic cycle, in which concerted Lewis-acid-activated oxygen transfer competes with stepwise epoxidation by the established Mn(V)(oxo) species. The experimental counterion effect is attributed to the computationally assessed ligand-dependent reaction profiles and stereoselectivities of the singlet, triplet, and quintet spin states available to the manganese species.