Cumulants and the moment algebra: tools for analysing weak measurements

Recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements. Here we consider the mathematical structure that underlies this, and find that it can be formulated in terms of what we call the moment algebra. Apart from resulting in simpler proofs, the flexibility of this structure allows generalizations of the original results to a number of weak measurement scenarios, including one where the weakly interacting pointers reach thermal equilibrium with the probed system.Comment: Journal reference added, minor correction

Iron catalyzed C-H activation and synthesis of novel ligands

Iron-catalyzed benzylic oxidations, nitrogen transfer reactions such as iminations of sulfides and sulfoxides, aziridinations of olefins and a-aminations of silyl enol ethers were developed. Also, the synthesis of novel nitrogen containing ligands such as 1,4,7-triazacyclononane (TACN) derivatives and dipyridylamine derivatives was undertaken.Iron-catalyzed benzylic oxidation was successfully performed by using a combination of iron(III) chloride (2 mol%) as a catalyst and tert-butyl hydroperoxide (TBHP) as an oxidant. Diarylmethane derivatives gave the corresponding ketones in excellent yields (up to >99%). Also, less reactive substrates bearing one annulated aryl group in a cyclic system, noncyclic compounds bearing one (hetero)aryl group and p-methoxytoluene led to the corresponding ketones or carboxylic acid in good yield. Even diphenylcarbinol was oxidized to afford benzophenone in 86% yield. Interestingly, the oxidation of triphenylmethane afforded tert-butyl triphenylmethyl peroxide in 91% yield instead of triphenylmethanol. Additionally, 1,4-dihydroxynaphthaline underwent oxidation in actonitrile at ambient temperature to afford 1,4-naphthoquinone in high yield (82%). Iron-catalyzed aziridination was achieved by using iron(II) triflate (2.5 mol%), MS 4A (20 mg), 20 equivs. of styrene and 1 equiv. of [N-(p-nitrobenzenesulfonyl)imino]phenyliodinane (PhINNs) affording phenylaziridine in 88% yield. Poorly reactive olefins such as a- or b-methyl styrene were aziridinated. A cyclic olefin such as cis-cyclooctene can be aziridinated efficiently by increasing the catalyst loading up to 20 mol%. These conditions were applied for the asymmetric aziridination of styrene. After ligand screening, an (S,S)-i-Pr-pybox ligand was discovered to be an efficient ligand, affording (R)-phenylaziridine in 83% yield with 40% ee. Also, aziridinations using in situ generated iminophenyliodinanes using 10-20 mol% of iron(II) triflate and a combination of the corresponding sulfonamide and iodobenzene diacetate (2 equivs.) with magnesium oxide (5 equivs.) as a base or iodosylbenzene (PhI=O) led to the corresponding phenylaziridines in high yields (up to 76%). Further investigation of the iron(II) triflate catalyzed aziridination of styrene, has led to the discovery that attempted iron(II) triflate catalyzed aziridinations of silyl enol ethers using the same conditions gave a-amino ketones, or esters. In the synthesis of novel ligands, aryl-TACN derivatives were synthesized by using palladium catalysis. Triple N-arylation of TACN was achieved by use of a catalyst comprised of tris(dibenzyllideneacetone)dipalladium(0) and 2-dicyclohexylphosphino-2’-(N,N-dimethylamino)biphenyl as a phosphine ligand in the presence of 4.2 equivs. of sodium tert-butoxide in toluene at 100 °C. Also, double N-phenylation of mono-tosyl TACN proceeded under the same conditions affording di-phenyl TACN in 70% yield. Furthermore, mono-N-phenylation of di-(Boc), (Cbz) or (Ts) protected TACN was successful by using a mixture of palladium(II) acetate and rac-BINAP affording mono-phenylated TACNs in good yields. Also, a synthesis of dipyridylamine ligands has been developed using palladium catalysis. As a representative example, N-arylation of 2-picolylamine with 2-bromopyridine was achieved by using tris(dibenzyllideneacetone)dipalladium(0) (2 mol%) and rac-BINAP (2 mol%) in the presence of sodium tert-butoxide (2.2 equivs.) in toluene at 100 °C affording 2,2’-N,N-dipyridyl-picolylamine in 80% yield