The Chemistry of Stable Carbenes. Part 2. Benzoin-type condensations of formaldehyde catalyzed by stable carbenes

Stable carbenes derived from thiazole, 1H-imidazole, and 4H-1,2,4-triazole are efficient catalysts for benzointype condensations of formaldehyde. Catalysts derived from N-substituted thiazolium salts trimerize formaldehyde to dihydroxyacetone ( II ). Catalysts based on 1,4-disubstituted 4H-1,2,4-triazol-1-ium salts give glycolaldehyde ( I ) as the main product and no II , whereas N,N′-disubstituted 1H-imidazol-3-ium salts yield mixtures of both products. The isolation of several intermediates in the catalytic cycle provide a better insight into the reaction mechanism.     

Theoretical calculation of the potential curves of the states of HF below 14 eV

Potential energy curves obtained through extensive ab initio configuration interaction calculations are reported for the states of HF below 14 eV as well as that for the ground state of HF⁺. Agreement with ultraviolet absorption data is obtained to within 0.2 eV. The disagreement between the reported UV absorption and electron scattering data is examined.     

Variations on polynomial subgroup growth

A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│ ^a subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ).