Equilibrium of homochiral oligomerization of a mixture of enantiomers. Its relevance to nonlinear effects in asymmetric catalysis.

The origin of nonlinear effects (no proportionality between enantiomeric excess (ee) of chiral auxiliary and ee of product) is first summarized in general terms, underlining the importance of the presence of molecular species bearing several moieties deriving from the chiral auxiliary. The presence of a heterochiral species, produced from enantioimpure chiral auxiliaries, usually explains well the deviation to linearity, especially asymmetric amplification. In this article it is shown that the absence of a heterochiral species is not incompatible with an asymmetric amplification. The demonstration has been done on a simple model, the equilibrium of homochiral dimerization. The monomers R and S being in equilibrium with the dimers R(2) and S(2), it was possible to calculate ee(monomer) and ee(dimer) as a function of the initial concentration and the initial ee of the monomer. The asymmetric amplification can be quite substantial for the dimer, while asymmetric depletion characterizes the residual monomers. Similar conclusions apply to homochiral tri- and tetramerizations. The extension to irreversible reactions was briefly analyzed as well as the use of these results.