Non-linear effects

In enantioselective synthesis, a non-linear effect refers to a process in which the enantiopurity of the catalyst (or the chiral auxiliary) does not correlate linearly with the enantiopurity of the product produced. This deviation from linearity is described as the non-linear effect, NLE. The linearity can be expressed mathematically, as shown in Equation 1. Stereoselection (i.e. the eeproduct) that is higher or lower than the enantiomeric excess of the catalyst (eecatalyst, relative to the equation) is considered non-routine behavior.

e e product = e e max e e catalyst {\displaystyle ee_{\text{product}}=ee_{\max }ee_{\text{catalyst}}}

For an ideal asymmetric reaction, the eeproduct may be described as the product of eemax multiplied by the eecatalyst. This is not the case for reactions exhibiting NLE's. In 1976, Wynberg and Feringa observed different chemical behavior in the reaction of an enantiopure and racemic substrate in a phenol coupling reaction. In 1981, Kagan and collaborators described the first non-linear effects in asymmetric catalysis and gave rational explanations for these phenomena. General definitions and mathematical models are essential for understanding nonlinear effects and their application to specific chemical reactions. In recent decades, the study of nonlinear effects has helped elucidate reaction mechanism and guide synthetic applications.

Types of non-linear effects

Positive non-linear effect, (+)-NLE A positive non-linear effect, (+)-NLE, is present in an asymmetric reaction which demonstrates a higher product ee (eeproduct ) than predicted by an ideal linear situation (Figure 1). It is often referred to as asymmetric amplification, a term coined by Oguni and co-workers. An example of a positive non-linear effect is observed in the case of Sharpless epoxidation with the substrate geraniol.In all cases of chemical reactivity exhibiting (+)-NLE, there is an innate tradeoff between overall reaction rate and enantioselectivity. The overall rate is slower and the enantioselectivity is higher relative to a linear behaving reaction.

Negative non-linear effect, (−)-NLE Referred to as asymmetric depletion, a negative non-linear effect is present when the eeproduct is lower than predicted by an ideal linear situation. In contrast to a (+)-NLE, a (−)-NLE results in a faster overall reaction rate and a decrease in enantioselectivity. Synthetically, a (−)-NLE effect could be beneficial with a reasonable assay for separating product enantiomers and a high output is necessary . An interesting example of a (−)-NLE effect has been reported in asymmetric sulfide oxidations.

Hyperpositive and enantiodivergent non-linear effect Beyond the positive or negative non-linear effects, there are atypical cases which are briefly described in this section. -A hyperpositive nonlinear effect refers to a case where the chiral catalyst, when not enantiopure, can be more enantioselective than its enantiopure counterpart. This case was first deduced from the theoretical models proposed by Henri Kagan in 1994 (i.e., ML3 model). The first experimental example of such non-linear effect was only observed in 2020 by S. Bellemin-Laponnaz, but with a mechanism that turns out to be different from Kagan's original proposal. -A catalytic system that generates either enantiomer of the product by modifying only the enantiomeric excess of the ligand (without changing the major enantiomer) is called an enantiodivergent non-linear effect. The first experimental example was described in 2002.The mechanism that could explain this type of behavior appears to be the same as for hyperpositive non-linear effects.

Modeling non-linear effects In 1986, Henri B. Kagan and coworkers observed a series of known reactions that followed a non-ideal behavior. A correction factor, f, was adapted to Equation 1 to fit the kinetic behavior of reactions with NLEs (Equation 2).

e e product = f e e max e e catalyst {\displaystyle ee_{\text{product}}=f\,ee_{\max }ee_{\text{catalyst}}}

Equation 2: A general mathematical equation that describes non-linear behavior Unfortunately, Equation 2 is too general to apply to specific chemical reactions. Due to this, Kagan and coworkers also developed simplified mathematical models to describe the behavior of catalysts which lead to non-linear effects. These models involve generic MLn species, based on a metal (M) bound to n number of enantiomeric ligands (L). The type of MLn model varies among asymmetric reactions, based on the goodness of fit with reaction data. With accurate modeling, NLE may elucidate mechanistic details of an enantioselective, catalytic reaction.

ML2 model

General description The simplest model to describe a non-linear effect, the ML2 model involves a metal system (M) with two chiral ligands, LR and LS. In addition to the catalyzed reaction of interest, the model accounts for a steady state equilibrium between the unbound and bound catalyst complexes. There are three possible catalytic complexes at equilibrium (MLSLR, MLSLS, MLRLR). The two enantiomerically pure complexes ( MLSLS, MLRLR) are referred to as homochiral complexes. The possible heterochiral complex, MLRLS, is often referred to as a meso-complex.

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