General description

The simplest model to describe a non-linear effect, the ML₂ model involves a metal system (M) with two chiral ligands, L_R and L_S. In addition to the catalyzed reaction of interest, the model accounts for a steady state equilibrium between the unbound and bound catalyst complexes.^[3] There are three possible catalytic complexes at equilibrium (ML_SL_R, ML_SL_S, ML_RL_R). The two enantiomerically pure complexes ( ML_SL_S, ML_RL_R) are referred to as homochiral complexes.^[3] The possible heterochiral complex, ML_RL_S, is often referred to as a meso-complex.^[3]

${\displaystyle {\begin{aligned}&{\begin{matrix}\mathrm {M} +\mathrm {L} _{R}+\mathrm {L} _{S}\quad \rightleftharpoons \quad &\mathrm {ML} _{R}\mathrm {L} _{R}&+&\mathrm {ML} _{S}\mathrm {L} _{S}&+&\mathrm {ML} _{R}\mathrm {L} _{S}&\\&x&&y&&z&\left\}\quad {\overset {}{\text{concentrations}}}\right.\\&\\&\left\downarrow {\begin{matrix}\\\mathrm {r} _{RR}\\\\\end{matrix}}\right.&&\left\downarrow {\begin{matrix}\\\mathrm {r} _{SS}\\\\\end{matrix}}\right.&&\left\downarrow {\begin{matrix}\\\mathrm {r} _{RS}\\\\\end{matrix}}\right.&\left\}\quad {\overset {}{\text{rates}}}\right.\qquad \qquad \\\end{matrix}}\\\\&\qquad \qquad \qquad \qquad \qquad \qquad \;\;\;\;\;{\text{reaction products}}\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\;\;\;{\text{ee}}_{\text{product}}\\\\&{\text{Equilibrium constant: }}K={\frac {z^{2}}{xy}}\end{aligned}}}$

The basic scheme of an ML₂ model developed by H. B. Kagan ^[8]

The equilibrium constant that describes this equilibrium, K, is presumably independent on the catalytic chemical reaction. In Kagan's model, K is determined by the amount of aggregation present in the chemical environment. A K=4 is considered to be the state at which there is a statistical distribution of ligands to each metal complex.^[3] In other words, there is no thermodynamic disadvantage or advantage to the formation of heterochiral complexes at K=4.^[4]

Obeying the same kinetic rate law, each of the three catalytic complexes catalyze the desired reaction to form product.^[8] As enantiomers of each other, the homochiral complexes catalyze the reaction at the same rate, although opposite absolute configuration of the product is induced (i.e. r_RR=r_SS). The heterochiral complex, however, forms a racemic product at a different rate constant (i.e. r_RS).^[9]

Mathematical model for the ML₂ Model

In order to describe the ML₂ model in quantitative parameters, Kagan and coworkers described the following formula:

${\displaystyle ee_{\text{product}}=ee_{\text{max}}ee_{\text{auxiliary}}(1+\beta )/(1+g\,\beta )}$

In the correction factor, Kagan and co-workers introduced two new parameters absent in Equation 1, β and g.^[9] In general, these parameters represent the concentration and activity of three catalytic complexes relative to each other. β represents the relative amount of the heterochiral complex (ML_RL_S) as shown in Equation 3.^[3] It is important to recognize that the equilibrium constant K is independent on both β and g.^[8] As described by Donna Blackmond at Scripps Research Institute, "the parameter K is an inherent property of the catalyst mixture, independent of the ee_catalyst. K is also independent of the catalytic reaction itself, and therefore independent of the parameter g."

${\displaystyle \beta =z/(x+y)}$

Equation 3: The correction factor, β, may be described as z, the heterochiral complex concentration, divided by x and y, the respective concentrations of the complex concentration divided by x and y, the respective concentrations of the homochiral complexes ^[3]

The parameter g represents the reactivity of the heterochiral complex relative to the homochiral complexes. As shown in Equation 5, this may be described in terms of rate constants. Since the homochiral complexes react at identical rates, g can then be described as the rate constant corresponding to the heterochiral complex divided by the rate constant corresponding to either homochiral complex.

${\displaystyle g=r_{rs}/r_{rr}}$

Equation 4: The correction parameter, g, can be described as the rate of product formation with the heterochiral catalyst ML_RL_S divided by the rate of product formation of the homochiral complex (ML_RL_R or ML_SL_S).

Interpretation of the mathematical results of the ML₂ Model

• If β=0 or g=1, the ML₂ equation simplifies to the Equation 1. No meso catalyst complex is present or active. Therefore, the simple additive properties should apply to such a scenario to establish a linear relationship between product enantioselectivity and the enantiopurity of the chiral catalyst.

• If the correction factor is greater than one, the reaction displays an asymmetric amplification, also known as a positive non-linear effect. Under the ML₂ model, a (+)-NLE infers a less reactive heterochiral catalyst. In this case, the equilibrium constant K also increases as the correction factor increases. Although the product enantioselectivity is relatively high compared to the enantiopurity of the chiral catalyst, this comes at a cost of the overall reaction rate. In order to achieve an asymmetric amplification, there must be a relatively large concentration of the heterochiral complex. In addition, this heterochiral complex must have a substantially slower rate of reactivity, r_RS. Therefore, the reactive catalytic species should decrease in concentration, leading to an overall slower reaction rate.

• If the correction factor is less than one, the reaction displays an asymmetric depletion, also known as a negative non-linear effect. In this scenario, the heterochiral catalyst is relatively more reactive than the homochiral catalyst complexes. In this case, the (−)-NLE may result in an overall faster although less selective product formation.

iv. Reaction Kinetics with the ML₂ Model: Following H.B. Kagan's publication of the ML₂ model, Professor Donna Blackmond at Scripps demonstrated how this model could be used to also calculate the overall reaction rates. With these relative reaction rates, Blackmond showed how the ML₂ model could be used to formulate kinetic predictions which could then be compared to experimental data. The overall rate equation, Equation 6, is shown below.^[8]

${\displaystyle r=(x+y+gz)r_{RR}}$

In addition to the goodness of fit to the model, kinetic information about the overall reaction may further validate the proposed reaction mechanism. For instance, a positive NLE in the ML₂ should result in an overall lower reaction rate.^[8] By solving the reaction rate from Equation 6, one can confirm if that is the case.

General description

Similar to the ML₂ model, this modified system involves chiral ligands binding to a metal center (M) to create a new center of chirality.^[4] There are four pairs of enantiomeric chiral complexes in the M*L₂ model, as shown in Figure 5.

${\displaystyle {\begin{matrix}{\begin{matrix}\mathrm {M} +\mathrm {L} _{R}+\mathrm {L} _{S}\quad \rightleftharpoons \rightleftharpoons \quad \\\\\\\\\\&\end{matrix}}{\begin{aligned}&{\begin{matrix}\mathrm {M} _{R}\mathrm {L} _{R}\mathrm {L} _{R}&+&\mathrm {M} _{R}\mathrm {L} _{R}\mathrm {L} _{S}&+&\mathrm {M} _{R}\mathrm {L} _{S}\mathrm {L} _{R}&+&\mathrm {M} _{R}\mathrm {L} _{S}\mathrm {L} _{S}\\\mathrm {M} _{S}\mathrm {L} _{S}\mathrm {L} _{S}&+&\mathrm {M} _{S}\mathrm {L} _{S}\mathrm {L} _{R}&+&\mathrm {M} _{S}\mathrm {L} _{R}\mathrm {L} _{S}&+&\mathrm {M} _{S}\mathrm {L} _{R}\mathrm {L} _{R}\\&\\\left\downarrow {\begin{matrix}k_{0}\\\pm EE_{0}\end{matrix}}\right.&&\left\downarrow {\begin{matrix}k_{1}\\\pm EE_{1}\end{matrix}}\right.&&\left\downarrow {\begin{matrix}k_{2}\\\pm EE_{2}\end{matrix}}\right.&&\left\downarrow {\begin{matrix}k_{3}\\\pm EE_{3}\end{matrix}}\right.\end{matrix}}\\\\&\qquad \qquad \qquad \qquad \qquad \,\,\,\,{\text{products}}\end{aligned}}\end{matrix}}}$

Figure 5: The equilibrium present in the M*L₂. This is very similar to the ML₂ model. However, due to the new center of chirality, there are four sets of enantiomers.

In this model, one can make the approximation that the dimeric complexes dissociate irreversibly to the monomeric species. In this case, the same mathematical equations apply to the ML*₂ model that applied to the ML₂ model.

General description

A higher level of modeling, the ML₃ model involves four active catalytic complexes: ML_RL_RL_R, ML_SL_SL_S, ML_RL_RL_S, ML_SL_SL_R. Unlike the ML₂ model, where only the two homochiral complexes reacted to form enantiomerically enriched product, all four of the catalytic complexes react enantioselectively. However, the same steady state assumption applies to the equilibrium between unbound and bound catalytic complexes as in the more simple ML₂ model. This relationship is shown below in Figure 7.

${\displaystyle {\ce {{M}+{L_{R}}+L_{S}<=>}}\ \underbrace {\ce {ML_{R}L_{R}L_{R}}} _{x}+\underbrace {\ce {ML_{S}L_{S}L_{S}}} _{y}+\underbrace {\ce {ML_{R}L_{R}L_{S}}} _{z}+\underbrace {\ce {ML_{S}L_{S}L_{R}}} _{w}}$

Figure 7: The equilibrium between the unbound species and the reactive catalytic species in the ML₃ model.

Mathematical modeling

Calculating the ee_product is considerably more challenging than in the simple ML₂ model. Each of the two heterochiral catalytic complexes should react at the same rate. The homochiral catalytic complexes, similar to the ML₂ case, should also react at the same rate. As such, the correction parameter g is still calculated as the rate of the heterochiral catalytic complex divided by the rate of the homochiral catalytic complex. However, since the heterochiral complexes lead to enantiomerically enriched product, the overall equation for calculating the ee_product becomes more difficult. In Figure 8., the mathematical formula for calculating enantioselectivity is shown.

${\displaystyle EE_{\text{prod}}=EE_{0}ee_{\text{aux}}{\frac {3+3\,g\,EE'_{0}/EE_{0}+(1-3\,g\,EE'_{0}/EE_{0})ee_{\text{aux}}^{2}}{1+3\,g+3(1-g)ee_{\text{aux}}^{2}}}}$

Figure 8: The mathematical formula describing an ML₃ system. The ee_product is calculated by multiplying the ee_max by the correction factor developed by Kagan and co-workers.^[4]

Interpretation of the ML₃ Model

In general, interpreting the correction parameter values of g to predict positive and negative non-linear effects is considerably more difficult. In the case where the heterochiral complexes ML_RL_RL_S and ML_SL_SL_R are less reactive than the homochiral complexes ML_SL_SL_S and ML_RL_RL_R, a kinetic behavior similar to the ML₂ model is observed (Figure 9). However, a substantially different behavior is observed in the case where the heterochiral complexes are more reactive than the homochiral complexes.

Figure 9: When the heterochiral complexes are more selective and reactive than the homochiral complexes.^[4]

Figure 10: When the homochiral complexes are more selective and reactive than the heterochiral complexes.^[4]

General Description

Often described adjacent or in collaboration with the ML₂ model, the reservoir effect describes the scenario in which part of the chiral ligand is allocated to a pool of inactive heterochiral catalytic complexes outside the catalytic cycle.^[4] A pool of unreactive heterochiral catalysts, described with an ee_pool, develops an equilibrium with the catalytically active homochiral complexes, described with an ee_effective.^[8] Depending on the concentration of the inactive pool of catalysts, one can calculate the enantiopurity of the active catalyst complexes. The general result of the reservoir effect is an asymmetric amplification, also known as a (+)-NLE.^[3]

Origin of the Reservoir Effect

The pool of unreactive catalytic complexes, as described in the reservoir effect, can be the result of several factors. One of these could potentially be an aggregation effect amongst the heterochiral catalytic complexes that takes place prior to the steady state equilibrium.^[3]