Negative nonlinear effects

It has recently been found that Et2Zn promotes the 1,3-dipolar cycloaddition of nitrile oxides to allyl alcohol in the presence of catalytic amounts of diisopropyl tartrate (DIPT). By this method, 2-isoxazlines are obtained in good yields and up to 96% ee (Eq. 8.73).124a A positive nonlinear effect (amplification of ee of the product) has been observed in this reaction. There is an excellent review on positive and negative nonlinear effects in asymmetric induction.124b... 

Catalytic amounts of this addend (4 equiv relative to Cu) increase the selectivity of the allylic oxidation when TBHP is used as the oxidant. No change was observed with terf-butyl perbenzoate. This observation suggests a dichotomy in the mechanism of this reaction when using the two oxidants. Furthermore, in the absence of anthraquinone, a small negative nonlinear effect (78) is observed while in its presence, a small positive nonlinear effect appears. The reasons for this reversal are not clear, although the authors observed that low enantiopurity catalysts lead to turbid... 

In an attempt to optimize the reaction conditions [21] it was realized that temperature had little influence on the enantioselectivity of the reaction, but the yields were optimum at approximately 0 °C. Also, a small negative nonlinear effect was observed as the relationship between the ee of the ligand and the ee values of the products, implying that aggregation takes place. In addition, the nature of the so-called precatalyst seems to play a significant role in the enantioselective catalytic cycle. This became obvious when Campagne et al. 

In order to achieve an amplification of chirality, it requires that/> 1. If P = 0 (no meso catalyst) or g = 1 (same reactivity of meso and homochiral catalysts), then/= 1. The condition/> 1 is achieved for 1 + p > 1 + g ), or g < 1. Thus the necessary condition for asymmetric amplification in the above model is for the heterochiral or meso catalyst to be less reactive than the homochiral catalyst. If the meso catalyst is more reactive, then/< 1, and hence a negative nonlinear effect is observed. The size of the asymmetric amplification is regulated by the value off, which increases as K does. The more meso catalyst (of the lowest possible reactivity) there is, the higher will be eeproduct. This is well illustrated by computed curves in Scheme 11. The variation of eeproduct with eeaux is represented for various values of g (the relative reactivity of the meso complex) with K = 4 (corresponding to a statistical distribution of ligands Scheme 11, top). The variation in the relative amounts of the three complexes with eeaux is also represented for a statistical distribution of ligands (Scheme 11, bottom). 

It was established for several examples that it was possible to observe some departure from the expected proportionality between the enantiomeric excess of the catalyst and the enantiomeric excess of the product. Nonlinear effects (NLE) are categorized as a positive nonlinear effect ((-i-)-NLE) if the curve ee(product) = f(ee(catalyst)) is above the straight line characterizing the expected proportionality between ee(product) and ee(catalyst). The (-i-)-NLE has also been named asymmetric amplification [92]. A negative nonlinear effect ((-)-NLE) means that the experimental curve ee(product) =f( ee(catalyst)) lies below the straight line of the linear correlation. The departure from linearity reflects the formation of diastereomeric species (catalytically active or not) which perturb the predictions based only on mixture of enantiomeric catalysts and the... 

It was also suggested that aggregation of the catalysts influenced the selectiv-ities in the Diels-Alder reactions, and the reaction of 3-(2-butenoyl)-l,3-oxazo-lidin-2-one with cyclopentadiene using ( R)-(+)-binaphthol in lower enantiomeric excesses was examined [84]. The results are shown in Fig. 1. Very interestingly, a positive nonlinear effect was observed in the chiral Sc catalyst. In the chiral Yb catalysts, on the other hand, the effect was dependent on the additives. The extent of asymmetric induction in catalyst A did not deviate from the enantiomeric excesses of ( R)-(+)-binaphthol in the range 60-100% ee [85], while a negative nonlinear effect was observed in catalyst B. These results can be ascribed to a difference in aggregation between the Sc catalyst, Yb catalyst A, and Yb catalyst B. 

In the chiral Yb catalysts, on the other hand, the effect was dependent on the additives. The extent of asymmetric induction in catalyst A did not deviate from the enantiomeric excesses of (f )-binaphthol in the range 60-100% ee1 while a negative nonlinear effect... 

Positive nonlinear effects (curve 3) or negative nonlinear effect (curve 2). Line 1 represents the proportionality between ee mx and ecproduct h is arbitrarily assumed here that enantiopure chiral auxiliary gives 100% ee in the asymmetric synthesis (ref. 61). 

Katsuki reported a much more catalyst derived from the diol (143) and EtAlCh (Scheme 6.138) [166]. Interestingly, solvent effect on the enantioselectivity was remarkable. Thus, both enantioselectivities of the mdo-adduct were obtained just by changing THE solvent to toluene. Although this solvent effect was not explained, a linear relationship between the optical purity of (143) and the optical purity of cycloadduct for the reaction in THE was observed. In contrast, a negative nonlinear effect was observed in CH2CI2. These results likely suggested a monomeric catalyst... 

Relationship between ee of product (y) and ligand (x) for positive (-I-) and negative (-) nonlinear effects (NLEs) [30, 33]... 

Fig. 3.19 Schematic illustration of the measurement geometry for Mossbauer spectrometers. In transmission geometry, the absorber (sample) is between the nuclear source of 14.4 keV y-rays (normally Co/Rh) and the detector. The peaks are negative features and the absorber should be thin with respect to absorption of the y-rays to minimize nonlinear effects. In emission (backscatter) Mossbauer spectroscopy, the radiation source and detector are on the same side of the sample. The peaks are positive features, corresponding to recoilless emission of 14.4 keV y-rays and conversion X-rays and electrons. For both measurement geometries Mossbauer spectra are counts per channel as a function of the Doppler velocity (normally in units of mm s relative to the mid-point of the spectrum of a-Fe in the case of Fe Mossbauer spectroscopy). MIMOS II operates in backscattering geometry circle), but the internal reference channel works in transmission mode...

Thirdly, when we separated equation 43-51 into two terms, we only worked with the first term. The second term, which we presented in equation 43-52B, was neglected. Is it possible that the nonlinear effects observed for equation 43-52A will also operate on equation 43-52B The answer is yes, it will, but... And the but..is this AEs is a random variable, just as AEr is. Furthermore, it is uncorrelated with AEx. Therefore, in order to evaluate the integral representing the variation of both AEs and AEr, it would be necessary to perform a double integration over both variables. Now, for each value of AEs, the nonlinearity caused by the presence of AEr in the denominator would apply. However, AEs is symmetrically distributed around zero, therefore for every positive value of AEs there is an equal but negative value that is subject to exactly the same nonlinear effect. The net result is that these pairs always form equal and opposite contributions to the integral, which therefore cancel, leaving no effect due to AEs. 

The presence of a nonlinear effect, either negative or positive, is a useful piece of information for the mechanistic study of a reaction. It implies that diastereomeric species are formed from the chiral auxiliary. If an asymmetric amplification is observed, it can be indicative of the formation of meso dimers (or tetramers etc.) of low reactivity. When the kinetic study of an asymmetric catalysis shows a rate second order with respect to catalyst concentration, it may be useful to investigate the possibility of nonlinear effects in the system. Jacobsen et al., for example, studied the... 

These states are assumed to be asymptotic values of state variables x at a very small duration of cycle (fc — 0). If periodic control exhibits a nonlinear effect on the state variables, the relaxed steady-state process indexes can substantially differ from those for the steady state, and a positive or a negative effect from periodic control can be produced. 

The standardization parameters are determined by a nonhnear least squares procedure. The a term mainly corresponds to the negative log of the activity coefficient of the [ff] at the working temperature and ionic strength. 5 refers to the Nemst slope. The /h term corrects pH readings for the nonlinear pH response due to liquid junction and asymmetry potentials in the acidic region (1.5-2.5), while /oh corrects for high-pH (11-12.2) nonlinear effects. = [ff] [OH ] is ionic product of water. AT varies as a... 

On the other hand, curious nonlinear effects have been observed in some polymeric nematics. Kiss and Porter found that with increasing shear rate, the first normal stress difference becomes negative and then becomes positive again. This effect is not explained by the present constitutive equation and some other physical reason appears to be needed. 

The most well-known nonlinear algorithm is error-squared. Since the controller works with a dimensionless error scaled between 0 and 1 (or 0 % and 1(X) %) the square of the error will have the same range. Strictly the error is not squared but multiplied by its absolute value, because we need to retain the sign if the error is negative. The effect is illustrated in Figure 4.13. 

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