Degenerate equilibria

The l3C NMR spectrum of the C4H7+ cation in superacid solution shows a single peak for the three methylene carbon atoms (72) This equivalence can be explained by a nonclassical single symmetric (three-fold) structure. However, studies on the solvolysis of labeled cyclopropylcarbinyl derivatives suggest a degenerate equilibrium among carbocations with lower symmetry, instead of the three-fold symmetrical species (13). A small temperature dependence of the l3C chemical shifts indicated the presence of two carbocations, one of them in small amounts but still in equilibrium with the major species (13). This conclusion was supported by isotope perturbation experiments performed by Saunders and Siehl (14). The classical cyclopropylcarbinyl cation and the nonclassical bicyclobutonium cation were considered as the most likely species participating in this equilibrium. 

Any observable effect of isotopic substitution on the rate or extent of a chemical/physical process. Equilibrium isotopic perturbation measurements can provide valuable information about kinetic isotope effects on enzymic catalysis. NMR shift difference measurements are also useful in detecting the effects of isotopic substitution on a fast (degenerate) equilibrium between two species differing only in their specific isotopic substitution . The... 

The twofold degenerate equilibrium in the case of the l-(2-methylcyclopropyl)ethyl cation 55 is also fast compared to that of the 1-methyl analogue. The static cation could not be frozen even at low temperatures. At-108 °C, the average cationic peaks at<5l3C 167.4... 

In contrast to classical tertiary and secondary cyclopropylmethyl cations (showing substantial charge delocalization into the cyclopropane ring but maintaining its identity), primary cyclopropylmethyl cations show completely o-delocalized nonclassical carbonium ion character (see Section 3.5.2.5). Also, some of the secondary cyclopropylmethyl cations undergo rapid degenerate equilibrium (see later discussion). 

From the dynamic NMR studies it was shown that the activation barrier for the degenerate equilibrium is 7.4 kcalmoT at -112 This activation barrier is similar to those observed for other 1-cyclopropylethyl cations. The l-(c/.s-l,2-dimethylcyclo-propyl)ethyl and l-(2-methylcyclopropyl)ethyl cations 51 and 55 have an activation barrier for the degenerate rearrangement of less than 5 kcalmol The degenerate equilibrium in all these cases may proceed through the substituted cations as intermediates vide suprdf. ... 

Isotopic perturbation studied by Saunders and Siehl (1980) also indicates that the bicyclobutonium ion [34] is the main species present (cf. Section 4) and the rapid degenerate equilibrium has a barrier less than 3 kcal mol . The C-nmr spectrum of ion [106], prepared from 4,4-dideuteriocyclopropyl-carbinol [107] showed that the two nondeuterated methylene carbons were... 

It is possible to synthesize [2]rotaxanes containing two identical recognition sites within their dumbbell component. The result is a degenerate equilibrium state in which the macrocyclic component can shuttle back and forth along the linear portion of the dumbbell. Such a system constitutes a molecular shuttle. Two examples [lla,b, 39] of [2]rotaxanes that behave as degenerate molecular shuttles are shown in Figure 12. 

When the synthesis of a [2]catenane leads to the interlocking of two different macrocycles each containing two identical recognition sites, then circumrotation of a macrocycle through the cavity of the other leads to degenerate equilibrium states. An example of a degenerate [2]catenane is shown in Figure 28, wherein the dynamic processes in solution are illustrated [23]. 

N-NMR spectroscopy provides a useful tool in tautomeric studies. Thus, the N-NMR spectrum of lb (R = R = Me) shows a broad average signal for the two equivalent nitrogen atoms of the A B degenerate equilibrium when drops of acid are added to accelerate the rate (86MRC444). 

Degenerate equilibrium reactions are the exception rather than the rule. For kinetically controlled reactions equilibrium data are often lacking, whereas structural and kinetic data may be available. This raises the question whether there are still correlations between structures and rate constants, analogous to those discussed in the preceding section and to those between equilibria and rate constants. As will be shown, very much the same conclusions may be reached as for degenerate reactions. 

For 2,3- and 1,2-dimethylnorbomyl ions the values of A5 are 287 and 274 ppm, respectively, lower than for classical 1,2-dimethylcyclopentyl and 1,2-dimethylcyclo-hexyl ions ( 350 ppm) being in a rapid degenerate equilibrium. These facts agree with the stmcture earlier ascribed by Olah to the l,2-dimethyl-2-norbomyl ion — a structure of partially c-delocalized, rapidly equilibrated ions (Wagner-Meerwein rearrangement) The same applies to Saunders s data on the 1,2-dimethylnor-bornyl cation in which one CHj is replaced by CDj (see below)... 

Thus, DCl addition to a mixture of exo/endo 14 gave the exo-6-chlor isomers 18a,b, exclusively. The corresponding entfo-6-chlor isomers 18c,d could not be detected, indicating the intermediate formation of either a bridged ion 20 or a fast degenerate equilibrium between two p-silyl-substituted carbocations 16, which both block the endo attack of a incoming nucleophil efficiently [14]. 

In [ C-l-methyl]-dj-l,4-dimethylcyclohexane [59] no isotopic perturbation was observed within experimental error. It was concluded that a C-isotope effect on the degenerate equilibrium (54) in m 1,4-dimethyl-cyclohexane is either too small to be detected or non-existent (Booth and Everett, 1980a). 

Pig. 9.2.4. A degenerate equilibrium state with hp+i 0. The center manifold is continued here in both directions. Such a bifurcation called a pitchfork is typical for systems where due to symmetry the first non-zero Lyapunov value at a degenerate equilibrium state is always of an odd order. 

Fig. C.2.3. A partial bifurcation diagram for the asymmetric Lorenz model. The point CP is a cusp, at BT the system has a double-degenerate equilibrium state with two zero characteristic exponents (see Sec. 13.2).

The Rossler system and the new Lorenz system (C.2,19) are remarkable in that both have a doubly degenerate equilibrium state with characteristic exponents equal to (0, iij). The feature of this bifurcation is that the unfolding may contain a torus bifurcation curve along with curves corresponding to homoclinic loops to saddle-foci, and therefore non-trivial dynamics may emerge instantly in a neighborhood of the bifurcating equilibria. ... 

C.4. 53. Below we present (following [185]) a list of asymptotic normal forms which describe the trajectory behavior of a triply-degenerate equilibrium state near a stability boundary in systems with discrete symmetry. We say there is a triple instability when a dynamical system has an equilibrium state such that the associated linearized problem has a triplet of zero eigenvalues. In such a case, the analysis is reduced to a three-dimensional system on the center manifold. Assuming that (x, y, z) are the coordinates in the three-dimensional center manifold and a bifurcating equilibrium state resides at the origin, we suppose also that our system is equivariant with respect to the transformation (x,y,z) <- (-X, -y, z). 

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