Bifurcation rearrangement

Table 3 Calculated Tunneling Splitting Patterns for the Water Trimer when the Single-flip and Bifurcation Rearrangements are Feasible, There are Two Po.ssible Patterns, Depending Upon the Number of Flips Accompanying the Bifurcation ...

The infinite potential barrier, shown schematically in figure 10 corresponds to a superselection rule that operates below the critical temperature [133]. Above the critical temperature the quantum-mechanical superposition principle applies, but below that temperature the system behaves classically. The system bifurcates spontaneously at the critical point. The bifurcation, like second-order phase transformation is caused by some interaction that becomes dominant at that point. In the case of chemical reactions the interaction leads to the rearrangement of chemical bonds. The essential difference between chemical reaction and second-order phase transition is therefore epitomized by the formation of chemically different species rather than different states of aggregation, when the symmetry is spontaneously broken at a critical point. 

The VRIs are not the only place where bifurcations can occur. The mechanism of the 1,2,6-heptatriene rearrangement that is summarized in Figure 21.2 also depicts a... 

Another hint toward understanding reaction dynamics on a bifurcating surface was supplied by Singleton in his study of the Diels-Alder cycloaddition of acrolein with methyl vinyl ketone (Reaction 8.12). Recognizing the interconversion of 87 and 88 through a Cope rearrangement along with careful kinetic analysis led to an estimate of the ratio of the rate of formation of 87 88 as 2.5 1. 

Siebert, M. R. Zhang, J. Addepalli, S. V. TantiUo, D. J. Hase, W. L. The need for enzymatic steering in abietic acid biosynthesis Gas-phase chemical dynamics simulations of carbocation rearrangements on a bifurcating potential energy surface, J. Am. Chem. Soc. 2011,133, 8335-8343. 

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

Dubreuil, D., Cleophax, J., De Almeida, M. V., Verre-Sebrie, C., Liaigre, J., Vass, G., Gero, S. D. Stereoselective synthesis of 6-deoxy and 3,6-dideoxy-D-myoinositol precursors of deoxy myoinositol phosphate analogs from D-galactose. Tetrahedron 1997, 53,16747-16766. Paquette, L. A., Kinney, M. J., Dullweber, U. Practical Synthesis of Spirocyclic Bis-C,C-glycosides. Mechanistic Models in Explanation of Rearrangement Stereoselectivity and the Bifurcation of Reaction Pathways. J. Org. Chem. 1997, 62,1713-1722. 

Upon the first phase transition (T > Tc), when a proton in a unit cell is bifurcated, the proton deforms the lattice and interacts with crystal vibrations. The interaction leads to the rearrangement of the phonon spectrum. Corresponding transformations of the to q) values can be obtained in the framework... 

A large part of the computational work has been influenced by the introduction of curvilinear coordinates, designed to take advantage of the topography of potential surfaces. These coordinates allow for a smooth change from reactant to product conformations and in effect transform the rearrangement problem into the much simpler one of inelastic collisions. The various treatments have employed reaction-path (or natural collision) coordinates less restricted reaction coordinates atom-transfer coordinates, somewhat analogous to those used for electron-transfer and, for planar and spatial motion, bifurcation coordinates. 

To illustrate the general procedure let us consider collinear reactions of type A + BC - AB + C. This means that the three atoms move on a line, e.g. the x-axis, and furthermore that velocities are also along the same line. Indicating relative atomic distances by xAB, xBC and xCA, we introduce centre-of-mass coordinates, which for reactants are x = xBC and X, the distance from A to the centre of mass of BC. Similar coordinates could be defined for products. Because of restrictions in the type of motion it is not possible to simultaneously account for the B + CA rearrangement. To do this one must proceed to planar or spatial motion and, for example, introduce bifurcation coordinates. 

Kinetic equation (6.35) will be rearranged to a simpler form using the substitution x = y + 1/3 and obtaining a standard (gradient) form of the cusp bifurcation... 

Beckmann rearrangement of oximes to amides can deviate to fragmentation to form nitriles and carbocations, if the latter possess reasonable stability. Both 1-substituted-phenyl-2-propanones and 3-substituted-phenyl-2-butanones in aqueous solvents give both products, and calculations have been used to probe the mechanisms. In borderline cases, a dynamic path bifurcation from a single transition state is claimed. 

Kinetic evidence for such a dynamic path bifurcation in the mechanism of the Beckmann rearrangement has also been presented for reactions of oxime sulfonates (43, R /R = H/Me). In addition to the rearrangement product (amide), these substrates can also fragment (to alcohols), with the bifurcation in the mechanism apparently occurring after the rate-determining transition state. 

Equation (7.15) predicts that the derivative of the solution with respect to the parameter will approach infinity as b + 2x approaches zero. Intuitively, as illustrated in Figure 7.9, it makes sense that an infinitely rapid change in the derivative would correspond to a qualitative change in the character of the solution. It is indeed the case that an infinite derivative indicates a bifurcation. We shall see shortly how to follow x asb changes, but a bit of algebra will tell us immediately what we want to know. We can rearrange eq. (7.13) in the form... 

---