Bifurcation reaction paths

There are two reasons why one needs to go beyond a quadratic expansion about the minimum-energy path. The first is anharmonicity, v/hich may be especially important for low-frequency modes, and which is essential for even a qualitatively correct treatment of bifurcating reaction paths. It is also very important for quantitative calculations of low-temperature rate constants to include anharmonicity of high-frequency modes. The second is tunneling in systems with intermediate and large... 

B3.5.7.3 BIFURCATION OF THE REACTION PATH AND VALLEY-RIDGE INFLECTION POINTS... 

The contour plot is given in fig. 43. As remarked by Miller [1983], the existence of more than one transition states and, therefore, the bifurcation of the reaction path, is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of this PES has been carried out by Benderskii et al. [1991b]. The... 

ET transition state is characterized by a looser structure, and accordingly, by a larger entropy than the SN2 transition state. It is interesting to trace the reaction paths from the reactant system to the S 2 and ET products, including the possible bifurcations. 

From the first transition state (TSl, Fig. 1), the reaction path leads to the tetrahedral intermediate 1 (INTI). In the latter, the proton transfer from methanol to the tertiary amine function is completed (from 1.183 to 1.059 A), and the negative charge at the former carbonyl oxygen atom reaches its maximum. This charge is compensated by a further shortening of the bifurcated hydrogen bonds to 2.040 A (-0.103 A) and 1.765 A (-0.096 A) (Fig. 1). The thiourea moiety thus forms an oxyanion hole similar to the amide groups of the serine protease backbone [41]. 

VRI, the actual reaction path bifurcates into two products before reaching TS7. The rate is determined by TS6, whereas the product ratio is controlled by the shape of the PES near the VRI point and TS7. Thus, a question arises how the product selection occurs when a subtle perturbation, such as isotopic substitution, is introduced and two symmetrical products become asymmetric ... 

Figure 6.20. (a) Projection of a three-dimensional PES K(p,p2,p3) for two-proton transfer in formic acid dimer onto the (p, p,) and (p, p3) planes. In contrast with points A and B, in points C and D the potential along the p3 coordinate is a double well resulting in bifurcation of the reaction path [from Shida et al., 1991b]. (b) The contour lines correspond to equilibrium value of p3 and potential (6.37) when V(Q) = V0(Q4 - 2Q2), V0 = 21 kcal/ mol, C = 5.()9V0, A = 5.351/, Qn = 0.5. When Q > Qc, two-dimensional tunneling trajectories exist in the shaded region between curves 1 and 2. Curve 3 corresponds to synchronous transfer. 

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

The potential energy surfaces of the reactions in equations 10 and 11 are more complex than that of equation 14 and involve reaction path bifurcation and low symmetry structures in each channel. The salient features of structures of these complexes and TSs are shown... 

Carrington and Miller (235) developed a method called the reaction-surface Hamiltonian for reactions with large amplitudes perpendicular to the reaction path and for some types of reactions with bifurcation of the reaction path. In contrast to the reaction-path Hamiltonian method, in the reaction-surface Hamiltonian method two coordinates are extracted from the complete coordinate set. One coordinate describes motion along the reaction path and the second one describes the large-amplitude motion. Potential energy in space of the remaining 3JV — 8 coordinates perpendicular to the two-dimensional reaction surface is approximated by quadratic functions. It... 

The third stage of our strategy is discussed in Sections IX and X. Our discussion is speculative, since quantitative analysis is lacking at present. In Section IX, we point out that, in reaction dynamics, breakdown of normal hyperbolicity would also play an important role. Such cases would include phase transitions in systems with a finite number of degrees of freedom. In Section X, we will discuss the possibility of bifurcation in the skeleton of reaction paths, and we point out that it corresponds to crisis in multidimensional chaos. This approach offers an interesting mechanism for chemical evolution. 

Figure 1.8 A PES with a bifurcation that takes place after TSl at the valley ridge inflection (VRI). The reaction path splits at the VRI leading in one direction to Product 1 and in the other direction to Product 2.

The possibility of bifurcation of the reaction path on the So surface corresponding to the various possible bond-forming schemes (indicated in Figure 7.42 by dark arrows) and the existence of several pathways in the S, state (light arrows) are the reasons for the great diversity of photoproducts from precalciferol. 

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. 

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