Negative trajectory

From a physicochemical point of view, we are only interested in positive trajectories, which in mathematics are called semitrajectories. However, in some cases, values of c(t, k, Co) on negative trajectories with t G [—oo, 0] and whole trajectories with t G [—00,00] are informative regarding the behavior in the physicochemically relevant (positive) concentration domain. 

The first energy derivative is called the gradient g and is the negative of the force F (with components along the a center denoted Fa) experienced by the atomic centers F = -g. These forces, as discussed in Chapter 16, can be used to carry out classical trajectory simulations of molecular collisions or other motions of large organic and biological molecules for which a quantum treatment of the nuclear motion is prohibitive. 

We counted the contribution of only those trajectories that have a positive momentum at the transition state. Trajectories with negative momentum at the transition state are moving from product to reactant. If any of those trajectories were deactivated as products, their contribution would need to be subtracted from the total. Why Because those trajectories are ones that originated from the product state, crossed the transition state twice, and were deactivated in the product state. In the TST approximation, only those trajectories that originate in the reactant well are deactivated as product and contribute to the reactive flux. We return to this point later in discussing dynamic corrections to TST. 

If the assumptions underlying the TST are satisfied, the trajectories with initially positive momenta will be trapped in the product well and those with initially negative momentum will be trapped in the reactant well. That will result in a value of k(t) = 1 and the rate constant k =... 

What is that negative contribution We can follow the trajectories backward in time to find the well from which they originated. Of the number of trajectories initially moving from product to reactant, a fraction P is deactivated as reactant and a fraction 1 — f recross the TST due to inertial motion or frequent collisions. A fraction P( — P) will then be deactivated as product, and the remaining (1 — P)- will recross. And so on. The total fraction that is deactivated as product is... 

Consider a trajectory S of a differential system issuing for t = tQ from a point close to the origin. It is dear that 8 will never intersect a surface F = Fc from inside to outside as dVfdt is negative. 

Figure 3. Classical trajectories of the champagne bottle Hamiltonian (Eq. (2)) at energies (a) below and (b) above the barrier maximum, (c) Trajectories of the local focus-focus Hamiltonian (Eq. (7)) at positive (dashed line) and negative (sohd line) energies, (d) The singular cusped orbit of the champagne bottle Hamiltonian with 8 = L = 0.

To complete the explanation of why GP effects cancel in the ICS, we need to explain why the 2-TS paths scatter into negative deflection angles. (It is well known that the 1-TS paths scatter into positive deflection angles via a direct recoil mechanism [55, 56].) We can explain this by following classical trajectories, which gives us the opportunity to illustrate a further useful consequence of the theory of Section II. 

Figure 14 shows a representative 2-TS trajectory, which demonstrates that the 2-TS paths follow a direct S-bend insertion mechanism. The trajectory passes through the middle of the molecule, and avoids the Cl this forces the products to scatter into negative deflection angles. The 2-TS QCT total reaction... 

In particular, the TS trajectory remains bounded for all times, which satisfies the general definition. The constants c and c in Eq. (39) depend on the specific choice of the TS trajectory. Because the saddle point of the autonomous system becomes a fixed point for large positive and negative times, one might envision an ideal choice to be one that allows the TS trajectory to come to rest at the saddle point both in the distant future and in the remote past. However, this is impossible in general because the driving force will transfer energy into or out of the bath modes in such a way that... 

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