Barrier crossing phases

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

Tsai, C.J. Jordan, K.D., Use of the histogram and jump-walking methods for overcoming slow barrier crossing behavior in Monte Carlo simulations applications to the phase transitions in the (Ar)i3 and (FbOjs clusters, J. Chem. Phys. 1993, 99, 6957... 

Figure 4 Two arbitrary potential energy surfaces in a two-dimensional coordinate space. All units are arbitrary. Panel A shows two minima connected by a path in phase space requiring correlated change in both degrees of freedom (labeled Path a). As is indicated, paths involving sequential change of the degrees of freedom encounter a large enthalpic barrier (labeled Path b). Panel B shows two minima separated by a barrier. No path with a small enthalpic barrier is available, and correlated, stepwise evolution of the system is not sufficient for barrier crossing.

In order to understand the problem of finding TS with three or more DOFs, it is useful to address the question of dimensionalities, in configuration and phase space. In classical, Hamiltonian dynamics, transition states are grounded on the idea that certain surfaces (more precisely, certain manifolds) act as barriers in phase space. It is possible to devise barriers in phase space, since in phase space, in contrast to configuration space, two trajectories never cross [uniqueness of solutions of ODEs, see Eq. (4)]. In order to construct a barrier in phase space, the first step is to construct a manifold if that is made of a set of trajectories [8]. 

Our focus so far was on unimolecular reactions and on solvent effects on the dynamics of barrier crossing. Another important manifestation of the interaction between the reaction system and the surrounding condensed phase comes into play in bimolecular reactions where the process by which the reactants approach each other needs to be considered. We can focus on this aspect of the process by considering bimolecular reactions characterized by the absence of an activation barrier, or by a barrier small relative to ke T. In this case the stage in which reactants approach each other becomes the rate determining step of the overall process. 

In Section III, we combine the analyses of Section II and the Appendix with the results of recent experiments [13] and simulations [14] to discuss the slow variable description [7-10] of liquid phase activated barrier crossing. In Section IV, using these same ideas we review our fast variable theory of liquid phase reaction dynamics [19-26] from the standpoint of the recent literature [6,11-17,27,28]. Finally, in Section V we summarize our main points. 

We next turn to the slow variable description of liquid phase activated barrier crossings. 

The plan of Section IV is as follows In section IV.A, we qualitatively outline the general picture of reaction dynamics that emerges from fast variable physics. Next, in section IV.B, we examine liquid phase-activated barrier crossing in the short time regime of Section II.C. In Section IV.C we note that the fast variable/slow bath timescale separation also applies to liquid phase vibrational energy relaxation and then discuss that process from the fast variable standpoint. Finally, in Section IV.D, we discuss some related work of others. 

The fast variable/slow bath timescale separation arises in liquid phase VER because of a frequency mismatch between the solute and solvent molecules, rather than because of a speed difference, as in typical reactions. Despite this, the fast variable timescale separation yields a picture of solute VER very similar to the picture of short time-activated barrier crossing reflected in Eq. (3.53). We expect similar pictures to emerge for other processes. 

Short time pictures of liquid phase vibrational energy relaxation, cage escape, and activated barrier crossing are described in S. A. Adelman, J. Stai. Phys. 42, 37 (1986). 

The short time treatment of liquid phase activated barrier crossing outlined in Section VB is described in detail in S. A. Adelman and R. Muralidhar, J. Chem. Phys. 95, 2752 (1991). 

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