Transition State in a Noisy Environment

Transition state theory overestimates reaction rates in solution because conventional dividing surfaces between reagents and products are crossed many times by the same reactive trajectory. We describe a recipe for constructing a time-dependent dividing surface free of such recrossings in the presence of noise. The no-recrossing limit of transition state theory thus becomes generally available for the description of reactions in a fluctuating environment.

Figure 1

Phase portrait of the one-dimensional noiseless damped dynamics corresponding to Eq. (3). The thick line indicates a possible choice for the no-recrossing surface. The simple relative dynamics illustrated here is exemplified by the trajectories in Fig. 2a.

Figure 2

A random instance of the TS trajectory (black) and a reactive trajectory (red) under the influence of the same noise in a system with $N=2$ degrees of freedom: the unstable reactive coordinate $q_{\mathrm{u}}$ and the stable transverse coordinate $q_{\mathrm{s}}$. The potential is $U(q_{\mathrm{u}},q_{\mathrm{s}})=-\frac{1}{2}{\omega }_{\mathrm{b}}^2{q}_{\mathrm{u}}^2+\frac{1}{2}{\omega }_{\mathrm{s}}^2{q}_{\mathrm{s}}^2$. Trajectories are projected onto (a) the reactive degree of freedom, (b) the transverse degree of freedom, and (c) configuration space. Units are chosen so that ${\omega }_{\mathrm{b}}=1$ and $k_{B}T=1$. The transverse frequency is ${\omega }_{\mathrm{s}}=1.5$, and the friction is isotropic, $\mathit{\Gamma }=\gamma \mathbf{I}$, with $\gamma =0.2$. The bottom of each column shows the projected trajectories in the corresponding space. Above this, their time evolution is illustrated using the same axes. The blue cut marks the unique reaction time $t_{\mathrm{react}}=8.936$ when the moving TS surface is crossed. Dotted lines in this cut, at $t=0$, and at $t=20$, indicate the moving coordinate axes centered on the TS trajectory. These axes are labeled explicitly only at the top face. Dashed lines in the cuts of column (a) show the moving invariant manifolds. No TS is indicated in column (b) because the $q_{\mathrm{s}}-v_{\mathrm{s}}$ subspace lies entirely within the moving TS surface. Thick green dots indicate repeated crossings of the stationary TS surface $q_{\mathrm{u}}=0$. Green lines in the top face of all three columns show the reactive trajectory in relative coordinates $\Delta \overrightarrow{q}$, $\Delta \overrightarrow{v}$ [not to scale in (a), for graphical reasons]. The typical behavior of a reactive trajectory shown in Fig. 1 is visible in the top face of column (a).