^{1}and Dmitrii E. Makarov

^{1,a)}

### Abstract

Motivated by recent experimental efforts to measure the time a molecular system spends in transit between the reactants and the products of a chemical reaction, here we study the properties of the distribution of such transit times for the case of conservative dynamics on a multidimensional energy landscape. Unlike reaction rates, transit times are not invariant with respect to the order parameter (a.k.a. the experimental signal) used to monitor the progress of a chemical reaction. Nevertheless, such order parameter dependence turns out to be relatively weak. Moreover, for several model systems we find that the probability distribution of transit times can be estimated analytically, with reasonable accuracy, by assuming that the order parameter coincides with the direction of the unstable normal mode at the transition state. Although this approximation tends to overestimate the actual mean transit time measured using other order parameters, it yields asymptotically correct long-time behavior of the transit time distribution, which is independent of the order parameter.

This work was supported by the National Science Foundation (Grant No. CHE-0848571) and Robert A. Welch Foundation (Grant No. F-1514). We are indebted to Bill Eaton, Irina Gopich, Graeme Henkelman, Anatoly Kolomeisky, Attila Szabo, Eric Vanden-Eijnden, and Noham Weinberg for many insightful discussions.

I. INTRODUCTION

II. GENERAL THEORY

A. Transit time distribution: Definition and computation

B. Some properties of transit times

III. PROPERTIES OF TRANSIT TIMES IN ONE DIMENSION

A. A flat potential maximizes the mean transit time

B. Mean transit time for a parabolic barrier

IV. TRANSIT TIMES FOR KRAMERS’ PROBLEM: THE ZWANZIG–CALDEIRA–LEGGET APPROACH

V. A CASE STUDY: PARABOLIC BARRIER COUPLED TO ONE HARMONIC OSCILLATOR

VI. CONCLUDING REMARKS

### Key Topics

- Normal modes
- 11.0
- Conformational dynamics
- 10.0
- Molecular conformation
- 9.0
- Time measurement
- 7.0
- Statistical properties
- 6.0

## Figures

Left: a long trajectory occasionally enters the transition region T between two basins of attraction, A and B, corresponding to the reactants and products of a chemical reaction. A transition path from A to B enters T from A and exits to B (bold segments of the trajectory). The corresponding transit time is the time interval between the moment the transition region is entered from A and the moment the trajectory finally arrives at B. Right: a reactive trajectory traveling between A and B may recross an intermediate dividing surface located inside the transition region T.

Left: a long trajectory occasionally enters the transition region T between two basins of attraction, A and B, corresponding to the reactants and products of a chemical reaction. A transition path from A to B enters T from A and exits to B (bold segments of the trajectory). The corresponding transit time is the time interval between the moment the transition region is entered from A and the moment the trajectory finally arrives at B. Right: a reactive trajectory traveling between A and B may recross an intermediate dividing surface located inside the transition region T.

A one dimensional potential , with a maximum value in the transition region .

A one dimensional potential , with a maximum value in the transition region .

Distribution of transit times computed numerically for the case of a parabolic barrier (solid line) and compared with Eq. (25) (dashed lines) for (a) and (b) .

Distribution of transit times computed numerically for the case of a parabolic barrier (solid line) and compared with Eq. (25) (dashed lines) for (a) and (b) .

(a) Contour plot of the potential of Eq. (27) with , , , and representative transition paths in this potential. The transition region is shaded. (b) Histogram of transit times obtained from simulations, compared with Eq. (25), where was estimated as the free energy barrier (solid line) and using Eq. (30) (dashed line). (c) Potentials profiles [ plotted vs ] traversed by the transition paths from (a). The thick line shows the potential .

(a) Contour plot of the potential of Eq. (27) with , , , and representative transition paths in this potential. The transition region is shaded. (b) Histogram of transit times obtained from simulations, compared with Eq. (25), where was estimated as the free energy barrier (solid line) and using Eq. (30) (dashed line). (c) Potentials profiles [ plotted vs ] traversed by the transition paths from (a). The thick line shows the potential .

(a) Contour plot of the potential of Eq. (27) with , , , and representative transition paths in this potential. The transition region is shaded. (b) Histogram of transit times obtained from simulations, compared with Eq. (25), where was estimated as the free energy barrier (solid line) and using Eq. (30) (dashed line). (c) Potentials profiles [ plotted vs ] traversed by the transition paths from (a). The thick line shows the potential .

(a) Contour plot of the potential of Eq. (27) with , , , and representative transition paths in this potential. The transition region is shaded. (b) Histogram of transit times obtained from simulations, compared with Eq. (25), where was estimated as the free energy barrier (solid line) and using Eq. (30) (dashed line). (c) Potential profiles [ plotted vs ] traversed by the transition paths from Fig. 4(a). The thick line shows the potential .

(a) Contour plot of the potential of Eq. (27) with , , , and representative transition paths in this potential. The transition region is shaded. (b) Histogram of transit times obtained from simulations, compared with Eq. (25), where was estimated as the free energy barrier (solid line) and using Eq. (30) (dashed line). (c) Potential profiles [ plotted vs ] traversed by the transition paths from Fig. 4(a). The thick line shows the potential .

Same data as in Fig. 4(b) but plotted on a logarithmic scale to show the exponential behavior of the distribution at long times.

Same data as in Fig. 4(b) but plotted on a logarithmic scale to show the exponential behavior of the distribution at long times.

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