^{1}and William H. Miller

^{1,a)}

### Abstract

One of the outstanding issues in the quantum instanton (QI) theory (or any transition-state-type theory) for thermal rate constants of chemical reactions is the choice of an appropriate “dividing surface” (DS) that separates reactants and products. (In the general version of the QI theory, there are actually two dividing surfaces involved.) This paper shows one simple and general way for choosing DSs for use in QI theory, namely, using the family of (hyper) planes normal to the minimum energy path on the potential energy surface at various distances along it. Here the *reaction coordinate* is not one of the dynamical coordinates of the system (which will in general be the Cartesian coordinates of the atoms), but rather simply a parameter which specifies the DS. It is also shown how this idea can be implemented for an atom system in three-dimensional space in a way that preserves overall translational and rotational invariance. Numerical application to a simple system (the collinear reaction) is presented to illustrate the procedure.

This work was supported in part by the Office of Naval Research Grant No. N00014-01-1-0236 and by the Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The authors also acknowledge a generous allocation of supercomputing time from the National Energy Research Scientific Computing Center (NERSC). The authors would like to thank Dr. Michele Ceotto for many useful discussions.

I. INTRODUCTION

II. GENERAL APPROACH

A. System without overall rotation and translation

B. atom system in 3D space

III. APPLICATION TO THE QI MODEL

A. The QI approximation of the canonical rate constant of chemical reaction

B. Path integral representation

IV. NUMERICAL TEST

V. CONCLUDING REMARKS

### Key Topics

- Transition state theory
- 20.0
- Chemical reaction theory
- 12.0
- Chemical reactions
- 8.0
- Potential energy surfaces
- 8.0
- Reaction rate constants
- 7.0

## Figures

A contour of potential energy surface. The dashed line indicates a MEP from the reactant to the product region. is a point on the MEP and is the tangent vector of the MEP. is the arc length from to the saddle point (indicated by a big black dot) along the MEP. The dividing surface determined by Eq. (2.3) is a line normal to the MEP at the point .

A contour of potential energy surface. The dashed line indicates a MEP from the reactant to the product region. is a point on the MEP and is the tangent vector of the MEP. is the arc length from to the saddle point (indicated by a big black dot) along the MEP. The dividing surface determined by Eq. (2.3) is a line normal to the MEP at the point .

(a) PES as a function of the radial coordinate . The top of the barrier is located at . (b) The DS in 2D Cartesian coordinates. The dashed line is a MEP passing through one particular saddle point. is a point at position along the MEP. The DS at the top of the barrier is a circle with the radius equal to . The dotted line indicates the planar DS defined by Eq. (2.3) at point .

(a) PES as a function of the radial coordinate . The top of the barrier is located at . (b) The DS in 2D Cartesian coordinates. The dashed line is a MEP passing through one particular saddle point. is a point at position along the MEP. The DS at the top of the barrier is a circle with the radius equal to . The dotted line indicates the planar DS defined by Eq. (2.3) at point .

PES of the collinear reaction. The dashed line shows the MEP on the reactant side of the saddle point.

PES of the collinear reaction. The dashed line shows the MEP on the reactant side of the saddle point.

Quantum free energy for the collinear reaction.

Quantum free energy for the collinear reaction.

for the collinear reaction.

for the collinear reaction.

for the collinear reaction.

for the collinear reaction.

The QI thermal rate constant for the collinear reaction as a function of the position of DS.

The QI thermal rate constant for the collinear reaction as a function of the position of DS.

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