^{1,a)}, William H. Miller

^{1,b)}, Jesús F. Castillo

^{2}and F. Javier Aoiz

^{2}

### Abstract

A general quantum-mechanical method for computing kinetic isotope effects is presented. The method is based on the quantum-instanton approximation for the rate constant and on the path-integral Metropolis–Monte Carlo evaluation of the Boltzmann operator matrix elements. It computes the kinetic isotope effect directly, using a thermodynamic integration with respect to the mass of the isotope, thus avoiding the more computationally expensive process of computing the individual rate constants. The method should be more accurate than variational transition-state theories or the semiclassical instanton method since it does not assume a single tunneling path and does not use a semiclassical approximation of the Boltzmann operator. While the general Monte Carlo implementation makes the method accessible to systems with a large number of atoms, we present numerical results for the Eckart barrier and for the collinear and full three-dimensional isotope variants of the hydrogen exchange reaction . In all seven test cases, for temperatures between 250 and 600 K, the error of the quantum instanton approximation for the kinetic isotope effects is less than .

This work was supported by the Director, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, U.S. Department of Energy under Contract No. DE-AC03-76SF00098, by the National Science Foundation Grant No. CHE-0345280, and by DGES of Spain (Project No. BQU2002-04627-C02-02). One of the authors (J.V.) would like to thank Yimin Li and the rest of Bill Miller’s group for many useful discussions. One of the authors (J.F.C.) acknowledges the support through the program “Ramon y Cajal” from the Ministry of Education and Culture of Spain.

I. INTRODUCTION

II. QUANTUM-INSTANTON APPROXIMATION FOR THE THERMAL RATE CONSTANT

III. APPLICATION TO THE KINETIC ISOTOPE EFFECTS

IV. PATH-INTEGRAL REPRESENTATION OF RELEVANT QUANTITIES

V. TRANSITION-STATE THEORY FRAMEWORKFOR THE KINETIC ISOTOPE EFFECTS

VI. EXACT QUANTUM-MECHANICAL METHOD

VII. NUMERICAL RESULTS AND DISCUSSION

A. Eckart barrier

B. Collinear reaction

C. Reaction in three spatial dimensions

1. Quantum-instanton calculation

2. Exact cumulative reaction probabilities and rate constants

3. Kinetic isotope effects

VIII. CONCLUSION

### Key Topics

- Reaction rate constants
- 28.0
- Hydrogen reactions
- 26.0
- Transition state theory
- 24.0
- Semiclassical theories
- 21.0
- Kinetic isotope effects
- 19.0

## Figures

The kinetic isotope effect for the Eckart barrier.

The kinetic isotope effect for the Eckart barrier.

The kinetic isotope effect for the collinear hydrogen exchange reaction: (a) , (b) , (c) , and (d) .

The kinetic isotope effect for the collinear hydrogen exchange reaction: (a) , (b) , (c) , and (d) .

QM *total* cumulative reaction probabilities calculated using the BKMP2 PES for the , , and reactions.

QM *total* cumulative reaction probabilities calculated using the BKMP2 PES for the , , and reactions.

QM rate constants calculated using the BKMP2 PES for the , and reactions.

QM rate constants calculated using the BKMP2 PES for the , and reactions.

The kinetic isotope effect for the hydrogen exchange reaction in three spatial dimensions: (a) and (b) .

The kinetic isotope effect for the hydrogen exchange reaction in three spatial dimensions: (a) and (b) .

## Tables

Size of the basis set employed in the exact QM scattering calculations for the reaction and isotopic variants on the BKMP2 PES.

Size of the basis set employed in the exact QM scattering calculations for the reaction and isotopic variants on the BKMP2 PES.

Kinetic isotope effect for the Eckart barrier: is the quantum-instanton result obtained from Eq. (3.3), the simplest quantum instanton from Eq. (7.2), the modified quantum instanton from Eq. (3.3) with replaced by Eq. (7.3), the high- expansion of the TST from Eq. (7.5), and is the ratio of rate constants obtained from Eqs. (2.1) and (7.6).

Kinetic isotope effect for the Eckart barrier: is the quantum-instanton result obtained from Eq. (3.3), the simplest quantum instanton from Eq. (7.2), the modified quantum instanton from Eq. (3.3) with replaced by Eq. (7.3), the high- expansion of the TST from Eq. (7.5), and is the ratio of rate constants obtained from Eqs. (2.1) and (7.6).

Kinetic isotope effect for the collinear reaction using the TK potential (Ref. 66). Here was obtained from Eq. (7.8), from Refs. 64 and 68. The meaning of the remaining quantities is the same as in Table II.

Kinetic isotope effect for the collinear reaction using the TK potential (Ref. 66). Here was obtained from Eq. (7.8), from Refs. 64 and 68. The meaning of the remaining quantities is the same as in Table II.

Kinetic isotope effect for the collinear reaction using the TK potential (Ref. 66). The meaning of various quantities is the same as in Table III.

Kinetic isotope effect for the collinear reaction using the TK potential (Ref. 66). The meaning of various quantities is the same as in Table III.

Kinetic isotope effect for the collinear reaction using the TK potential (Ref. 66). The meaning of various quantities is the same as in Table III.

Exact QM thermal rate constants for the reaction and isotopic variants as a function of temperature calculated on the BKMP2 PES. The rate constants were obtained from Eq. (6.1). They are in units of , the figures in parentheses denote the power of 10.

Exact QM thermal rate constants for the reaction and isotopic variants as a function of temperature calculated on the BKMP2 PES. The rate constants were obtained from Eq. (6.1). They are in units of , the figures in parentheses denote the power of 10.

Kinetic isotope effect for the reaction in three spatial dimensions, using the BKMP2 potential (Refs. 81 and 82) except for which uses the DMBE potential (Refs. 92 and 93). Here is the high- expansion of TST from Eq. (7.11), is the result of canonical variational TST with semiclassical tunneling corrections from Ref. 56 and is the quantum-mechanical result from Eq. (6.1). The meaning of remaining quantities is the same as in Table III.

Kinetic isotope effect for the reaction in three spatial dimensions, using the BKMP2 potential (Refs. 81 and 82) except for which uses the DMBE potential (Refs. 92 and 93). Here is the high- expansion of TST from Eq. (7.11), is the result of canonical variational TST with semiclassical tunneling corrections from Ref. 56 and is the quantum-mechanical result from Eq. (6.1). The meaning of remaining quantities is the same as in Table III.

Kinetic isotope effect using the BKMP2 potential (Refs. 81 and 82) except for which uses the DMBE potential (Refs. 92 and 93). The meaning of various quantities is the same as in Table VIII.

Kinetic isotope effect using the BKMP2 potential (Refs. 81 and 82) except for which uses the DMBE potential (Refs. 92 and 93). The meaning of various quantities is the same as in Table VIII.

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