^{1}, Yan Zhao

^{1}, Hai Lin

^{1,a)}and Donald G. Truhlar

^{1,b)}

### Abstract

This article presents a multifaceted study of the reaction and three of its deuterium-substituted isotopologs. First we present high-level electronic structure calculations by the W1, G3SX, MCG3-MPWB, CBS-APNO, and MC-QCISD/3 methods that lead to a best estimate of the barrier height of . Then we obtain a specific reaction parameter for the MPW density functional in order that it reproduces the best estimate of the barrier height; this yields the MPW54 functional. The MPW54 functional, as well as the MPW60 functional that was previously parametrized for the reaction, is used with canonical variational theory with small-curvature tunneling to calculate the rate constants for all four ethane reactions from 200 to 2000 K. The final MPW54 calculations are based on curvilinear-coordinate generalized-normal-mode analysis along the reaction path, and they include scaled frequencies and an anharmonic C–C bond torsion. They agree with experiment within 31% for 467–826 K except for a 38% deviation at 748 K; the results for the isotopologs are predictions since these rate constants have never been measured. The kinetic isotope effects (KIEs) are analyzed to reveal the contributions from subsets of vibrational partition functions and from tunneling, which conspire to yield a nonmonotonic temperature dependence for one of the KIEs. The stationary points and reaction-path potential of the MPW54 potential-energysurface are then used to parametrize a new kind of analytical potential-energysurface that combines a semiempirical valence bond formalism for the reactive part of the molecule with a standard molecular mechanics force field for the rest; this may be considered to be either an extension of molecular mechanics to treat a reactive potential-energysurface or a new kind of combined quantum-mechanical/molecular mechanical (QM/MM) method in which the QM part is semiempirical valence bondtheory; that is, the new potential-energysurface is a combined valence bond molecular mechanics (CVBMM) surface.Rate constants calculated with the CVBMM surface agree with the MPW54 rate constants within 12% for 534–2000 K and within 23% for 200–491 K. The full CVBMM potential-energysurface is now available for use in variety of dynamics calculations, and it provides a prototype for developing CVBMM potential-energysurfaces for other reactions.

This work was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences.

I. INTRODUCTION

II. CVBMM POTENTIAL-ENERGYSURFACE

A. Molecular mechanical terms

B. VB/MM interaction

C. Reactive part

III. VARIATIONAL TRANSITION STATE THEORY

A. Reaction-path potentials and rate constants

B. Hindered rotator approximation

IV. ELECTRONIC STRUCTURE THEORY

A. High-level methods

B. DFT

C. Parametrization of the DFT-SRP functional

V. SOFTWARE

VI. RESULTS FROM DIRECT DYNAMICS CALCULATIONS

A. Details of the calculations

B. Energetics

C. Transition state geometries and imaginary frequencies

D. Torsional potential

E. Barrier shape

F. Reaction-rate constants

G. Kinetic isotope effects

VII. PARAMETRIZATION OF THE CVBMM POTENTIAL-ENERGYSURFACE

A. Energetics and geometries

B. Frequencies

C. Barrier shape and rate constants

VIII. CONCLUDING REMARKS

### Key Topics

- Reaction rate constants
- 52.0
- Potential energy surfaces
- 34.0
- Surface dynamics
- 29.0
- Transition state theory
- 27.0
- Hydrogen reactions
- 25.0

## Figures

Transition state geometry for the reaction.

Transition state geometry for the reaction.

Rotational barrier (in ) of the transition state for the reaction at the MPW54/6-31 level.

Rotational barrier (in ) of the transition state for the reaction at the MPW54/6-31 level.

for the four reactions as a function of reaction coordinate obtained using the MPW54/6-31 level with the HR approximation.

for the four reactions as a function of reaction coordinate obtained using the MPW54/6-31 level with the HR approximation.

Logarithm of the deuterium KIE for reactions (R1) and (R2) vs obtained using the MPW54/6-31 level with the HR approximation. is for and is for .

Logarithm of the deuterium KIE for reactions (R1) and (R2) vs obtained using the MPW54/6-31 level with the HR approximation. is for and is for .

Logarithm of the deuterium KIE for reactions (R1) and (R3) vs obtained using the MPW54/6-31 level with the HR approximation. is for and is for .

Logarithm of the deuterium KIE for reactions (R1) and (R3) vs obtained using the MPW54/6-31 level with the HR approximation. is for and is for .

Logarithm of the deuterium KIE for reactions (R1) and (R4) vs obtained using the MPW54/6-31 level with the HR approximation. is for and is for .

Logarithm of the deuterium KIE for reactions (R1) and (R4) vs obtained using the MPW54/6-31 level with the HR approximation. is for and is for .

Comparison of the plot of for the reaction as a function of reaction coordinate obtained by using the CVBMM surface and the MPW54/6-31 level.

Comparison of the plot of for the reaction as a function of reaction coordinate obtained by using the CVBMM surface and the MPW54/6-31 level.

Comparison of the plot of for the reaction as a function of reaction coordinate obtained by using the CVBMM surface and the MPW54/6-31 level.

## Tables

Energetics for the reaction . is the forward classical barrier height, is the reverse classical barrier height, and is the classical energy of reaction.

Energetics for the reaction . is the forward classical barrier height, is the reverse classical barrier height, and is the classical energy of reaction.

ZPEs, differences in ZPE between saddle point and reactants and between saddle point and products, ground-state vibrationally adiabatic barrier height at the saddle point, and enthalpy of reaction at 0 K.

ZPEs, differences in ZPE between saddle point and reactants and between saddle point and products, ground-state vibrationally adiabatic barrier height at the saddle point, and enthalpy of reaction at 0 K.

Comparison of calculated barrier heights for and reactions.

Comparison of calculated barrier heights for and reactions.

Geometries and imaginary frequencies for the saddle point. Distances are in Å, bond angles are in deg, and frequencies are in .

Geometries and imaginary frequencies for the saddle point. Distances are in Å, bond angles are in deg, and frequencies are in .

Direct dynamics rate constants for the reaction. The 6-31 basis set used. HO denotes the harmonic-oscillator approximation for all modes; HR denotes that the lowest-frequency mode is treated as a hindered rotator.

Direct dynamics rate constants for the reaction. The 6-31 basis set used. HO denotes the harmonic-oscillator approximation for all modes; HR denotes that the lowest-frequency mode is treated as a hindered rotator.

Direct dynamics rate constants for the , and reactions All calculations in this table use the MPW 54/6-31 electronic structure level and use the HR approximation for the lowest-frequency mode.

Direct dynamics rate constants for the , and reactions All calculations in this table use the MPW 54/6-31 electronic structure level and use the HR approximation for the lowest-frequency mode.

KIEs and factors for the . All calculations in this table use the MPW 54/6-31 electronic structure level with the HR approximation for the torsion mode.

KIEs and factors for the . All calculations in this table use the MPW 54/6-31 electronic structure level with the HR approximation for the torsion mode.

Optimized geometries of the stationary points for the reaction obtained from the CVBMM potential-energy surface. Distances are in Å and angles are in deg.

Optimized geometries of the stationary points for the reaction obtained from the CVBMM potential-energy surface. Distances are in Å and angles are in deg.

Normal-mode analysis of the stationary points of the CVBMM potential-energy surface of the reaction.

Normal-mode analysis of the stationary points of the CVBMM potential-energy surface of the reaction.

Rate constants obtained using the CVBMM surface for the reaction.

Rate constants obtained using the CVBMM surface for the reaction.

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