^{1,a)}, Joel M. Bowman

^{1,b)}and David J. Nesbitt

^{2,c)}

### Abstract

We report tunneling splittings associated with the large amplitude 1,2 H-atom migration to the global minima in the vinyl radical. These are obtained using a recent full-dimensional *ab initio*potential energy surface (PES) [A. R. Sharma, B. J. Braams, S. Carter, B. C. Shepler, and J. M. Bowman, J. Chem. Phys. **130**(17), 174301 (2009)] and independently, directly calculated “reaction paths.” The PES is a multidimensional fit to coupled cluster single and double and perturbative treatment of triple excitations coupled-cluster single double triple (CCSD(T)) with the augmented correlation consistent triple zeta basis set (aug-cc-pVTZ). The reaction path potentials are obtained from a series of CCSD(T)/aug-cc-pVnTZ calculations extrapolated to the complete basis set limit. Approximate 1D calculations of the tunneling splitting for these 1,2-H atom migrations are obtained using each of these potentials as well as quite different 1D Hamiltonians. The splittings are calculated over a large energy ranges, with results from the two sets of calculations in excellent agreement. Though negligibly slow (>1 s) for the vibrational ground state, this work predicts tunneling-promoted 1,2 hydride shift dynamics in vinyl to exhibit exponential growth with internal vibrational excitation, specifically achieving rates on the sub-*μ*s time scale at energies above *E* ≈ 7500 cm^{−1}. Most importantly, these results begin to elucidate the possible role of quantum isomerization through barriers without dissociation, in competition with the more conventional picture of classical roaming permitted over a much narrower window of energies immediately below the bond dissociation limit. Furthermore, when integrated over a Boltzmann distribution of thermal energies, these microcanonical tunneling rates are consistent with sub-μs time scales for 1,2 hydride shift dynamics at *T* > 1400 K. These results have potential relevance for combustion modeling of low-pressure flames, as well as recent observations of nuclear spin statistical mixing from high-resolution IR/microwave spectroscopy on vinyl radical.

This work was supported by the U.S. Department of Energy (D.J.N. from DE-SC0002123 and J.M.B. from DE-FG02-97ER14782), Office of Science (A.R.S. from DE-FG02-07ER54914). A.R.S. also wishes to acknowledge the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, under Contract No. DE-AC02- 06CH11357 as part of the Argonne-Sandia Consortium on High-Pressure Combustion Chemistry; FWP# 2009 ANL 59044. In particular, we would like to thank Dr. Larry Harding for his extremely helpful discussions, as well as calculations of the intrinsic reaction path for 1,2 hydride shift dynamics in vinyl.

I. INTRODUCTION

II. POTENTIAL ENERGY SURFACES AND SADDLE POINT CONFIGURATIONS

III. 1D POTENTIALS, TUNNELING PATHS, AND HAMILTONIAN TREATMENTS

IV. EIGENFUNCTIONS, EIGENVALUES, AND TUNNELING SPLITTINGS

V. DISCUSSION

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Tunneling
- 90.0
- Isomerization
- 14.0
- Dissociation
- 12.0
- Ab initio calculations
- 11.0
- Combustion
- 11.0

## Figures

Important stationary point geometries in vinyl radical. Upper left: global minimum; upper right: the lowest energy transition state saddle point (Y-planar C_{2v}); lower left: the transition state for 1,2, hydride shift in plane migration; lower right: the second lowest energy transition state for 1,2 hydride shift, this time corresponding to a non-planar migration pathway.

Important stationary point geometries in vinyl radical. Upper left: global minimum; upper right: the lowest energy transition state saddle point (Y-planar C_{2v}); lower left: the transition state for 1,2, hydride shift in plane migration; lower right: the second lowest energy transition state for 1,2 hydride shift, this time corresponding to a non-planar migration pathway.

Two slices through the multidimensional potential energy surface for 1,2, H shift migration, corresponding to the (i) planar (blue) and (ii) non-planar (red) transition state minimum energy pathways. The non-planar barrier is close in energy but slightly higher than the planar barrier pathway, with both conformations found near ≈17 800 cm^{−1}. Note that the non-planar barrier is significantly broader.

Two slices through the multidimensional potential energy surface for 1,2, H shift migration, corresponding to the (i) planar (blue) and (ii) non-planar (red) transition state minimum energy pathways. The non-planar barrier is close in energy but slightly higher than the planar barrier pathway, with both conformations found near ≈17 800 cm^{−1}. Note that the non-planar barrier is significantly broader.

The vinyl 1,2 H shift tunneling splittings predicted from the Bowman PES, both for the in-plane (blue) and out-of-plane (green) tunneling coordinates. The nearly two orders of magnitude smaller tunneling splittings observed for the out-of-plane vs in-plane coordinate are predominated by a much longer double minimum potential path for H atom out-of-plane tunneling.

The vinyl 1,2 H shift tunneling splittings predicted from the Bowman PES, both for the in-plane (blue) and out-of-plane (green) tunneling coordinates. The nearly two orders of magnitude smaller tunneling splittings observed for the out-of-plane vs in-plane coordinate are predominated by a much longer double minimum potential path for H atom out-of-plane tunneling.

Two *ab initio* minimum energy path potential surfaces for 1,2 H shift dynamics in vinyl: (i) CCSD(T)-aug-cc-pVnZ-CBS (blue x's), and (ii) full-dimensional fitted surface (solid red line). Both 1D potentials are plotted as a function of the coordinate *q*, defined by the in-plane angle with respect to the CC bisector axis (see inset) and are in excellent agreement.

Two *ab initio* minimum energy path potential surfaces for 1,2 H shift dynamics in vinyl: (i) CCSD(T)-aug-cc-pVnZ-CBS (blue x's), and (ii) full-dimensional fitted surface (solid red line). Both 1D potentials are plotted as a function of the coordinate *q*, defined by the in-plane angle with respect to the CC bisector axis (see inset) and are in excellent agreement.

The effective moment of inertia, *μ*(*q*), for planar 1,2 hydride shift tunneling dynamics in vinyl vs. *q*. Note the dramatic changes in *μ*(*q*) as a function of the tunneling coordinate, with a nearly 2.5 fold difference between the transition state (TS) and global minimum behavior. This reflects that the local tunneling coordinate involves motion of *multiple* atoms and not simply the tunneling H species, but also that the radius of H atom motion increases substantially between the TS and global minimum. Taking such changes in effective moment of inertia into account is critical in calculating high accuracy energy levels and tunneling splittings in the HBJ and Rush-Wiberg formulation of large amplitude dynamics.^{12,26}

The effective moment of inertia, *μ*(*q*), for planar 1,2 hydride shift tunneling dynamics in vinyl vs. *q*. Note the dramatic changes in *μ*(*q*) as a function of the tunneling coordinate, with a nearly 2.5 fold difference between the transition state (TS) and global minimum behavior. This reflects that the local tunneling coordinate involves motion of *multiple* atoms and not simply the tunneling H species, but also that the radius of H atom motion increases substantially between the TS and global minimum. Taking such changes in effective moment of inertia into account is critical in calculating high accuracy energy levels and tunneling splittings in the HBJ and Rush-Wiberg formulation of large amplitude dynamics.^{12,26}

The lowest 18 even symmetry vinyl 1,2 hydride shift tunneling wave functions, plotted along with the CCSD(T)-aug-cc-pVnZ-CBS minimum energy potential as a function of the tunneling coordinate *q*. In the interest of clarity, the accompanying odd symmetry wavefunctions are shown only starting from the 13th even symmetry state upward. The tunneling splittings are calculated between the adjacent even and odd symmetry eigenenergies, which are only visible in the two highest energy pairs of even/odd eigenfunctions.

The lowest 18 even symmetry vinyl 1,2 hydride shift tunneling wave functions, plotted along with the CCSD(T)-aug-cc-pVnZ-CBS minimum energy potential as a function of the tunneling coordinate *q*. In the interest of clarity, the accompanying odd symmetry wavefunctions are shown only starting from the 13th even symmetry state upward. The tunneling splittings are calculated between the adjacent even and odd symmetry eigenenergies, which are only visible in the two highest energy pairs of even/odd eigenfunctions.

Logarithmic plot (base 10) of the predicted tunneling splitting (cm^{−1}) from both the CCSD(T)-aug-cc-pVnZ-CBS potential (Nesbitt, red circles) and the *V*(*Q* _{im}) minimum energy path potential (Bowman, green triangles), vs the energy below the transition state barrier. Note the outstanding agreement obtained over 11 orders of magnitude dynamic range.

Logarithmic plot (base 10) of the predicted tunneling splitting (cm^{−1}) from both the CCSD(T)-aug-cc-pVnZ-CBS potential (Nesbitt, red circles) and the *V*(*Q* _{im}) minimum energy path potential (Bowman, green triangles), vs the energy below the transition state barrier. Note the outstanding agreement obtained over 11 orders of magnitude dynamic range.

Logarithmic plot (base 10) of the 1,2, H atom shift tunneling rate constant (*k* _{tun}(*E*)) from the CCSD(T)-aug-cc-pVnZ-CBS potential versus energy above the global minimum, with (red circles) and without (green diamonds) inclusion of zero point energy corrections in the perpendicular vibrational coordinates (see text for details). For *E* ≈ 7500 cm^{−1}, the tunneling time is on the *μ*s time scale, which begins to be relevant in modeling low-pressure flame conditions, which rapidly decreases down to ps time scales at the H + HCCH dissociation limit near *E* _{diss} = 14 265 cm^{−1}. The energy window (>20 kcal/mol) between 7500 cm^{−1} and 14 265 cm^{−1}, therefore, represents a “quantum isomerization regime” in which quantum effects can make appreciable contributions to the overall 1,2, H atom shift dynamics under typical flame conditions. Note the >20-fold smaller window of energies (<1 kcal/mol) in the immediate vicinity of the dissociation limit in which classical roaming is able to contribute significantly.

Logarithmic plot (base 10) of the 1,2, H atom shift tunneling rate constant (*k* _{tun}(*E*)) from the CCSD(T)-aug-cc-pVnZ-CBS potential versus energy above the global minimum, with (red circles) and without (green diamonds) inclusion of zero point energy corrections in the perpendicular vibrational coordinates (see text for details). For *E* ≈ 7500 cm^{−1}, the tunneling time is on the *μ*s time scale, which begins to be relevant in modeling low-pressure flame conditions, which rapidly decreases down to ps time scales at the H + HCCH dissociation limit near *E* _{diss} = 14 265 cm^{−1}. The energy window (>20 kcal/mol) between 7500 cm^{−1} and 14 265 cm^{−1}, therefore, represents a “quantum isomerization regime” in which quantum effects can make appreciable contributions to the overall 1,2, H atom shift dynamics under typical flame conditions. Note the >20-fold smaller window of energies (<1 kcal/mol) in the immediate vicinity of the dissociation limit in which classical roaming is able to contribute significantly.

Corresponding logarithmic plot (base 10) of the 1,2, H atom shift tunneling rate constant, *k* _{tun}(*T*), from the CCSD(T)-aug-cc-pVnZ-CBS potential as a function of temperature (see text for details). At temperatures >1300 K, the tunneling rate *k* _{tun}(*T*) is already > 10^{6} (s^{−1}), i.e., sufficiently large to make such “quantum isomerization” dynamics potentially relevant under typical flame conditions.

Corresponding logarithmic plot (base 10) of the 1,2, H atom shift tunneling rate constant, *k* _{tun}(*T*), from the CCSD(T)-aug-cc-pVnZ-CBS potential as a function of temperature (see text for details). At temperatures >1300 K, the tunneling rate *k* _{tun}(*T*) is already > 10^{6} (s^{−1}), i.e., sufficiently large to make such “quantum isomerization” dynamics potentially relevant under typical flame conditions.

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