Important stationary point geometries in vinyl radical. Upper left: global minimum; upper right: the lowest energy transition state saddle point (Y-planar C2v); 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.
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.
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.
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 μs 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 > 106 (s−1), i.e., sufficiently large to make such “quantum isomerization” dynamics potentially relevant under typical flame conditions.
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