Approaches to Proton Tunneling Dynamics

In general, proton transfer occurs via a combination of over-barrier and through-barrier pathways. The rate constant of over-barrier transfer is usually calculated by standard transition state theory (TST) [22] by separating the reaction coordinate from the remaining degrees of freedom. If tunneling effects and the curvature of the reaction path are neglected, this leads to the expression 

The instanton method defines the thermal rate constant for tunneling transfer in terms of the action S (T) (expressed hereafter in units h) along this extremal path  

The one-dimensional potential along the tunneling coordinate, represented by t/c(x) in Eq. (29.20), is a crude-adiabatic potential evaluated with the heavy atoms fixed in the equilibrium configuration, i.e. with y = Ay ,y5 = Ay it is equivalent to the potential along the linear reaction path. This symmetric doubleminimum potential has a maximum U (0) = Ug at x = 0, minima U( ( Ax) = 0 at X = +Ax, and a curvature in the minima given by the effective frequency Qq which accounts for the contribution of the normal modes of the minima to the reaction coordinate [27]. Eor the shape of the potential in the intermediate points we use an interpolation formula based on the calculated energies and curvatures near the stationary points. We have found that in many cases the simple quartic potential of the form 

To calculate the parameters governing the Hamiltonian, we use an approximation that amounts to separating the transverse modes into high-frequency (HF) modes, treated adiabatically, and low-frequency (LF) modes, treated in the sudden approximation. This separation is based on the value of the zeta factor [27] 

Coupling to HF modes leads to an effective one-dimensional motion with renormalized potential U (x) and coordinate-dependent mass m =ff(x). Since each HF mode y is assumed to follow the reaction coordinate x adiabatically, we have dUldyi = 0, so that y = C x jail and y F = C xlto. Substitution in the Hamiltonian (29.20) provides a correction to the mass of the tunneling particle and thus modifies the kinetic energy operator  

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