Tunneling Wigner approximation

In the discussion of TST in Chapter 4 tunneling was introduced as a multiplicative correction factor to the TST rate constant. TST tunneling is usually discussed in one of three approximations. In first order (for small tunneling), Wigner has shown the correction is given by the u2/24 law... 

In Equation 21, T is the absolute temperature, h is Planck s constant, is Boltzmann constant, and AG is the free energy barrier height relative to infinitely-separated reactants. The temperature-dependent factor r(7) represents quantum mechanical tunneling and the Wigner approximation to tunneling through an inverted parabolic barrier ... 

The Breit-Wigner approximation allows one to represent the resonance tunneling spectrum as superposition of Lorentzian lines. In the limit of narrow electronic levels I o 0- the resonance transparency obeys the equation... 

The need to include quantum mechanical effects in reaction rate constants was realized early in the development of rate theories. Wigner [8] considered the lowest order terms in an -expansion of the phase-space probability distribution function around the saddle point, resulting in a separable approximation, in which bound modes are quantized and a correction is included for quantum motion along the reaction coordinate - the so-called Wigner tunneling correction. This separable approximation was adopted in the standard ad hoc procedure for quan-... 

The traditional treatment of KIEs is based upon conventional TST [29, 30], in which the dividing surface is placed at the saddle point of the reaction, with tunneling effects generally included by a separable approximation such as the Wigner... 

At somewhat lower temperatures, when quantum effects start to play a role, the basic TST expression (3.29) can be improved in several ways First, classical partition functions 2r and can be replaced by their exact quantum analogues for a harmonic potential. (In fact, this is usually done automatically in TST.) Second, quantum tunneling can be included approximately via the Wigner tunneling correction (3.7) ... 

Methods used Arrhenius law, transition state theory, Wigner tunneling correction, quanrnm instanton approximation, and exact analytical solution... 

If information on the reaction path is available, as, for instance, in variational transition state theory, this can be used to calculate k [69,70]. In transition state theory, only the knowledge of the energy and its first and second derivatives at the reactant and transition state locations is needed and the barrier is typically approximated by a simple functional form. One possibility is to describe the reaction barrier by an Eckart potential [75] (also called a sech potential, depending on the literature source), k in Eq. (7.19) is defined as the ratio of transmitted quantum particles to classical particles and the resulting integral for the Eckart potential can be solved numerically. An approximate solution is the Wigner tunneling correction ... 

Rate expressions are recommended to be used as relative quantities only. PPL, phenethyl phenyl ether a-PPE", a-radical of PPL )8-PPE, )8-radical of PPL TST, transition state theory including anharmonic corrections TST, transition state theory including anharmonic corrections and Wigner tunneling correction SS, steady-state approximation for intermediate. 

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