Nuclear frequency factor

For adiabatic reaction pathways (i.e. Kel = 1) the nuclear frequency factor, vn (s 1), represents the rate at which reacting species in the vicinity of the transition state is transformed into products. This frequency will be influenced by a combination of the various motions associated with the passage of the system over the barrier, approximately weighted according to their relative contributions to the activation energy. These motions usually involve bond vibrations and solvent motion, associated with the characteristic inner- and outer-shell frequencies, vis and vos, respectively. A simple formula for vn which has been employed recently is [la, 7] 

This formula follows from the transition-state theory (TST) of unimole-cular reactions [42], Since it is commonly anticipated that vIS P vos, eqn. (22) predicts that the effective frequency is dominated by the inner-shell frequency even when the outer-shell barrier provides a substantial contribution to AG t. For metal-ligand, and other typical inner-shell vibrations, vis ss 1013s-1. Indeed, for the common circumstance where AG [ AG S, we expect that vn vis. This is intuitively reasonable since, on the basis of TST, we generally expect that the fastest motion along the reaction coordinate will control the frequency factor [28], 

Recent theoretical treatments, however, suggest instead that the dynamics of solvent reorganization can play an important and even dominant role in determining vn, at least when the inner-shell barrier is relatively small [43-45]. The effective value of vos can often be determined by the so-called longitudinal (or constant charge ) solvent relaxation time, rL [43, 44]. This quantity is related to the experimental Debye relaxation time, rD, obtained from dielectric loss measurements using [43] 

Although viB will still dominate vn if AG is sufficiently large, it is anticipated that vn vP8 for exchange reactions when the inequality 

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Nuclear frequency factors are calculated directly from the calculated molecular vibrational frequencies and the reorganizational energies, and these, in conjunction with the calculated Hab values lead to values for the electronic transmission coefficient, Kep... 

Where AG is the activation energy of the process, and T are the Boltzmann constant and the absolute temperature, respectively, v is the nuclear frequency factor, and is the transmission coefficient, a parameter that expresses the probability of the system to evolve from the reactant to the product configuration once the crossing of the potential energy curves along the reaction coordinate has been reached (Fig. 17.5). 

The remaining two terms in equation (1), vn and rel are, respectively, the nuclear frequency factor and the electronic transmission coefficient. The frequency factor gives the frequency with which reaction trajectories reach the avoided crossing region, and rel gives the probability that, once a trajectory has reached the avoided crossing region, it will pass into the product well, rather than be deflected back into the reactant well. 

Here, k is the electronic transmission coefficient (k = 1 for adiabatic electron transfer) and v x the nuclear frequency factor, whereas is the equilibrium constant for assembly of a precursor state and effectively includes any coulombic work and medium (Debye-Hiickel) terms [4, 5]. Following the approach taken by Stranks [7], the observed volume of activation AV for a simple, adiabatic, outer-sphere, bimolecular electron transfer reaction can be represented as... 

It is common to write an activationless ET rate as a product of a nuclear frequency factor, v ( 10 s ), and an electronic factor Ke ... 

Solvent reorganization energy Internal reorganization energy Photoexcited triplet state of metalloporphyrin Nuclear frequency factor... 

The clear dependence of spectral coalescence on solvent dynamics implies that the origin of the coalescence is itself dynamic. In a semiclassical electron transfer theoretical framework (Equation (2)), this could be explained in terms of electron transfers with essentially zero activation energy. The rates would then converge to the nuclear frequency factor, Vn, which in this case would appear to be dominated by solvent dipolar relaxation frequencies. In more recent solvent dynamical models, the observed effects could be considered to result from solvent friction limiting the rate of exploration of the electron transfer reaction coordinate. 

In equations 1-4, Aq is the pre-exponential factor, and AG is and AG os are the inner-and outer-shell free energies of activation. Ae is defined as the product of Kel, the electron tunneling probability, F, the nuclear tunneling factOT, Vn, the nuclear frequency factor and Kwhich efficient tunneling between the electrode and reactant occurs. Nominally, Ae = 10 cm s l at room temperature. AG is is defined in... 

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