Transmission coefficient electronic

Relatively little attention has been given in the literature to the electronic transmission coefficient for electrochemical reactions. On the basis of the conventional collisional treatment of the pre-exponential factor for outer-sphere reactions, Kel has commonly been assumed to equal unity, i.e. adiabatic reaction pathways are followed. Nevertheless, as noted above, the dependence of xei upon the spatial position of the transition state is of key significance in the encounter pre-equilibrium treatment embodied in eqns. (13) and (14). Thus, the manner in which Kel varies with the reactant-electrode separation for outer-sphere reactions will influence the integral of reaction sites that effectively contribute to the overall measured rate constant and hence the effective electron-tunneling distance, Srx, in eqn. (14). 

Other theoretical activity has centered on the dependence of reaction non-adiabaticity upon the structure of the intervening medium as well as the donor-acceptor separation for intramolecular electron transfer [50], i.e. between donor and acceptor sites contained within a single species such as a binuclear complex. The electron-tunneling probability is predicted to be enhanced substantially by the presence of delocalized electron groups, such as aromatic ligands, between the reacting centers [50]. This is consistent with experimental studies of thermal and optically induced electron transfer within binuclear complexes [51]. 

It is important to point out that for the reactions in which the species are chemiadsorbed on the electrode surface (e.g., H2 and O2 evolutim reactions), the interaction V12 is very strong, the value of the transmission coefficient is expected to be close to unity, and the reaction will be mostly adiabatic. 

Newton made a quantum chemical computation of transmission coefficient and reported a value of 1.1 x 10 for an ET reaction between Fe(H20)f redox couple in a homogeneous solution using the 

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The smallness of the electron transmission coefficient for the transition from individual energy levels does not mean that aU electrochemical electron transfer reactions should be nonadiabatic. If the inequality opposite to Eq. (34.33) is fulfilled. 

Approximate calculation of the integral over 8 in Eq. (34.27) shows that the ejfective electron transmission coefficient for nonadiabatic reactions is equal to... 

FIGURE 34.7 Dependence of the effective electron transmission coefficient on the electrochemical Landau-Zener parameter. 

Sautet P, Joachim C (1988) Electronic transmission coefficient for the single-impurity problem in the scattering-matrix approach. Phys Rev B 38 12238... 

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... 

R to P is slow, even when the isoenergetic conditions in the solvent allow the ET via the Franck-Condon principle. The TST rate for this case contains in its prefactor an electronic transmission coefficient Kd, which is proportional to the square of the small electronic coupling [28], But as first described by Zusman [32], if the solvation dynamics are sufficiently slow, the passage up to (and down from [33]) the nonadiabatic curve intersection can influence the rate. This has to do with solvent dynamics in the solvent wells (this is opposed to the barrier top description given above). We say no more about this here [8,11,32-36]. 

The preexponential factors k t and k may therefore be factorized by introduction of an electronic transmission coefficient, Ke ... 

A recently proposed semiclassical model, in which an electronic transmission coefficient and a nuclear tunneling factor are introduced as corrections to the classical activated-complex expression, is described. The nuclear tunneling corrections are shown to be important only at low temperatures or when the electron transfer is very exothermic. By contrast, corrections for nonadiabaticity may be significant for most outer-sphere reactions of metal complexes. The rate constants for the Fe(H20)6 +-Fe(H20)6 +> Ru(NH3)62+-Ru(NH3)63+ and Ru(bpy)32+-Ru(bpy)33+ electron exchange reactions predicted by the semiclassical model are in very good agreement with the observed values. The implications of the model for optically-induced electron transfer in mixed-valence systems are noted. 

Electronic Transmission Coefficient. The probability that the electron transfer will occur in the intersection region (in other words, the probability that the system will remain on the lower adiabatic surface on passing through the intersection region) is given by (36)... 

To summarize, in this article we have discussed some aspects of a semiclassical electron-transfer model (13) in which quantum-mechanical effects associated with the inner-sphere are allowed for through a nuclear tunneling factor, and electronic factors are incorporated through an electronic transmission coefficient or adiabaticity factor. We focussed on the various time scales that characterize the electron transfer process and we presented one example to indicate how considerations of the time scales can be used in understanding nonequilibrium phenomena. 

The magnitude of the electronic interaction term, J, is also critical in determining the degree of electronic non-adia-baticity [such as Newton has been discussing] that may be present. The electronic transmission coefficient, K, can be expressed as (21) ... 

Finally, xel is the electronic transmission coefficient that accounts for the probability of electron transfer upon reaching the configuration of the transition state. Note that it has been assumed that the electronic interaction between the redox species and the energy levels in the electrode is independent of the energy of such level x, so the electronic and nuclear factors can be treated separately. 

The effect of the reaction adiabaticity on the electronic transmission coefficient can be estimated by making use of an extension of the Landau-Zener formalism [43—47] ... 

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. 

Fig. 5 Adiabatic and non-adiabatic ET processes. In the adiabatic process (Fig. 5a), Vel > 200 cm and the large majority of reaction trajectories (depicted as solid arrows) which reach the avoided crossing region remain on the lower energy surface and lead to ET and to the formation of product (i.e., the electronic transmission coefficient is unity). In contrast, non-adiabatic ET is associated with Vel values <200 cm-1, in which case the majority of reaction trajectories which reach the avoided crossing region undergo non-adiabatic transitions (surface hops) to the upper surface. These trajectories rebound off the right-hand wall of the upper surface, enter the avoided crossing region where they are likely to undergo a non-adiabatic quantum transition to the lower surface. However, the conservation of momentum dictates that these trajectories will re-enter the reactant well, rather than the product well. Non-adiabatic ET is therefore associated with an electronic transmission coefficient which is less than unity.

Another evident mechanism for energy transfer to activated ions may be by bimolecular collisions between water molecules and solvated ion reactants, for which the collision number is n(ri+ r2)2(87tkT/p )l/2> where n is the water molecule concentration, ri and r2 are the radii of the solvated ion and water molecule of reduced mass p. With ri, r2 = 3.4 and 1.4 A, this is 1.5 x 1013 s"1. The Soviet theoreticians believed that the appropriate frequency should be for water dipole librations, which they took to be equal 10n s 1. This in fact corresponds to a frequency much lower than that of the classical continuum in water.78 Under FC conditions, the net rate of formation of activated molecules (the rate of formation minus rate of deactivation) multiplied by the electron transmission coefficient under nonadiabatic transfer conditions, will determine the preexponential factor. If a one-electron redox reaction has an exchange current of 10 3 A/cm2 at 1.0 M concentration, the extreme values of the frequency factors (106 and 4.9 x 103 cm 2 s 1) correspond to activation energies of 62.6 and 49.4 kJ/mole respectively under equilibrium conditions for adiabatic FC electron transfer. 

In equation (5), is the equilibrium constant for the outer-sphere association of the donor and acceptor, is the electronic transmission coefficient (the probability that products form once the nuclear configuration of the transition state is achieved), Vnu is the effective frequency for nuclear motion along the reaction coordinate in the neighborhood of the transition state, and the nuclear transmission coefficient nu is the classical exponential function of the activation energy. The weak-coupling limit corresponds to the limit in which Kei < 1, and for the strong-coupling limit /Cei = 1. 

The K parameter, the electronic transmission coefficient, is related to the extent of overlap between the donor and acceptor orbitals. When this overlap is very small, electron tunneling frequency determines the pre-exponential factor, the reaction is nonadiabatic and k <1. Such overlap may be diminished if the electrode-reactant distance, in the course of the charge transfer, is increased due, for instance, to the presence of a blocking film on the electrode. On the other hand, when the overlap is relatively large, k is close to 1. Only when the reactant is near the electrode surface does significant overlap of donor and acceptor orbitals occur. 

This close distance determines also the dr value. Therefore, the product K°dr may be considered as an effective reaction zone thickness [139], where k° is the electronic transmission coefficient at a distance of closest approach of the reactant to the electrode surface. For adiabatic reactions the value of k° should approach 1. 

In contrast to the experimentally based work discussed above, in the most recent comprehensive theoretical discussion [21d], Bixon and Jortner state that the question of whether non-adiabatic or adiabatic algorithms describe electron-transfer reactions was settled in the 1960s, and that the majority of outer-sphere electron-transfer reactions are non-adiabatic. This is certainly true for the reactions that occur in the Marcus inverted region in which these authors are interested, but we think the question of whether reactions in the normal region are best treated by adiabatic theory that includes an electronic transmission coefficient or by non-adiabatic equations remains to be established. 

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