Electron tunnelling

We saw in Section 14.2 that, according to the Franck–Condon principle, electronic transitions are so fast that they can be regarded as taking place in a stationary nuclear framework. This principle also applies to an electron transfer process in which an electron migrates from one energy surface, representing the dependence of the energy of DA on its geometry, to another representing the energy of D+A−. We can represent the potential energy (and the Gibbs energy) surfaces of the two complexes (the re actant complex, DA, and the product complex, D+A−) by the parabolas characteristic of harmonic oscillators, with the displacement coordinate corresponding to the changing geometries (Fig. 24.27). This coordinate represents a collective mode of the donor, acceptor, and solvent. According to the Franck–Condon principle, the nuclei do not have time to move when the system passes from the reactant to the product surface as a result of the transfer of an electron. Therefore, electron transfer can occur only after thermal fluctuations bring the geometry of DA to q* in Fig. 24.27, the value of the nuclear coordinate at which the two parabolas intersect. The factor κν is a measure of the probability that the system will convert from reactants (DA) to products (D⁺A⁻) at q* by electron transfer within the thermally excited DA complex. To understand the process, we must turn our attention to the effect that the rearrangement of nuclear coordinates has on electronic energy levels of DA and D+A− for a given distance r between D and A (Fig. 24.28). Initially, the elec tron to be transferred occupies the HOMO of D, and the overall energy of DA is lower than that of D+A− (Fig. 24.28a). As the nuclei rearrange to a configuration represented by q* in Fig. 24.28b, the highest occupied electronic level of DA and the lowest un occupied electronic level of D⁺A⁻ become degenerate and electron transfer becomes energetically feasible. Over reasonably short distances r, the main mechanism of electron transfer is tunnelling through the potential energy barrier depicted in Fig. 24.28b. The height of the barrier increases with the ionization energies of the DA and D+A− complexes. After an electron moves from the HOMO of D to the LUMO of A, the system relaxes to the configuration represented by q0 P in Fig. 24.28c. As shown in the illustration, now the energy of D⁺A⁻ is lower than that of DA, reflecting the thermo dynamic tendency for A to remain reduced and for D to remain oxidized. The tunnelling event responsible for electron transfer is similar to that described in Section 9.3, except that in this case the electron tunnels from an electronic level of D, with wavefunction ψD, to an electronic level of A, with wavefunction ψA. We saw in Section 9.3 that the rate of an electronic transition from a level described by the wave function ψD to a level described by ψA is proportional to the square of the integral

where HDA is a hamiltonian that describes the coupling of the electronic wavefunctions. It turns out that in cases where the coupling is relatively weak we may write:

(HAD)²=(H°DA)²e^−βr

where r is the edge-to-edge distance between D and A, β is a parameter that measures the sensitivity of the electronic coupling matrix element to distance, and (H°DA)² is the value of the electronic coupling matrix element when D and A are in contact (r = 0). The exponential dependence on distance in eqn 24.80 is essentially the same as the exponential decrease in transmission probability through a potential energy barrier described in Section 9.3.

Fig. 24.27 The Gibbs energy surfaces of the complexes DA and D+A− involved in an electron transfer process are represented by parabolas characteristic of harmonic oscillators, with the displacement coordinate q corresponding to the changing geometries of the system. In the plot, q0^R and q0^P are the values of q at which the minima of the reactant and product parabolas occur, respectively. The parabolas intersect at q = q*. The plots also portray the Gibbs energy of activation, ∆‡G, the standard reaction Gibbs energy, ∆_rG⁰, and the reorganization energy, λ (discussed in Section 24.11b).

Fig. 24.28 Correspondence between the electronic energy levels (shown on the left) and the nuclear energy levels (shown on the right) for the DA and D+A− complexes involved in an electron transfer process. (a) At the nuclear configuration denoted by q0 R, the electron to be transferred in DA is in an occupied electronic energy level (denoted by a blue circle) and the lowest unoccupied energy level of D⁺A⁻ (denoted by an unfilled circle) is of too high an energy to be a good electron acceptor. (b) As the nuclei rearrange to a configuration represented by q*, DA and D+A−become degenerate and electron transfer occurs by tunnelling through the barrier of height V and width r, the edge-to-edge distance between donor and acceptor. (c) The system relaxes to the equilibrium nuclear configuration of D⁺A⁻ denoted by q0 P, in which the lowest unoccupied electronic level of DA is higher in energy than the highest occupied electronic level of D⁺A⁻. (Adapted from R.A. Marcus and N. Sutin, Biochim. Biophys. Acta 811, 265 (1985).)

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مواضيع ذات صلة

Surface composition

Surface growth

Experimental results

Controlling chemical reactions with lasers

Spectroscopic observation of the activated complex

Quantum mechanical scattering theory

Classical trajectories

Attractive and repulsive surfaces

The direction of attack and separation

Potential energy surfaces

State-to-state dynamics

Experimental probes of reactive collisions