Semi-classical Marcus theory

Thus, the semi-classical Marcus theory of non-adiabatic ET expresses the ET rate constant in terms of three important quantities, namely Vel, A, and AG°. It therefore follows that an understanding of ET reactions entails an understanding of how these three variables are dependent on factors such as the electronic properties of the donor and acceptor chromophores, the nature of the intervening medium and the inter-chromophore separation and orientation. 

To conclude this section, it is important to note that, in its classical form, Marcus s theory only implies two parameters the activation energy to the free energy of the reaction and the reorganization energy. It does not explicitly depend on the importance of the energetic coupling between the initial (reactant) and final (product) state. This effect is explored in the semi-classical Marcus theory [88] and will not be detailed here. 

The semi-classical extension of electron transfer theory evolved from models developed by Landau, Zener, Marcus, and Hush [3,6]. The semi-classical Marcus-Hush... 

The semi-classical Marcus equation derives from quantum-mechanical treatments of the Marcus model, which consider in wave-mechanical terms the overlap of electronic wave-functions in the donor-acceptor system, and the effects of this overlap on electronic and nuclear motions (see Section 9.1.2.8 above). Such treatments are essential for a satisfactory theory of D-A systems in which the interaction between the reactant and product free-energy profiles is relatively weak, such as non-adiabatic reactions. A full quantum-mechanical treatment, unfortunately, is cumbrous and (since the wave-functions are not accurately known) difficult to relate to experimental measurements but one can usefully test equations based on simplified versions. In a well-known treatment of this type, leading to the semi-classical Marcus equation introduced in Section 9.1.2.8, the vibrational motions of the atomic nuclei in the reactant molecule (as well as the motions of the transferring electron) are treated wave-mechanically, while the solvent vibrations (usually of low frequency) are treated classically. The resulting equation, already quoted (Equation (9.25)), is identical in form with the classical equation (9.16) (Section 9.1.2.5), except that the factor... 

Since the Marcus model was initially a classical or later a semi-classical theory, the introduction of quantum effects was considered to account for these observations. In particular, tunnelling pathways of e.t. would increase the rates of some reactions. A closer analysis has, however, led to the conclusion that this could not be a general explanation [80]. 

Fig. 1. The classical (a) and the semi-classical (b-d) representations of the Marcus theory for X — 1.0 eV at T = 300 K. In the classical expression (Eq. 1), X determines both the position of the maximum and the breadth of the parabola. The maximum keI is determined by the frequency factor (Z, here taken as 6 x 10 1 s ) in the Eyring expression (ket = KZexp( — AGlJkbT) where k is the transmission coefficient, usually taken to be unity). In the semi-classical approach the reorganization energy is explicitly divided into Xh (here equal 0.2 eV) and 2a (0.8 eV). The value of V is chosen to...

In this section, we summarize the Marcus theory in its classical formulation. We would like to draw the reader s attention to aspects that are not generally presented in the literature and that, in our opinion, are valuable to note. We show by a few examples how Marcus ideas can be successfully applied to the elucidation of practical problems in conducting, semi-conducting, photo- and electro-luminescent systems. 

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