Electron transfer semi-classical model

In the (semi-)classical models of ETR (Marcus the Russian school), redox orbitals of reactants overlap at a close separation, followed by swift electron transfer. The activated complex, considered in equilibrium with the reactants, consists of these overlapping orbitals. In the tunneling model, the electron penetrates... 

Most electron-transfer systems belong to classes I and IIA. In terms of the semi-classical model the rate constant for electron transfer in these systems is given by ... 

Classical and semi-classical theories of electron transfer provide quantitative models for determining the reaction pathway. Of particular importance is the theory of nonequilibrium solvent polarization based on the dielectric continuum model.5 From these theories Eqs. 

A semi-classical treatment171-175 of the model depicted in Fig. 15, based on the Morse curve theory of thermal dissociative electron transfer described earlier, allows the prediction of the quantum yield as a function of the electronic matrix coupling element, H.54 The various states to be considered in the region where the zero-order potential energy curves cross each other are shown in the insert of Fig. 15. The treatment of the whole kinetics leads to the expression of the complete quenching fragmentation quantum yield, oc, given in equation (61)... 

Some authors have described the time evolution of the system by more general methods than time-dependent perturbation theory. For example, War-shel and co-workers have attempted to calculate the evolution of the function /(r, Q, t) defined by Eq. (3) by a semi-classical method [44, 96] the probability for the system to occupy state v]/, is obtained by considering the fluctuations of the energy gap between and 11, which are induced by the trajectories of all the atoms of the system. These trajectories are generated through molecular dynamics models based on classical equations of motion. This method was in particular applied to simulate the kinetics of the primary electron transfer process in the bacterial reaction center [97]. Mikkelsen and Ratner have recently proposed a very different approach to the electron transfer problem, in which the time evolution of the system is described by a time-dependent statistical density operator [98, 99]. 

Enthalpies of activation, transition-state geometries, and primary semi-classical (without tunneling) kinetic isotope effects (KIEs) have been calculated for 11 bimolecu-lar identity proton-transfer reactions, four intramolecular proton transfers, four nonidentity proton-transfer reactions, 11 identity hydride transfers, and two 1,2-intramole-cular hydride shifts at the HF/6-311+G, MP2/6-311+G, and B3LYP/6-311+-1-G levels.134 It has been found that the KIEs are systematically smaller for hydride transfers than for proton transfers. The differences between proton and hydride transfers have been rationalized by modeling the central C H- C- unit of a proton-transfer transition state as a four-electron, three-centre (4-e 3-c) system and the same unit of a hydride-transfer transition state as a 2-e 3-c system. 

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 formaldehyde disproportionation has been examined by semi-empirical MO methods (Rzepa and Miller, 1985). With the MNDO procedure, transfer of hydride from hydrate mono-anion to formaldehyde is exothermic by 109 kJ mol-1, and the transition structure [29], corresponding to near symmetrical transfer of hydride, lies 72 kJ mol -1 above the separated reactants. Inclusion of two water molecules, to model solvation effects, stabilizes reactants and transition structures equally. Hydride transfer from the hydrate dianion was found to have a less symmetrical transition structure [30] not unexpected for a more exothermic reaction, but the calculated activation energy, 213 kJ mol-1, is unexpectedly high. Semi-classical primary kinetic isotope effects, kH/kD = 2.864 and 3.941 respectively, have been calculated. Pathways involving electron or atom transfers have also been examined, and these are predicted to be competitive with concerted hydride transfers in reactions of aromatic aldehydes. Experimental evidence for these alternatives is discussed later. 

Classical, semi-classical, and quantum mechanical procedures have been developed to rationalize and predict the rates of electron transfer. In summary, the observed rate of a self-exchange reaction can be calculated as a function of interatomic distances, force constants, electronic coupling matrix element, and solvent parameters. These model parameters are either calculated, estimated, or determined by experiment, in each case with a corresponding standard deviation. Error propagation immediately demonstrates that calculated rates have error ranges of roughly two orders of magnitude, independent of the level of sophistication in the numerical procedures. 

If we can successfully predict the influence of basic molecular properties such as size or mass on the rate constant k of electron-transfer reactions, we can in principle calculate the value of k in terms of those properties, using some model (classical, semi-classical or quantum-mechanical). The comparison of such theoretical values with experimental ones... 

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... 

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