Spin-boson model, electron-transfer

The semiclassical mapping approach outlined above, as well as the equivalent formulation that is obtained by requantizing the classical electron-analog model of Meyer and Miller [112], has been successfully applied to various examples of nonadiabatic dynamics including bound-state dynamics of several spin-boson-type electron-transfer models with up to three vibrational modes [99, 100], a series of scattering-type test problems [112, 118, 120], a model for laser-driven... 

Electron transfer processes, more generally transitions that involve charge reorganization in dielectric solvents, are thus shown to fall within the general category of shifted harmonic oscillator models for the thennal enviromnent that were discussed at length in Chapter 12. This is a result of linear dielectric response theory, which moreover implies that the dielectric response frequency a>s does not depend on the electronic charge distribution, namely on the electronic state. This rationalizes the result (16.59) of the dielectric theoiy of electron transfer, which is identical to the rate (12.69) obtained from what we now find to be an equivalent spin-boson model. 

A series of electron donors and acceptors together can form a charge transfer chain (CTC). Examples of these are found in some electron transfer proteins, Electron transport in ID CTCs has been studied using the spin-boson model with multiple tight-binding electronic states. The nature of the transport depends on the strength of the dissipation of the environment. 

Abstract Photoinduced processes in extended molecular systems are often ultrafast and involve strong electron-vibration (vibronic) coupling effects which necessitate a non-perturbative treatment. In the approach presented here, high-dimensional vibrational subspaces are expressed in terms of effective modes, and hierarchical chains of such modes which sequentially resolve the dynamics as a function of time. This permits introducing systematic reduction procedures, both for discretized vibrational distributions and for continuous distributions characterized by spectral densities. In the latter case, a sequence of spectral densities is obtained from a Mori/Rubin-type continued fraction representation. The approach is suitable to describe nonadiabatic processes at conical intersections, excitation energy transfer in molecular aggregates, and related transport phenomena that can be described by generalized spin-boson models. 

A detailed review of the spin-boson model can be found in [13]. In case of electron transfer in proteins, the spin-boson model can be related to a simple microscopic picture, namely, the well-known Marcus energy diagram[14, 15]. In this diagram, the free energy of both reactant and product states is described by a one-dimensional harmonic potential with identical force constants /. We assume the reactant and product free energy curves have the functional form. 

In the above equations, q represents schematically the nuclear configuration of the protein and qp o represent the shift of the equilibrium position after the electron transfer. As pointed out in [3] and [7], the potential functions originate from a dependence on thousands of nuclear coordinates, which define a many-dimensional potential-energy surface. The spin-boson model goes beyond the Marcus model in that it allows one to represent the multitude of degrees of freedom coupled to the electron transfer through an ensemble of harmonic oscillators of various frequencies. 

AE(t) represents the energy gap between reactant and product states which depends strongly on the thermal motion of the protein. In the spin-boson model, the two states of electron transfer are described as the spin operator, while thermal vibrations of the protein are accounted for by the boson operators. 

Figure 2. Comparison of electron transfer rates k e T) shown as a function of e evaluated in the framework of the spin-boson model (solid lines) and by Marcus theory (dashed lines) at temperatures 10 K and 300 K. The functions are centered approximately around m-...

The key new aspect of our investigation is two-fold first, we base all model parameters on molecular dynamics simulations second, the spin-boson model allows one to account for a very large number of vibrations quantum mechanically. We have demonstrated that the spin-boson model is well suited to describe the coupling between protein motion and electron transfer in biological redox systems. The model, through the spectral function can be matched to correlation functions of the redox energy... 

The main result regarding the electron transfer rates evaluated is that for a spectral function consistent with molecular dynamics simulations the spin-boson model at physiological temperatures predicts transfer rates in close agreement with those predicted by the Marcus theory. However, at low temperatures deviations from the Marcus theory arise. The resulting low temperature rates are in qualitative agreement with observations. The spin-boson model explains, in particular, in a very simple and natural way the slow rise of transfer rates with decreasing temperature, as well as the asymmetric dependence of the redox energy. 

Dong Xu and Klaus Schulten. Coupling of protein motion to electron transfer in a photosynthetic reaction center Investigating the low temperature behaviour in the framework of the spin-boson model. Chem, Phys., 1991. submitted. 

The empirical models are of two kinds. The course of organic reaction mechanisms is mapped out by curved arrows that represent the transfer of electron pairs. Electrochemical processes, on the other hand are always analyzed in terms of single electron transfers. There is a non-trivial difference involving electron spin, between the two models. An electron pair has no spin and behaves like a boson, for instance in the theory of superconductivity. An electron is a fermion. The theoretical mobilities of bosons and fermions are fundamentally different and so is their distribution in quantized potential fields. 

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