Marcus theory, electron transfer, nonadiabatic

The ZN formulas can also be utihzed to formulate a theory for the direct evaluation of thermal rate constant of electronically nonadiabatic chemical reactions based on the idea of transition state theory [27]. This formulation can be further utilized to formulate a theory of electron transfer and an improvement of the celebrated Marcus formula can be done [28]. 

In general, for the triads and other types of linked donor-acceptor systems to be discussed, electron transfer is assumed to occur in the nonadiabatic regime. That is, the mixing between the electronic state of the donor and acceptor before electron transfer occurs and the corresponding state after electron transfer is weak ( IcaT) [33]. The electron transfer event is assumed to be fast compared to the time scale of nuclear motions. The electron transfer theory proposed by Marcus [35,38] states that the electron transfer rate constant is given by Eq. (2). 

The nontraditional example of applying the AMSA theory is connected with the treatment of electrolyte effects in intramolecular electron transfer (ET) reactions [21, 22], Usually the process of the transfer of the electron from donor (D) to acceptor (A) in solutions is strongly nonadiabatic. The standard description of this process in connected with semiclassical Marcus theory [35], which reduces a complex dynamical problem of ET to a simple expression of electron... 

We now turn to the hierarchy of electron-transfer rate theories that have developed since the 1950s, starting with classical Marcus theory of homogeneous reactions and the development of eq. 4.4. In later sections we shall consider theories of nonadiabatic ET, which allow the identification and evaluation of the prefactor A in eq. 4.4, and also electrochemical ET, which differs from homogeneous reactions in that an electronic conductor is one of the reactants . 

The Marcus theory, as described above, is a transition state theory (TST, see Section 14.3) by which the rate of an electron transfer process (in both the adiabatic and nonadiabatic limits) is assumed to be determined by the probability to reach a subset of solvent configurations defined by a certain value of the reaction coordinate. The rate expressions (16.50) for adiabatic, and (16.59) or (16.51) for nonadiabatic electron transfer were obtained by making the TST assumptions that (1) the probability to reach transition state configuration(s) is thermal, and (2) once the reaction coordinate reaches its transition state value, the electron transfer reaction proceeds to completion. Both assumptions rely on the supposition that the overall reaction is slow relative to the thermal relaxation of the nuclear environment. We have seen in Sections 14.4.2 and 14.4.4 that the breakdown of this picture leads to dynamic solvent effects, that in the Markovian limit can be characterized by a friction coefficient y The rate is proportional to y in the low friction, y 0, limit where assumption (1) breaks down, and varies like y when y oo and assumption (2) does. What stands in common to these situations is that in these opposing limits the solvent affects dynamically the reaction rate. Solvent effects in TST appear only through its effect on the free energy surface of the reactant subspace. 

In the case where the electron transfer is faster than molecular reorganization (nonadiabatic transfer), the path for electron transfer in the dimer coordinate representation illustrated in Figure 2.2.7 can be decomposed in a vertical activation from the minimum of Vp to the Vp curve, followed by a relaxation to the equilibrium configuration of the product. Accordingly, the Marcus theory of electron transfer [23] introduces the reorganization energy X... 

III. RELEVANT ELECTRON TRANSFER THEORY MARCUS S DESCRIPTION OF HETEROGENEOUS NONADIABATIC ELECTRON TRANSFER REACTIONS... 

The research groups of Lewis and Wasielewski estimated the rate constants of the charge separation (kcs) and recombination (kcR) between the nucleobase, which acts as the electron donor, and the electron acceptor at the loop position of the DNA hairpins, and also investigated the free energy dependence of the electron transfer rate. " It was found that the single-step electron transfer in DNA mediated by nucleobases can be described by the Marcus theory (4) developed for nonadiabatic electron transfer system. 

In summary, the hole transfer in DNA by tunneling and hopping mechanisms was summarized in this section. The hole transfer proceeding via the tunneling mechanism has been successfully described by the Marcus theory, the electron transfer theory for the nonadiabatic condition, in which the electronic interaction between the donor and acceptor is small or moderate. The electron transfer rate decreased with exponential of the distance as described by the f value around 0.7. This value lies between those of the con-... 

We have presented several approaches to calculate the rate constants of electron transfer occurring in solvent from the weak to strong electronic couplings. In the fast solvent relaxation limit, the approach based on the nonadiabatic transition state theory can be adopted. It is related to the Marcus formula by a prefactor and referred as a modified Marcus formula. When the solvent dynamics begin to play a role, the quantum Kramers-like theory is applicable. For the case where the intramoleeular vibrational motions are much faster than the solvent motion, the extended Sumi-Mareus theory is a better ehoice. As the coherent motion of eleetron is ineorporated, such as in the organic semiconductors, the time-dependent wavepaeket diffusion approach is proposed. Several applications show that the proposed approaches, together with electronic structure calculations for the faetors eontrolling eleetron transfer, can be used to theoretically predict electron transfer rates correctly. 

Figure 9.6 Pressure-effect on rates of some self-exchange electron-transfer reactions between metal ions comparison of observed volumes of activation with values calculated from classical Marcus theory for adiabatic reactions. The plot shows calculated and observed AP values (cm mol ) at mid-range of pressure (100 MPa, except 70 MPa for Fe(H20)g ) for adiabatic (filled symbols) and nonadiabatic (open circles) self-exchange in couples with rigid ligands. Solvents (o, ) water ( ) CD3CN (A) (CD3)2CO (V) CD3OD. Key (A,B) (C,D) Cu(dmp)2 (E-G) Ru(hfac)j (H) Fe(C5H5)2 (I-K) Mn(CN-t-Bu)g ...

The Marcus equation for nonadiabatic electron-transfer reactions (Eq. B5.3.4), and the Forster theory that we discussed in Chap. 7 apply only to systems with weak intermolecular interactions, which we now can define more precisely as meaning that H21 lh steady-state approximation to the stochastic Liouville equation for a two-state reaction in this limit From Eqs. (BIO.1.15), (10.29a), (10.29b), and (10.30), we have... 

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