Nuclear reorganization free energy

We address briefly the two central quantities, the nuclear reorganization free energy and the electronic turmeling factor. 

The quadratic form, however, rests on the very general assumption that whatever structural features characterize the environment, conformational, polarization, density, or other physical properties respond linearly to the field changes, that is, changes in electric field, pressure, or other forces. The quadratic form of the activation free energy is thus broadly vahd independently of the nature of the solvation and reorganization. 

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In addition to the driving force or overpotential, the activation free energy contains the nuclear reorganization free energy, r, addressed further in Section 2.2.2. 

All these areas are covered in a broad literature, overviewed, for example in refs. 24 and 25. We do not here address all these elements of molecular charge transfer theory. Instead we discuss the two central factors in the interfacial (bio)electrochemical electron transfer process, first the nuclear reorganization (free) energy and then the electronic tunneling factor. 

The reorganization free energy /.R represents the electronic-vibrational coupling, ( and y are fractions of the overpotential r] and of the bias voltage bias at the site of the redox center, e is the elementary charge, kB the Boltzmann constant, and coeff a characteristic nuclear vibration frequency, k and p represent, respectively, the microscopic transmission coefficient and the density of electronic levels in the metal leads, which are assumed to be identical for both the reduction and the oxidation of the intermediate redox group. Tmax and r max are the current and the overvoltage at the maximum. 

In the classical treatment electron transfer is assumed to be adiabatic and the rate depends exponentially on the reorganization free energy (Table 1). The latter varies as the square of the displacement along the reaction coordinate. This quadratic dependence is a consequence of the assumptions that the inner-sphere reorganization energy depends on the square of the nuclear displacement from equilibrium, and that the solvent-polarization energy depends on the square of the difference between the hypotheti-... 

So far, only the nuclear reorganization energy attending electron transfer has been discussed, yielding the expressions above of the free energy of activation in the framework of classical transition state theory. A second series of important factors are those that govern the preexponential factor, k, raising in particular the question of the adiabaticity or nonadiabaticity of electron transfer between a molecule and the electronic states in the electrode. 

AG is the free energy variation, and and the contributions to the reorganization energy arising from the dielectric medium and the oscillator, respectively. When the temperature is sufficiently high so that all the nuclear motions can be treated classically, the following expression appUes [4, 33] ... 

The Marcus classical free energy of activation is AG , the adiabatic preexponential factor A may be taken from Eyring s Transition State Theory as (kg T /h), and Kel is a dimensionless transmission coefficient (0 < k l < 1) which includes the entire efiFect of electronic interactions between the donor and acceptor, and which becomes crucial at long range. With Kel set to unity the rate expression has only nuclear factors and in particular the inner sphere and outer sphere reorganization energies mentioned in the introduction are dominant parameters controlling AG and hence the rate. It is assumed here that the rate constant may be taken as a unimolecular rate constant, and if needed the associated bimolecular rate constant may be constructed by incorporation of diffusional processes as ... 

In semiclassical ET theory, three parameters govern the reaction rates the electronic couphng between the donor and acceptor (%) the free-energy change for the reaction (AG°) and a parameter (X.) related to the extent of inner-shell and solvent nuclear reorganization accompanying the ET reaction [29]. Additionally, when intrinsic ET barriers are small, the dynamics of nuclear motion can limit ET rates through the frequency factor v. These parameters describe the rate of electron transfer between a donor and acceptor held at a fixed distance and orientation (Eq. 1),... 

The inner-sphere component of the reorganization energy represents the minimum energy required to change the internal structure of the redox center to its nuclear transition state configuration. Equation (2.3) is derived from the classical harmonic oscillator model and is an expression of the free energy associated with... 

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