Potential-energy surfaces electronic factor

Figure 2.1 Schematic representation of the ground and electronic excited potential energy surfaces (PESs) and the corresponding absorption spectra of the parent molecule, resulting from the reflection of different initial wavefunctions on a directly dissociative PES (a) absorption from a vibrationless ground state consists of a broad continuum and (b) absorption from a vibrationally excited state shows that extended regions are accessed, leading to a structured spectrum with intensities of the features being dependent on the Franck-Condon factors. Reproduced with permission from Ref. [34]. Reproduced by permission of lOP Publishing.

The rate of a given reaction depends on the thermal activation conditions of the particle in donor and acceptor, factors which are accounted for in the Marcus model [6,7] or models where the vibrational wave functions are included [8-10], The reaction rate is derived in rather much the same way as for ordinary chemical reactions, using the concept of potential energy surfaces (PES s) [6]. The electronic factor is introduced either as a matrix element H]2 or as an... 

Let us end this section by a very important observation about the expression for the rate constant in Eq. (6.8) the exponential dependence on Eo in Eq. (6.8) implies that the rate constant is very sensitive to small changes in Eo. For example, assume that an energy 5E is added to Eo (say, corresponding to a numerical error in the determination of the electronic energy). The rate constant will then be multiplied by the factor exp(—5E/(kBT)). If 5E = ksT, an error corresponding to the factor 2.7 will be introduced. At T = 298 K, = 4.1 x 10-21 J = 0.026 eV 2.5 kJ/mol. Thus, the potential energy surface must be highly accurate in order to provide the basis for an accurate determination of a macroscopic rate constant k(T). 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

Figure 2.1(a) above illustrates the potential energy surface for a diabatic electron transfer process. In a diabatic (or non-adiabatic) reaction, the electronic coupling between donor and acceptor is weak and, consequently, the probability of crossover between the product and reactant surfaces will be small, i.e. for diabatic electron transfer /cei, the electronic transmission factor, is transition state appears as a sharp cusp and the system must cross over the transition state onto a new potential energy surface in order for electron transfer to occur. Longdistance electron transfers tend to be diabatic because of the reduced coupling between donor and acceptor components this is discussed in more detail below in Section 2.2.2. 

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