The Marcus parabolas

It is possible to define the reaction coordinate as an explicit function of the solvent structure in the following way. Let F0(R 7 ab) be the potential energy surface of the system, a function of the nuclear configuration = (Ri. Rv) (N is the number of all nuclear centers, including those associated with the donor and acceptor species), when d and B are held at a fixed distance 7 ab and when the electronic charge distribution is pe r), Eq. (16.21). define a new variable, the difference in potential energies. 

We now define A as the reaction coordinate for the electron transfer reaction for any given 7 ab- In what follows we suppress the label J ab but the presence of this parameter as an important factor in the transfer reaction should be kept in mind. Note that a given fixed 7 ab is also implicit in the calculations of Section 16.3.6 and 16.4. By the discussion of Section 16.3.4, the functions 

For = 0 oiN =1 this is the Bom-Oppenheimer nuclear potential surface of the corresponding electronic state. We assume that we can define an analogous potential surface for any 0. In fact, the treatment here makes use only of the standard surfaces Ig and Tj. 

As defined, the new reaction coordinate A is different from the reaction coordinate d defined and used in Sections 16.3.5 and 16.3.6. However, if the potential surfaces lFo(A) and Wi(X) also turned out to be shifted parabolas as the corresponding functions of 6 were in Eqs (16.46) and (16.47) it has to follow that X and 3 are proportional to each other, so they can be used equivalently as reaction coordinates. This argument can be reversed The validity of the Marcus theory, which relies heavily on the parabolic fonn of Marcus free energy surfaces, implies that lTo(A) and Wi X) should be quadratic functions of A. 

The parabolic form of the Marcus surfaces was obtained from a linear response theory applied to a dielectric continuum model, and we are now in a position to verify this form by using the microscopic definition (16.76) of the reaction coordinate, that is, by verifying that ln(P(A)), where P X) is defined by (16.77), is quadratic in A. Evaluating PIA) is relatively simple in systems where the initial and final charge distributions po and pi are well localized at the donor and acceptor sites so that /)o(r) = - a) + 7b 5(r - rs) and pi (r) = - fa) + 

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In the Marcus model the important part of the system is donor and acceptor and the process takes place on the potential energy surface of the ground state of the total system, while the first excited state corresponds to the remaining, upper parts of the Marcus parabolas. If eq.(4) is solved, a number of excited states correspond to excitations of the bridge. It is interesting to... 

Table 9.4 represents the calculated AG values for the charge separation and charge recombination processes. Hereby, the charge recombination falls into the inverted regime of the Marcus parabola. With these values in hand, it was possible to place the different possible reaction pathways in a state diagram (Fig. 9.25). 

Experimental slope is given first the calculated one (within parentheses) is based on the Marcus parabola, approximated as a straight line in the AG° region involved... 

The diabatic free-energy profiles of the reactant and product states provide the microscopic equivalent of the Marcus parabolas.26,27 For example, in the case of the (Cl- + CH3-CI —> CICH3 + Cl-) Sn2 reaction, one obtains23 the results shown in Fig. 2. 

The diabatic free energy profiles of the reactant and product states provide the microscopic equivalent of the Marcus parabolas [29, 30]. 

Fig. 31 Schematic drawings of the Marcus parabolas for dimer and monomer...

Fig. 14.27. Electron transfer in the reaction DA -> D+A , as well as the relation of the Marcus parabolas to the concepts of the conical intersection, diabatic and adiabatic states, entrance and exit channels and the reaction barrier. Panel (a) shows two diabatic surfaces as functions of the and 2 variables that describe the deviation from the comical intersection point (within the...

The Marcus parabolas (Fig. 14.27d) represent a special section (along the collective variable) of the hypersurfaces passing through the eonieal intersection (parabolas Vr and Vp). Each parabola represents a diabatie state, so a part of each reactant parabola is on the lower hypersurface, while the other one is on the upper hypersurface. We see that the parabolas are only an approximation to the hypersurface profile. The reaction is of a thermal character, and as a consequence, the parabolas should not pass through the conical intersection, because it corresponds to high energy, instead it passes through one of the saddle points. 

Fig. 14.25. Electron transfer in the reaction DA- -D+A " as well as the relation of the Marcus parabolas to the concepts of the conical intersection, diabatic and adiabatic states, entrance and exit channels and the reaction barrier. Fig. (a) shows two diabatic (and adiabatic) surfaces of the electronic energy as functions of the f and 2 variables that describe the deviation from the conical intersection point (cf. p. 262). Both diabatic surfaces are shown schematically in the form of the two paraboloids one for the reactants (DA), the second for products (D+A ). The region of the conical intersection is also indicated. Fig. (b) also shows the conical intersection, but the surfaces are presented more realistically. The upper and lower parts of Fig. (b) touch at the conical intersection point. On the lower part of the surface we can see two reaction channels each with its reaction barrier (see the text), on the upper part (b) an energy valley is shown that symbolizes a bound state that is separated from the conical intersection by a reaction barrier.

In the most common case, the ET reaction is asymmetric with a driving force AG < 0. If we assume that the reaction starts from the left energy minimum, the right minimum will be lower, since the Marcus parabolas are supposed to be free energy PES (Figure 10.10). The rate is obtained via an Arrhenius equation. It is important to multiply the exponential with an electronic factor that accounts for the probability to pass the avoided crossing region. 

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