Reaction coordinate parabolas

Figure 2.2 Reaction coordinate parabolas for diabatic electron transfer, illustrating the relationship between AG° and k for (a) the normal region, where AG° < k (b) the activationless electron transfer region, where AG° = k (c) the inverted region, where AG° > k...

Let us assume a reaction coordinate x running from 0 (reactant) to 1 (product). The energy of the reactant as a function of x is taken as a simple parabola with a force constant of a. The energy of the product is also taken as a parabola with the same force constant, but offset by the reaction energy AEq- The position of the TS (jc ) is taken as the point where the two parabola intersect, as shown in Figure 15.27. The TS position is calculated by equating the two energy expressions. 

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

Fig. 1.15 Schematic of the energy curves in the Marcus-Hush model with a single, global reaction coordinate q such that the potential energy hypersurface reduces to two parabolas and the activation energy can he calculated from the intersection point between them. The electronic coupling (Sect. 1.7.2.2) and the continuum of electronic levels in the metal electrode (Sect. 1.7.2.1) are not shown...

See Appendix 1 (Section A 1.2) for graphical demonstrations of movement perpendicular and parallel to the reaction coordinate at the transition structure in surfaces approximating to parabolas. 

Figure 4 Reaction coordinate considered as system of intersecting parabolas ...

Continuum dielectric-linear response theory yields a free energy surface for dielectric fluctuations which is a parabola in the reaction coordinate 3. This harmonic oscillator property is quite nonobvious and very significant. 

This theory gives us the force constant associated with this parabola in terms of physical parameters The dielectric response parameters Sg and Es, the initial and final charge distributions and a geometric factor, here the ionic radius a. Note that the dimensionality of this force constant is energy, in correspondence with the dimensionless nature of the reaction coordinate 3. 

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. 

In order to visualize the energy barrier between reactants and products, it is assumed that each system can be represented as a classical harmonic oscillator along the reaction coordinate. This is illustrated in fig. 7.15. The left-hand parabola gives the Gibbs energy of the reactants and the right-hand parabola, that of... 

Using the properties of the parabolas shown in fig. 7.15, one can determine a relationship between the Gibbs energy of activation and the shape and position of the parabolas on the reaction coordinate. The equation giving the Gibbs energy of the reactant system is... 

The Marcus equation was first formulated to model the dependence of rate constants for electron transfer on the reaction driving force [47-49]. Marcus assumed in his treatment that the energy of the transition state for electron transfer can be calculated from the position of the intersection of parabolas that describe the reactant and product states (Fig. 1.2A). This equation may be generalized to proton transfer (Fig. 1.2A) [46, 50, 51], carbocation-nucleophile addition [52], bimolecular nucleophilic substitution [53, 54] and other reactions [55-57] by assuming that their reaction coordinate profiles may also be constructed from the intersection of... 

Figure 1.2. A, Reaction coordinate profiles for proton transfer at carbon constructed from the intersection of parabolas for the reactant and product states. B, The reaction coordinate profile for a reaction where AC° = 0 and ACt is equal to the Marcus intrinsic barrier A...

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