Parabolas, intersecting, reaction

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. 

A schematic diagram of the free energy changes in an electron exchange reaction, showing the intersection of two parabolas. The lighter curve represents n,c the darker one, eng, ncg. 

Past the transition state of the photoinduced reaction, J, the system goes down the repulsive product surface, P, until P crosses the intersection with the reactant surface R. These two surfaces are sketched in Fig. 16a. Their intersection is a parabola (the central line in Fig. 16c) whose projection in the X -Yi plane (straight line in Fig. 16b) is defined by equation (67). 

The minimum on the intersection parabola is the saddle point corresponding to the transition state of the dark reaction, denoted J in Figs 16b and 16c. The first-order potential energy surfaces involve an upper surface associating the portions of the R and P zero-order potential energy surfaces situated above the intersection parabola and a lower surface associating the portions of the R and P zero-order potential energy surfaces situated below the intersection parabola. 

The parameter re characterizes the distance between the two minimum points of the intersecting parabolas. This parameter is equal to the sum of the elongation of the two transformed bonds in the TS of the reaction. 

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...

Fig. 3 (A) Free energy reaction profdes, constructed from intersecting parabolas, for addition of water to a simple carbocation that show the change in reaction barrier with changing reaction driving force. (B) Free energy profile for thermoneutral addition of water to a carbocation for which the observed activation barrier is equal to the intrinsic barrier A.

Hine (1966a) also addressed the question of the physical origin of least motion effects. He considered them to arise from bond stretches and deformations in the substrate molecule, and treated the energetic consequences of these stretches and deformations in terms of intersecting Morse curves which could be approximated by parabolas. PLNM could thus be recast in terms of energetics. The energy required for a molecular deformation was considered to vary as the square of the displacement from a stable equilibrium geometry, be it reactant or product. As a consequence, reactions with very early or very late transition states were anticipated to show only small least motion effects, which should be most pronounced in reactions with central transition states. 

Since there is no rupture or formation of a chemical bond, the extensive computation normally needed for quantum chemistry calculations for such bond breaking-bond-forming reactions is now absent. For reactions that do involve a concerted bond rupture and formation, the intersecting parabolas of Fig. 1.2 are normally inappropriate. 

Only if there were a transfer of, say, an H+, H, or H in a reaction, AH + B - A + HB (charges not indicated), at a fairly large AB separation distance, would the situation be rather analogous to that of ETs. The H transfer would occur at an approximately fixed position of A and B, fixed because of the substantially larger masses of A and B compared with that of H. That is, an approximate version of the Franck-Condon principle would apply. Under such conditions of an H transfer, the description of the reaction via two intersecting approximate parabolas would be a reasonable first approximation. 

Marcus, 1969) by assuming that the point which represents the transition state on the potential energy profile is the intersection of two parabolas of equal curvature. It was also considered that the proton abstraction reaction can be divided up [eqn (7)] into three discrete steps (0 encounter of the reactants, (ii)... 

Equation (112) was originally proposed for electron transfer reactions and modified for proton transfer. It has been derived in other ways [192, 201, 202], for example from [192] Leffler s Principle (Sect. 3.2.2) and by assuming a model for reaction of AH with B in which the energies of AH and BH+ are represented by two intersecting parabolae [202]. ... 

There are a number of ways to describe this FC term. An early way of describing the nuclear position and free energy-dependent FC term was proposed in Nobel prize-winning work by Marcus [9, 10]. Marcus approximated the reactant and product, before and after electron transfer, as simple harmonic oscillators with intersecting parabolic potential surfaces. As the driving force of the reaction increases and the product potential surface drops further down in energy, the barrier that must be crossed in going from the bottom of the reactant parabola to the bottom of... 

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

The intersecting parabolas model is consistent with the Hammond postulate that two states (such as reactant and transition structures) occurring consecutively in a reaction and having nearly the same energy content will involve interconversions with only a small reorganisation of the molecular structure. Consideration of Equation (11) reveals that for a... 

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