Model of two intersecting parabolas

Figure 2 Two parameters defining the Marcus-Hush model of two intersecting parabolas the equilibrium free energy gap AFo and the classical reorganization energy Xci- The parabolas curvature is l/(2Xd).

Fig. 5.10. Reaction profile, model of two intersecting parabolas. The activation energy is if the reaction energy AE is zero (solid parabola on the right). If the reaction energy AE is negative (dotted parabola on the right) AE is smaller than AEq and the transition state, i.e. the intersection of parabolas, is closer to the reactants. A modification of the simple model in which the two potentials interact (resonance energy H) is shown as dash-dotted line...

The molecular concerted decomposition of nitroalkanes has been analysed using the model of two intersecting parabolas. This approach and two elementary event... 

The rationalization of PER by Marcus is based on a simple model of the reaction profile, that of two intersecting parabolas (Pigure 5.10) [35]. In most applications of the Marcus equilibrium-rate theory, the reaction coordinate is a normalized quantity between 0 and 1, measuring in a generalized way the progress of reaction it is usually poorly defined from a geometrical, structural point of view. Indeed, when the word structure is used in works on PER, it refers mainly to the connectivity... 

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]. ... 

Fig. 16. The configuration coordinate model of Fig. 15c on an expanded scale. Various relevant energies (see text) have been defined. Also illustrated are the three primary types of return to the electronic ground state (1) the direct transition from the upper minimum, (2) the tunneling (horizontal) transition to an excited vibrational state of the electronic ground state, and (3) the thermally activated transition to the intersection of the two 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... 

G. The unified model for barrier crossing in solution, (a) Construct the two intersecting two-dimensional parabolas, each being a function of the solvation coordinate r and the solute coordinate q. (b) Determine the reaction coordinate and the lowest barrier, (c) Determine the intrinsic barrier, when the reaction is symmetrical, (d) Derive Eq. (11.29) for the free energy at the barrier. 

The same relations axe more easily obtained from a very simple one-dimensional model, in which only one degree of freedom is considered in this case the two potential energy surfaces reduce to parabolas, and the energy of activation is simply calculated from their intersection point (see Problem 1). 

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...

Marcus was the first to investigate how the energy profile might vary with AG (equivalent to ApAT) the assumption was that the intersecting curves are two parabolae and the effect of changing AG is to shift the vertical relationship [23,24]. Using this model the relationship (Eqn. 33) was obtained. 

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