The Morse Curve Model

ELECTRON TRANSFER, BOND BREAKING, AND BOND FORMATION 

The coordinate pertaining to solvent reorganization, z, is the same fictitious charge number as already considered in the Hush-Marcus model of outer-sphere electron transfer (Section 1.4.2), and so is the definition of 2q [equation (1.27)] and the difference between the Hush and Marcus estimation of this parameter. The coordinated describing the cleavage of the bond is the bond length, y, referred to its equilibrium value in the reactant, yRX. Db is the bond dissociation energy and the shape factor ft is defined as 

The equations above for the reactant and product hypersurfaces may be recast as 

The transition state is obtained as the saddle point on the intersection between the two hypersurfaces in the framework of the classical activation-complex theory in a manner similar to that in Section 1.4.2, thus leading to the following equations, which summarize the predictions of the model  

In cyclic voltammetry, the current-potential curves are completely irreversible whatever the scan rate, since the electron transfer/bond-breaking reaction is itself totally irreversible. In most cases, dissociative electron transfers are followed by immediate reduction of R, as discussed in Section 2.6, giving rise to a two-electron stoichiometry. The rate-determining step remains the first dissociative electron transfer, which allows one to derive its kinetic characteristics from the cyclic voltammetric response, ignoring the second transfer step aside from the doubling of the current. 

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Thermodynamics. Microscopic reversibility 120 The Morse curve model 123... 

FIGURE 3.2. Variation of the rate constants of dissociative electron transfer from aromatic anion radicals to butyl and benzyl halides as a function of steric hindrance. Data points from reference 10. Solid lines, best-fit parabola dashed lines, prediction of the Morse curve model, logAf-1 s-1). Adapted from Figure 3 of reference 6b, with permission from the American Chemical Society. 

FIGURE 3.30. Reaction of iron(0) and iron(I) pophyrins with n-, s-, and r-butyl bromides. The chart shows the various porphyrins and their symbolic designations. iron porphyrins, aromatic anion radical, lines best-fitting parabolas through the aromatic anion radicals data. Dashed lines outer-sphere curves obtained by use of the Morse curve model (Section 3.2.2). Adapted from Figure 4 in reference 47b, with permission from the American Chemical Society. 

In the stepwise case, the intermediate ion radical cleaves in a second step. Adaptation of the Morse curve model to the dynamics of ion radical cleavages, viewed as intramolecular dissociative electron transfers. Besides the prediction of the cleavage rate constants, this adaptation opens the possibility of predicting the rate constants for the reverse reaction (i.e., the reaction of radicals with nucleophiles). The latter is the key step of SrnI chemistry, in which electrons (e.g., electrons from an electrode) may be used as catalysts of a chemical reaction. A final section of the chapter deals... 

Fig. 1 Morse curve modeling of the contribution of bond-breaking to the dynamics of dissociative electron transfer...

The kinetics of the electron transfer reaction leading to the homolytically dissociating primary radical is also a question of interest. It may be modeled using the Morse curve for the reactant and the Morse curve shown in Fig. 10 representing the homolytic dissociation of the primary radical. This point will be discussed in detail in Section 5. 

A semi-classical treatment171-175 of the model depicted in Fig. 15, based on the Morse curve theory of thermal dissociative electron transfer described earlier, allows the prediction of the quantum yield as a function of the electronic matrix coupling element, H.54 The various states to be considered in the region where the zero-order potential energy curves cross each other are shown in the insert of Fig. 15. The treatment of the whole kinetics leads to the expression of the complete quenching fragmentation quantum yield, oc, given in equation (61)... 

The classical Morse curve model of intramolecular dissociative electron transfer, leading to equations (3.23) to (3.27), involves the following free energy surfaces for the reactant (Grx-) and product (Gr +x ) systems, respectively ... 

Fig. 9.11. For the sake of simplicity, the Morse curves can be linearized, and the model thus obtained can be considered sufficiently real for general (if approximate) considerations.

Point a corresponds to X at large distances from YZ. Interaction energy between X and YZ is virtually zero, and the side view of the model shows the typical Morse PE curve for the diatomic molecule YZ. Point d represents products, giving the PE for molecule XY with Z at a large distance, the side view giving the Morse curve for XY. 

If one substitutes 1 V cm forX, 3 x 10 cmfor S, and the usual values of F and R at room temperature, the exponential in the integral can become less than 10 , a rather small number, and the exponential can be linearized as approximately 1 +x We can use this model to fix the relative positions of the minima of the Morse curves. When the two minima are fixed, we can see from Fig. 4.127 that the two Morse... 

The new type of nonadiabatic transition between asymptotically degenerate states has been fully analyzed with use of the Morse potential model, and the analytical expression of the scattering matrix has been derived. The j)resent analysis warns that we have to be careful about this type of nonadiabatic transition whenever we encounter asymi)totic degeneracy of potential energy curves. The new mechanism presented in this paper works whenever the limiting value at. t —oo of the inverse Massey type parameter is a finite quantity ... 

The problem is to formulate these relationships quantitatively. One approach, called the interacting state model (ISM) uses the Morse curves for the C-H and H-X bonds as the starting point and describes the TSs in terms of the length of the C-H and C-X bonds at the The total bond order is taken to be 1.0 unless one of the... 

The Hooke s law functional form is a reasonable approximation to the shape of the potential energy curve at the bottom of the potential well, at distances that correspond to bonding in ground-state molecules. It is less accurate away from equilibrium (Figure 4.5). To model the Morse curve more accurately, cubic and higher terms can be included and the bondstretching potential can be written as follows ... 

If atoms are bonnd together, quantum mechanics and experiment both show that the bond between than is well described by a Morse potential. In the MM model, the VDW interaction is considered to just be lumped into the Morse curve and does not need to be considered separately. (It is much weaker than the normal bonding interaction, and the distance dependence curve has the same general shape.)... 

Explicit forms for the potential energy in the terms Hi and Hf have been proposed by Saveant [1993], who has developed a semiclassical version, along the lines of the Marcus theory, and applied it successfully to several reactions. In his model, the potential curve for the reactants is a Morse curve, and that for the products is the repulsive branch of a Morse curve ... 

There is thus an apparent continuity between the kinetics of an electron transfer leading to a stable product and a dissociative electron transfer. The reason for this continuity is the use of a Morse curve to model the stretching of a bond in a stable product in the first case and the use of a Morse curve also to model a weak charge-dipole interaction in the second case. We will come back later to the distinction between stepwise and concerted mechanisms in the framework of this continuity of kinetic behavior. 

Figure 2.3 shows the potential energy curve for a diatomic molecule, often referred to as a Morse curve, which models the way in which the potential energy of the molecule changes with its bond length. 

The conclusions of Ciraci et al. (1990a) can be understood with a simple independent-atom model, that is, by considering the tip as a single atom, and the total force is the sum of the forces on every atom on the sample surface. Assuming that the force between individual pairs of atoms can be represented as a Morse curve, the z component of the force to the nth atom at a point in space, r, is ... 

As mentioned above, HOSi(OA)3 may be taken as the simplest cluster model of the terminal hydroxyl group in silicas. Indeed, even with this cluster CNDO/BW provided a quite satisfactory description of the lower part of the curve representing potential energy as a function of the OH stretching vibration coordinate ROH (Fig. 2) (48,49). The respective experimental curve was plotted by Kazansky et al. (49) based on the analysis of the fundamental frequency vOH and the first overtone of the characteristic OH stretching vibration in terms of the Morse potential function. The frequencies of the second and third overtones were also determined in that work, and it was shown that the Morse potential reproduced well the potential curve within a rather wide range of ROH. 

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