Crossing point heights

Fig. 11. Schematic representation of the relationship between the free energy change (difference between the product and reactant minima) and free energy of activation (height of the crossing point with respect to the reactant minimum) for the corss reactions between a single reductant (A ) and a homogeneous series of oxidants (A2a, A2b> A2c) having variable oxidation potential. The curve corresponding to the formation of Ain an excited state is also shown...

This expression shows that the height of the crossing point depends on the triplet repulsive interactions between the bonded atom in the center (e.g., H) and the two terminal groups X. 

Once the curves are calculated, one has the promotion energy, G, and the height of the crossing point, AEc, which defines the / factor, / = AEc/G. In a separate calculation, one uses the entire VB structure set, (1-8), and calculates the adiabatic state, which is the curve in bold in Fig. 10.1. These calculations provide the energy barrier and the resonance energy of the TS. For this reaction, it was shown that the VBCISD//cc-pVTZ leads to an accurate barrier (10 kcal/mol), and B and G values that match semiempirical estimates. 

Note that Ea(.R ) is the height of the curve-crossing energy above the bottom of the reactant well a. It is also noteworthy that in the non-adiabatic limit the rate depends explicitly on the interstate coupling Vab (in additional to its dependence on the characteristic frequency a>a and the slopes at the crossing point via AF). In the adiabatic limit dependence on Vab enters only indirectly, through its effect on the adiabatic barrier height. 

These simple considerations yield several corollaries, sometimes known together as the Bell-Evans-Polanyi (BEP) principle [14]. First, there is an approximately linear relation between the barrier height and the reaction energy this is the basis of the Bronsted relation (and other LFERs). Second, the proportionality constant a in Eq. (19.2) tends to be smaller for exothermic reactions (but larger for endothermic reactions). Third, the position of the crossing point between the curves lies closer to the reactants for more exothermic reactions this is the basis of the Hammond postulate, that the TS for a more exothermic reaction more closely resembles the reactants (and that for a more endothermic reaction more closely resembles the products). 

Alternatively, the height of the crossing point can be calculated with any MO-based method, by determining the energy of the reactant wave function at zero iteration (see Appendix 23A), by constrained optimization of block-localized wavefunctions [44], or by an energy decomposition scheme of the Morokuma-analysis type [45]. Lastly, the height of the crossing point can be computed by means of molecular mechanical methods [46], or related empirical VB calculations [47,48]. 

Except for VB theory that calculates B explicitly, in all other methods this quantity is obtained as the difference between the energy of transition state and the computed height of the crossing point. In a few cases it is possible to use analytical formulas to derive expressions for the parameters/and B [7,11,23,49]. 

Fig. 23.14. VBCMDs, like in Fig. 23.1b, for the Srn2 and mechanisms. Below the VBCMDs we show the respective triple-ionic configuration that affect the height of the crossing point of the reactant and product curves (adapted from Ref. [11] with permission of Wiley-VCH STM-Copyright and Licenses). The three electron bonds in the charge transfer states are indicated here by a combination of a line and a dot.

Furthermore, the height of the crossing point can be most simply related to the gap, Gr, as a fraction (/) of this gap, that is ... 

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