Transition state theory landscapes

Consider the energy landscape given in eqn (12.73). (a) Derive the transition rate for this potential based on conventional transition state theory, (b) Use a bias potential of the type described in the text and reproduce results like those given in fig. 12.15. Compare the results of the two sets of calculations. In addition, plot the hyperdynamics boost factor as a function of the parameter Vo introduced in the text. 

The transition state theory and the assumption of complex and rugged free-energy landscapes are still under debate. One reason is the inherent difficulty to identify the true relevant degrees of freedom Q which are typically highly system-specific. The problem is that the kinetic barrier frequently depends on the choice of Q which makes an experimental verification of a theoretically proposed free-energy landscape more difficult. Nonetheless, the free-energy landscape concept is helpful in understanding details of phase transitions (such as, e g., the occurrence of barriers that slow down the transition process) and to quantity these. 

There is a difference between experimentalists and theoreticians experimentalists observe the minima and maxima in free energy profiles—the experimental entities of intermediates and transition states—whereas theoreticians wish to calculate the entire energy surface of a reaction. Experimentalists talk about pathways, theoreticians about energy landscapes. Experiment and theory touch base around the ground and transition states that provide the milestones in the energy landscapes for the theoreticians to benchmark their calculations. The two views are reconciled in section G. 

The second tenet of this theory is that the breakdown from the transition state to products is the result of a vibration at the frequency v between two moieties in a locally flat energy profile at the saddle point corresponding to the transition from the reactant part to the product part of the free energy landscape. 

Fig. 8.1 Density functional theory calculations of electronic versus geometric effects, (a) Eneigy landscape successive steps of the dissociation of N2. The activation energy for the dissociation, Fa, corresponds to the eneigy of the transition state (TS). The atoms of the dissociated molecule, 2N, have a chemisorption eneigy, (b) Density functional theory (DPT) calculations show that...

Fig. 2 Disconnectivity graphs for minima and transition states in the five lowest Stone-Wales stacks [3] of Cso calculated using a plane-wave implementation of density functional theory [2j. The graphs on the left and right correspond to the LDA and BLYP functionals, respectively. The vertical energy scales are the same and the energy zero has been shifted to buckminsterfullerene in both cases. The structures of six minima are indicated, including buckminsterfullerene and the next-lowest structure with Cav symmetry. Reproduced with permission from D. J. Wales, Energy Landscapes, Cambridge University Press, Cambridge (2003)...

---