Potential energy surfaces bottlenecks

The 7-shifting method depends on our ability to identify a unique bottleneck geometry and is particularly well suited to reactions that have a barrier in the entrance channel. For cases where there is no barrier to reaction in the potential energy surface, a capture model [149,150,152] approach has been developed. In this approach the energy of the centrifugal barrier in an effective onedimensional potential is used to define the energy shift needed in Eq. (4.41). For the case of Ai = 0, we define the one-dimensional effective potential as (see Ref. 150 for the case of AT > 0)... 

Even when the harmonic approximation is not quantitatively justified it provides a convenient starting point for exact treatments. Thus, even if the potential energy surface is anharmonic in the bottleneck, it is often smooth enough for there to be a principal saddle point that can be found by minimizing IVU 2. 

As mentioned in Section 10.2, saddle points on the potential energy surface frequently correspond to the transition states that constitute bottlenecks to reaction. Finding these saddle points can provide a remarkable level of information about the mechanism. Such information about TS structure is not readily available in direct form from experiment. Calculation is then highly complementary with experiment and can be used to confirm a predicted mechanism, cast insight into observed substituent effects, and so on. 

The bottleneck of very short lifetimes of resonace states (10 14s) becomes less severe once one assumes that the primary role of resonance states is to provide doorways to bound valence anionic states, with lifetimes determined by kinetics of the following chemical reactions [36], The reactions might proceed on these regions of potential energy surfaces, at which valence anions are bound with respect to the neutral species. The rates of these chemical transformations, e.g., the SSB formation, do not have to compete with short lifetimes of resonance states. It is worth noting that even for a kinetic barrier of ca. 20 kcal/mol, the half lifetime amounts (at 298 K) to about 30 seconds. Hence, if the kinetic barrier for SSB formation were lower than 20-23 kcal/mol, all nucleotides that could form stable anions would have enough time to cleave the C-O bond on the timescale of the electrophoretic assay of DNA damage. 

A different strategy to approach such problems is to search for the dynamical bottlenecks through which the system passes during a transition between metastable states. If the dynamics of the system is dominated by energetic effects (as opposed to entropic effects), such bottlenecks can be identified with saddle points in the potential energy surface. In this case, saddle points are transition states, activated states from which the system can access different stable states through small fluctuations. Comparing stable states with transition states one can often infer the mechanism of the reaction. Reaction rate constants, which are very important because they are directly comparable to... 

Calculations of reaction rates with variationally determined dynamical bottlenecks and realistic treatments of tunneling require knowledge of an appreciable, but still manageably localized, region of the potential energy surface [33[. In this chapter we assume that such potentials are available or can be modeled or calculated by direct dynamics, and we focus attention on the dynamical methods. 

A computational bottleneck is given by the inversion of the D matrix (eq.23) and this is a reason to keep the dimension T of this matrix low. For calculations at fixed geometries divergences near the minimum rarely occour (they may appear when sizeable portions of the potential energy surface are sampled), on the contrary events of this type, interrupting the search of the critical point, are more probably when the number T of tesserae is larger. Divergences are here connected to the occurrence of very small tesserae. 

Figure 1.1. Prototypical potential energy surface of a simple system (a) and of a complex system (b). In a simple, low-dimensional system, dynamical bottlenecks for transitions between long-lived stable states most often coincide with saddle points on the potential energy surface. Locating these stationary points reveals the reaction mechanism. In a typical complex system, the potential energy surface is rugged and has countless local minima and saddle points. Nevertheless, there can be well-defined long-lived stable states and rare transitions between them. Such transitions can occur via a multitude of different transition pathways.

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