Dynamical bottleneck

Transition-state theory is based on two assumptions, the existence of both a dynamic bottleneck and a preceding equilibrium between a transition-state complex and reactants. Eq. (2.4) results, where k denotes the observed reaction rate constant, k the transmission coefficient, and v the mean frequency of crossing the barrier. 

D.C. Chatfield, S.L. Mielke, T.C. Allison, D.G. Truhlar, Quantum dynamical bottlenecks and transition state control of the reaction of D with H2 Effect of varying the total angular momentum, J. Chem. Phys. 112 (2000) 8387. 

How does the solvent influence a chemical reaction rate There are three ways [1,2]. The first is by affecting the attainment of equilibrium in the phase space (space of coordinates and momenta of all the atoms) or quantum state space of reactants. The second is by affecting the probability that reactants with a given distribution in phase space or quantum state space will reach the dynamical bottleneck of a chemical reaction, which is the variational transition state. The third is by affecting the probability that a system, having reached the dynamical bottleneck, will proceed to products. We will consider these three factors next. 

A new issue arises when one makes a solute-solvent separation. If the solvent enters the theory only in that V(R) is replaced by TT(R), the treatment is called equilibrium solvation. In such a treatment only the coordinates in the set R can enter into the definition of the transition state. This limits the quality of the dynamical bottleneck that one can define depending on the system, this limitation may cause small quantitative errors or larger more qualitative ones, even possibly missing the most essential part of a reaction coordinate (in a solvent-driven reaction). Going beyond the equilibrium solvation approximation is called nonequilibrium solvation or solvent friction [4,26-28], This is discussed further in Section 3.3.2. 

In two-mode [17] and three-mode [20] systems it was shown that dynamical bottlenecks exist to intramolecular energy transfer that is, cantori are buried in the reactant basin, which form partial barriers between irregular regions of phase space. This brought about multiply... 

As illustrated in the previous section, the kinetics associated with an ET process may be complex when diffusion or relaxation processes create dynamic bottlenecks. In limiting cases, however, a simple model based on transition state theory (TST) suffices. According to TST, the system maintains thermal equilibrium between different positions along the reaction coordinate [87]. We consider the TST rate constant for electron transfer after some preliminary comments about state manifolds and energetics. 

It is of considerable interest to examine the extent to which solvent (or other medium) modes may lead to dynamical bottlenecks requiring a departure from the TST framework [35]. This effect can be represented by a transmission factor, k [84, 87], where... 

Cell system Replicating liposome with internal reaction network Dynamic bottleneck in autocatalytic reaction system Evolvability and recursiveness for growth... 

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... 

Since the reactive trajectories will in general slow down near the dynamical bottlenecks of the reaction, this allows one to identify the transition state regions roughly as the regions where mAsix) is peaked. Observe however that these regions can be multiple (i.e. there may be more than one dynamical bottleneck for a reaction) and quite wide (i.e. the dynamical bottleneck may be a rather extended region in state-space). 

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. 

The quantization of transition state energy levels is not simply a mathematical device to add quantum effects to the partition functions. The quantized levels actually show up as structure in the exact quantum mechanical rate constants as functions of total energy [51]. The interpretation of this structure provides clear evidence for quantized dynamical bottlenecks, both near to and distant from the saddle points, as reviewed elsewhere [52]. Quantized variational transition states have also been observed in molecular beam scattering experiments [53]. Analysis of the reactive flux in state-to-state terms from reactant states to transition state levels to product states provides the ultimate limit of resolution allowed by quantum mechanics [53,54]. Quantized energy levels of the variational transition state have been used to rederive TST using the language of quantum mechanical resonance scattering theory [55]. 

The function p7( ) is a symmetric bell-shaped curve centered at T, and pT( ) is narrower when the effective potential barrier is wider. For an ideal dynamical bottleneck kt is unity deviations from unity indicate that recrossing or other multidimensional effects are important. 

The value of the transmission coefficient kt is shown for each feature in Table 2. (The value of kt for the last feature is greater than 1 because it includes contributions from higher energy transition states that have not been included in the fit.) Many of the values of the transmission coefficients are very close to unity, suggesting that these features correspond to quantized transition states that are nearly ideal dynamical bottlenecks to the reactive flux. Several of the values of kt deviate from unity this could be the result of the assumption of parabolic effective potential barriers or from recrossing or other multidimensional effects. 

Fitting the quantal density by a sum of terms KTpT( ) is difficult because of the large number of transition states for 7 = 4. However, quantized transition state control of chemical reactivity can be assessed for 7 = 4 without identifying all of the individual contributions to the total density by comparing the accurate values of N4(E) with those in the next to last column of Table 4. If the transition states were ideal (kt = 1), the two numbers would be equal. Up to 1.228 eV, the energy of the sixth peak, the numbers are very close at 1.228 eV the accurate value of N4(E) is 24. Thus, the quantized transition states up to 1.228 eV are nearly ideal dynamical bottlenecks. Above 1.228 eV the quantal N4(E) is somewhat smaller than the predicted value, but even at 1.570 eV the difference is only 15%. This difference may be due to the inaccuracy of Eq. (25) at high v2 or to... 

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