Barrier dynamics

Smith, B. B., Staib, A. and Hynes, J. T. Well and barrier dynamics and electron transfer rates. A molecular dynamics study, Chem.Phys., 176(1993), 521-537... 

Quasiclassical direct dynamics trajectories at the various levels of theory were later calculated to study the central barrier dynamics for the C1 I CH3C1, Cr + C2H5C1, C1- + CH3I, F +CH3C1, OH +CH3C1, and other Sn2 reactions.31,32,47,97 108 The effect of initial reaction conditions, such as energy injection, substrate orientations, and the mode of collision, on the fate of the reaction, product, and energy distribution, was analyzed. Some of these trajectory calculations required serious modification in RRKM and TST for... 

First of all, liquid-phase studies generally do not obtain data which allows static and dynamic solvent effects to be separated [96,97], Static solvent effects produce changes in activation barriers. Dynamic solvent effects induce barrier recrossing and can lead to modification of rate constants without changing the barrier height. Dynamic solvent effects are temperature and viscosity dependent. In some cases they can cause a breakdown in transition state theory [96]. 

To summarize this section, for the Diels-Alder reaction, HF and CASSCF vastly overestimate the reaction barrier. Dynamical correlation is essential for the description of the Diels-Alder transition state. MP2 underestimates the barrier. MP4 and Cl methods both provide very good results, but triples configurations must be included. The preferred method, when one combines both computational efficiency and accuracy, is clearly DFT. It is likely the strong performance of B3LYP with pericyclic reactions, typified by the Diels-Alder results described here, that propelled this method to be one of the most widely used among computational organic chemists. 

Finally we assume that the well dynamics region (determined by the time evolution of Ej) and the barrier dynamics region (governed by the onedimensional flux across the saddle point) overlap somewhere below the barrier. Furthermore we assume that the reactive mode (defined near the barrier) keeps its identity below the barrier, at least down to this overlap region. The latter assumption is trivially always valid in a one-dimensional model. 

Most of the ingredients of the model described above have been formerly postulated in treatments of unimolecular reactions. In particular, the model for the barrier dynamics is inherent in the usual TST for unimolecular reactions involving polyatomic molecules, while taking the total molecular energy Ej as the important dynamic variable in the well is the underlying assumption in theories that use a master or a diffusion equation for Ej- as their starting point. 

Consider first the barrier dynamics problem, which is defined by replacing the potential barrier by an inverted parabola, Eq. (2.4), and by looking for a steady-state probability distribution which satisfies Eqs. (2.6) and (2.8). Here we follow the treatment of Hanggi and Mojtabai. Equations (5.1) and (5.2), with K(x) = Eg — are used to obtain (with a procedure due to... 

In order to make contact later with the barrier dynamics, we need P " ( ), the probability that the energy of the reactive mode is E. We note in passing that P ( ) is meaningful only provided that the reactive mode (defined near the barrier) keeps its identity in the well region. We show below that it is enough that this will be so high in the well, below the barrier region. 

Equations (6.72)-(6.74) show that t decreases dramatically for an increasing number of molecular degrees of freedom. This results both because of the larger Icvr expected for larger molecules and because of the n-dependent correction in Eqs. (6.72)-(6.74). Equation (6.55) then implies that the turnover from well dynamics to barrier dynamics dominated rate occurs for large molecules at much smaller solvent viscosities (or pressure in the gas phase) than for small molecules. This point was discussed in the literature and was the subject of several recent experimental investigations. Since for this small friction the barrier dominated rate is identical to the TST rate, it may be concluded that for large molecules a plateau in the rate versus solvent friction, where k = should be observed. 

Given the qualitative character of the theory with regard to realistic situations, such ad hoc approach is both reasonable and practical. A rigorous theoretical treatment is based on the normal mode approach to the barrier dynamics (see Section 14.5.3 below) supplemented by incorporating the rate at which the reactive barrier mode exchanges energy with other modes in the well. It yields the expression... 

The parabolic barrier plays a special role in rate theory. The GLE (with space-independent friction) may be solved analytically using Laplace transforms. The two-dimensional Fok-ker-Planck equation derived from the Langevin equation may be solved analytically, as was done by Kramers in his famous paper of 1940. In this section we present some of the analytic results for the parabolic barrier dynamics. These results are important from both a conceptual and a practical point of view. Later we shall see how one returns to the parabolic barrier case as a source of comprehension, approximation, etc. 

Static barriers (different layers of cornea, sclera, and retina, including blood aqueous and blood-retinal barriers), dynamic barriers (choroidal and conjunctival blood flow, lymphatic clearance, and tear dilution), and efflux pumps in conjunction pose a significant challenge for delivery of a drug alone or in dosage form, especially to the posterior segment. 

The above description assumes that an intermediate is formed with statistical classical dynamics and pooling of zero-point energy. If the dynamics of the intermediate is nonstatistical (i.e. as for Cl ---CH3C1 °), the intermediate s lifetime and product energy distribution may agree with experiment. A discussion of the applicability of classical mechanics for studying the central barrier dynamics of the [C1---CH3---C1] moiety is given below. 

Sn2 Nucleophilic Substitution. 8. Central Barrier Dynamics for Gas Phase CD + CH3CI. 

It can be shown [14] for general barrier dynamics that the rate can be written as... 

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