Numerical simulation of barrier crossing

Why (said the queen), sometimes I ve believed as many as six impossible things before breakfast. (Alice s Adventures in Wonderland—Lewis Carrol) 

One way to bridge the gap between simple models used for insight, in the present case the Kramers model and its extensions, and realistic systems, is to use numerical simulations. Given a suitable force field for the molecule, the solvent, and their interaction we could run molecular dynamic simulations hoping to reproduce experimental results like those discussed in the previous section. Numerical simulations are also often used to test approximate solutions to model problems, for 

To generate the trajectories that result from stochastic equations of motion (14.39) and (14.40) one needs to be able to properly address the stochastic input. For Eqs (14.39) and (14.40) we have to move the particle Linder the influence of the potential T(.v), the friction force—yvm and a time-dependent random force R(t). The latter is obtained by generating a Gaussian random variable at each time step. Algorithms for generating realizations of such variables are available in the applied mathematics or numerical methods hterature. The needed input for these algorithms are the two moments, (2J) and In our case (7 ) = 0, and (cf. Eq. (8.19)) = liiiyk/jT/At. where Ai is the time interval 

R(t) and average every calculated result over this ensemble of solutions. 

As indicated in the solution to Problem 14.3, for high barrier, Eq kgT, the starting point xq and the end point xj (where exit is decided) can be taken anywhere weU in the reactant region, and well hr the product region, respectively. WeU in these region imply a position. x at which the potential is considerably lower (relative to kjjT i than its value at the barrier top. Variations in xq and X] that adhere to this condition affect the computed rate only marginally. 

Besides indicating areas for future concern, these questions exemplify the everpresent tension between our desire to explain observations by the most general and generic models, and between the ever-present system-specific and experiment-specific features. 

A way to overcome this difficulty is to realize that the factor that makes exit events extremely rare has no dynamical origin. To put this observation to practice we reformulate our problem Rather than attempting to calculate the rate k we look for the correction a to the TST rate 

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Molecular dynamic simulations are very useful for solvation dynamic studies. In contrast to the difficulties described in applying numerical methods to the problems of vibrational relaxation (Section 13.6) and barrier crossing (Section 14.7), solvation dynamics is a short-time downhill process that takes place (in pure simple solvents) on timescales easily accessible to numerical work. 

Barrier crossing phenomena are fundamental in chemistry. In many situations barrier crossing events correspond to first-order kinetics characterized by a rate constant. Because the dynamics of such processes corresponds to infrequent events, direct simulation of a reaction from reactants to products can be extremely inefficient- A much better approach is offered by the reactive flux formalism which expresses the reaction rate in terms of the flux crossing a fictitious dividing surface that separates reactants from products. Even though the reactive flux method offers a numerically advantageous approach compared with direct simulation of the reaction itself, fully... 

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