Transition path sampling

Diffusion of Isobutane in Silicalite studied by Transition Path Sampling 

In this section, we will briefly summarize the transition path sampling method for deterministic paths which has been developed by David Chandler and co-workers based on earlier ideas of Pratt [235]. This method is not only able to calculate the hopping rate (and therefore also the diffusion coefficient) between two stable sites (here the intersections of Silicalite). For a more complete discussion about this simulation technique, the reader is referred to refs. [227,232,234]. 

Consider a dynamical system with two stable states, A and B, in which transitions from A to B are rare. This could be, for example, intersections of the zeolite Siliccdite in which a branched alkane is preferentially adsorbed. The transition rate k from A to B can be calculated from the time derivative of an autocorrelation function C (t). 

Since the function xt is fully determined by the initial condition xo, the ensemble averages in equation 6.1 can be written as an integration over the initial conditions weighted with the equilibrium distribution A/ (xo) 

We can also look at this equation as the ensemble average of Kb (xt) weighted with the equilibrium distribution A lxo) x (xo). In other words, C (t) is the fraction of trajectories that start in A with distribution M (xo) and reach B after time t. Since we are sampling over paths this ensemble is called the path ensemble. A procedure to sample this ensemble would be to 

A different approach for such systems can be considered, however, that invokes a different set of methodologies that attempt to compute trajectories connecting conformations from the reactant state to conformations of the product state, i.e., the reaction path. Transition path sampling, MaxFlux, discrete path sampling, string methods, and optimization of actions are examples of methodologies that search for these transition paths. We now will review briefly the first four methods and then present the theory and implementation of the action formalism in more detail. 

Flere x is a phase space vector, Hb(T) = maxo i T B(Xi), and -)ahb(T) an average over the ensembles of paths that start in A and go to B at least once during a fixed length T. An order parameter is introduced to describe 

The molecular processes typically studied with TPS involve a transition over a single, albeit significant barrier. TPS is more efficient than standard MD because the reactive trajectories (computed by TPS) are much shorter through phase space than the time it takes between successive transitions more (reactive) trajectories are therefore computed with TPS than with normal MD methods. 

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Dellago C, Boihuis P G, Csajka F S and Chandler D 1998 Transition path sampling and the calculation of rate constants J. Chem. Phys. 108 1964-77... 

C. Dellago, P.G. Bolhuis, F.S. Csajka, and D. Chandler, Transition path sampling and the calculation of rate constants , a preprint. 

P. G. Bolhuis, D. Chandler, C. Dellago, and P. Geissler, Transition path sampling throwing ropes over mountain passes, in the dark, Arum. Rev. Phys. Chem. 53, 291 (2002). 

Dellago C, Bolhuis PC (2007) Transition Path Sampling Simulations of Biological Systems. 268 291-317... 

Transition Path Sampling and the Calculation of Free Energies... 

Transition path sampling can also be helpful in the calculation of free energies in the context of fast-switching methods described in Chap. 5. As shown by Jarzynski [12], equilibrium free energies can be computed from the work performed on a system in repeated transformations carried out arbitrarily far from equilibrium. From a computational point of view, this remarkable theorem is attractive because it promises efficient free energy calculations due to the reduced cost of... 

The basis of the transition path sampling method is the statistical description of dynamical pathways in terms of a probability distribution. To define such a distribution consider a molecular system evolving in time and imagine that we take snapshots of this system at regularly spaced times fj separated by the time step At. Each of these snapshots, or states, consists of a complete description z of the system in terms of the positions q = <71, <72, , [Pg.252]

In the transition path sampling method we are interested in trajectories that start in a certain region of configuration space, which we will call region si, and end in another region, 38. We call such trajectories reactive. Accordingly, we restrict the probability distribution from (7.3) to reactive trajectories only (see Fig. 7.2)... 

The efficiency of a transition path sampling simulation crucially depends on how new pathways are generated from old ones. Various schemes to do that are possible. 

Free Energies from Transition Path Sampling Simulations... 

In most applications of transition path sampling it is sufficient to define the stable regions si and 88 in terms of configurational coordinates without reference to the momenta. The transition path sampling formalism, however, can be also applied to situations in which si and 88 also depend on the atomic momenta. 

Note that the transition path sampling method can be also used for the calculation of activation energies (as opposed to activation free energies) [29]. This approach is useful for systems in which it is not possible to identify transition states. 

Jarzynski [12], Although this identity is an exact result, statistical sampling problems arise if the transformation moves the system too far from equilibrium. In this section we will explain the origin of these difficulties and show how transition path sampling can be used to overcome them. 

The calculation of reaction rate constants with the transition path sampling methods does not require understanding of the reaction mechanism, for instance in the form of an appropriate reaction coordinate. If such information is available other methods such as the reactive flux formalism are likely to yield reaction rate constants at a lower computational cost than transition path sampling. 

The slope of (7(f) in the time regime rmoi < f forward reaction rate constant. Thus, for the calculation of reaction rate constants it is sufficient to determine the time correlation function (7(f). In the following paragraphs we will show how to do that in the transition path sampling formalism. 

Finally, in Sect. 7.6, we have discussed how various free energy calculation methods can be applied to determine free energies of ensembles of pathways rather than ensembles of trajectories. In the transition path sampling framework such path free energies are related to the time correlation function from which rate constants can be extracted. Thus, free energy methods can be used to study the kinetics of rare transitions between stable states such as chemical reactions, phase transitions of condensed materials or biomolecular isomerizations. 

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