Reaction coordinate/committor

The path ensemble, as created by the transition path sampling methodology, is a statistically representative collection of trajectories leading from a reactant region to a product region. Further analysis of this ensemble of pathways is necessary to obtain rate constants, reaction mechanisms, reaction coordinates, transition state structures etc. In this section we will describe how to analyze the path ensemble by determining transition state ensembles, and how to test proposed reaction coordinates using committor distributions. 

Consider the situation illustrated in Fig. 10 for proton transfer in the pro-tonated water trimer. Suppose one has postulated the variable Ar as the reaction coordinate for the transition. Then, the question arises if Ar is a relevant and sufficient description of the reaction mechanism. In the case of the Fig. 10 it is clearly not. If one drives the transition by controlling Ar, hysteresis will occur, indicating that the variable Ar is not sufficient for a dynamical description of the reaction. Observing hysteresis is a crude way of testing reaction coordinates and a more precise procedure is called for. Calculating distributions of committors for constraint ensembles can be precisely the powerful tool we need to test the correctness of the proposed reaction coordinate. This diagnostic tool is not restricted to TPS but can be applied... 

Fig. 11. Three scenarios leading to different committor distributions, (a) The variable g is a good reaction coordinate. Configurations constrained at g = g produce a distribution of committors peaked at ps = 0.5. (b) The variabie g is insufficient to describe the reaction properly. As a results the committor distribution for configurations with g = g is peaked at zero and unity. For a correct description of the transition the variable g must be taken into account, (c) The transition occurs diffusively in the direction of q. In this case the committor distribution is flat...

Here, P pb) is the probability (density) for finding the committor ps in the ensemble g = g. If this distribution is peaked around pb = 0.5, the constraint ensemble g = g is located on the separatrix and coincides with the transition state ensemble. In this case, g is a good reaction coordinate, at least in the neighborhood of the separatrix. This is illustrated in Fig. 11a. Other possible scenarios for the underlying free energy landscape result in different committor distributions, and are also illustrated in Fig. 11. The committor distribution can thus be used to estimate how far a postulated reaction coordinate is removed from the correct reaction coordinate. An application of this methodology can be found in [33], where the reaction coordinate of the crystallization of a Lennard-Jones fluid has been resolved by analysis of committor distributions. 

The committor pg is a direct statistical indicator for the progress of transitions from A to B. In this sense, it is an ideal reaction coordinate. But interpreting this highly nonlinear function of atomic coordinates in terms of molecular motions and intuitively meaningful fields is not straightforward. Understanding a transition s mechanism basically amounts to identifying... 

Distributions of committor values are a powerful diagnostic for differentiating coordinates that drive a transition from those that are simply correlated with it. Consider an order parameter q whose potential of mean force w q) has a maximum at = q. If q serves as a reaction coordinate, then the ensemble of configurations with q = q coincides with the separatrix [see Fig. 1.18(a)]. The committor distribution for this ensemble. 

Figure 1.18. Three different free energy landscapes w q, q"), the free energy w(q, q") for q = q and its corresponding committor distribution P(p - (a) The reaction is correctly described by q and the committor distribution of the constrained ensemble with q=q peaks at pg = 0.5. (b) q plays a significant additional role as a reaction coordinate, indicated by the additional barrier in w q, q ) and the bimodal shape of P(pg). (c) Similar to case (fc), but now the committor distribution is flat, suggesting diffusive barrier crossing along q. ...

We can make the concept of the quality of a reaction coordinate q r) more precise by considering the so-called commitment probability, or conunittor. The conunit-tor pb (r) is defined as the probability that a trajectory started at configuration r with random momenta reaches state B before it reaches state A (see Fig. 10). (The commitment probability for state A is defined analogously.) The commitment probability was introduced as splitting probability already by Onsager, who used this concept to analyze ion pair recombination [262]. It has proven very useful in theoretical studies of protein folding, where the committor is known as pfou [263], and even in experimental work on liquid-solid nucleation [264]. Calculation of the probability pb r) involves a MaxweU-Boltzmann average over momentum space... 

It has been shown that for diffusive barrier crossing under certain not unduly restrictive conditions the reaction coordinate that is optimum in the TST-sense is orthogonal to the committor-1/2 surface [270]. 

As discussed above, a good reaction coordinate needs to parametrize the committor, i.e., configurations with a particular value of the reaction coordinate should all have the same committor. Therefore, isosnrfaces of the reaction coordinate defined by q r) = constant should coincide, at least where they are mostly populated, with the corresponding isocommittor surfaces. This is something that can be easily tested by determining the probability distribution P(pb) of the committor for equilibrium-weighted configurations with a particular value q of the reaction coordinate [25] ... 

As discussed in previous sections, the committor is the ideal reaction coordinate in the sense that it exactly quantifies how far a reaction has proceeded. This concept also provides the basis for transition path theory (TPT) [33,278], a probabilistic framework developed by Vanden-Eijnden and collaborators to study the statistical properties of rare event trajectories. In TPT, isocommittor surfaces, i.e., surfaces on which all points have the same conunittor value, play a prominent role. Trajectories initiated from any point of an isocommittor surface have the same probability to reach the final rather than the initial state first. It can be shown [31] that the distribution of points where reactive trajectories pierce a given isocommittor surface is identical to the equiUbrium distribution confined to that surface. From the committor and the equilibrium distribution one can determine the distribution of reactive trajectories, so-called reaction tubes, which contain entire reaction pathways with high probabUity, as well as the reaction rates, providing useful statistical information about the reaction mechanism. 

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