Committor distributions

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. 

To analyze this situation further we construct a histogram of the committors, a committor distribution P(p ),... 

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. 

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. ...

This methodology has been also used to extract subtle mechanistic details of the dissociation mechanism of an NaCl ion pair in liquid water [12]. In this case, committor distributions revealed that rearrangement of solvent molecules around the dissociating ion pair significantly contributes to the free energy barrier separating the contact state from the dissociated state. 

State. The committor values calculated in this way are histogrammed yielding an estimate of the committor disffibution P pb). A detailed statistical analysis of the computation of committor histograms has been carried out by Peters [272], Initially introduced to study ionic dissociation in water [25], committor distribution analysis was subsequently applied to elucidate the mechanism of various complex reactions [26,28,29,199,200,202,273]. 

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