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Next: Summary Up: PROPERTIES OF REARRANGEMENT PATHWAYS Previous: Cooperativity   Contents

Applications to LJ$ _{13}$ and LJ$ _{75}$ Clusters and BLJ$ _{256}$ Liquid

Figure 3.2 shows results for the most cooperative and uncooperative processes we have found for the LJ$ _{75}$ cluster that are localised mainly on two atoms. In these calculations we have used the database of transition states that was found previously as the result of a discrete path sampling calculation conducted for this system [10,8]. The cooperative rearrangement [Figure 3.2(a,c)] is the one with the maximum two-overlap $ \Theta _2$. For this pathway $ \Theta_2=0.7$, $ N_p=3.4$, and $ N_c=7.7$. The values of $ N_p$ and $ N_c$ both reflect the fact that the motion of the two atoms is accompanied by a slight distortion of the cluster core. This example shows that while $ N_p$ and $ N_c$ allow us to quantify localisation and cooperativity, and correctly reflect the number of atoms that participate and move cooperatively in ideal cases, there will not generally be a simple correspondence between their values and the number of atoms that move. This complication is due to the fact that small displacements of atoms in the core will generally occur, no matter how localised the rearrangement is. In addition, the data reduction performed in Equation 3.7 and Equation 3.13 means that a range of pathways can give the same value for $ N_p$ or $ N_c$. Since the size of the contribution from a large number of small displacements depends on the shape of the displacement distribution function the number of possibilities grows with the size of the index.

The uncooperative rearrangement depicted in the Figure 3.2(b,d) was harder to identify. In principle we could have selected the pathway that maximises $ \Theta _1$ from all the rearrangements localised on two atoms. However, this approach picks out rearrangements localised on one atom, where distortion of the core accounts for the value of $ N_p>1$. Instead, we first selected from all the rearrangements with $ N_p < 4$ those where two atoms move by approximately the same amount, while the displacement of any other atom is significantly smaller. These are the rearrangements that maximise $ 1/ \gamma (\{d_1,d_2\})$, where $ d_1$ and $ d_2$ are the total displacements of the two atoms that move the most. After this procedure we selected the rearrangement with the maximum value of $ \Theta _1$. Figure 3.2 (b) shows that this rearrangement features the displacement of one atom at a time, and the atom that moves first also moves last. For this pathway the values of $ \Theta _1$, $ N_p$ and $ N_c$ are $ 0.7$, $ 3.8$ and $ 5.3$, respectively. Further illustrations and movies of the corresponding rearrangements are available online [181].

Figure 3.2 illustrates several general trends that we have observed for cluster rearrangements. Firstly, we have found that the barrier height is smaller for the cooperative rearrangements [Figure 3.2(c,d)]. Usually atoms that move cooperatively are neighbours. Rearrangements generally involve a change of the environment for the atoms that move. Cooperative motion can reduce this perturbation since for any of the participants the local environment is partly preserved because it moves with the atom in question. Flatter points on the energy profile [circled in Figure 3.2(d)] usually signify a change in the mechanism, i.e. one group of atoms stops moving and another group starts. By comparing (b) and (d) in Figure 3.2 we conclude that flatter points on the energy profile correlate with the most cooperative parts of this rearrangement.

A simple correlation between barrier heights and $ N_p$ and $ N_c$ does not seem to exist. The barrier height is not a function of cooperativity alone, but also of the energetics of the participating atoms. The way the $ N_p$ and $ N_c$ indices have been defined can make them insensitive to details of the rearrangements that will affect the energetics. For instance, neither index depends on the location of the participating atoms or the directionality of their motion. In most cases cooperatively moving particles are adjacent, i.e. localised in space; however, long-distance correlations of atomic displacements also occur. One such case is depicted in Figure 3.5 (a). This path is nearly symmetric with respect to the integrated path length ( $ \pi = 0.01$), but is very asymmetric with respect to the uphill and downhill barrier heights ( $ \beta = 0.91$). This rearrangement has $ N_p = N_c = 10$. Interestingly, $ N_p$ and $ N_c$ calculated separately for both sides of the pathway are very similar, i.e. the two steepest-descent pathways cannot be distinguished using these indices. Close inspection of this rearrangement reveals that one side of the pathway involves the rearrangement of two atomic triplets that share a vertex, while the other side involves the drift of all five atoms on the surface of the cluster [see the insets in Figure 3.5 (a)]. Although $ N_c$ does not distinguish these cases, the motion in the second side of the path is more cooperative. The participating atoms move together, which results in a significantly lower downhill barrier.

Figure: Two limitations of the cooperativity index $ N_c$. (a) $ N_c$ is not sensitive to the spatial positions of the cooperatively moving atoms, nor to the directionality of their motion. The energy profile is depicted for a rearrangement of the LJ$ _{75}$ cluster, which is very asymmetric with respect to barrier height but has similar integrated path lengths on either side of the transition state. The cooperativity index $ N_c$ evaluated separately for the two sides is about $ 10$ in both cases. The motion of the five atoms that displace the most is shown schematically relative to a reference atom (black). (b) The displacement of two (left) and three (right) atoms $ d$ as a function of the frame number $ j$ is shown schematically for a hypothetical pathway. The rearrangement on the left is more cooperative because two atoms move together over a longer region of the path. However, the current definition of $ N_c$ does not distinguish between these two cases.
\begin{psfrags} \psfrag{e1} [br][br]{$-391.0$} \psfrag{e2} [br][br]{$-390.5$... ...ine{\includegraphics[width=.56\textheight]{coop/limitations.eps}} \end{psfrags}

$ N_p$ and $ N_c$ also describe properties of the whole pathway. A significant number of pathways that we observed were rather non-uniform, i.e. very cooperative phases alternated with uncooperative ones. To distinguish such pathways in the LJ$ _{75}$ database we calculated a set $ \{N_p\}$ containing $ N_p$'s evaluated for each pair of adjacent frames. Then a selection of pathways was made with $ m_2/(m'_1)^2 < 0.01$, where $ m_2/(m'_1)^2$ is the moment ratio evaluated for the set $ \{N_p\}$. While this procedure ensured that $ N_p$ corresponds closely to the number of atoms that moves between any two snapshots of the rearrangement, it did not distinguish cases where different atoms contribute to the value of $ N_p$ in different frames [see Figure 3.5 (b)]. The average uphill and downhill barriers for this subset of rearrangements are $ 100$ times smaller than the average barriers for the complete LJ$ _{75}$ database (Table 3.1). Figure 3.6 shows that $ N_p$ and $ N_c$ calculated for these rearrangements are highly correlated and span a range of values, implying that widely different pathways are represented. Finally, all the selected pathways are an order of magnitude shorter than the average path length for the whole database, even though this database contains many short rearrangements localised on one atom.


Table: Average uphill and downhill barriers, average integrated path length for the LJ$ _{75}$ rearrangement pathway database. Values are given for the whole database containing $ 31,342$ paths, and for a subset containing the $ 57$ most cooperative paths. The units of energy and distance are $ \epsilon $ and $ \sigma $, respectively.
  All Cooperative
Uphill barrier 3.03 0.06
Downhill barrier 0.97 0.03
Path length 3.08 0.58

Figure: $ N_c$ as a function of $ N_p$ calculated for the $ 57$ most cooperative rearrangements from the LJ$ _{75}$ pathway database.
\begin{psfrags} \psfrag{5} [bc][bc]{$5$} \psfrag{10} [bc][bc]{$10$} \psfra... ...centerline{\includegraphics[width=.46\textheight]{coop/coop.eps}} \end{psfrags}

Figure 3.7 shows the values of the $ N_p$ and $ N_c$ indices plotted against each other for pathway databases calculated for LJ$ _{13}$, LJ$ _{75}$ and BLJ$ _{256}$. The two databases for BLJ$ _{256}$ labelled as $ 1$ and $ 12$ are taken from Reference middletonw01 and correspond to databases BLJ1 and BLJ12 in that paper. BLJ1 and BLJ12 were obtained using two different sampling schemes intended to provide an overview of a wide range of configuration space and a thorough probe of a smaller region, respectively. These databases were constructed by systematic exploration of the PES, and we refer the reader to the original work for further details [171]. Each BLJ$ _{256}$ database contains $ 10,000$ transition states. The LJ$ _{13}$ and LJ$ _{75}$ databases were obtained in discrete path sampling (DPS) studies [10,8]. and contain $ 28,756$ and $ 31,342$ transition states, respectively. Figure 3.7 is a density plot where darker shading signifies a higher concentration of data points. The outlying points are connected by a solid line to define the area in which all the points lie. Figure 3.7 shows that as $ N_p$ grows the allowed range of $ N_c$ increases, especially for LJ$ _{13}$. For the LJ$ _{75}$ database rearrangements with $ N_c > N/2$ appear to be very rare or poorly sampled. Figure 3.7 also shows that for all these systems rearrangements localised on two or three atoms dominate. This result may be an intrinsic property. However, it may also be due to the geometric perturbation scheme used in producing the starting points for the transition state searches employed in generating these databases. For databases BLJ1 and BLJ12 there are significantly more rearrangements with larger values of $ N_p$ and $ N_c$ compared to LJ$ _{75}$, which suggests that the abundance of very localised rearrangements for clusters may be a surface effect. The apparent absence of cooperative rearrangements in LJ$ _{75}$ database for large values of $ N_p$ may be due to the fact that only the pathways between compact phases of this cluster were sampled thoroughly, i.e. rearrangements involving liquid-like structures that are expected to have larger values of $ N_c$ are probably underrepresented.

Figure: $ N_c$ as a function of $ N_p$ calculated from four pathway databases for LJ$ _{13}$, LJ$ _{75}$ and BLJ$ _{256}$ systems. Due to the large number of data points we employ a density plot, where the darkest shading corresponds to the highest concentration of points. Outlying points are connected to illustrate the boundaries of the data area. The two BLJ databases are taken from Reference middletonw01.
\begin{psfrags} \psfrag{0} [bc][bc]{$0$} \psfrag{2} [bc][bc]{$2$} \psfrag{... ...terline{\includegraphics[width=.71\textheight]{coop/ntildes.eps}} \end{psfrags}

Figure 3.8 depicts the average barrier as a function of the participation and cooperativity indices. $ N_p$ and $ N_c$ were calculated separately for both sides of the pathway and the corresponding barriers (uphill or downhill) were averaged to produce a density plot of barrier height. Data boundaries do not coincide with those shown in Figure 3.7 due to numerical imprecision in preparing Figure 3.7. Figure 3.8 illustrates that for each system cooperative rearrangements have the lowest barriers, irrespective of the value of $ N_p$. For clusters, cooperative rearrangements have lower barriers than uncooperative rearrangements with $ N_p$ as small as $ 1-3$, while for bulk barriers corresponding to rearrangements with low $ N_p$ become comparable to these for very cooperative rearrangements with high $ N_p$.

Figure: Average barrier height as a function of $ N_p$ and $ N_c$ calculated for the same LJ$ _{13}$, LJ$ _{75}$ and BLJ$ _{256}$ pathway databases used in Figure 3.7. The indices were calculated separately for the two sides of each path. In this case the darkest shading corresponds to the highest barriers. Outlying points are connected to illustrate the boundaries of the data area. The two BLJ databases are taken from Reference middletonw01.
\begin{psfrags} \psfrag{0} [bc][bc]{$0$} \psfrag{2} [bc][bc]{$2$} \psfrag{... ...erline{\includegraphics[width=.71\textheight]{coop/barriers.eps}} \end{psfrags}

In further computational experiments we found that attempts to connect the endpoints of uncooperative pathways using the algorithm described in Chapter 2 either required more images and iterations or converged to an alternative pathway. In some cases additional difficulties arose, such as convergence to a higher index saddle instead of a transition state, which can happen if the linear interpolation guess conserves a symmetry plane. Figure 3.9 shows $ N_c$ calculated from Equation 3.13 plotted against $ N_p$ for the LJ$ _{75}$ pathway database. Knowing the integrated path length, $ s$, for each pathway we started doubly nudged elastic band calculations with three images per unit of distance and $ 30$ iterations per image. Most of the points that correspond to runs that failed or converged to an alternative pathways are concentrated in the region of small values for $ N_c/N_p$.

Figure: $ N_c$ as a function of $ N_p$ calculated for the LJ$ _{75}$ pathway database. For each pathway we conducted DNEB calculations [7] assuming prior knowledge of the path. Every DNEB calculation employed $ 3s$ images and $ 90s$ iterations in each case, where $ s$ is the integrated path length. Out of $ 31,342$ DNEB runs $ 25,158$ yielded a connected pathway while the rest did not (FAILED). Connected pathways are classified further as one-step pathways involving the correct transition state (OK) or an alternative transition state (ALT), or as multi-step pathways (MULTI), which involve more than one transition state. For each set of data points best fit straight lines obtained from linear regression are shown and labelled appropriately.
\begin{psfrags} \psfrag{5} [bc][bc]{$5$} \psfrag{10} [bc][bc]{$10$} \psfra... ...ne{\includegraphics[width=.46\textheight]{coop/dnebproblems.eps}} \end{psfrags}

As can be seen from Figure 3.7, the LJ$ _{13}$ database contains significantly more pathways with large values of $ N_p$ and $ N_c$ compared to LJ$ _{75}$, where most of the rearrangements are localised on two atoms. The fact that the LJ$ _{13}$ database is almost exhaustive then suggests that localised rearrangements either start to dominate as the system size increases or that the sampling scheme used for LJ$ _{75}$ was biased towards such mechanisms. Systematic sampling of the configuration space for stationary points often employs perturbations of every degree of freedom followed by minimisation [162]. The LJ$ _{13}$ database was obtained in this fashion, while the LJ$ _{75}$ database was generated during the DPS approach [8]. In this procedure discrete paths are perturbed by replacing local minima with structures obtained after perturbing all the coordinates and minimising. To investigate whether the perturbation scheme can affect the resulting database of stationary points in more detail we consider the case of LJ$ _{13}$, since nearly all the transition states are known. Figure 3.10 presents the results of two independent runs aimed at locating most of the transition states for this system. Every cycle a perturbation was applied to a randomly selected transition state from the database and the resulting geometry was used as a starting point for a new transition state search using eigenvector-following [22,23,72,24,19]. Only distinct permutation-inversion isomers were saved. In the first run (bottom curve) every degree of freedom was perturbed by $ 0.4 x$, where $ x$ is a random number in the interval $ [-1,1]$ [162]. For the second run (top curve) we introduced a perturbation scheme including correlation. $ 2 \leqslant K \leqslant N/2$ atoms out of $ N$ were displaced by a vector $ 0.4(x_1, x_2, x_3)$, where the components $ x_1$, $ x_2$ and $ x_3$ are again random numbers drawn from $ [-1,1]$. The $ K$ atoms to be displaced were selected based on their relative positions in the cluster. One atom was first selected at random, while the remaining $ K-1$ were chosen to be its closest neighbours. The top curve was generated from a run with $ K=6$. Both runs required approximately the same time to produce two nearly identical databases, each containing about $ 29,000$ pathways. However, as can be seen from Figure 3.7, random perturbation of all the degrees of freedom results in uncooperative rearrangements being found first, while employing correlated perturbations has the opposite effect.

Figure: The average value of $ N_c$ for LJ$ _{13}$ pathway databases as new paths are added. 6-atom correlated perturbations (top curve) and random perturbations of every degree of freedom (bottom curve) were used to produce starting points for refinement by eigenvector-following [22,23,72,24,19]. Average values were calculated every time $ 100$ new pathways were added.
\begin{psfrags} \psfrag{5} [bc][bc]{$5$} \psfrag{10} [bc][bc]{$10$} \psfra... ...erline{\includegraphics[width=.43\textheight]{coop/sampling.eps}} \end{psfrags}

next up previous contents
Next: Summary Up: PROPERTIES OF REARRANGEMENT PATHWAYS Previous: Cooperativity   Contents
Semen A Trygubenko 2006-04-10