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$ N_p$ atoms can participate in a rearrangement according to a continuous range of cooperativity. At one end of the scale there are rearrangements where $ N_p$ atoms all move simultaneously [see Figure 3.2 (a)]. Although these paths exhibit the highest degree of correlated atomic motion they do not usually pose a problem for double-ended transition state search algorithms [7,30]. Linear interpolation between the two minima tends to generate initial guesses that lie close to the true pathway, particularly if $ \alpha \sim 0$. At the opposite extreme, atoms can move almost one at a time, following a `domino' pattern [see Figure 3.2 (b)]. Locating a transition state for such rearrangements may require a better initial guess, since linear interpolation effectively assumes that all the coordinates change at the same rate.

Figure: Comparison of cooperative (a) and uncooperative (b) rearrangements of the LJ$ _{75}$ cluster, for mechanisms that are localised mainly on two atoms. The displacement $ d$ as a function of the frame number $ j$ is shown for the two atoms that move the most. Panels (c) and (d) illustrate the potential energy $ V$ as a function of frame number $ j$ for the rearrangements in (a) and (b), respectively. Dashed circles indicate flatter parts of the energy profile, which correspond to the most cooperative regions of the pathway.
\psfrag{y1} [bc][bc]{$0.5$}
\psfrag{t1} [bc][bc]{$\Theta_1=0...{\includegraphics[width=.75\textheight]{coop/displacement.eps}}

The degree of correlation in the atomic displacements can be quantified by considering the displacement `overlap'

$\displaystyle O_{k} = \sum_{j=2}^{N_f} O_{k}\left(j\right) = \sum_{j=2}^{N_f} {\rm min}\Bigl[ d_{c(1)}(j), d_{c(2)}(j), ..., d_{c(k)}(j)\Bigr],$ (5.11)

where the index $ k$ indicates that $ O$ was calculated for $ k$ atoms numbered $ c(1)$, $ c(2)$,...,$ c(k)$. $ {\bf c}$ is a $ k$-dimensional vector that represents a particular choice of $ k$ atoms from $ N$, and hence there are $ C_N^k = N!/k!(N-k)!$ possible values of $ O_{k}$. The index $ O$ can be thought of as a measure of how the displacements of the atoms $ c(1)$, $ c(2)$, etc. overlap along the pathway. For example, if two atoms move at different times then $ O_2$ is small for this pair because the minimum displacement in Equation 3.11 is always small. However, if both atoms move in the same region of the path then $ O_2$ is larger.

We now explain how the statistics of the overlaps, $ O_k$, can be used to extract a measure of cooperativity (Figure 3.3). Suppose that $ m$ atoms move simultaneously in a hypothetical rearrangement. Then all the overlaps $ O_k$ for $ k > m$ will be relatively small, because one or more atoms are included in the calculation whose motion is uncorrelated with the others. For overlaps $ O_k$ with $ k \leqslant m$ the set of $ O_k$ for all possible choices of $ k$ atoms from $ N$ will exhibit some large values and some small. The large values occur when all the chosen atoms are members of the set that move cooperatively, while other choices give small values of $ O_k$. Hence the kurtosis of the set $ \{O_k\}$, $ \gamma'(O_k)$, calculated from moments taken about the origin, will be large for $ k \leqslant m$, and small for $ k > m$.

To obtain a measure of how many atoms move cooperatively we could therefore calculate $ \gamma '(O_2)$, $ \gamma'(O_3)$, etc. and look for the value of $ k$ where $ \gamma'(O_k)$ falls in magnitude. However, to avoid an arbitrary cut-off, it is better to calculate the kurtosis of the set $ \{\gamma'(O_2)$, $ \gamma'(O_3),...,\gamma'(O_k)\}$, or $ \gamma'[\gamma'(O)]$ for short. There are $ N-2$ members of this set, and by analogy with the definition of $ N_p = N/\gamma'(d)$, we could define a cooperativity index $ N_c = (N-2)/\gamma'[\gamma'(O)] + 1$. Then, if $ \gamma '(O_2)$ is large, and all the other $ \gamma'(O_k)$ are small, we obtain $ \gamma'(\gamma'(O)) \sim N-2$ and $ N_c \sim 2$, correctly reflecting the number of atoms that move together.

Figure: $ \gamma '(O_2)$ plotted against $ \gamma '(d)$ for the LJ$ _{75}$ pathway database. The figure shows how $ \gamma '(O_2)$ can discriminate between rearrangements that have similar values of $ \gamma '(d)$ but different cooperativity. The data point for the most cooperative rearrangement localised on two atoms has $ N_p=N_c=2$.
\psfrag{10} [bc][bc]{$10$}
\psfrag{20} [bc][bc]{$20$}

In practice, there are several problems with the above definition of $ N_c$. Calculating $ N_c$ in this way quickly becomes costly as the number of atoms and/or number of frames in the pathway increases, because the number of elements in the set $ \{O_k\}$ varies combinatorially with $ k$. Secondly, as $ k$ approaches $ N$ the distribution of all the possible values for $ O_{k}$ becomes more and more uniform. Under these circumstances deviations from the mean that are negligible in comparison with the overall displacement can produce large kurtosises. Instead, we suggest a modified (and simpler) definition of $ N_c$, which better satisfies our objectives.

We first define the overlap of atomic displacements in a different manner. It can be seen from Equation 3.11 that the simultaneous displacement of $ l$ atoms is included in each set of overlaps $ \{O_k\}$ with $ k \leqslant l$. For example, if three atoms move cooperatively then both the $ \{O_2\}$ and $ \{O_3\}$ sets will include large elements corresponding to these contributions. Another redundancy is present within $ \{O_k\}$, since values in this set are calculated for all possible subsets of $ k$ atoms and the displacement of each atom is therefore considered more than once. However, we can avoid this redundancy by defining a single $ k-$overlap, rather than dealing with $ C_N^k$ different values.

Recall that $ d_i(j)$ is the displacement of atom $ i$ between frames $ j-1$ and $ j$. The ordering of the atoms is arbitrary but remains the same for each frame number $ j$. We now define $ \Delta_i(j)$ as the displacement of atom $ i$ in frame $ j$, where index $ i$ numbers the atoms in frame $ j$ in descending order, according to the magnitude of $ d_i(j)$, e.g. atom $ 1$ in frame $ 2$ is now the atom with the maximum displacement between frames $ 1$ and $ 2$, atom $ 2$ has the second largest displacement etc. As the ordering may vary from frame to frame, the atoms labelled $ i$ in different frames can now be different. This relabelling greatly simplifies the notation we are about to introduce. Consider the $ k$-overlap defined as 

$\displaystyle \Theta_k = \frac{1}{\Delta_{tot}}\sum_{j=2}^{N_f} \Bigl[ \Delta_k(j) - \Delta_{k+1}(j)\Bigr],$ (5.12)

where $ k$ ranges from $ 1$ to $ N$, $ \Delta_{tot}=\sum_{j=2}^{N_f} \Delta_1(j)$ and $ \Delta_{N+1}(j)$ is defined to be zero for all $ j$. A schematic illustration of this construct is presented in Figure 3.4. For example, if only two atoms move in the course of the rearrangement, and both are displaced by the same amount (which may vary from frame to frame), the only non-zero overlap will be $ \Theta _2$.

Figure: The $ \Theta $ indices for a hypothetical rearrangement localised on three atoms. For each of these atoms the displacement $ d$ as a function of frame number $ j$ is shown. The $ d_i$ values in successive frames are connected with dotted ($ d_1$), dashed ($ d_2$) and solid ($ d_3$) lines. The corresponding contributions to $ \Theta _1$, $ \Theta _2$, and $ \Theta _3$ are shown as shaded squares and are labelled accordingly. If the remaining $ N-3$ atoms do not participate, and the area of one square is $ S$, the only non-zero overlaps will be $ \Theta _1$, $ \Theta _2$, and $ \Theta _3$ with values $ 5/9$, $ 3/9$ and $ 1/9$, respectively.
\psfrag{d} [bc][bc]{$d_i(j)$}
\psfrag{j} [bc][bc]{$j$}

We can now define an index to quantify the number of atoms that move cooperatively as

$\displaystyle N_c = \sum_{k=1}^{N} k \Theta_k.$ (5.13)

If only one atom moves during the rearrangement then $ N_c = 1$, while if $ K$ atoms displace cooperatively during the rearrangement then $ N_c = K$. This definition is independent of the total displacement, the integrated path length, and the number of atoms, which makes it possible to compare $ N_c$ indices calculated for different systems.

next up previous contents
Next: Applications to LJ and Up: PROPERTIES OF REARRANGEMENT PATHWAYS Previous: Localisation   Contents
Semen A Trygubenko 2006-04-10