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The outcome of a pathway calculation for an atomic system will generally be a set of intermediate geometries, and the corresponding energies, for points along the two unique steepest-descent paths that link a transition state to two local minima. This discrete representation is a convenient starting point for our analysis of localisation and cooperativity. We number the structures along the path $ j=1,\2,\ldots,\ N_f$  starting from one of the two minima and reversing the other steepest-descent path, so that structure $ N_f$ corresponds to the other minimum. The transition state then lies somewhere between frames $ 1$ and $ N_f$. We define the three-dimensional vector $ {\bf r}_i(j)$  to contain the Cartesian coordinates of atom $ i$ for structure $ j$, i.e.  $ {\bf r}_i(j)= \Bigl(X_i(j),Y_i(j),Z_i(j)\Bigr)$, where $ X_i(j)$ is the $ X$ coordinate of atom $ i$ in structure $ j$, etc.

For each atom $ i$ we also define the displacement between structures $ j-1$ and $ j$ as 

$\displaystyle d_i(j) = \Bigl\vert {\bf r}_i(j) - {\bf r}_i(j-1) \Bigr\vert.$ (5.1)

Hence the sum of displacements 

$\displaystyle d_i = \sum_{j=2}^{N_f} d_i(j),$ (5.2)

is an approximation to the integrated path length for atom $ i$, which becomes increasingly accurate for smaller step sizes. The total integrated path length, $ s$, is approximated as

$\displaystyle s = \sum_{j=2}^{N_f} \sqrt{\sum_{i=1}^N d_i\left(j\right)^2},$ (5.3)

where $ N$ is the total number of atoms. $ s$ is a characteristic property of the complete path, and is expected to correlate with parameters such as the curvature and barrier height for short paths [178,9,177].

The set $ \{d_1,d_2,\ldots,d_N\}$ containing all $ N$ values of $ d_i$ will be denoted $ \{d\}$, and analogous notation will be used for other sets below. We will also refer to the frequency distribution function, which can provide an alternative representation of such data [179]. For example, the frequency distribution function $ \mF $ for a given continuous variable, $ x$, tells us that $ x$ occurs in a certain interval $ \mF (x)$ times.

Our objective in the present analysis is to provide a more detailed description of the degree of `localisation' and `cooperativity' corresponding to a given pathway. The first index we consider is $ N_p$, which is designed to provide an estimate of how many atoms participate in the rearrangement. We will refer to a rearrangement as localised if a small fraction of the atoms participate in the rearrangement, and as delocalised in the opposite limit. The second index we define, $ N_c$,is intended to characterise the number of atoms that move simultaneously, i.e. cooperatively. We will refer to a rearrangement as cooperative if most of the atoms that participate in the rearrangement move simultaneously, and as uncooperative otherwise.

The $ n$th moment about the mean for a data set $ \{x_1, x_2, ... ,x_M \}$ is the expectation value of $ (x_i - \langle x\rangle )^n$, where $ \langle x \rangle =\sum_{i=1}^M x_i /M$ and $ M$ is the number of elements in the set. Hence for the set $ \{d\}$ defined above we define the moments, $ m_n$, as

$\displaystyle m_n = \dfrac{1}{N}\sum_{i=1}^N \left(d_i - \langle d \rangle \right)^n.$ (5.4)

The kurtosis of the set $ \{d\}$ is then defined as the dimensionless ratio 

$\displaystyle \gamma(d) = \dfrac{m_4}{(m_2)^2},$ (5.5)

and provides a measure of the shape of the frequency distribution function corresponding to $ \{d\}$. If only one of the atoms moves, or all atoms except one move by the same amount, then $ \gamma(d) = N$. Alternatively, if half the atoms move by the same amount whilst the others are stationary, then $ \gamma(d) = 1$. Hence, a distribution with a broad peak and rapidly decaying tails will have a small kurtosis, $ \gamma\sim{\cal O}(1)$, while a distribution with a sharp peak and slowly decaying tails will have a larger value (Figure 3.1). The kurtosis can therefore identify distributions that contain large deviations from the average value [179]. For comparison, a Gaussian distribution has $ \gamma=3$ and a uniform distribution has $ \gamma=1.8$.

Figure: Two frequency distribution functions $ \mF _1$ and $ \mF _2$ of a continuous variable $ x$ are contrasted. Both functions have the same average $ m'_1$ and standard deviation $ \sqrt m_2$. However, due to the long tails, $ \mF _1$ has a significantly larger fourth moment $ m_4$ and, hence, a larger kurtosis, $ \gamma $. If $ \gamma >3$ the distribution is said to be peaked or leptocurtic. Distributions with $ \gamma >3$ are known as platycurtic or heavy-tailed.
\psfrag{f} [bc][bc]{$\mF _1, \mF _2$}
\psfrag{f1} [bc][bc]{$...

The above results show that the kurtosis $ \gamma(d)$ can be used to quantify the degree of localisation or delocalisation of a given rearrangement. However, it has the serious disadvantage that highly localised and delocalised mechanisms both have large values of $ \gamma(d)$. Since we are interested in estimating the number of atoms that move relative to the number with small or zero displacements, a better approach is to use moments taken about the origin, rather than about the mean, i.e.

$\displaystyle m'_n = \dfrac{1}{N} \dsum _{i=1}^N d_i^n,$ (5.6)

following Stillinger and Weber [152]. Note that while $ m_1=0$, the first moment $ m'_1$ is the mean value. We therefore estimate the number of atoms that participate in the rearrangement, $ N_p$, as

$\displaystyle N_p = \frac{N}{\gamma'(d)},$ (5.7)


$\displaystyle \gamma'(d) = \dfrac{m'_4}{\left(m'_2\right)^2}.$ (5.8)

For the system with $ N$ atoms, if only one atom moves $ N_p=1$, while if $ K$ atoms move by the same amount, $ N_p=K$.

A similar index to $ N_p$ has been employed in previous work [9,180,19] using only the displacements between the two local minima, which corresponds to taking $ N_f=2$ in Equation 3.2. Using $ d_i$ values based upon a sum of displacements that approximates the integrated path length for atom $ i$, rather than the overall displacement between the two minima, better reflects the character of the rearrangement, as it can account for the nonlinearity of the pathway. To describe this property more precisely we introduce a pathway nonlinearity index defined by 

$\displaystyle \alpha = \dfrac{s - D}{s},$ (5.9)

where $ D$ is the Euclidean distance  between the endpoints,

$\displaystyle D = \sqrt{\sum_{i=1}^{N} \left( {\bf r}_i(N_f)-{\bf r}_i(1) \right)^2}.$ (5.10)

We calculated the $ \alpha $ values for a database of 31,342 single transition state pathways of LJ$ _{75}$ (hereafter referred to as the LJ$ _{75}$ database). The average value of $ \alpha $ was $ 0.4$ with a standard deviation of $ 0.2$, and, hence, there is a significant loss of information if $ \gamma'$ is calculated only from the endpoints using $ N_f=2$. Comparison of the two indices for the LJ$ _{75}$ database revealed many examples where neglect of intermediate structures produces a misleading impression of the number of atoms that move. The definition in Equation 3.7 is therefore suggested as an improvement on previous indices of localisation [9,152,180,19].

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