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 starting from one of the two minima and reversing the other steepest-descent path, so that structure corresponds to the other minimum. The transition state then lies somewhere between frames and . We define the three-dimensional vector to contain the Cartesian coordinates of atom for structure , i.e. , where is the coordinate of atom in structure , etc.

For each atom we also define the displacement between structures and as

(5.1) |

Hence the sum of displacements

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

(5.3) |

where is the total number of atoms. 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 containing all values of will be denoted , 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 for a given continuous variable, , tells us that occurs in a certain interval 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 , 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, ,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 th moment about the mean for a data set is the expectation value of , where and is the number of elements in the set. Hence for the set defined above we define the moments, , as

(5.4) |

The kurtosis of the set is then defined as the dimensionless ratio

(5.5) |

and provides a measure of the shape of the frequency distribution function corresponding to . If only one of the atoms moves, or all atoms except one move by the same amount, then . Alternatively, if half the atoms move by the same amount whilst the others are stationary, then . Hence, a distribution with a broad peak and rapidly decaying tails will have a small kurtosis, , 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 and a uniform distribution has .

The above results show that the kurtosis 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 . 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.

(5.6) |

following Stillinger and Weber [152]. Note that while , the first moment is the mean value. We therefore estimate the number of atoms that participate in the rearrangement, , as

where

(5.8) |

For the system with atoms, if only one atom moves , while if atoms move by the same amount, .

A similar index to has been employed in previous work [9,180,19] using only the displacements between the two local minima, which corresponds to taking in Equation 3.2. Using values based upon a sum of displacements that approximates the integrated path length for atom , 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

(5.9) |

where is the Euclidean distance between the endpoints,

(5.10) |

We calculated the values for a database of 31,342 single transition state pathways of LJ (hereafter referred to as the LJ database). The average value of was with a standard deviation of , and, hence, there is a significant loss of information if is calculated only from the endpoints using . Comparison of the two indices for the LJ 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].