The number of elementary rearrangements increases exponentially with system size as for the number of transition states. For instance, there are approximately 30,000 such pathways on the PES of the 13-atom cluster bound by the Lennard-Jones potential. When permutation-inversion isomers are included, this number increases by a factor of order 
[9]. For PES's that support such a large number of stationary points a whole range of properties is spanned by the corresponding rearrangement pathways. Understanding these properties can be helpful in answering questions such as why some rearrangement pathways are harder to find than others and whether there is any correlation between these properties that we could potentially utilise in studying PES's.
Two activation barriers can be defined for each pathway in terms of the energy difference between the transition state and each of the minima. For non-degenerate rearrangements [9,148] the two sides of the path are termed uphill and downhill, where the uphill barrier is the larger one, which leads to the higher minimum. The barriers and the normal modes of the minima and transition states can be used to calculate rate constants using harmonic transition state theory [48,45,46]. More sophisticated treatments based on anharmonic densities of states are possible but can be hard to reconcile with the coarse-grained view of the landscape adopted here, as a detailed knowledge of the basins of attraction is required [169,168].
For each local minimum a catchment basin can be defined in terms of all the configurations from which steepest-descent paths lead to that minimum [170]. Some of these paths originate from transition states on the boundary of the catchment basin, which connect a given minimum to adjacent minima. The integrated path length for such rearrangements provides a measure of the separation between local minima, and may be related to the density of stationary points in configuration space. The integrated path length is usually approximated as the sum of Euclidean distances between configurations sampled along appropriate steepest-descent paths [9]. It provides a convenient coordinate for monitoring the progress of the reaction.
Calculated pathways can always be further classified mechanistically. For example, some rearrangements preserve the nearest-neighbour coordination shell for all the atoms. In previous studies of bulk models these cage-preserving pathways were generally found to outnumber the more localised cage-breaking processes, which are necessary for atomic transport [171]. It was found that the barriers for cage-breaking and cage-preserving processes were similar for bulk LJ systems, while the cage-breaking mechanisms have significantly higher barriers for bulk silicon modelled by the Stillinger-Weber potential [171].
For minima separated by increasing distances in configuration space, the pathways that connect them are likely to involve more and more elementary steps, and are not unique. Finding such paths in high-dimensional systems can become a challenging task [7,8]. Some difficulties have been attributed to instabilities and inefficiencies in transition state searching algorithms [107,7], as well as the existence of very different barrier height and path length scales [7]. A new algorithm for locating multi-step pathways in such cases was presented in the previous chapter.
In the present work we have used the doubly nudged elastic band (DNEB) method [7] in conjunction with eigenvector-following (EF) algorithms [17,18,12,14,24,13,19,16,22,21,23,20,15] to locate rearrangement pathways in various systems. The LJ potential was used to describe the 
- and 
-atom Lennard-Jones clusters, LJ
and LJ
. We have also considered a binary LJ (BLJ) system with parameters 
, 
, 
, 
, 
, 
, where 
and 
are atom types. The mixture with 
ratio 
provides a popular model bulk glass-former [172,171]. We employed a periodically repeated cubic cell containing 
atoms and 
atoms. The density was fixed at 
and the Stoddard-Ford scheme was used to prevent discontinuities [173].
The motivation for this work was our observation that construction of some multi-step pathways using the connection algorithm described in Chapter 2 is particularly difficult. We were unable to relate these difficulties to simple properties of the pathways such as the integrated path length, the uphill and downhill barriers, or the barrier and path length asymmetries. Instead, more precise measures of localisation and cooperativity are required, as shown in the following sections. It also seems likely that such tools may prove useful in analysing the dynamics of supercooled liquids, where processes such as intrabasin oscillations and interbasin hopping have been associated with different rearrangement mechanisms [174]. In particular, cooperativity is believed to play an important role at low temperatures in glass-forming systems [175], and dynamic heterogeneity may result in decoupling between structural relaxation and transport properties for supercooled liquids [176].