Summary of Contributions

In this thesis I have tried to improve upon existing double-ended methods for finding rearrangement pathways as well as methods for extracting kinetic information from pathway ensembles. The main accomplishments are as follows:

(I)

We presented a graph transformation (GT) method, which can be used to calculate the total transition probabilities and mean escape times for arbitrary digraphs with arbitrary sets of sources and sinks that are allowed to overlap. At low temperatures the GT method becomes the method of choice, outperforming kinetic Monte Carlo and matrix multiplication methods.

(II)

We have suggested a version of the GT method (SDGT) that can take full advantage of the sparsity of the problem. Apart from switching to the standard sparse-optimised adjacency-list-based data structure, the modifications were the implementation of Fibonacci-heap-based min-priority queue to ensure that nodes with smaller degrees are detached first, and an algorithm that monitors the graph density and switches to the dense-optimised version of the method when it becomes computationally cost-effective.

(III)

The stability of the NEB method was improved by introducing a portion of the spring gradient component perpendicular to the path back into the NEB gradient.

(IV)

The efficiency of the DNEB method was improved by eliminating the removal of the overall rotation and translation and employing a quasi-Newton method (L-BFGS) for minimisation of the band.

(V)

The efficiency and stability of the QVV minimiser was increased by finding the optimal point in time of quenching the velocity.

(VI)

We have devised a method for finding rearrangement pathways between distant local minima, which is based on the consecutive DNEB searches and uses the Euclidean distance as a measure of separation in configuration space. Part of this work that concerned the Dijkstra-based selector was performed in collaboration with Dr. Joanne M. Carr [167].

(VII)

A new cooperativity index, introduced in Chapter 3, enabled us to find a correlation between the cooperativity of an atomic rearrangement and the energy barrier. We showed that cooperative rearrangements of LJ clusters and the BLJ liquid have lower energy barriers irrespective of the degree of localisation.

(VIII)

We have demonstrated that it is possible to control the overall cooperativity of the pathway sample, and outlined a technique for sampling cooperative pathways using single-ended transition state searching methods.

(IX)

We have described the edge weight function that allows us to find the path with the largest DPS non-recrossing rate constant using the Dijkstra algorithm. We have also described an algorithm for sampling for the fastest paths.

(X)

We have devised a method for computing a recrossing contribution to the DPS rate constant exactly in linear time with constant memory requirements.

(XI)

We have obtained recursive expressions for the total transition probabilities from an arbitrary digraph by considering the corresponding pathway ensembles.

Because random walk has applications in many areas of sciences, from Brownian motion [120] and diffusion [121] in physics to dynamics of stock markets in economics [281] and tumour angiogenesis in medicine [282], we expect methods developed in Chapter 4 to be relevant to a much wider domain.