Acknowledgements

I am glad I have the opportunity to express my sincere gratitude to all the people that accompanied and supported me during my study.

I had the pleasure of being supervised by Dr. David Wales and would like to thank him for the high-quality guidance I received in the duration of the Ph.D. course.

The second person I gratefully acknowledge is Dr. Catherine Pitt who introduced me to the Debian GNU/Linux operating system and various scientific applications that were heavily used throughout the development of this work.

My appreciation also goes to Dr. Thomas Middleton for providing transition state databases for the binary Lennard-Jones system, Dr. David Evans for his Dijkstra-based utilities that locate shortest paths in connected databases of minima and transition states, Dr. Mark Miller for many KMC-related discussions as well as his programs for creating and manipulating disconnectivity graphs, and Dr. Dmytro Hovorun and Dr. Pavel Hobza who greatly influenced my decision to do a Ph.D. in this area.

The chain of my gratitude would definitely be incomplete without special thanks to Tetyana Bogdan, Dr. Joanne Carr and Tim James, for many stimulating discussions we had on various aspects of my research, the Life, the Universe and everything.

Lastly, I would like to thank Cambridge Commonwealth Trust, Cambridge Overseas Trust and Darwin College for financial support. Most of the calculations were performed using computational facilities funded by the Isaac Newton Trust.

To My

Teachers

Abstract

A modification of the nudged elastic band (NEB) method is presented that enables stable optimisations to be run using both the limited-memory quasi-Newton (L-BFGS) and slow-response quenched velocity Verlet minimisers. The performance of this new ‘doubly nudged’ DNEB method is analysed in conjunction with both minimisers and compared with previous NEB formulations. We find that the fastest DNEB approach (DNEB/L-BFGS) can be quicker by up to two orders of magnitude. Applications to permutational rearrangements of the seven-atom Lennard-Jones cluster (LJ${}_{7}$) and highly cooperative rearrangements of LJ${}_{38}$ and LJ${}_{75}$ are presented.

Secondly, we propose new measures of localisation and cooperativity for the analysis of atomic rearrangements. We show that for both clusters and bulk material cooperative rearrangements usually have significantly lower barriers than uncooperative ones, irrespective of the degree of localisation. We also find that previous methods used to sample stationary points are biased towards rearrangements of particular types. Linear interpolation between local minima in double-ended transition state searches tends to produce cooperative rearrangements, while random perturbations of all the coordinates, as sometimes used in single-ended searches, has the opposite effect.

Thirdly, we report a new algorithm for constructing pathways between local minima that involve a large number of intervening transition states on the PES. A significant improvement in efficiency has been achieved by changing the strategy for choosing successive pairs of local minima that serve as endpoints for the next search. We employ Dijkstra’s algorithm to identify the ‘shortest’ path corresponding to missing connections within an evolving database of local minima and the transition states that connect them.

Finally, we describe an exact approach for calculating the total transition probabilities in finite-state discrete-time Markov processes. All the states and the rules for transitions between them must be known in advance. We can then calculate averages over a given ensemble of paths for both additive and multiplicative properties in a non-stochastic and non-iterative fashion. In particular, we can calculate the mean first passage time between arbitrary groups of stationary points for discrete path sampling databases, and hence extract phenomenological rate constants. We present a number of examples to demonstrate the efficiency and robustness of this approach.