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Working with a Coarse-grained Model

In Chapter 2 and Chapter 4 of this thesis coarse-grained models of a PES are discussed in graph-theoretical terms. Nowadays a flourishing branch of mathematics and computer science, graph theory arguably started in the year of 1736 with Leonhard Euler's paper on the seven bridges of Königsberg, where he abstracted from landmasses and bridges to highlight the connectivity, and proved that it is impossible to cross every bridge exactly once in a single walk that starts and ends at the same point.

A graph[1]In modern literature the term `network' is often used synonymously with the term `graph'. is defined as a set of nodes with connections between them called edges [111]. A coarse-grained picture of a PES therefore naturally fits into this definition, with nodes representing minima and edges representing the transition states that connect them. This approach was adopted in a number of previous energy landscapes studies, examples being the characterisation of dynamics in a region of a PES [113,112,114,8] and detailed topological analyses of semi-complete PES samples [115].

A directed edge is defined by its origin and destination nodes and can be travelled in one direction only. A graph composed of directed edges is termed a directed graph (digraph). Although every transition state facilitates both forward and reverse reactions, these are usually inequivalent, which leads to a symmetric digraph representation of a PES, i.e. to a digraph that is composed of pairs of complementary edges that share the same endpoints. Studies aimed at elucidating the global topology of a PES usually do not make such a distinction and deal with undirected graphs.

Interesting properties of the PES's of small Lennard-Jones clusters were recently discovered by Doye and Massen in a study of the corresponding undirected graph representations [115]. It appears that such graphs have features similar to both small-world and scale-free graphs. (A graph is said to have a small-world property if most pairs of nodes are connected by relatively short paths [111,116,117]. Such a graph can be easily obtained via a random rewiring of regular grid or lattice. The degree of a node is defined as the total number of its neighbours, and a graph is termed scale-free if its distribution of degrees follows a power law [111,116].) While scale-free graphs are usually obtained via preferential attachment during growth [118], the origin of scale-free topology in graphs corresponding to PES's may lie in an Apollonian-like [119] packing of the basins of attraction [115]. Following the `inherent structure' PES partitioning due to Stillinger and Weber [41,42], a basin of attraction of a minimum can be defined as a set of points in configuration space connected to that minimum via steepest-descent paths. Topological studies of PES's are important because they can provide further insights into the PES connectivity and, consequently, increase our understanding of the relationship between the structural organisation of the PES and the observed physical and chemical properties.

Describing physical phenomena such as, for example, Brownian motion [120] and diffusion [121], requires more sophisticated graph models that allow for different types of edges. In edge-weighted graphs a label (weight or cost) is associated with every edge. Such graphs will be used in this thesis for various purposes. For example, an important part of the path-finding method described in Chapter 2 is the undirected edge-weighted graph representation, where every node is connected to every other node via an edge with a weight that is a function of Euclidean distance. In Appendix E we explain how a weighted graph approach can be used to identify the fastest reaction pathways.

Both Brownian motion and diffusion were extensively studied in the past with the help of stochastic processes such as the random walk [121]. In mathematics and physics, a random walk is a formalisation of the intuitive idea of taking successive steps, each in a random direction [33]. If the number of possible directions is predefined and finite, a random walk is very easy to realize on an edge-weighted graph. A walk on a graph is defined as a sequence of edges such that the second node of each edge (except for the last edge) is the first node of the next edge. If the weights of the outgoing edges for every graph node add up to unity the graph is called probabilistic. A single step of a random walk confined to a probabilistic graph constitutes choosing the next graph node from the neighbours of the current node with a probability that equals the weight of the corresponding edge.

In Chapter 4 we will use random walks to model unimolecular chemical reactions [122]. An elementary reaction pathway is described as a transition from one state to a neighbouring state via a single transition state. Valid predictions of the sequence of such steps, known as the reaction mechanism, and a time scale associated with it are the holy grails of modern theoretical chemistry. Node-weighted probabilistic graphs are needed to address the time scale issue, because the time spent by a system before the transition occurs is likely to vary from state to state. An alternative description based on a probabilistic graph where each node is allowed to have a connection to itself is a more general approach to achieve this objective [123]. Both these models are employed in Chapter 4 in calculations of the average time for the reaction known as the mean first passage time.

The aforementioned graph model with self-connections is known by scientists in the fields of probability and stochastic processes as a discrete-time Markov chain -- a stochastic process with a discrete state space [123]. The question of the first moment of the distribution of first passage times in stochastic processes is one of the most basic ones and is over two hundred years old. The first mean first passage time calculation was probably done by Jacob Bernoulli in the beginning of eighteenth century. In his ground-breaking work on probability titled `The art of conjecture', which was posthumously published by his nephew Nicholas Bernoulli in 1713, he describes the techniques for calculating the duration of various games of chance [124]. A number of methods for efficient calculation of the mean first passage times was developed since then, some of which are yet to be fully utilised in chemical applications.

In this thesis I will attempt to address two important stages in an energy-landscapes-based approach to the analysis of chemical reactivity, namely, finding reaction (or rearrangement) pathways and extracting the kinetic information from the obtained pathway ensemble. As the meaning of the term pathway (or path) varies from chapter to chapter I will spend some time in the introductory sections clarifying the terminology.

next up previous contents
Next: Thesis Overview Up: INTRODUCTION Previous: Creating a Coarse-grained Model   Contents
Semen A Trygubenko 2006-04-10