We describe a global optimization technique using 'basin-hopping' in which the potential energy surface is transformed into a collection of interpenetrating staircases. This method has been designed to exploit the features which recent work suggests must be present in an energy landscape for efficient relaxation to the global minimum. The transformation associates any point in configuration space with the local minimum obtained by a geometry optimization started from that point, effectively removing transition state regions from the problem. However, unlike other methods based upon hypersurface deformation, this transformation does not change the global minimum. The lowest known structures are located for all Lennard-Jones clusters up to 110 atoms, including a number that have never been found before in unbiased searches.

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Finding the optimal solution to a complex optimization problem is of great importance in many fields, ranging from protein structure prediction to the design of microprocessor circuitry. Some recent progress in finding the global minima of potential energy functions is described, focusing on applications of the simple "basin-hopping" approach to atomic and molecular clusters and more complicated hypersurface deformation techniques for crystals and biomolecules. These methods have produced promising results and should enable larger and more complex systems to be treated in the future.

The study of energy landscapes holds the key to resolving some of the most important contemporary problems in chemical physics. Many groups are now attempting to understand the properties of clusters, glasses and proteins in terms of the underlying potential energy surface. The aim of this book is to define and unify the field of energy landscapes in a reasonably self-contained exposition. This is the first book to cover this active field. The book begins with an overview of each area in an attempt to make the subject matter accessible to workers in different disciplines. The basic theoretical groundwork for describing and exploring energy landscapes is then introduced followed by applications to clusters, biomolecules and glasses in the final chapters. Beautifully illustrated in full colour throughout, this book is aimed at graduate students and workers in the field.

Energy landscapes hold the key to understanding a wide range of molecular phenomena. The problem of how a denatured protein re-folds to its active state (Levinthal's paradox 1 ) has been addressed in terms of the underlying energy landscape 2±7 , as has the widely used`strong' and`fragile' classi®cation of liquids 8,9 . Here we show how three archetypal energy landscapes for clusters of atoms or molecules can be characterized in terms of the disconnectivity graphs 10 of their energy minimaÐthat is, in terms of the pathways that connect minima at different threshold energies. First we consider a cluster of 38 Lennard±Jones particles, whose energy landscape is a`double funnel' on which relaxation to the global minimum is diverted into a set of competing structures. Then we characterize the energy landscape associated with the annealing of C 60 cages to buckministerfullerene, and show that it provides experimentally accessible clues to the relaxation pathway. Finally we show a very different landscape morphology, that of a model water cluster (H 2 O) 20 , and show how it exhibits features expected for a`strong' liquid. These three examples do not exhaust the possibilities, and might constitute substructures of still more complex landscapes.We use disconnectivity graphs 10 to visualize the energy landscapes, based upon samples of pathways linking local minima via transition states. At any given total energy we elucidate which of the local minima in our sample are connected by pathways that lie below the energy threshold. At ®nite energy the minima are divided into disconnected sets of mutually accessible structures, separated by insurmountable barriers. The resulting graphs are clearest when we present the results as the total energy increases in regular steps along the vertical axis. Each line begins from a different local minimum at a vertical height determined by the potential energy at the bottom of the corresponding well. A node joins lines at the lowest energy for which the minima become mutually accessible. We are free to choose the horizontal displacements to give the most helpful representation of the resulting graph.Graphs such as these, which are connected but contain no cycles, are known as`trees' for reasons which should be clear from Fig. 1. The gentle`funnel' with high barriers in Fig. 1a produces a graph which looks like a weeping willow. The graph for the ef®cient funnel with lower barriers in Fig. 1b reminds us of a palm tree, whilst the graph for the rough landscape in Fig. 1c resembles a banyan tree with its branches planted into the ground.We ®rst focus on a cluster of 38 atoms bound by the Lennard± Jones potential, represented here by (LJ) 38 . The lowest-energy icosahedral minimum lies signi®cantly above the truncated octahedron 11 , and the two minima are well separated in con®gura-tion space. Relaxation to the global minimum is hampered because the vast majority of local minima are associated with the liquidlike state of the cluster and have polytetrahedral character 11 . Minima based upon ic...