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...

The 38-atom Lennard-Jones cluster has a paradigmatic double-funnel energy landscape. One funnel ends in the global minimum, a face-centred-cubic (fcc) truncated octahedron. At the bottom of the other funnel is the second lowest energy minimum which is an incomplete Mackay icosahedron. We characterize the energy landscape in two ways. Firstly, from a large sample of minima and transition states we construct a disconnectivity tree showing which minima are connected below certain energy thresholds. Secondly we compute the free energy as a function of a bond-order parameter. The free energy profile has two minima, one which corresponds to the fcc funnel and the other which at low temperature corresponds to the icosahedral funnel and at higher temperatures to the liquid-like state. These two approaches show that the greater width of the icosahedral funnel, and the greater structural similarity between the icosahedral structures and those associated with the liquid-like state, are the cause of the smaller free energy barrier for entering the icosahedral funnel from the liquid-like state and therefore of the cluster's preferential entry into this funnel on relaxation down the energy landscape. Furthermore, the large free energy barrier between the fcc and icosahedral funnels, which is energetic in origin, causes the cluster to be trapped in one of the funnels at low temperature. These results explain in detail the link between the double-funnel energy landscape and the difficulty of global optimization for this cluster.

We systematically study the design of simple patchy sphere models that reversibly self-assemble into monodisperse icosahedral clusters. We find that the optimal patch width is a compromise between structural specificity (the patches must be narrow enough to energetically select the desired clusters) and kinetic accessibility (they must be sufficiently wide to avoid kinetic traps). Similarly, for good yields the temperature must be low enough for the clusters to be thermodynamically stable, but the clusters must also have enough thermal energy to allow incorrectly formed bonds to be broken. Ordered clusters can form through a number of different dynamic pathways, including direct nucleation and indirect pathways involving large disordered intermediates. The latter pathway is related to a reentrant liquid-to-gas transition that occurs for intermediate patch widths upon lowering the temperature. We also find that the assembly process is robust to inaccurate patch placement up to a certain threshold and that it is possible to replace the five discrete patches with a single ring patch with no significant loss in yield.

Disconnectivity graphs are used to characterize the potential energy surfaces of Lennard-Jones clusters containing 13, 19, 31, 38, 55 and 75 atoms. This set includes members which exhibit either one or two 'funnels' whose low-energy regions may be dominated by a single deep minimum or contain a number of competing structures. The graphs evolve in size due to these specific size effects and an exponential increase in the number of local minima with the number of atoms. To combat the vast number of minima we investigate the use of monotonic sequence basins as the fundamental topographical unit. Finally, we examine disconnectivity graphs for a transformed energy landscape to explain why the transformation provides a useful approach to the global optimization problem.

Using a combination of Monte Carlo techniques, we locate the liquid-vapor critical point of adhesive hard spheres. We find that the critical point lies deep inside the gel region of the phase diagram. The (reduced) critical temperature and density are τc = 0.1133 ± 0.0005 and ρc = 0.508 ± 0.01. We compare these results with the available theoretical predictions. Using a finite-size scaling analysis, we verify that the critical behavior of the adhesive hard sphere model is consistent with that of the 3D Ising universality class.PACS numbers: 61.20.Ja, 64.70.Ja The structure of a simple liquid is well described by that of a system of hard spheres at the same effective density. To a good approximation, the effect of attractive interactions on the liquid structure can be ignored. This feature of simple liquids is implicit in the Van der Waals theory for the liquid-vapor transition, and has been made explicit in the highly successful thermodynamic perturbation theories for simple liquids [1]. The perturbation approach becomes exact as the range of the attractive interaction tends to infinity while its integrated strength remains constant [3]. We refer to this limit as the 'Van der Waals' (VDW) limit [2]. Conversely, as the attractive forces become shorter-ranged and stronger, the perturbation approach is likely to break down. Fluids with strong, short-ranged attraction (so-called 'energetic' fluids [4]) are of growing importance in the area of complex liquids. For example, short-range attractions are thought to be responsible for the transition from a 'repulsive' to an 'attractive' glass [4], which has recently been observed experimentally in PMMA (polymethylmethacrylate) dispersions [6].In this Letter, we consider a model system that can be considered as the prototypical energetic fluid: a fluid of adhesive hard spheres (AHS). Introduced in 1968 [7], the AHS model is a reference system for particles with short range attractions. The pair potential consists of an impenetrable core plus a surface adhesion term that favors configurations where spheres are in contact. At larger separations, there is no interaction. The AHS model can be considered as the 'anti Van der Waals' limit.Baxter showed [7] that the Percus-Yevick (PY) equation can be solved analytically for adhesive hard spheres. In fact, Baxter's solution is often used to to analyze experimental results for systems as diverse as silica suspensions [8], copolymer micelles [9], and the fluid phase of lysozyme [10].One important feature of the AHS model is that its phase diagram contains a liquid-vapor coexistence region [11]. The PY equation offers different routes to estimate the location of the liquid-vapor critical point. However, the 'compressibility route' [7] and the 'energy route' [12] lead to estimates for the critical temperature that differ by some 20%, while the estimates for the critical density differ by almost a factor of three. For the analysis of experimental data, it is important to know the location of the critical point more accurately. The ...