The 38-atom Lennard-Jones cluster has a paradigmatic double-funnel energy landscape. One funnel ends in the global minimum, a face-centered-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. First, from a large sample of minima and transition states we construct a disconnectivity graph showing which minima are connected below certain energy thresholds. Second, 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 liquidlike 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 liquidlike state, are the cause of the smaller free energy barrier for entering the icosahedral funnel from the liquidlike 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.

Theoretical design of global optimization algorithms can profitably utilize recent statistical mechanical treatments of potential energy surfaces (PES’s). Here we analyze the basin-hopping algorithm to explain its success in locating the global minima of Lennard-Jones (LJ) clusters, even those such as ${\mathrm{LJ}}_{38}$ for which the PES has a multiple-funnel topography, where trapping in local minima with different morphologies is expected. We find that a key factor in overcoming trapping is the transformation applied to the PES which broadens the thermodynamic transitions. The global minimum then has a significant probability of occupation at temperatures where the free energy barriers between funnels are surmountable.