The double-funnel energy landscape of the 38-atom Lennard-Jones cluster

Abstract

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.

On the Use of Different Potential Energy Functions in Rare-Gas Cluster Optimization by Genetic Algorithms: Application to Argon Clusters

“…This improvement offers the possibility to enlarge narrow funnels in the potential energy surface, and it enhances the performance of the GA. This is particularly patented for the double-funnel Ar 38 cluster, where the improved version of the GA leads to 100% of success against ∼10% with the original algorithm. Also, the new GA could discover the global minima of all "difficult" cases for global optimization, that is, Ar 65 (with Pot I), Ar 75 , Ar 76 , Ar 77 , and Ar 78 (with Pot I, Pot III, and Pot IV).…”

“…The general trend consists of a fast increase of the threebody contribution up to N around 30; then, it drops down at N ) 31, followed by a slower increase for larger clusters. However, small oscillations occur in the curves of Figure 6 for particular nuclearities, the maxima being associated with less compact icosahedral structures (e.g., Ar 13 or Ar 55 ), while the minima appear for structures with a certain degree of compactness like the nonicosahedral Ar 38 cluster. Note that the triple- since the deformation indices always fulfill the relations -e 0 and + g 0; hence, when -and + are simultaneously zero, the two corresponding curves touch each other, and the structure is a spherical top.…”

“…Whereas for most of the Lennard-Jones clusters the topography of the PES is characterized by a single funnel coming from the high-energy regime to the low-energy icosahedral-based minima 36 (which contains the global minimum), the N ) 38 and 75-77 clusters (just to mention those in the range studied here) present, as well, a much narrower funnel leading to the global minimum. 37,38 Thus, the probability of reaching the basin of attraction of the global minimum is extremely low in such cases, while the large energy barriers between the two funnels prevent traditional searching methods (e.g., simulated annealing) from being successful. Indeed, even more powerful approaches like the "basin-hopping" 39 or GAs 30, 40 have difficulties to find minima for those double-funnel clusters.…”

“…We note that the implementation of the two-phase optimization procedure in our GA is straightforward (only the local optimization step of the flowchart in Figure 2 needs to be modified), being that the transformation of the PES (eqs 6 and 7) is applied only to the two-body potentials; even in the cases of Pot III and Pot IV, we expect this transformation to be efficient since the magnitude of the three-body terms is very small in comparison with that of two-body ones. As a testing ground, we have applied this new GA to search the global minimum for Ar 38 . The rate of success increases from ∼10% (as mentioned above) to 100% for all of the potentials.…”

A conformational factorisation approach for estimating the binding free energies of macromolecules

“…This approach to binding free energy calculations is employed in the mining minima algorithm, 35 which has been successfully applied to various biomacromolecular systems, especially small host-guest systems. Benchmark superposition calculations for atomic and molecular clusters show that the energy landscape approach can be much faster than MD or MC based methods, especially for cases of broken ergodicity, 36,37,[37][38][39] since the superposition partition function is explicitly ergodic.…”