Searching the potential energy surface

The principal method that we have used to generate candidate structures for the global minima makes use of the physical insight gained from the last section. We have simply attempted to construct geometries that maximize n_nn for the three ordered morphologies.[7] The resulting structures were then optimized by either conjugate gradient[58] or eigenvector-following[59] methods. A similar approach was successfully used by Northby to generate low energy icosahedra for LJ clusters.[6] The effectiveness of this method is demonstrated by how few of Northby's lowest energy structures have been superseded and by the length of time that it has taken to find these exceptions.[7,37,38,60,61,62,63] Our approach, however, depends on the imagination of the practitioner to conceive of all the possible ways that a structure with a large value of n_nn could be obtained. Furthermore, this method will always fail to find the global minimum if the latter is not based on one of the ordered structures, as is the case for the larger clusters we have considered at low values of $\rho_0$.

To complement the above approach two global optimization techniques were used to try to find structures that might have been missed. Firstly, we used a method in which eigenvector-following is employed to take steps directly between minima on the PES.[64,65] If low temperature Metropolis Monte Carlo is used in this space of minima, the system will walk down to the bottom of a basin containing many minima. This technique avoids the difficulties associated with trapping in local minima that can occur for methods which take steps directly in configuration space. Secondly, we used a `basin hopping' approach, which has proved to be effective for LJ clusters;[63,66] it is the only unbiased global optimization method to have found the global minima that are Marks decahedra. For Morse clusters it was able to find all the lowest energy minima at $\rho_0$=3, 6, 10 and all but twelve at $\rho_0$=14; this included some structures that were lower than any of those we had constructed. The fact that we found most of the minima both by unbiased global optimization and by construction makes us confident that our lowest energy structures are truly global minima.

The results for the basin-hopping algorithm are impressive because the global optimization task for Morse clusters is a difficult one. The size of the configuration space for the larger clusters is compounded for short-ranged potentials by the nature of the PES. Firstly, the PES becomes more rugged as the range is decreased.[42] The physical reason for the larger number of minima at short range is the loss of accessible configuration space as the potential wells become narrower, producing barriers where there are none at long range. Stillinger and Stillinger[67] and Bytheway and Kepert[40] both found that minimizations performed from random starting configurations are much less likely to find the global minimum for a short-ranged potential.

Similarly, barrier heights are also likely to become higher and rearrangements more localized for a shorter-ranged potential. These trends have been observed in comparisons of the rearrangements of 55-particle $\rm C_{60}$ and LJ clusters.[68] Hence the range of the potential is likely to have a significant impact on the dynamics, making escape from local minima much more difficult. This effect has been observed by Rose and Berry who have shown that the rate at which the ground state structure of a potassium chloride cluster is found upon cooling can be significantly decreased by using a shielded Coulomb potential to reduce the range of the interactions.[69]

Furthermore, as a result of the competition between the decahedral and the various types of close-packed structures for short-ranged potentials, it is more likely that there are a number of low energy minima which are very close in energy but are structurally dissimilar. Each of these minima lie at the bottom of their own funnel on the PES. This multiple funnel topography can lead to cases where optimization is extremely difficult because the free energy barriers for transitions between the funnels can be large, thus leading to trapping. The worst cases are when relaxation down the PES preferentially takes the system into a funnel which does not end at the global minimum,[70] and when the global minimum only becomes the state with the lowest free energy at low temperatures.[66]

Finally, fcc and decahedral minima are more structurally dissimilar from minima typical of the liquid-like state than the icosahedral structures. Therefore, the paths from the liquid to these structures are likely to be fewer and longer than those leading to the icosahedral structures,[64] and so relaxation down the PES from the liquid-like state to fcc and decahedral global minima is harder.

For a number of reasons Morse clusters are an ideal system with which to test global optimization methods designed for configurational problems. Firstly, in this paper we provide very good estimates for the energies of the global minima. Secondly, this system represents a much more general--the global minima have a variety of structural types--and tougher examination than is provided by LJ clusters, a much-used test system for global optimization algorithms.[63] And finally, the results for our basin-hopping algorithm provide a benchmark that any would-be global optimization method should try to beat.