## Appendix: calculation of the structure and bond energy of clustersThe stable equilibrium configuration of a cluster with As is well known, such an optimization problem with a large number of variables cannot be solved directly by analytical methods and the task of solving the problem by numerical methods becomes extremely difficult and time consuming if the surfaces have a complicated geometric structure. Up to now, no general method is known which allows one - to find definitely
*all*local extrema of an optimization problem and - to decide whether the real optimum is contained within the set of extrema found by the applied method.
Numerous methods, however, have been developed which allow one to find in a certain sense a best result, which means that for test cases with a known absolute minimum the algorithm always finds this minimum or at least a local minimum not much different from the absolute one. One method, which has been applied in the past successfully to several difficult optimization problems like that of the travelling salesman is the "evolution strategy" developed by Rechenberg (1973 , 1989). We apply this method to determine the possible equilibrium configurations of ionic clusters. The special variant of the method used in our computation is the
(1,10)-evolution strategy (Rechenberg 1973 , 1989), which proceeds in
the following manner: Starting from an initial vector
in the variable space (called The best offspring, i.e. that offspring whose quality function is nearest to the optimum, is declared as the new parent of the next generation and its step length is declared as the new step length of the next generation. In our case the quality function is the cluster potential and the best offspring is that which results in the lowest potential energy. The basic idea of the whole method is to approach the optimum by an
intelligent trial and error strategy, in which the experience with
respect to the optimum step sizes is left from generation to
generation in the "evolution" process. Since the step length itself is
varied by a random process, from time to time "mutations" of the step
lengths occur if a big sidestep improves the quality. This prevents
the process from being trapped into a deep local isolated minimum. Its
numerical efficiency stems from the fact that only the quality
function has to be evaluated a few times in each generation. Its
derivatives are The search for the absolute minimum is stopped if the step lengths drops below some prescribed (small) limit because this indicates that further "mutations" do not improve the result (cf. Fig. 6 for an example). All local deep minima encountered during the search are stored because these are required for determining the isomers. Details with respect to the application of the evolution strategy to the calculation of cluster structures are described in Köhler (1988 , 1989).
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |