Genetic operators for a two-dimensional bonded molecular model
Introduction
Recently, genetic algorithms have been successfully applied to molecular conformation problems involving clusters of carbon atoms (Deaven and Ho, 1995), pure atomic clusters (Deaven et al., 1996), mixed atomic clusters (Pullan, 1997) and clusters of benzene and water molecules (Pullan, 1996). These genetic algorithms restricted the search domain through the use of local (deterministic) optimisers and also utilised domain-specific genetic crossover operators operating in the problem (phenotype) domain rather then the more usual parameter (genotype) domain. Unfortunately these phenotype genetic crossover operators do not directly translate to bonded molecular structures so, as an initial step in developing similar genetic crossover operators for bonded molecular structures, this paper describes the implementation of phenotype genetic operators for a simple two-dimensional bonded molecular model.
Section 2of this paper describes the molecular model used in this study and reviews previous optimisation studies of this model. Section 3provides an overview of the genetic algorithm developed in this paper with particular reference to the genetic operators specifically designed for the model problem. Results are presented and discussed in Section 4while a review and future directions are contained in Section 5.
Section snippets
Model molecular problem
The model molecular problem consists of N atoms connected by rigid bonds of unit length and constrained to two dimensions. The potential energy, V, of this system is given by the pairwise addition of scaled Lennard–Jones potentials, that iswhere rij is the Euclidean distance between atoms i and j. The scaled Lennard–Jones potential is commonly used in molecular structure problems and has the form shown in Fig. 1.
While this model problem is a very simple
Optimisation method
The optimisation method used in this study to optimise the model molecule was a parallel real-coded genetic algorithm (GA). GAs use ideas from the process of nature selection where the proportion of “fitter” individuals in a population tends to increase at each generation (Goldberg, 1989). Standard GAs operate on a binary encoded version of the parameters of a problem and use genetic operators such as elitism, rank-based selection (based on a fitness factor), random mutation and crossover to
Results
The GA was successfully applied to the model molecular structure and found the minima shown in Table 2, Table 3. Of note is that all packed hexagonal energies in the range N=2, …, 42 were found and some in the range N=43, …, 55. In addition the GA achieved the best results to date for the N=61 molecule. When the GA was unable to find the global minimum, it was always able to find a value near the global minimum. Representative structures found are shown in Fig. 4, Fig. 5.
Conclusion
The genetic algorithm developed here successfully found most global minima for molecules consisting of between 2 and 55 atoms. In addition, it is the only optimisation method to successfully optimise the N=37 molecule and achieved the best results to date for the N=61 molecule. The GA was able to find these minima using random starting conformations. While tailored genetic operators were used, no a priori knowledge of the global minimisers was incorporated in the function of these genetic
Acknowledgements
The author would like to thank Graham Wood for his many critical and constructive comments while preparing this paper. For a substantial portion of this study the author was a visitor at the Computer Science Department, University of Colorado at Boulder and would particularly like to thank Robert Schnabel, Richard Byrd and Elizabeth Eskow for the opportunity to work with their group and for sharing their knowledge of global optimisation.
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