Global optimisation applied to molecular architecture

Examines the problem of "identifying the configurations of molecular structures which correspond to the globally minimum potential energy for that structure".. This thesis addresses the problem of identifying configurations of molecular structures which correspond to the globally minimum potential energy for that structure. Molecular structures arise as a result of non-bonded and bonded atomic interactions and experimental evidence shows that, in the great majority of cases, the potential energy global minimum corresponds to the most stable configuration of the molecular structure. This configuration is of particular importance as it dictates most of the physical properties of the molecular structure. The potential energy of a molecular structure may be calculated, as a function of the atomic positions, using appropriate molecular models. However, as these give rise to potential energy functions that are typically nonconvex with many local minima, finding the global minima is an extremely difficult problem. For many years this problem has been investigated by chemists and physicists however, in more recent years, researchers from optimisation and computer science have also become involved and, in fact, the minimisation of non-convex potential energy functions arising from molecular conformation or protein folding problems has become one of the rnost important interdisciplinary problems [43]. This thesis develops and analyses a molecular structure global optimisation method using both deterministic local and stochastic global optimisation techniques within a genetic algorithm based environment. By incorporating different genetic operators, the one basic method was able to globally optimise a numlber of different types of molecular structures. From an experimental point of view, the method was particularly successful and found all currently accepted global minima for scaled Lennard-Jones atomic clusters of 2 to 80 atoms. two new global minima for 77 and 78 atom scaled Lennard-Jones atomic clusters . all currently accepted and some improved global minima for mixed argon-xenon atomic clusters of 7, 13 and 19 atoms. In addition, minima were determined for all remaining clusters in the 2, .... ,20 atom range. all currently accepted global minima for clusters of benzene molecules of 2 to 6 molecules and new minima for clusters of 8 to 12 molecules. all currently accepted global minima for a two-dimensional model molecular structure where the number of atoms ranged from 3 to 42. currently accepted global minima for a number of small molecules. Of particular importance is that, in determining these global minima, the method always started from randomly generated initial configurations and at no stage used any heuristic information to accelerate the search. From a theoretical point of view, this thesis presents an analytical comlparisonof the phenotype crossover operators used in the method with the more standard (genotype) crossover operators normally used in genetic algorithms. This analysis is confirmed with experimental results. In addition, a proof of convergence for the stochastic global optimisation technique used within the genetic algorithm environment and analytical evaluation of all potential energy gradients required by the deterministic local optimiser are presented. Chapter 1 of this thesis describes the molecular architecture problem and presents a review of local and global optimisation techniques. Chapter 2 describes the development of APSE, the stochastic global optimisation technique used in this study while the results obtained by applying APSE to the pure atomic cluster problem are presented in Chapter 3. Chapter 4 describes the development of GEM*, the major computational method used in this study. GEM* implements a combination of local optimisation and APSE probabilistic searches witrlin a genetic algorithm based environment. The results obtained by applying GEM* to the pure atomic cluster problem and a theoretical comparison of phenotype genetic crossover operators with rnore standard genetic crossover operators are presented in Chapter 5. The results obtained by applying GEM* to mixed argon-xenon atomic cluster problems are described in Chapter 6 while the optimisation of clusters of benzene and water molecules by GEM* is discussed in Chapter 7. Chapter 8 describes the GEM* optimisation results obtained for a model molecular structure and Chapter 9 presents the GEM* optimisation results for a number of small molecules. A summary and future research directions are presented in Chapter 10 while the appendices contain the analytical derivation of the potential energy gradients required for the implementation of the BFGS local optimiser and tables describing the structures obtained for mixed atomic clusters. Within this thesis Chapter 2 and Sections 3.3.1 and 3.4.1 appeared in the Australian Computer Journal, Vol. 28, No.4, November 1996. Chapter 6 has been accepted for publication by the Journal of Computational Chemistry. Sections 4.2, 5.2, 5.3 and 5.4 have been submitted to the Journal ofGlobal Optimization. Section 3.3.2 appeared as Technical Report 95 - 010, Department of Mathematics and Computing, Central Queensland University. Chapter 7 appeared as Technical Report 96 - 005, Department of Mathematics and Computing, Central Queensland University. Chapters 8 and 9 appeared as Technical Report 96 - 006, Department of Mathmatics and Computing, Central Queensland University.