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Optimization of Carbon and Silicon Cluster Geometry for Tersoff Potential using Differential Evolution

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Optimization in Computational Chemistry and Molecular Biology

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 40))

Abstract

In this paper we propose a new version of the Differential Evolution (DE) Algorithm for large scale optimization problems. The new algorithm, for exploration and localization of search, periodically uses topographical information on the objective function, in particular the k g -nearest neighbour graph. The algorithm is tested on hard practical problems from computational chemistry. These are the problems of semi-empirical many-body potential energy functions considered for carbon-carbon and silicon-silicon atomic interactions. The minimum binding energies of both carbon and silicon clusters consisting of upto 15 particles are reported.

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References

  1. Garrison, B. J., Winograd, N., Deavon, D. M., Reimann, C. T., Lo, D. Y., Tombrello, T. A., Harrison D. E. Jr. and Shapiro, M. H. (1988), “Many-body Embedded-atom Potential for describing the Energy and Angular distributions of Rh atoms distorted from Ion-bombarded Rh111”, Physics Review B, Vol. 37, 7197–7204.

    Article  Google Scholar 

  2. Smith, R. (1992), “A Semi-empirical Many-body interatomic Potential for Modelling Dynamical Processes in Gallium Arsenide”, Nuclear Instruments and Methods in Physics Research, Section B, Vol. 67, 335–339.

    Article  Google Scholar 

  3. Smith, R., Harrison, D. E. Jr, and Garrison, B. J. (1989), “keV Particle Bombardment of Semiconductors: A Molecular Dynamic Simulation”, Physics Review B, Vol. 40, 93–107.

    Article  Google Scholar 

  4. Ali, M. M. and Smith, R. (1993), “The Structure of Small Clusters Ejected by Ion Bombardment of Solids”, Vacuum, Vol. 44, 377–379.

    Article  Google Scholar 

  5. Scheraga, H. A. (1996), “Recent Developments in the Theory of Protein Folding: Searching for the Global Minimum”, Biophysical Chemistry, Vol. 59, 329–339.

    Article  Google Scholar 

  6. Carlsson, A. E. (1990), “Beyond Pair Potentials in Elemental Transition Metals and Semiconductors”, Solid State Physics, Vol. 43, 1–90.

    Article  Google Scholar 

  7. Beardmore, K. and Smith, R. (1996), “Empirical Potentials for C — Si — H Systems with Application to C 60 interactions with Si Crystal Surface”,Philosophical Magazine A, Vol. 74, 1439–1466.

    Article  Google Scholar 

  8. Tersoff, J. (1988), “New Empirical approach for the Structure and Energy of Covalent Systems”, Physics Review B, Vol. 37, 6991–7000.

    Article  Google Scholar 

  9. Tersoff, J. (1988), “Empirical Interatomic Potential for Silicon with improved Elastic Properties”, Physics Review B, Vol. 38, 9902–9905.

    Article  Google Scholar 

  10. Pardalos, P. M., Shalloway, D. and Xue, G. L. (1994), “Optimization Methods for Computing Global Minima of Non-convex Potential Energy Function”, Journal of Global Optimization, Vol. 4, 117–133.

    Article  MathSciNet  MATH  Google Scholar 

  11. Moré, J. J. and Wu, Z. (1995), “Global Smoothing and Continuation for Large Scale Molecular Optimization”, MCS-P539–1095.

    Google Scholar 

  12. Kostrowicki, J., Piela, L., Cherayil, B. J. and Scheraga, A. (1991), “Performance of the Diffusion Equation Method in Searches for Optimum Structures of Clusters of LennardJones Atoms”, Journal of Physical Chemistry, Vol. 95, 4113–4119.

    Article  Google Scholar 

  13. Xue, G. L. (1994), “Molecular Conformation on the CM-5 by Parallel Two-level Simulated Annealing”, Journal of Global Optimization, Vol. 4, 187–208.

    Article  MATH  Google Scholar 

  14. Deaven, D. M., Tit, N., Morris, J. R. and Ho, K. M. (1996), “Structural Optimization of Lennard-Jones Clusters by a Genetic Algorithm”, Chemical Physics Letters, Vol. 256, 195–198.

    Article  Google Scholar 

  15. Ali, M. M., Storey, C. and Törn, A. (1997), “Application of Stochastic Global Optimization Algorithms to Practical Problems”, Journal of Optimization Theory and Applications, Vol. 95, 545–563.

    Article  MathSciNet  MATH  Google Scholar 

  16. Hobday, S. (1998), “Artificial Intelligence and Simulations Applied to Interatomic Potentials”, PhD Thesis, Department of Mathematical Sciences, Loughborough University of Technology, UK.

    Google Scholar 

  17. Hobday, S. and Smith, R. (1997), “Optimization of Carbon Cluster Geometry Using a Genetic Algorithm”, Journal of Chemical Society, Faraday Transactions, Vol. 93, 3919–3926.

    Article  Google Scholar 

  18. Brenner, D. W. (1990), “Empirical Interatomic Potential for Hydrocarbon for Use in Simulating the Chemical Vapor Deposition of Diamond Films”, Physics Review B, Vol. 42, 9458–9471.

    Article  Google Scholar 

  19. Tersoff, J. (1988), “Empirical Interatomic Potential for Carbon with Applications to Amorphous Carbon”, Physical Review Letters, Vol. 21, 2879–2882.

    Article  Google Scholar 

  20. Tersoff, J. (1989), “Modelling Solid-State Chemistry: Interatomic Potential for Multicomponent Systems”, Physical Review B, Vol. 39, 5566–5568.

    Article  Google Scholar 

  21. Tersoff, J. (1990), “Carbon Defects and Defect Reactions in Silicon”, Physical Review Letters, Vol. 64, 1757–1760.

    Article  Google Scholar 

  22. Storn, R. and Price, K. (1997), “Differential Evolution — A Simple and Efficient Heuristic for Global Optimization over Continuous Spaces,” Journal of Global Optimization, Vol. 11, 341–359.

    Article  MathSciNet  MATH  Google Scholar 

  23. Ali, M. M. and Törn, A. (1999), “Evolution based Global Optimization Techniques and the Controlled Random Search Algorithm: Proposed Modifications and Numerical Studies,” submitted to the Journal of Global Optimization.

    Google Scholar 

  24. Törn, A. and Viitanen, S. (1992), “Topographical Global Optimization,” in Recent Advances in Global Optimization, Edited by A. Floudas and M. Pardalos, Princeton University Press, USA.

    Google Scholar 

  25. Lui, D. C. and Nocedal, J. (1989), “On the Limited Memory BFGS Method for Large Scale Optimization”, Mathematical Programming, Vol. 45, 503–528.

    Article  MathSciNet  Google Scholar 

  26. Ali, M. M. and Storey, C. (1995), “Modified Controlled Random Search Algorithms”, International Journal of Computer Mathematics, Vol. 54, 229–235.

    Google Scholar 

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Ali, M.M., Törn, A. (2000). Optimization of Carbon and Silicon Cluster Geometry for Tersoff Potential using Differential Evolution. In: Floudas, C.A., Pardalos, P.M. (eds) Optimization in Computational Chemistry and Molecular Biology. Nonconvex Optimization and Its Applications, vol 40. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3218-4_17

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  • DOI: https://doi.org/10.1007/978-1-4757-3218-4_17

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4826-7

  • Online ISBN: 978-1-4757-3218-4

  • eBook Packages: Springer Book Archive

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