1. Genetic algorithm - Monte Carlo hybrid method for finding stable geometries of atomic clusters Application to carbon clusters Nazım Dugan, Şakir Erkoç METU, Physics Department

2. Outline Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion

3. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion

4. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion GA Overview Atomic systems >> 3N degrees of freedom (with empirical potentials) Minimum potential energy >> maximum stability Find global minimum on the Potential energy hypersurface Fitness criteria >> Total potential energy of the system Encoding : 111010001010100101000101 CTGA....

5. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion Genetic Operations Mutation Natural selection Reproduction Crossover 100110010010100110101100........ 101110010010100100101100........ 0011001001 00011001000 10011001001 11010001001 10011001101

6. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion -33 eV -28 eV-32 eV-31 eV-26 eV Genetic Operations Mutation Natural selection Reproduction Crossover 111010001010100101000101

7. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion Genetic Operations Mutation Natural selection Reproduction Crossover

8. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion 100110010010100110101100 001110100011010000101001 100110010010010000101001 Genetic Operations Mutation Natural selection Reproduction Crossover

9. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion We have to decide on Population size too low >> not enough diversity !!! too high >> not necessary !!! Mutation rate too low >> lose diversity !!! too high >> mutants !!! Reproduction type asexual sexual >> how to combine individuals Natural selection type not big deal (select better with higher probability) elitism

10. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion

11. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion random walk of atoms if (ΔE < 0) accept if (ΔE > 0) check convergence if rand(0-1) < exp(- ΔE / kT) accept LOOP

12. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion

13. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion search over all phase space >> local optimization between GA steps >> search over local minima geometric mutation operations Rotation Atom permutation Geometric crossover (Deaven - Ho)

14. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion

15. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion Generate N clusters randomly MC local optimization Natural selection -29.6342 eV-28.9163 eV-33.3887 eV-28.8877

16. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion Reproduction Mutation (Rotation) Mutation (Shrink) MC local optimization (1st step)

17. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion C 20 animation

18. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion Parallelization Distribute individuals to available nodes MC local optimization takes much more time >> all nodes do MC GA operations takes no time >> master node gathers individuals and do GA ~100 % efficiency !!!

19. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion

20. Genetic Algorithms (GAs) Classical Monte Carlo (CMC) Application of GAs to atomic clusters Method overview Results and discussion n = 11n = 16 n = 20 shrink 0.5 n = 38 Shrink 0.85 Carbon clusters (Tersoff - Brenner potential energy function) n = 12n = 14 n = 22 shrink 0.5 n = 32 Shrink 0.8 n = 19

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