Conformational Optimization of Silicon Cluster System by Simulated Annealing

1. Conformational Optimization of Silicon Cluster System by Simulated Annealing Maria Okuniewski Nuclear Engineering Dept. Folusho Oyerokun Material Science & Eng. Dept. Rui Qiao Mechanical Engineering Dept.

2. Project Goals • Apply Simulated Annealing to Si Cluster Optimization • Incorporate Adaptive Cooling Schedule • Compare Results with Genetic Algorithm

3. Motivation • Properties of clusters are directly related to their conformation • Quantum chemistry calculations are very expensive • Traditional optimization techniques do not perform well when applied to cluster optimization problems

4. What is simulated annealing (SA) ? A Monte-Carlo approach for minimizing multi-variate functions • Advantages • Ability to find the global minimum independent of initial configuration • Less likely to get trapped in local minima • Mimics physical annealing process

5. Cooling Schedules • Initial Value of Temperature • Final Value of Temperature • Decrement Rule for Temperature • Markov Chain Length at Each Temperature (Quasi-Equilibrium)

6. Initial Value of Temperature • Issues • Too High Results in Wasted Computer Time • Too Low Might Get You Trapped in Local Minimum • Selection Based on Acceptance Ratio • Select Low T and Compute AR • Increase T until AR >= 0.8

7. Final Temperature • Issues • Zero Kelvin Not Feasible! • Possibility of Stopping Before Global Conformation is Found if T is Too High • Wasted Computer Time After Global Minimum Has Been Found if T Too Low • Selection Based on Minimum AR and Box Size

8. Plot of Cv for Fixed Decrement Rate Tn+1= f (Cv, Tn) ?

9. Decrement Rule • Adaptive Cooling Based on Cv • Two Schemes Chosen (Modified Aarts and van Laarhoven) (Huang et al.)

10. Markov Chain Properties • Mathematical Requirement for Convergence • Infinite Markov Length • Transition Matrix Must Satisfy the Following Requirements:

11. Markov Chain (Contd) • Practical Implementation • Finite Chain Length Based On Acceptable Variance • Detailed Balance Sufficient for Irreducibility Requirement of the Markov Chain

12. Initialize configuration X0 Initialize temperature T0 Perturb atom position once Metropolis algorithm Record lowest energy E_good = f ( X_good ) Tn+1 = f(Tn, Cv) Adjust box size D based on accept ratio k*natom times? N Y update configuration: X = Xgood N Y Accept # < M Calculate Cv N D < Dmin ? Y END

13. 6 5 4 3 z y 1 2 Initial Configuration x

14. Robustness of algorithm tested for different initial configurations Initial Configuration (n=12) Final Configuration (n=12)

15. Gong Potential • Gong is a Modified SW Potential • Two Body Term • Three Body Term where

16. Global Minima for Clusters Units are in ε = 2.17eV.

17. Structural Evolution 4 atoms 10 atoms 12 atoms 8 atoms Initial Final

18. Comparison of Temperature Decrement Rules

19. Inner Loop Sensitivity • Small inner loop – difficult to reach convergence • Large inner loop – helps to improve convergence, but slows algorithm

20. Parametric study: Delta Sensitivity • Regions of robustness for choice of delta

21. Genetic Algorithm Basics • Developed by John Holland in the 1960’s • Incorporates principles based on Darwin’s evolution theories • Survival of the fittest – selects candidate solutions (coordinates of the cluster structures) from total population (all available cluster structures) • Candidate solutions compete with each other for survival • Breeding, selection, and mutations – fittest individuals pass on their genetics to subsequent generations • After several generations – fittest individual obtained (global potential energy minimum)

22. Function Cost Evaluation • Exponential function is less costly for larger clusters (6-12) • GA is less costly for small clusters • Exponential - O(n1.2) • GA – O(n8.2)

23. Achievements • Developed SA Algorithm Based on Adaptive Cooling Schedule • Implemented Adaptive Box Size • Found Global Minimum Energy State for Cluster up to 14 atoms • Highlighted Sensitivity of Algorithm to Choice of Parameters • Compared Results with GA