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 Selection Based on Minimum AR and Box Size
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
k*natom times? N Y
Metropolis algorithm
Record lowest energy E_good = f ( X_good )
Accept # < M
Calculate Cv
D < Dmin ?
END
update configuration:
X = Xgood
Tn+1 = f(Tn, Cv)
Adjust box size D based on accept ratio

13. Initial Configuration1 2 3 5 4 6 y z 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
Cluster size
Energy per atom
3 atoms
9 atoms
4 atoms
10 atoms
5 atoms
11 atoms
6 atoms
12 atoms
7 atoms
13 atoms
8 atoms
14 atoms
Units are in ε = 2.17eV.

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

18. Comparison of Temperature Decrement Rules
Exponential
Fixed
Inverse
natoms
Function cost
error
function cost 4
65646
30371
326400
32883
78900
9200 6
72148
33860
323124
41990
204490
9535 10
139974
45833
591374
17846
481550
113470

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