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.

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

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

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

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

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

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

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

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

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

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

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

Initial Configuration 1 2 3 5 4 6 y z x

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

Gong Potential Gong is a Modified SW Potential Two Body Term Three Body Term where

Global Minima for Clusters Cluster size Energy per atom 3 atoms -0.8085 9 atoms -1.5649 4 atoms -1.0004 10 atoms -1.5800 5 atoms -1.2783 11 atoms -1.6661 6 atoms -1.3749 12 atoms -1.6245 7 atoms -1.4418 13 atoms -1.6494 8 atoms -1.5111 14 atoms -1.6464 Units are in ε = 2.17eV.

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

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

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

Parametric study: Delta Sensitivity Regions of robustness for choice of delta

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)

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)

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