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– 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

nv

n TC

T/1

11 (Modified Aarts and van Laarhoven)

nC

n TeT v )/(1

(Huang et al.)

Markov Chain Properties

Mathematical Requirement for Convergence– Infinite Markov Length– Transition Matrix Must Satisfy the Following

Requirements:

jiji P

1 ijP

1max P

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 # < MY

Calculate Cv

< min ?

Y

N

END

update configuration: X = Xgood

N

Tn+1 = f(Tn, Cv)

Adjust box size based on accept ratio

1 2

3

54

6

x

yz

Initial Configuration

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

),,,(),(),...,2,1( 32 kjivjivnn

kji

n

ji

ar

arrBrAjiv

ij

ijq

ijp

ij

],)exp[()(),( 12

),,(),(),(),,(3 jkikijkjkiji rrhrrhrrhkjiv

ar

ar

ccararrrh

ki

ij

ojikjikkiijkiji

]).[(cos)3/1.)](cos)()((exp[),( 12211

where

Global Minima for Clusters

Cluster size Energy per atom 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.

4 atoms 8 atoms 10 atoms 12 atoms

Initial

Final

Structural Evolution

Comparison of Temperature Decrement Rules

  Exponential   Fixed   Inverse  

natoms Function cost error function cost error function cost error

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

8 atoms, T=50, delta = .001; dx = .9

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

0 100 200 300 400 500 600 700 800 900

Loops

Func

tion

Cos

t inverse

exponential

fixed step

Parametric study: Delta Sensitivity

Regions of robustness for choice of delta

Delta Dependence, 8 atoms, T=50

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

0 0.002 0.004 0.006 0.008 0.01 0.012

Delta

Func

tion

Cost

Inverse

Exponential

Inverse - non-converge

Exponential - non-converge

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)

Function Costs of Simulated Annealing and Genetic Algorithms

y = 0.2188x8.1629

R2 = 0.9682

y = 10974x1.1759

R2 = 0.8795

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+00 1.E+01 1.E+02

number of atoms in cluster

func

tion

eval

uatio

ns

GA (Sastry & Xiao, 2000)

Exponential

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