Genetic algorithm - Monte Carlo hybrid method for finding stable geometries of atomic clusters

Application to carbon clusters

Nazım Dugan, Şakir ErkoçMETU, Physics Department

Outline

Genetic Algorithms (GAs)

Classical Monte Carlo (CMC)

Application of GAs to atomic clusters

Method overview

Results and discussion

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults 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....

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Genetic Operations

MutationNatural selectionReproductionCrossover

100110010010100110101100 ........

101110010010100100101100 ........

0011001001

00011001000

1001

1001

001

10011001001

11010001001

10011001101

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

-33 eV

-28 eV -32 eV -31 eV -26 eV

Genetic Operations

MutationNatural selectionReproductionCrossover

111010001010100101000101

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Genetic Operations

MutationNatural selectionReproductionCrossover

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

100110010010100110101100 001110100011010000101001

100110010010010000101001

Genetic Operations

MutationNatural selectionReproductionCrossover

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

We have to decide on

Population sizetoo low >> not enough diversity !!!too high >> not necessary !!!

Mutation ratetoo low >> lose diversity !!!too high >> mutants !!!

Reproduction typeasexualsexual >> how to combine individuals

Natural selection typenot big deal (select better with higher probability)elitism

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

random walkof atoms

if (ΔE < 0)accept if (ΔE > 0)

check convergence

if rand(0-1) < exp(- ΔE / kT)accept

LOOP

LOOP

2 4 6 8 10

-20

20

40

60

80

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

search over all phase space >> local optimization between GA steps >> search over local minima

geometric mutation operations

RotationAtom permutation

Geometric crossover(Deaven - Ho)

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Generate N clusters randomly

MC local optimization

Natural selection

-29.6342 eV-28.9163 eV -33.3887 eV -28.8877

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Reproduction

Mutation (Rotation)

Mutation (Shrink)

MC local optimization (1st step)

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

C20 animation

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults 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 !!!

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

Genetic Algorithms (GAs)Classical Monte Carlo (CMC)Application of GAs to atomic clustersMethod overviewResults and discussion

n = 11 n = 16

n = 20shrink 0.5

n = 38Shrink 0.85

Carbon clusters (Tersoff - Brenner potential energy function)

n = 12 n = 14

n = 22shrink 0.5

n = 32Shrink 0.8

n = 19

References

R.L. Johnston, Dalton Trans., 4193(2003)

B. Hartke, Chem. Phys. Lett. 240, 560(1995)

S.K. Gregurik, M.K. Alexander, B. Hartke, J. Chem Phys. 104, 2684(1996)

S. Hobday, R. Smith, J. Chem. Soc., Faraday Trans. 93, 3919(1997)

M. Iwamatsu, J. Chem Phys. 112, 10976(2000)

J. Zhao, R. Xie, J. Comp. Theoretical Nanoscience 1, 117(2004)

D.M. Deaven, K.M. Ho, Phys. Rev. Lett. 75, 288(1995)

Thank you