The European Physical Journal D pp 235-241 | Cite as

# Genetic algorithms for determining the topological structure of metallic clusters

## Abstract

Genetic algorithms (GA) are applied for the optimization of the structure of metallic clusters by the calculation of the ground-state energies from a tight-binding (Hückel) Hamiltonian. The optimum topology or graph is searche by the use of the adjacency matrix A_{ ij } as a natural coding. The initial population for *N*-atom clusters are generated from a representative group of fit cluster structures having *N* — 1 atoms by the addition Of random connections or hoppings between the *N*th atom and the rest of the cluster atoms (A_{ iN } = 0 or 1). The diversity of geometries is enlarged by 20% with fully random structures. Several crossover strategies are proposed for the genetic evolution that combine the “parent” clusters while trying to preserve or transmit the physical characteristics of the parents’ topologies. The performance of the different procedures is tested. For *N*≤ 13, the present GA yield topological structures that are in agreement with previous geometry optimizations performed using an enumerative search (*N* ≤ 9) or simulated annealing Monte Carlo (10 ≤ *N* ≤ 13) methods. Limitations and extensions for *N* ≥ 14 are discussed.

## PACS

61.46.+w Clusters, nanoparticles and nanocrystalline materials 36.40.Cg Electronic and magnetic properties of clusters 02.60.Pn Numerical optimization## Preview

Unable to display preview. Download preview PDF.

## References

- 1.D.E. Goldberg:
*Genetic algorithms in search*,*optimization and machine learning*(Addison-Wesley, Reading, Massachusetts 1989 )Google Scholar - 2.D.M. Deaven, KAI. Ho: Phys. Rev. Lett.
**75**, 288 (1995)ADSCrossRefGoogle Scholar - 3.See, for instance, S.R. Gregurick, M.H. Alexander, E. Hartke: J. Chem. Phys. 104, 2684 (1996); W-1. Pullan: J. Comp. Chem.
**18**, 1096 (1997) and references thereinGoogle Scholar - 4.M.D. Wolf, U. Landman: J. Phys. Chem. A
**102**, 6129 (1998)CrossRefGoogle Scholar - 5.A graph
*G =*(V, E) is defined as a set V of sites or vertices i = I,, N and a set*E*of connections, bonds or edges*(i*,j) with i, j E V. On physical grounds (see (1)) we restrict ourselves to graphs that are simple (i.e., there is no more than one connection between two given sites), connected (i.e., for all i,*j*E V there is a a sequence, with lj = i and iK =*j*such that*(it*E*E*for all*k K)*and without on-site loops (i.e., if*(i*,,j) E*E*thenGoogle Scholar - 6.Y. Wang, T.F. George, D.M. Lindsay, A.C. Beri: J. Chem, Phys. 86, 3493 (1987); D.M. Lindsay, Y. Wang, T.F. George: J. Chem. Phys.
**86**, 3500 (1987)ADSGoogle Scholar - 7.A. Yoshida, T. Dossing, M. Manninen: J. Chem, Phys.
**101**, 3041 (1994) 9.Google Scholar - 8.For example, a = 7 corresponds to a triangle and a = 3, 5 10. or 6 to a linear chain (N -= 3). Notice that there are different matrices
_{Au}and integers a(A), which yield the same graph but with permuted site indices. However, a canonical form may be defined, for example, by the choice of the smallest a (among the equivalent ones). Physically, this yields a cluster where the atoms having the largest i have the smallest local coordination numbers, and vice versaGoogle Scholar - 9.Further details will be published elsewhereGoogle Scholar
- 10.G.M. Pastor, R. Hirsch, B. Miihischlegel: Phys. Rev. Lett.
**72**, 3879 (1994); Phys. Rev. B**53**, 10 382 (1996)Google Scholar - 11.R. Poteau, F. Spiegelmann: J. Chem. Phys.
**98**, 6540 (1993)ADSCrossRefGoogle Scholar - 12.K. Michaelian: Chem. Phys, Lett.
**293**, 202 (1998)ADSCrossRefGoogle Scholar