Computer Physics Communications 142 (2001) 214–218 www.elsevier.com/locate/cpc
Applying evolutionary programming to structural optimization of atomic clusters Masao Iwamatsu Department of Information & Computer Engineering, Kisarazu National College of Technology, Kisarazu City, Chiba 292-0041, Japan
Abstract Evolutionary programming is used to study the structural optimization of atomic clusters. We find that this algorithm can successfully find the lowest energy structure of clusters efficiently without relying on any traditional local minimization methods or scheduled simulated annealing methods. 2001 Published by Elsevier Science B.V. PACS: 02.70.Lq; 02.70.Rw; 89.80+h Keywords: Evolutionary programming; Structural optimization; Cluster
1. Introduction Application of evolutionary algorithms (EA) [1], such as genetic algorithm (GA), evolutionary strategy (ES), and evolutionary programming (EP), to numerical optimization problems in multi-dimensional space has attracted much attention recently. In physics and chemistry, the numerical optimization problem appears typically in finding the lowest energy structure of atomic and molecular clusters. For these problems, simulated annealing (SA) [2] has traditionally been employed by physicists and chemists. There is, however, a cumbersome problem of temperature scheduling in SA. Recently, genetic algorithm has been shown to be more powerful than the traditional simulated annealing method. However, it was also recognized that this genetic algorithm should always be augmented with traditional local optimization methods [3] such as conjugate gradient or simplex methods.
E-mail address:
[email protected] (M. Iwamatsu).
In this paper, we employ EP [4] to study the lowestenergy structures of atomic clusters. This method is attractive because it does not rely on any gradient information about the potential energy surfaces and because the narrowing of the search space is automatically realized. Neither the cumbersome planning of a cooling schedule in SA nor the relaxation of clusters by local optimization in GA is necessary. Among various variants of EP, we have employed the algorithm compiled by Bäck and Schwefel [1] and tested extensively by Yao et al. [5] for many artificial test functions. In this paper we will show the experimental results of the EP applied for atomic silicon clusters and examine the effectiveness of the self-adaptation of mutation compared with the prescheduled annealing. 2. Algorithm The algorithm is as follows: (1) Generate initial population of M individuals (clusters consist of N atoms). Each individual i is
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coded as a pair of real valued vectors (xi , ηi ) (i = 1, 2, . . . , M), which represent the Cartesian coordinates xi (j ) (j = 1, 2, . . . , 3N − 6) of atoms and their standard deviations ηi (j ) for Gaussian mutation known as the “strategy parameter”. Initial coordinates xi (j ) are randomly chosen from the interval (0, 1), and the same initial strategy parameters ηi = 1 instead of ηi = 3 [1,5] are used, because the length is scaled such that the average interatomic distance is approximately unity. (2) Evaluate the fitness (total energy) f (xi ) of each individual of the population. (3) Each parent (xi , ηi ) creates a single offspring (xi , ηi ) for each component xi (j ) and ηi (j ), j = 1, 2, . . . , 3N − 6, by mutation:
(4)
(5) (6)
(7)
(8)
xi (j ) = xi (j ) + ηi (j )Nj (0, 1),
(1)
ηi (j ) = ηi (j ) exp τ N(0, 1) + τ Nj (0, 1) ,
(2)
where N(0, 1) denotes a random number of Gaussian distribution with mean zero and standard deviation one. Nj (0, 1) indicates that the Gaussian random number is generated anew for each com are commonly set ponent j. The factor τ and τ√ √ −1 to τ = ( 2 n) and τ = ( 2n)−1 , where n = 3N − 6 is the number of degree of freedom [1]. If ηi (j ) is smaller than the minimum threshold (ηi (j ) < THRESHOLD), then ηi (j ) is replaced by the THRESHOLD. This replacement is necessary to avoid the premature convergence to the metastable minimum [4,6]. Evaluate the fitness of offspring f (xi ). Conduct pairwise comparison (tournament) over the union (2M) of parents (xi , ηi ) and offsprings (xi , ηi ). For each individual in the union, q opponents are chosen randomly from the 2M − 1 other individuals. For each comparison, if the individual’s total energy (fitness) is not larger than the opponent’s fitness, he receives a winning point. Select M individuals out of the union of parents (xi , ηi ) and offsprings (xi , ηi ) that have more winning point. Stop if the halting condition (maximum generation G) is met, otherwise go to step (3).
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3. Experiments We have calculated the lowest energy structures of several silicon clusters using the above EP algorithm. We have used the three-body empirical potential of Gong et al. [7] for silicon clusters because it was shown [3] to give more realistic cluster structures than the popular Stillinger–Weber potential. The analytic form of this three-body empirical potential is given in Gong et al. [7]. The lowest-energy structures of silicon clusters using this empirical potential have been determined recently by the genetic algorithm by the present author [3]. We have used the EP algorithm of Section 2 with the following parameters (1) Population size M = 30. (2) Tournament size q = 10. (3) Maximum generation G = 103 ∼ 104 . (4) Resolution THRESHOLD = 10−4 . This algorithm is repeated 50 times using above parameters. From the 50 lowest energies produced from
Fig. 1. The lowest energy among the lowest energies obtained from 50 trials (solid line) and the average of the lowest energies (dashed line) over 50 trials for Si11 plotted as a function of the number of generation. The lowest energy E = −1.5800 obtained from the EP reproduces the result obtained previously from the genetic algorithm [3].
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each trial, we have determined the lowest structural energy and atomic configuration of silicon clusters with N = 3–11 atoms. In Fig. 1, we show the lowest energy among the 50 lowest energies obtained from each trial and the average of the lowest energies over 50 trials for Si11 plotted as a function of the number of generation. We find that the average energy as well as the lowest energy decreases as the function of the number of generation without invoking any searching schedule. Therefore, the self-adaptation of searching parameter by Eq. (2) is effective in this cluster optimization problem. The lowest energy thus obtained from the EP is E = −1.5800 in unit of = 2.17 eV, which repro-
duces the result obtained previously from the GA [3]. The lowest energy structure of this Si11 cluster obtained from this EP is shown in Fig. 2. The structure is icosahedron with one vertex missing (C5v ), which is exactly the same as the one determined from the GA [3]. The lowest energy of clusters and structures of smaller clusters are also obtained successfully by the EP algorithm of Section 2. Table 1 lists the structures and energies of the silicon clusters obtained from this EP algorithm. These results reproduce the previous results obtained from the GA [3]. Fig. 3 shows the distribution of the lowest energy obtained from 50 trials for Si11 . Four trials out of 50 can successfully find
Fig. 2. The lowest energy structure of Si11 cluster obtained from this EP. The structure is icosahedron with one vertex missing (C5v ), which is the same as the one determined from the GA [3].
Table 1 Ground state structures and their energies of silicon clusters Structure
Energy (in unit of = 2.17 eV)
Si3
Equilateral triangle
−0.8058
Si4
Tetrahedron
−1.0004
Si5
Compressed trigonal bipyramid
−1.1504
Si6
Octahedron
−1.2783
Si7
Pentagonal bipyramid
−1.3749
Si8
Unicapped distorted pentagonal bipyramid
−1.4418
Si9
Tricapped trigonal prism
−1.5111
Si10
Bicapped tetragonal antiprism
−1.5649
Si11
Icosahedron with one vertex missing
−1.5800
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Fig. 3. The distribution of the lowest energy obtained from 50 trials for Si11 . Four trials out of 50 can successfully find out the lowest energy structure.
out the lowest energy structure. Therefore the success probability is roughly 10% and it is necessary to repeat the trial at least more than ten times.
4. Effectiveness of self-adaptation of mutation In order to check the effectiveness of the selfadaptation of mutation for ηi (j ) given by Eq. (2) of the EP, we have changed the algorithm to mimic the scheduled annealing employed in simulated annealing (SA) algorithm as √ ηi (j ) = T , (3) where T has the meaning of temperature and is scheduled to decrease as a function of generation k. We have used the schedule [2] T = T0 f k ,
(4)
where T0 (= 10) is the initial temperature and f is the multiplication factor which is chosen so that the final temperature becomes THRESHOLD (resolution) of the EP at the end of the annealing. This algorithm is based on the parallel search of the original EP and, therefore, can be regarded as the parallel simulated annealing (SA) with tournament selection where various clusters started from different initial configurations are compared and selected. Fig. 4 shows the lowest energy of Si11 clusters among 50 trials both for EP and
Fig. 4. The effectiveness of the self-adaptation in the EP compared with the scheduled annealing of SA-like algorithm. The lowest energy among the lowest energies obtained from 50 trials from EP (solid line) and from SA (dashed line) for Si11 plotted as a function of the number of generation.
SA algorithms as a function of the number of generation. The self-adaptation of searching behavior of EP seems more effective compared with the prescheduled searching of SA.
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5. Conclusion We have tested the evolutionary programming (EP) algorithm using silicon clusters bound by an empirical three-body potential. We have found that the evolutionary programming can successfully find the lowest energy structures of these atomic clusters determined previously using the genetic algorithm. Furthermore the self-adaptation of mutation (search space narrowing) seems effective compared with the traditional scheduled annealing used in simulated annealing (SA) method. Therefore, by employing EP, we can avoid both the cumbersome planning of a cooling schedule as in SA and the relaxation of clusters by local optimization as in GA. The EP is such a robust algorithm that many optimization problems in physics and chemistry can be handled effectively. Acknowledgements The author is grateful to the Department of Physics, Tokyo Metropolitan University for providing him a
visiting fellowship during the course of this work. The author is particularly grateful to Professor Y. Okabe of Tokyo Metropolitan University for his constant interest and encouragement. References [1] T. Bäck, H.-P. Schwefel, Evol. Comp. 1 (1993) 1. [2] W.H. Press, S.A. Teukolsky, W.T. Vatterling, B.P. Flannery, Numerical Recipes in C, Cambridge University Press, New York, 1992, Chapter 10. [3] M. Iwamatsu, J. Chem. Phys. 112 (2000) 10 976. [4] D.B. Fogel, Evolutionary Computation, 2nd edn., IEEE Press, Piscataway, 2000. [5] X. Yao, Y. Liu, G. Lin, IEEE Trans. Evol. Comp. 3 (1999) 82. [6] K.-H. Lian, X. Yao, Y. Liu, C. Newton, D. Hoffmann, in: V.W. Porto, N. Saravanan, D. Waagen, A.E. Eiben (Eds.), Evolutionary Programming VII, Lecture Notes in Computer Science, Vol. 1447, Springer, Berlin, Heidelberg, 1998, pp. 291–300. [7] X.G. Gong, Q.Q. Zheng, Y.Z. He, J. Phys.: Condens. Matter 7 (1995) 577.
