A Study of the Cu Clusters Using Gray-Coded Genetic Algorithms and Differential Evolution N. Chakraborti, P. Mishra, and S ¸ . Erkoç (Submitted 26 August 2002; in revised form 9 October 2003) Energy minimization studies were carried out for a number of Cu clusters using binary and Gray-coded genetic algorithms along with real coded differential evolution, and their optimized ground state geometries are presented. The potential energy function is constructed using a two-body interaction methodology, involving both attractive and repulsive pair-potential terms. The results obtained through the evolutionary algorithms are compared against those obtained earlier using a Monte Carlo technique. 1. Introduction Genetic algorithms (GAs) tend to mimic several biologi- cal processes in the realm of function optimization. [1-5] Their usage in determining the ground state configuration of various clusters and molecules is becoming increasingly successful and popular. [6-15] In their most common forms, the GAs use a binary representation of variables, and the emulated genetic operators, crossover, and mutation, for example, are made to act on it. As elaborated in our earlier work, [7] this binary representation often suffers from the so-called Hamming Cliff problem. The Hamming distance indicates the number of bits that are different between two binary strings. In a Hamming Cliff situation, a very large perturbation in binary space would cause only a small change in integer space. For example, the decoded value of the binary string 01 111 111 is 127 in the integer space, while the string 10 000 000, which is at a large Hamming distance apart, since all of its corresponding bits are differ- ent from the previous one, decodes as 128, the next integer. In such a situation, the usual binary representation is often pushed to the limits of its efficacy, and the GAs tend to become stagnant. The real coded differential evolution (DE), [16] as demonstrated earlier, [7] becomes quite handy in a situation like this. Another option is to use the phenotype operation of creep mutation, where the binary strings are mapped back to the real space, perturbed slightly, and re- converted back to binary. In this study, along with those two procedures, we have experimented with another option, the so-called Gray Coding [1-2] of the binary variables. We ap- plied our methodology on a number of copper clusters and compared the present results with our earlier investigations conducted through Monte Carlo simulation and other meth- ods, [17-18] in which the success of an evolutionary approach became quite apparent. The details of binary GAs and DE are provided in our earlier papers [3,5-8,12] and are not repeated here. We begin with a brief overview of the Gray Coding technique. 2. The Elements of Gray Coding Gray coding ingeniously uses the Exclusive OR (XOR) operator between the binary bits. The XOR differs from the more conventional OR operator, as shown in Table 1 using two logical variables,  and , both of which could be either TRUE or FALSE and thus could be assigned a bit value of either 1 or 0. It is evident from Table 1 that the OR opera- tion returns a TRUE value when any one, or both the op- erands,  and , are TRUE. However, in order for an XOR operation to return a TRUE value, one of the operands necessarily has to be TRUE, while the other one needs to remain FALSE. To convert a binary string, say 10 000 to its Gray equiva- lent, the first step is to transfer the leftmost bit in the binary, 1 in the present case, unchanged to the same location in the Gray representation. The resultant of an XOR operation between the next bit in the binary and its left-hand side neighbor fills its corresponding position in the Gray string, and the XOR operation continues till all the bit locations are filled up. The binary number 10000 thus translates into 11000 in Gray encoding, and the Gray encoded GAs would use the number as such. The major attraction of the Gray encoding is its unique property that any two adjacent inte- gers, if Gray coded, will have only one corresponding bit different from each other. The Hamming Distance between any two adjacent Gray coded integers is therefore always unity, and this reduces the possibility of getting stranded in a Hamming Cliff, which is quite commonplace in the ordi- N. Chakraborti and P. Mishra, Department of Metallurgical & Ma- terials Engineering, Indian Institute of Technology, Kharagpur (W.B) 721 302, India; and S ¸ . Erkoç, Department of Physics, Middle East Technical University, TR 06531, Ankara, Turkey. Contact e-mail: nchakrab@metal.iitkgp.ernet.in. Table 1 Truth Table for OR and XOR Operators    OR   XOR  1 1 1 0 0 0 0 0 1 0 1 1 0 1 1 1 JPEDAV (2004) 25:16–21 DOI: 10.1361/10549710417650 1547-7037/$19.00 ©ASM International Section I: Basic and Applied Research 16 Journal of Phase Equilibria and Diffusion Vol. 25 No. 1 2004