## 4 Single parent parallel Lamarckian GA implementation
### Application on carbon clusters  
The single parent parallel GA implementation summarized in previous section has been applied to moderate sized carbon clusters for benchmarking purposes [20]. Carbon clusters are suitable for testing the method since it is well known that the global energy minimum geometry of carbon clusters can not be found by local relaxation. Directional bonds between carbon atoms results in high energy barriers between different isomers of the molecule [19] making the optimization process tricky also for global optimization methods. Comparison of the single parent GA with a GA facilitating crossover between individuals suggested that single parent GAs are also promising for the geometry optimization problem of nanoparticles.  
Brenner potential energy function (PEF) [60] has been used to define the two-body and three-body interactions between carbon atoms forming clusters ranging from 3 atoms to 38 atoms. Population size of the GA has been chosen to be 32 individual clusters for the reasons of both this number being close to the optimal population size for this problem and for it is being suitable for parallelization. Multiplication factors varying between 0.5 and 1.0 are used in the shrinkage mutation operation. While a multiplication factor of 0.5 shrinks the cluster to half its size, a factor 1.0 leaves it unchanged. Shrinkage operation is found to be efficiency increasing for the carbon clusters having more than 20 atoms since they form cage like structures and the optimal value of multiplication factor increases as the cluster size goes up from 20 atoms. Images of some optimized carbon clusters are given in Figure 4 as examples together with their symmetry groups if applicable. The rest of the images can be seen in the original article of the current authors cited above [20].  
Optimized geometries of carbon clusters have been compared with the results of another GA study [43] and the results of these two studies agree on the fullerene geometries of the clusters with number of atoms even and larger than 20 except the case of 36 atoms for which different isomers of the corresponding fullerene structure is found by the two methods. This agreement on the fullerene structures suggests that the GA implementation described in the previous section works fine for moderate sized carbon clusters. There is some inconsistency in the results of the two methods for small carbon clusters as a consequence of the slight difference in the PEF used which was discussed in the original article.  
A very small number of GA steps are observed in the application of carbon clusters for the single parent Lamarckian GA used. Lamarckian property of the GA is responsible for reduction of GA step number since it reduces the search space a great amount. Comparison of the number of GA steps for the C\({}_{20}\) cluster with the other GA method mentioned above which also has the Lamarckian property (13 in  
Figure 3: Illustration of the single parent GA implementation showing evolution of the fittest individual in C\({}_{20}\) optimization. Three GA step samples are shown in separate lines ignoring some intermediate GA steps. 1. Column shows the cluster just after the shrinkage mutation operation (initial geometry in top line), 2. Column shows the result of local optimization and 3. Column shows the cluster after the piece rotation mutation step together with the cutting plane.  
authors work and 800 in the comparison work) suggests that MC optimization used in the current GA may have some advantages compared to deterministic local optimization routines even though single MC local optimization is more time consuming.