Data_Sheet_2_Modified Particle Swarm Optimization Algorithms for the Generation of Stable Structures of Carbon Clusters, Cn (n = 3–6, 10).doc

Particle Swarm Optimization (PSO), a population based technique for stochastic search in a multidimensional space, has so far been employed successfully for solving a variety of optimization problems including many multifaceted problems, where other popular methods like steepest descent, gradient descent, conjugate gradient, Newton method, etc. do not give satisfactory results. Herein, we propose a modified PSO algorithm for unbiased global minima search by integrating with density functional theory which turns out to be superior to the other evolutionary methods such as simulated annealing, basin hopping and genetic algorithm. The present PSO code combines evolutionary algorithm with a variational optimization technique through interfacing of PSO with the Gaussian software, where the latter is used for single point energy calculation in each iteration step of PSO. Pure carbon and carbon containing systems have been of great interest for several decades due to their important role in the evolution of life as well as wide applications in various research fields. Our study shows how arbitrary and randomly generated small Cn clusters (n = 3–6, 10) can be transformed into the corresponding global minimum structure. The detailed results signify that the proposed technique is quite promising in finding the best global solution for small population size clusters.

粒子群优化算法（PSO），一种基于群体的随机搜索多维度空间中的技术，迄今为止已在解决多种优化问题中取得成功，包括众多复杂问题，在这些问题中，其他流行的如最速下降法、梯度下降法、共轭梯度法、牛顿法等传统方法均无法得到令人满意的结果。本研究中，我们提出了一种改进的PSO算法，通过结合密度泛函理论以实现无偏全局最小值搜索，该算法在与其他进化方法如模拟退火、盆地跳跃和遗传算法相比时表现出显著优势。当前PSO代码通过将PSO与高斯软件接口，实现了进化算法与变分优化技术的结合，其中高斯软件在PSO的每一步迭代中用于单点能量计算。纯碳和含碳系统由于其重要角色——在生命的进化中以及广泛应用于各个研究领域——已引起数十年的极大关注。我们的研究揭示了如何将任意生成的随机小Cn簇（n = 3-6，10）转化为相应的全局最小结构。详细的结果表明，所提出的技术在寻找小种群大小簇的最佳全局解方面具有极大的潜力。