GLOBAL GEOMETRY OPTIMIZATION OF SILICON CLUSTERS EMPLOYING EMPIRICAL POTENTIALS, DENSITY FUNCTIONALS, AND AB INITIO CALCULATIONS

Abstract

Sin clusters in the size range n = 4-30 have been investigated using a combination of global structure optimization methods with DFT and ab initio calculations. One of the central aims is to provide explanations for the structural transition from prolate to spherical outer shapes at about n = 25, as observed in ion mobility measurements. Firstly, several existing empirical potentials for silicon and a newly generated variant of one of them were better adapted to small silicon clusters, by global optimization of their parameters. The best resulting empirical potentials were then employed in global cluster structure optimizations. The most promising structures from this stage were relaxed further at the DFT level with the hybrid B3LYP functional. For the resulting structures, single point energies have been calculated at the LMP2 level with a reasonable medium-sized basis set, cc-pVTZ. These DFT and LMP2 calculations were also carried out for the best structures proposed in the literature, including the most recent ones, to obtain the currently best and most complete overall picture of the structural preferences of silicon clusters. In agreement with recent findings, results obtained at the DFT level do support the shape transition from prolate to spherical structures, beginning with Si 26 (albeit not completely without problems). In stark contrast, at the LMP2 level, the dominance of spherical structures after the transition region could not be confirmed. Instead, just as below the transition region, prolate isomers are obtained as the lowestenergy structures for n ≤ 29. We conclude that higher (probably multireference) levels of theoretical treatments are needed before the puzzle of the silicon cluster shape transition at n = 25 can safely be considered as explained.

Efficient global optimization of reactive force‐field parameters

“…[40,41] It combines threadlevel and MPI-level parallelism to achieve high scalability on shared memory as well as distributed memory architectures. The OGOLEM framework embodies our accumulated knowledge on nondeterministic global optimization in general and on EA s in particular [42,43] for various applications: cluster structures, [44][45][46][47][48][49][50][51][52][53][54] protein folding, [55] potential fitting, [34,35,[56][57][58][59][60] molecular design, [61] and abstract benchmarks. [62] EAs [19] borrow nomenclature from natural selection and evolution processes.…”

“…It combines thread‐level and MPI‐level parallelism to achieve high scalability on shared memory as well as distributed memory architectures. The ogolem framework embodies our accumulated knowledge on nondeterministic global optimization in general and on EA s in particular for various applications: cluster structures, protein folding, potential fitting, molecular design, and abstract benchmarks …”

A Fukui function‐guided genetic algorithm. Assessment on structural prediction of Sin (n = 12–20) clusters

“…It is worth mentioning that for the Si 12 cluster, the global minimum is identified among the Fukui-isomers, similarly to the previously studied cases Si 4 -Si 8 . [33] However, for the larger studied clusters (Si 13 -Si 20 ) [69][70][71][72][73][74][75] the Fukui function does not lead to the global minimum (vide infra).…”

Efficiency of random search procedures along the silicon cluster series: Sin (n = 5–10, 15, and 20)

“…The structures of atomic nanoclusters frequently cannot be derived based solely on chemical intuition,1, 2 even for simple neutral homoatomic species (e.g., Au 20 ,3–6 Si 4–25 ,7–24 Hg 2–8 ,25 Y 3–15 ,26 Pb 2–20 ,27 Ge 2–20 ,28 Rh 2–20 , and Ru 2–20 ,29 or Zr 2–10 30). Experimental structural determination of neutral species is limited by the scarcity of suitable analytical techniques and by the low amounts of purified clusters.…”

“…Pure silicon wafers on which the semiconductor industry has placed tremendous effort are now approaching nano sizes. In this context, the establishment of key structure‐property relationships associated with silicon clusters has attracted much interest 7–24. Bare silicon clusters, therefore, have been commonly used to validate new implementations of widely reviewed31–34 search algorithms such as basin‐hopping,23 genetic algorithm,7, 11, 12, 14, 19, 21, 24 and simulated annealing,15 as well as less known methods such as dual minima hopping,13, 17 and taboo search9 (which both avoid visiting twice the same minimum either by forbidding PES regions or by escaping them by a short Molecular Dynamics run), the big bang algorithm (that generates millions of supercompressed random structures before density‐functional‐based tight‐binding optimization),8 or the “GAGA” approach (which performs a global optimization using empirical potentials that are, at the same time, readapted to ab initio data at each newly found low‐lying minima) 10…”