Building Clusters Atom-by-Atom: From Local Order to Global Order

Principles of isomer stability in small clusters

In this work we study isomers of several representative small clusters to find principles for their stability. Our conclusions about the principles underlying the structure of clusters are based on a huge database of 44 000 isomers generated for 58 different clusters on the density functional theory level by Minima Hopping. We explore the potential energy surface of small neutral, anionic and cationic isomers, moving left to right across the third period of the periodic table and varying the number of atoms n and the cluster charge state q (X q n , with X = {Na, Mg, Al, Si, Ge}, q = -1, 0, 1, 2). We use structural descriptors such as bond lengths and atomic coordination numbers, the surface to volume ratios and the shape factor as well as electronic descriptors such as shell filling and hardness to detect correlations with the stability of clusters. The isomers of metallic clusters are found to be structure seekers with a strong tendency to adopt compact shapes. However certain numbers of atoms can suppress the formation of nearly spherical metallic clusters. Small non-metallic clusters typically also do not adopt compact spherical shapes for their lowest energy structures. In both cases spherical jellium models are not any more applicable. Nevertheless for many structures, that frequently have a high degree of symmetry, the Kohn-Sham eigenvalues are bunched into shells and if the available electrons can completely fill such shells, a particularly stable structure can result. We call such a cluster whose shape gives rise to shells that can be completely filled by the number of available electrons an optimally matched cluster, since both the structure and the number of electrons must be special and match. In this way we can also explain the stability trends for covalent silicon and germanium cluster isomers, whose stability was previously explained by the presence of certain structural motifs. Thus we propose a unified framework to explain trends in the stability of isomers and to predict their structure for a wide range of small clusters.

Density functional investigations on structural and electronic properties of anionic and neutral sodium clusters Na(N) (N = 40-147): comparison with the experimental photoelectron spectra

Density functional calculations have been carried out to obtain low energy equilibrium geometries of anionic and neutral sodium clusters over a wide range of sizes 40 ≤ N ≤ 147, where N is the number of atoms. An exhaustive search for the low energy equilibrium geometries has been carried out. The density of states of the lowest energy geometries are compared with the experimental photoelectron spectra (Huber et al 2009 Phys. Rev. B 80 235425; Kostko et al 2007 Phys. Rev. Lett.98 043401) for N > 41. The agreement between theory and experiment is good for almost all the clusters and the changes in the spectrum with size correlate very well with the changes in the shapes as observed in the evolutionary trend of the ground state geometries.

Melting and Freezing of Metal Clusters

Recent developments allow heat capacities to be measured for size-selected clusters isolated in the gas phase. For clusters with tens to hundreds of atoms, the heat capacities determined as a function of temperature usually have a single peak attributed to a melting transition. The melting temperatures and latent heats show large size-dependent fluctuations. In some cases, the melting temperatures change by hundreds of degrees with the addition of a single atom. Theory has played a critical role in understanding the origin of the size-dependent fluctuations, and in understanding the properties of the liquid-like and solid-like states. In some cases, the heat capacities have extra features (an additional peak or a dip) that reveal a more complex behavior than simple melting. In this article we provide a description of the methods used to measure the heat capacities and provide an overview of the experimental and theoretical results obtained for sodium and aluminum clusters.

Metal clusters with hidden ground states: Melting and structural transitions in Al115(+), Al116(+), and Al117(+)

Heat capacities measured as a function of temperature for Al(115)(+), Al(116)(+), and Al(117)(+) show two well-resolved peaks, at around 450 and 600 K. After being annealed to 523 K (a temperature between the two peaks) or to 773 K (well above both peaks), the high temperature peak remains unchanged but the low temperature peak disappears. After considering the possible explanations, the low temperature peak is attributed to a structural transition and the high temperature peak to the melting of the higher enthalpy structure generated by the structural transition. The annealing results show that the liquid clusters freeze exclusively into the higher enthalpy structure and that the lower enthalpy structure is not accessible from the higher enthalpy one on the timescale of the experiments. We suggest that the low enthalpy structure observed before annealing results from epitaxy, where the smaller clusters act as a nucleus and follow a growth pattern that provides access to the low enthalpy structure. The solid-to-solid transition that leads to the low temperature peak in the heat capacity does not occur under equilibrium but requires a superheated solid.