A quantitative tool to establish magic number clusters, ε3, applied in small silicon clusters, Si2-11

Abstract

The present work focuses on establishing a function to rank the stability of small silicon clusters to characterize their magic numbers. This function is composed by a thermodynamic descriptor, the atomization Gibbs free energy, and indirect kinetic descriptors, the highest occupied molecular orbital energy and the lowest excitation energy of each system. The silicon clusters geometries were optimized using density functional theory within a hybrid meta-GGA approximation (M06), while the electronic energy was corrected by single-point calculation using CASPT2 level of theory to obtain the molecular properties. Both methodologies were combined with polarized diffused triple zeta, 6-311++G(3df,3pd), basis set for all atoms. Some molecular properties and their combinations were considered to create the aforementioned function to represent the clusters chemical stability and their magic numbers. The chosen stability ranking function, called ε, presents results in agreement with the previous mass spectrometry experimental data identifying 4, 6, 7 and 10 as magic numbers for small silicon clusters. We believe this stability ranking function can be useful to study other intramolecular atomic and molecular clusters. Graphical abstract Stability ranking function, ε, applied on Si (n = 2 - 11) clusters showing Fukui's functions for the Si (n = 2 - 11) obtained by the electronic density difference through CASPT2//M06/6-311++G(3df,3pd) with an isosurface value equal to 0.003.

Stability and Reactivity of Silicon Magic Numbers Doped with Aluminum and Phosphorus Atoms

“…These results could be expected, based on the fact that the Si 11 is one of the less stable on the Si 2−11 series and the Si 6 , Si 7 , and Si 10 are the most stable ones. 37 In general, the double substitution (AlP) produces more stable clusters than the monodoping (Al or P) once they combine both aluminum and phosphorus stability properties: increase of the strength of the weakest chemical bond and increase of the cohesion energy, respectively. From Fukui's functions, Figure 7, in general, one can infer that the chemical substitution tends to increase the Si 6 , Si 7 , and Si 10 reactivity due to the significant increment of the electronic density volume.…”

“…Also, point group symmetry was considered to construct structures, 27−36 which were not taken into account by the substitution or addition routes when incorporating aluminum and phosphorus atoms on the pure silicon clusters. 37 For each starting structure, the allowed spin multiplicities were also considered. Single doped clusters exhibit an odd number of electrons, then only doublet and quartet spin multiplicity were considered.…”

“…Alternatively, an initial space of isomers can also be obtained by simpler techniques, as it was employed in the present work, by performing substitution ,,,, and addition (exohedral) routes, ,,,, as shown in Figure . Also, point group symmetry was considered to construct structures, − which were not taken into account by the substitution or addition routes when incorporating aluminum and phosphorus atoms on the pure silicon clusters . For each starting structure, the allowed spin multiplicities were also considered.…”

“…Silicon nanoclusters, Si 2–11 , were extensively studied by previous theoretical and experimental works, − focusing on the global minimum, electronic structure, and magic numbers ( n = 4, 6, 7, and 10) characterization. However, none of these works have systematically sought to characterize cage-like silicon clusters since their experimental formation is difficult to obtain because of their sp 3 hybridization. − Cage-like structure was experimentally observed on small fullerene cluster (C 28 ) by uranium addition inside the carbon cluster .…”

“…The multiconfigurational CASPT2 method was applied to compute the vertical excitation energy ( T e ) and the electronic density variation due to the doping. The chemical stability was probed by means of our previously developed ε 3 function, which encompasses both thermodynamical and kinetic aspects of the systems based on the HOMO energy, vertical excitation energy, and atomization free energy. Also, the reactivity was studied using Fukui’s functions − calculated by electronic density of the neutral and ionic species.…”