Figure 2.11 Initial crystal structure for (a) body‐centered‐cubic (bcc), (b) rock salt (rs), and (c) zinc‐blende (zb) phases of (CH₃)₄N⁺Al₁₃⁻ bulk. (d) Optimized structures of (CH₃)₄N⁺Al₁₃⁻ show the coalescence of Al₁₃ clusters when forming a bulk material.

      Source: Huang [36]. © American Chemical Society.

      On the contrary, jellium shell closure rule has been effectively used to explain the stability of ligated metal clusters, particularly ligated Cu, Ag, and Au clusters. As the number of atoms in the metal core as well as the number and type of ligands can be varied independently, one is gifted with considerable flexibility to design ligated metal clusters as superatoms. Consider, for example a metal core consisting of N_C number of core atoms and N_S number of surface atoms. The surface atoms are the ones that bind to the ligands forming either an ionic or a covalent bond. If the number of valence electrons in the core atoms correspond to shell closure, it is likely that such a cluster can gain unusual stability. Hakkinen and coworkers [37, 38] used this approach to explain the unusual stability of some ligated Al and Au clusters. For example, the unusual stability of Al₅₀Cp*₁₂ can be explained by realizing that 12 Cp* ligands withdraw 12 electrons from the Al₅₀ core, thus leaving 50 × 3 − 12 = 138 electrons. This is just enough electrons to close the 1I superatom jellium shell. Similarly, the authors showed that ligand‐protected Au clusters, satisfying electron shell closure at 8, 34, and 58, can indeed be synthesized.

That stable ligated metal clusters consistent with the jellium shell closure rule can be synthesized has been verified by experiments. Some of these clusters include Au₂₅(SR)₁₈ ⁻ [39] and As₇ and As₁₁, with cryptated alkali atoms [40, 41]. However, it has been pointed out that the strength of the ligand interaction and its effect on the geometry on the metal core play an important role in the electron counting [33, 42, 43]. Examples of some ligated clusters consistent with the jellium model are given in Table 2.1.

      2.2.2 Octet Rule

The octet rule was developed in early 1900s to account for the chemistry of low atomic number (<20) elements [8–10]. The noble gas atoms, with their outer electron configuration of ns ² np ⁶, have closed electronic shells and hence, a large HOMO–LUMO gap, high ionization potential, and low electron affinity. Thus, they are very stable and chemically inert. A variety of superatomic clusters containing elements with low atomic number can be designed such that constituent atoms satisfy the octet shell closure. We begin with the description of Group 1 alkali and Group 17 halogen atoms.

Table 2.1 Examples of ligated clusters with effective number (n_e) of valence electrons for jellium shell closure.