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One of the early works using the jellium rule to study clusters as “giant atoms” was due to Saito and Ohnishi [20]. The authors studied if a closed shell Na cluster will interact weakly as noble gas atoms do and if an open shell Na cluster will be reactive. They showed that two Na₈ clusters interact weakly just as two noble gas atoms do, thus implying that Na₈ clusters with 1S² 1P⁶ closed electronic shells are chemically inert. In a similar fashion, they showed that a Na₁₉ cluster with electronic configuration of 1S² 1P⁶ 1D¹⁰ 2S¹ can be viewed as an alkali atom as both need one extra electron to close the s‐shell. In Figure 2.2 we show the binding energy of two Na₁₉ clusters as a function of distance computed by these authors. Note that there is an initial attraction leading to the formation of a Na₁₉ dimer with the centers of the Na₁₉ clusters 17 a. u. apart. As the distance between the two Na₁₉ jellium clusters is further reduced, the clusters face a significant energy barrier and eventually coalesce to form a Na₃₈ jellium cluster that is magnetic with two unpaired spins. But, does Na₁₉ cluster mimic the chemistry of a Na atom? To understand this, we plot in Figure 2.3 the binding energy as function of distance between two Na atoms using the Gaussian 16 code [21] and the density functional theory with the Perdew, Burke, Ernzerhof (PBE) form for the generalized gradient approximation [22] and 6‐31 + G* basis function. As can be seen, the energy profile in Figure 2.3 is very different from that in Figure 2.2. Clearly, Na₁₉ cannot be regarded as a superatom mimicking the chemistry of a Na atom.

Figure 2.2 Binding‐energy curves of (Na₁₉)₂ for two different electronic configurations, (N_↑, N_↓ = 20, 18) (filled circles) and (N_↑, N_↓ = 19,19) (open circles). For several inter‐jellium distances, schematic pictures for positive background are shown.

Source: Saito and Ohnishi [20]. © American Physical Society.

Figure 2.3 Binding energy as a function of distance between two Na atoms. The computed bond length (3.0 Å) of the Na₂ dimer agrees well with the experimental value of 3.08 Å.

To understand the effect of geometry of a cluster on its electronic structure, we focus on Na₂₀, which is a closed shell cluster in the jellium model. In Figure 2.4 we show the ground state geometry of Na₂₀ calculated by Sun et al. [23]. Clearly, its geometry is not spherical. However, the molecular orbitals of Na₂₀ (Figure 2.5) show strong resemblance with that in the jellium model. The nondegenerate highest occupied molecular orbital (HOMO) is primarily a 2S orbital and HOMO‐q (q = 1–5) are d‐type, q = 6–8 are p‐type, and q = 9 is s‐type, just as the case in the jellium model. In addition, a HOMO–lowest unoccupied molecular orbital (LUMO) gap of 1.43 eV is indicative of a chemically inert behavior of Na₂₀ cluster.

Figure 2.4 Ground‐state geometry of Na₂₀.

Source: Sun et al. [23]. © American Chemical Society.

Figure 2.5 Molecular orbitals and energy levels of neutral Na₂₀ cluster. The HOMO–LUMO energy gap is indicated (in green).

Source: Sun et al. [23]. © American Chemical Society.

To what extent can a jellium model describe the interaction between two real clusters was further investigated by Hakkinen and Manninen [24] by taking into account the geometries and electronic structure of clusters, explicitly. Using molecular dynamics and density functional theory, they considered a Na₈ cluster in a variety of surroundings. In the gas phase, Na₈ cluster was found to retain its geometry even up to 600 K. But, when two Na₈ clusters are brought together (see Figure 2.6), they collapse forming a deformed Na₁₆ cluster and the electronic shell structure is destroyed. They further showed that Na₈ cluster forms an epitaxial layer (Figure 2.7) when supported on a Na (100) surface. This shows that Na₈ is a magic cluster only when it is held in isolation.

Figure 2.6 Reaction between two Na₈ clusters in vacuum. (a) Time evolution of the potential energy relative to its value in the initial configuration (solid curve, scale on the left) and the center‐of‐mass distance of the two clusters (dotted curve, scale on the right). The two snapshots indicate the initial configuration (left) and the configuration at 2.6 ps (right). (b) Time evolution of the Kohn‐Sham eigen values. The dotted curves indicate empty states.

Source: Hakkinen and Manninen [24]. © American Physical Society.

While Na₈ was found to see its geometry destroyed when interacting with another Na₈ cluster or when supported on a Na (100) surface, the result for Au₂₀ is different. Note that according to the jellium model, Au₂₀ is also a closed shell cluster. Although it has a pyramidal geometry (Figure 2.8), its molecular orbitals show the same pattern as in the jellium model [25]. As shown in Figure 2.9, Au₂₀ maintains its pyramidal structure when deposited on a carbon substrate [26]. However, it is not clear if Au₂₀ would continue to maintain its gas phase geometry when interacting with each other or when supported on an Au substrate? No studies have yet been done to make any conclusion. However, based on extensive studies of Al₁₃, another free electron meal cluster [27–36], it is unlikely that Au₂₀ would maintain its virgin geometry in the above situation.

With 39 valence electrons and an electronic configuration of 1S² 1P⁶ 1D¹⁰ 2S² 1F¹⁴ 2P⁵, Al₁₃ is known to mimic the chemistry of a halogen atom. Indeed, its electron affinity of 3.57 eV is almost identical to that of the Cl atom. It was theoretically predicted [30] and experimentally verified [31] that KAl₁₃ is an ionically bonded cluster where an electron is transferred from K to Al₁₃. Evidence that Al₁₃ behaves like a halogen also came from an experiment of Bergeron et al. who showed that Al₁₃I₂ ⁻ can be viewed as Al₁₃ ⁻.2I, making it look like a triiodide (I₃ ⁻) ion [32]. Similarly, Al₁₄I₃ ⁻ can be viewed as Al₁₄ ²⁺.3I⁻ with Al₁₄ behaving like an alkaline earth element. From the above results, the authors concluded that Al₁₃ and Al₁₄ exhibit a new form of superatom chemistry in which superatoms behave like atoms when they react with other atoms/molecules. However, a different conclusion was reached by Han and Jung who examined whether Al _n clusters exhibit multiple atomic characteristics depending upon n by studying halogenated Al _n (n = 11–15) complexes and plotted the charge (Q) distribution in MX and MX₂ systems (M = Al₁₁–Al₁₅, X = F, Cl, Br, I) vs electronegativity, η of X [33, 34]. The results are presented in Figure 2.10. Noting that the charge transfer Q(M) is nearly independent of n in Al _n clusters in both the systems, the authors concluded that “there is no evidence of an alkaline earth superatom in the Al₁₄ clusters” and that “there are no theoretical grounds to regard Al₁₃I₂ ⁻ as Al₁₃ ⁻.2I.”

Figure 2.7 The initial (left) and the final (right) configuration of the collapse of Na₈ on Na (100). Note the epitaxial arrangement of the adatoms at the end of the run (at 2.8 ps). Both side and top views of the two configurations are shown.

Source: Hakkinen and Manninen [24]. © American Physical Society.

Figure 2.8 (a) Structure and super atomic‐molecule models of Au₂₀ (TAu4). (b) Schematic representation for the superatom−atom D³S−s bonding of Au₂₀ (TAu4).

Source: Cheng et al. [25]. © Royal Society of Chemistry.

Figure 2.9 Direct atomic imaging and dynamical fluctuations of the tetrahedral Au20 cluster soft‐landed on amorphous carbon substrate.

Source: Adapted with permission from Wang et al. [26]. © Royal Society of Chemistry.

Figure 2.10 Q(M) versus η (X = F, Cl, Br, I) for (a) MX (M = Al₁₁–Al₁₅, Al, halogen atoms) and (b) MX₂ (M = Al₁₁–Al₁₅, Al, Si, alkaline earth atoms). The data of Al₁₁ through Al₁₅ basically coincide without revealing any exceptions for Al₁₃ or Al₁₄.

Source: Han and Jung [33]. © American Chemical Society.

For superatoms to be used as building blocks of cluster‐assembled materials, it is important that not only they be stable and mimic the chemistry of atoms but also they should remain in their virgin form when forming a crystal. Liu et al. [35] studied the stability of a KAl₁₃ crystal confined to the CsCl structure. The hypothesis was that Al₁₃ ⁻ clusters will stay apart from each other due to the negative charge they carry. On the contrary, they found that the Al atoms in neighboring Al₁₃ clusters interact and KAl₁₃ as a crystal was unstable. To determine whether changing the cation from a metal to a nonmetallic one would result in stabilizing the Al₁₃ ⁻ icosahedral geometry, Huang et al. [36] recently studied the stability of [(CH₃)₄N⁺][Al₁₃ ⁻] crystal. Note that the ionization potential of (CH₃)₄N⁺ is 3.27 eV, which is even smaller than that of a K atom, namely, 4.34 eV. In addition, the diameter of (CH₃)₄N⁺ is 4.2 Å, which is comparable to the diameter of Al₁₃, namely 5.3 Å. The binding energy of [(CH₃)₄N⁺][Al₁₃ ⁻] cluster, namely 2.68 eV, is also larger than that of KAl₁₃ cluster, which is 2.49 eV. Expecting that a crystal of [(CH₃)₄N⁺][Al₁₃ ⁻] may be stable with both the cation and the anion maintaining their individual geometry, Huang et al. confined the initial crystal structure to three forms – (i) body‐centered‐cubic, (ii) rock salt, and (iii) zinc‐blende phases, which are common crystal structures of binary salts (e.g., CsCl, NaCl, and ZnO) (see Figure 2.11). After optimization, the results show that while the (CH₃)₄N maintains its structure in all of the above systems, Al₁₃ clusters coalesce, ceasing to remain as individual clusters. It is, thus, safe to conclude that stable metallic clusters with electronic closed shells are not good candidates for cluster‐assembled materials.

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.