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СКАЧАТЬ a spherical geometry and the charges on the positive ion cores are distributed uniformly, they showed that the stability of the magic Na clusters is due to shell closures of their electronic orbitals such as 1S², 1S² 1P⁶, 1S² 1P⁶ 1D¹⁰ 2S², 1S² 1P⁶ 1D¹⁰ 2S² 1F¹⁴ 2P⁶, . . . .

      In the following, we first study clusters of simple metals whose stability can be well explained by the jellium model and see if magic clusters can be assembled to make a bulk material. Next, we explore a number of other electron‐counting rules such as the octet rule for sp elements [8–10], 18‐electron rule for transition metal elements [11], 32‐electron rule for rare earth elements, Hückel's aromaticity rule for organic molecules [12, 13], and Wade‐Mingos rule [14–17] for boron‐based clusters and Zintl ions [18, 19]. We focus not only on neutral but also on charged clusters that can be stabilized by using any one of the above rules and combinations thereof.

2.2 Electron‐Counting Rules

      2.2.1 Jellium Rule

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 СКАЧАТЬ