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Figure 2.16 (a) Isolated (CH₃)₄N⁺Al₁₃⁻ cluster. (b) Optimized body‐centered‐cubic phase of (CH₃)₄N⁺Al₁₃⁻ crystal with T_d symmetry (same as the molecular point‐group of the respective ions). (c) Total energy and lattice parameter (lattice constants a, b, and c in Å, interaxial angles α, β, and γ in degrees, and volume in ų) fluctuations during the ab initio molecular dynamics simulation. (d) Snapshot of the structure obtained from the ab initio molecular dynamics (AIMD) simulation for 10 ps after 4 ps to allow the system to reach thermal equilibrium.

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

      2.2.3 18‐Electron Rule

      Stability of transition metal compounds can be accounted for by using the 18‐electron rule where 18 electrons are needed to fill ns ² np ⁶ and (n−1)d ¹⁰ orbitals [11]. Classic examples of complexes stabilized by the 18‐electron rule are chromium bisbenzene [Cr(C₆H₆)₂] and ferrocene [Fe(C₅H₅)₂]. C₆H₆ and C₅H₅ have six and five π electrons each. As Cr and Fe have outer electronic configuration of 3d ⁵ 4s ¹ and 3d ⁶ 4s², respectively, one can see that both Cr(C₆H₆)₂ and Fe(C₅H₅)₂ are 18‐electron systems.

Pykko and Runenberg [55] showed that an all‐metal cluster, Au₁₂W, with a HOMO–LUMO gap of 3.0 eV is very stable due to the 18‐electron rule. Here, 12 Au atoms contribute 12 electrons while W atom (3d ⁵ 4s ¹) contributes 6 electrons. This prediction was later verified in photoelectron spectroscopy experiment by Wang and collaborators [56]. A further proof of the 18‐electron rule can also be seen by measuring the electron affinity of Ta@Au₁₂. Note that with 17‐valence electrons, Ta@Au₁₂ needs one extra electron to satisfy the 18‐electron shell closure rule. Indeed, the measured electron affinity of 3.76 eV makes Ta@Au₁₂ an all‐metal superhalogen [57]. Chen et al. calculated the structure and stability of M@Au₁₂ ²⁻ (M = Ti, Zr, Hf) to see if these clusters can be stable and thus can be regarded as superchalcogens. The results are given in Figure 2.17. Note that all these structures are dynamically stable. However, Ti@Au₁₂ ²⁻ is unstable against an electron loss by 0.23 eV while M@Au₁₂ ²⁻ (M = Zr, Hf) dianions are stable against the second electron loss by 0.05 eV, due to their increased size.

Figure 2.17 (a)–(c) are the optimized geometries for MAu₁₂^0,1−,2− (M = Ti, Zr, and Hf) clusters, respectively. Yellow, dark red, purple, and blue spheres stand for Au, Ti, Zr, and Hf atoms, respectively.

      Source: Chen et al. [54]. © American Chemical Society.

      2.2.4 32‐Electron Rule

The 32‐electron rule applies to clusters containing early 5f elements where complete shell closure of s ² p ⁶ d ¹⁰ f ¹⁴ orbitals give them stability and chemical inertness. The discovery of empty icosahedral Zintl ions such as Pb₁₂ ²⁻ and Sn₁₂ ²⁻ [58, 59] motivated Dognon et al. [60] to study the stability of endohedral clusters Pu@Pb₁₂. With an electronic configuration of [Rn] 5f ⁶ 7s ² for Pu and [Xe] 4f ¹⁴ 5d ¹⁰ 6s ² 6p ² for Pb, Pu@Pb₁₂ constitutes a 32‐electron system. The calculated HOMO–LUMO gap of 1.93 eV and the binding energy of 22.17 eV measured against the ionic dissociation limit confirm that the stability of Pu@Pb₁₂ arises due to the 32‐electron shell closure. In Figure 2.18 we compare the orbital energies of Pu@Pb₁₂ with that of Pb₁₂ ²⁻. One can see the strong participation of the central atom orbitals in bonding.

A few years later, Ghanty and coworkers [61] showed that the stability of Pu embedded in a C₂₄ cage also follows the 32‐electron rule, with 8 electrons contributed by Pu and 24 π electrons contributed by the C₂₄ fullerene. The authors found that the C₂ symmetry of the empty C₂₄ fullerene transforms to D_6d symmetry, once encapsulated with the Pu atom. The HOMO–LUMO gap of 1.83 eV of the bare C₂₄ cage changes to 3.26 eV following Pu encapsulation. The binding energy of the Pu@C₂₄ clusters measured with respect to atomic fragments is 6.77 eV. Other 32‐electron systems studied recently include An@C₂₈ [62], Pu@Sn₁₂ [63], (U@Si₂₀)⁶⁻ [64], and actinide‐encapsulated fullerene systems [65], U@C₂₈ [66], Ln(CO)₈ ⁻ (Ln = Tm, Yb, Lu) [67], and superatomic CBe₈H₁₂ cluster [68].

Figure 2.18 Orbital energies of Pu@Pb₁₂

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