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Figure 2.21 Left panel: photoelectron spectra at 355 nm (3.496 eV) for (a) LiAl₄⁻, (b) NaAl₄⁻, and (c) CuAl₄⁻ and at 266 nm (4.661 eV) for (d) LiAl₄⁻, (e) NaAl₄⁻, (f) CuAl₄⁻, and (g) square planar Al₄^2– cluster. Right panel: optimized structures of LiAl₄⁻, NaAl₄⁻, Al₄²⁻ [at the CCSD(T)/6‐311+G* level of theory], and CuAl₄⁻.

      Source: Li et al. [76]. © American Association for the Advancement of Science.

      2.2.6 Wade‐Mingos Rule

Polyhedral skeletal electron pair theory (PSEPT), commonly known as the Wade‐Mingos rule, was developed by Wade [14, 15] and Mingos [16, 17] to account for the structure and bonding of polyhedral borane clusters (B _n H _n ). Because of electron deficiency, B cannot form conventional covalent bonds between adjacent pairs of atoms. Instead, it forms multicenter bonds with the valence electrons separated into external and skeletal electrons. The former forms traditional covalent bonds with ligands while the latter contributes to cage bonding. Consider a borane polyhedron, B _n H _n with n being the number of vertices. According to the Wade‐Mingos rule, 2n + 2, 2n + 4, and 2n + 8 electrons are needed to form closo, nido, and archano boranes, respectively. B₁₂H₁₂ ²⁻ is a well‐known example of a closo‐borane having an icosahedral geometry (Figure 2.22). Here, n = 12. Of the three valence electrons of B, one is involved in the covalent bonding with H while the other two are contributed to the cage bonding. Because 2n + 2 = 2 × 12 + 2 = 26 electrons are needed to stabilize the cage, B₁₂H₁₂ ²⁻ is stable as a dianion. Indeed, the electron affinities of the first and second electron are 4.56 and 0.9 eV, respectively.

If one of the B atoms in B _n H _n is replaced by a C atom, according to the Wade‐Mingos rule only one extra electron will be needed to satisfy the stability criterion. Hence, CB _{n − 1}H _n ⁻ should be a stable cluster. Indeed, this is the case and mono‐carboranes with n = 7–12 have been isolated and structurally characterized [77–80]. Pathak et al. [81] hypothesized that CB₁₁H₁₂ should also behave like a halogen atom. Using density functional theory, the authors calculated the structure and found the electron affinity of CB₁₁H₁₂ to be 5.39 eV, establishing it as a superhalogen. They also showed that MB₁₂H₁₂ (M = Li, Na, K, Rb, and Cs) clusters are superhalogens as they only need one electron to satisfy the Wade‐Mingos rule. In addition, they showed that CB₁₁H₁₂ can be used as the building blocks of hyperhalogens.