Now, the B12Li4 will be examined. Removal of four Li atoms from B12Li8
formally leads to B12Li4 which has 40 valence electrons. This is a magic number associated with a high thermodynamic stability of many spherical or subspherical, especially tetrahedral structures [137–140], and well defined by the PSM. In the present case, the most stable spherical structure found is the 4C isomer with a C3
symmetry after geometry relaxing from a structure with Td symmetry having a three-fold negative frequency of 360i cm-1. Its large HLG (3.2 eV) does not make 4C the global minimum of B12Li4, but instead the conical 4A does. This is a form
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similar to its isoelectronic B13Li [128] which is a new member of the tetrahedral- typed B13 ligand half-surround cluster (cf. structures given in Figure 3.27).
Figure 3.27. The lowest-energy structures of the isoelectronic B13Li and B12Li4
clusters.
Such a conical structure is far different from a spherical shape. Therefore, the electronic structure of 4A cannot be explained in terms of spherical models as in the case of 4C (cf. Figure 3.23). While the solution of the Schrửdinger equation for a free particle moving in a cone is not yet available, the circular disk model (DM) [101, 117, 141] is the most adaptable for this cone-shaped structure. The DM eigenstates are usually classified according to nα with n and α being two principal quantum numbers, namely, n = 1, 2, 3… is a radial number, and α = 0, ±1, ±2, ±3
… and usually denoted by Greek letter as α = σ, π, δ, ϕ, … is a rotational quantum number. An eigenstate defined by a none-zero value for α is twofold degenerate.
Both structures B13Li and 4A have similar MO shapes, in such a way that they are both suitable to be modelled by the DM. For the sake of convenience, the B13Li structure is discussed in some detail here because it is more symmetrical than 4A and therefore, the nodes in its MOs are more easily recognized and assigned.
The valence MOs and LUMO of 4A are represented in Figure 3.28 including two different sets, namely, a) the σ set, and b) the π set of B13Li. The 1γ-orbital is
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found to have only a single MO (HOMO-7) because it meets the maximum number of cylindrical nodes condition [141]. Accordingly, the latter consists of a perfect sign alternation in every atom of the outer ring of the cylinder. Although a molecular conical structure differs much from a perfectly planar disk structure in which the atoms are concaved from the centre out to the edge of the disk, the MOs observed in a cone also have such properties. Observation of the 2σ and 3σ-orbitals of the planar disk B202- (ref. [142]) and B13Li under the top view and side view as shown in Figure 3.30 can clearly emphasize this similarity. This transformation makes these MOs appear earlier in the corresponding electronic wavefunction.
Specifically, the σ electronic configuration for the DM of B202- is [1σ21π41δ42σ21ϕ42π41γ41η42δ43σ2], whereas that of B13Li (cf. Figure 3.28) is [1σ21π42σ21δ42π41ϕ43σ21γ23π42δ4].
Figure 3.28. Valence MOs and LUMO of the B13Li cluster assigned within the circular disk model. H stands for HOMO and L stands for LUMO.
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For the sake of a clearer view, a look back at the π electron configuration of the bowl B30, [1σ21π41δ42σ22π41ϕ4] [117], is necessary, which was assigned as a disk aromatic species, even though it is also bowl or slightly conical with a wider angle than those of the B13Li counterpart. Phenomenologically, a mixed cone-disk model can be proposed according to which the MOs spectrum changes the earlier appearance of the MOs with large radial and small rotational quantum numbers. The smaller the apex angle, the larger the change.
Figure 3.29. The total current density maps of a) B13Li and b) B12Li4 and the c) π and d) σ current density maps of B13Li. External magnetic field vector is present by the blue arrow. Vectors are plotted on a surface having the cone shape at 1 Å inside
B-cone framework.
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Similarly, the electronic configuration of the π set for B13Li is [1σ21π42σ2].
In particular, the 2σ(π)-orbital is the HOMO of B13Li. Because such an assignment for the HOMO can be confusing, we make a representation of the 2σ(π)-orbital from the disk model and compare it to the HOMO of B13Li in a C4v symmetry; this is shown in Figure 3.28. Interestingly, the natural appearance of such HOMO has been observed in other stable structures such as the 5σ1-orbital in the disk Ge104+, the tube B14, the 6σ1-orbital in B12Si2by the cylinder model (cf. ref. [25]), and the (2 0 2)-orbital in NiB14 and Ni2B202- (cf. ref. [143]), Li2FeB14 (cf. ref. [144]) by a hollow cylinder model [18]. This type of HOMO plays a role in the separation of two different electron shells in a molecule, which ultimately increases significantly the thermodynamic stability of that molecule.
Figure 3.30. 2σ and 3σ-orbitals of B202- and B13Li under the top view and side view.
The diatropic flows of the current density maps of B13Li and B12Li4 4A depicted in Figure 3.29 point out that they are strongly aromatic species. The sets of π and σ MOs of B13Li exhibit diatropic ring currents, implying that this structure is characterized by a double aromaticity character.