The procedure for determining the most stable isomers of B₁₂H _n was very similar to that reported in 12 , namely, we started with optimized icosahedral and quasi-planar B₁₂ clusters, and for a given n, we have calculated the total energies of all possible clusters that resulted from adding hydrogen atoms to the vertices of the distorted icosahedron or to the outer boron atoms of the quasi-planar structure; 2D clusters with an even n have been considered in our previous work 6 , and here, we have extended the investigation to an odd n. The energetically most favorable 2D and 3D B₁₂H _n structures are shown in Figure 1. The minimum-energy cluster structures of B₁₂F _n , shown in Figure 2, have been found by replacing the hydrogen atoms of the low-lying B₁₂H _n isomers by fluorine atoms. Interestingly enough, the resulting structures are similar to those found for B₁₂H _n . One of the small differences is that the B-F bonds are on average 13% longer than the B-H bonds.

In Figure 3, we have plotted the total energy difference between quasi-planar (or fully planar) and icosahedral B₁₂X _n (X = H, F) clusters as a function of n, the number of H or F atoms in the cluster. As can be seen from the figure, the quasi-planar clusters with up to four hydrogen atoms are more stable than the corresponding icosahedral structures (a similar result has been recently reported 11 for B₁₂H _n ^0/- clusters). The same is true for the fully planar B₁₂F₆ molecule, which is 0.63 eV lower in energy than the 3D cluster. The 2D and 3D B₁₂F₅ isomers are almost degenerated in energy. From Figure 3, we can also see that the energy difference, E _2D - E _3D, increases monotonically with n, with the exception of the two 'minima' for B₁₂F₄ and B₁₂F₆. These two minima may suggest an additional stabilization of the 2D structures over the 3D counterparts due to the presence of aromatic stabilization energy.

Similar results to those presented in Figure 3 were reported for the icosahedral and quasi-planar B₁₂H _n ⁺ structures 12 . However, in their recent work, Ohishi et al. 14 have used the PBE0 functional instead of the B3LYP functional to determine the energies of the B₁₂H _n ⁺ clusters. The authors' choice was motivated by the fact that the B3LYP functional may overestimate the energy difference between 2D and 3D structures. To address this problem, we calculated the energy difference between the 2D and 3D structures of B₁₂, B₁₂H₆, and B₁₂F₆ using the very accurate DMC approach. The DMC E _2D - E _3D values are -5.13, 0.79, and -0.47 eV for B₁₂, B₁₂H₆, and B₁₂F₆, respectively, whereas the corresponding B3LYP values are -5.34, 0.41, and -0.63 eV, respectively (see Figure 3). This means that the DMC values are shifted up by a value not larger than about 0.4 eV with respect to the B3LYP values. This, however, does not affect the conclusions that are drawn from Figure 3, since even if we shift up the curves by 0.4 eV, the quasi-planar B₁₂H _n and B₁₂F _n (n = 0 to 4), and the fully planar B₁₂F₆ clusters still remain energetically favorable.

As calculated here and also reported in 11 , the fully planar B₁₂H₆ cluster corresponds to a local minimum of energy, whereas the D _3h-B₁₂F₆ structure wins the competition with other 2D and 3D isomers and corresponds to a global minimum of energy. Many properties of the B₁₂H₆ cluster have been previously described in 6 , but for consistency purposes, we have repeated some of those calculations at the B3LYP/6-311++G(d, p) level of theory. The highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gaps of the planar B₁₂H₆ and B₁₂F₆ structures are 3.54 and 4.39 eV, respectively, whereas the HOMO-LUMO gaps of the corresponding 3D clusters are the same and equal to 2.73 eV. The B-H and B-F interatomic distances are 1.179 and 1.326 Å in B₁₂H₆ and B₁₂F₆, respectively. For comparison, the computed B-H and B-F bond lengths in borane (BH₃) and boron trifluoride (BF₃) are 1.190 and 1.318 Å, respectively.

While both 2D structures, B₁₂H₆ and B₁₂F₆, have similar shape and size, they exhibit quite different magnetic properties that are directly related to aromaticity. First, we have computed the anisotropy of magnetic susceptibility. The values for B₁₂H₆ and B₁₂F₆ are -208.1 and -125.8 cgs ppm, respectively. The isotropic values of the magnetic susceptibility are -91.9 and -118.2 cgs ppm for B₁₂H₆ and B₁₂F₆, respectively. These results suggest that the induced ring current is stronger for B₁₂H₆ than for B₁₂F₆. Similarly, as reported in 6 for B₁₂H₆, the central part of the B₁₂F₆ molecule has a paratropic current flowing inside the inner B₃ triangle. The antiaromaticity of the inner triangle is, however, smaller for B₁₂F₆ than for B₁₂H₆ since the NICS(0) values are 3.9 and 13.3 ppm, respectively. A global aromatic current is dominant above the B₁₂F₆ (B₁₂H₆) molecule since the NICS values are negative, NICS(1) = -5.5 ppm (-3.6 ppm) and NICS(2) = -4.8 ppm (-5.0 ppm), 1 and 2 Å above and below the center of the cluster.

To examine the influence of charge on the structure and stability of the fully planar clusters, we have also studied charged 2D and 3D B₁₂X₆ (X = H, F) structures. The lowest energy 2D structures identified for B₁₂H₆ and B₁₂F₆ were used as initial structures for the structural optimization at a given charge state. For the 3D structures, we have made a search over all possible configurations of the hydrogen or fluorine atoms. For the lowest energy structures with an even number of electrons, a singlet multiplicity has been assumed, whereas doublet and quartet multiplicities were considered for clusters with an odd number of electrons. In the later case, clusters with lower multiplicity were energetically more favorable. The structures of the 2D and 3D charged clusters are shown in Figure 4. It has been previously reported that the fully planar D _3h-B₁₂H₆ cluster undergoes structural distortions if charged with one electron, although the quasi-planarity is preserved 6 . In general, all the charged 2D B₁₂X₆ (X = H, F) clusters are quasi-planar rather than fully planar, as can be seen in Figure 4. In Figure 5, we have plotted the energy difference between 2D and 3D [B₁₂X₆] ^q (X = H, F) structures as a function of the cluster charge state q. We have found that the addition of one or two electrons to fully planar B₁₂H₆ and B₁₂F₆ clusters (or the removal of one electron from them) makes those structures even less energetically favorable with respect to the corresponding 3D isomers. This is, however, less pronounced for B₁₂H₆ than for B₁₂F₆ as shown in Figure 5. Finally, it should be noted that the quasi-planar B₁₂F₆ ²⁺ cluster is much more stable than its 3D isomer. Finally, all structures and energies are provided in Additional file 1.