Origin of the Unusual Stability of B<sub>12</sub> and B<sub>13</sub><sup>+</sup> Clusters

A novel way to analyze the boron clusters is proposed in which the cluster is partitioned as inner and outer rings. Fragment molecular orbital analysis, based on this fragmentation, reveals that the delocalized valence electrons in B<sub>12</sub> and B<sub>13</sub><sup>+</sup> clusters can be trifurcated as 6π−6σ<sub>delo</sub>−6σ<sub>3-ring</sub>, leading to triple aromaticity, which is unique to these clusters.