The pairing principle in tensor surface harmonic theory: Electron counts of large closo boranes

Tetratomic tetrahedral main-group clusters have four or six (but not five) skeletal pairs, an exception to the (n + 1) rule for closo clusters. A general group-theoretical criterion, based on the pairing principle in Stone's tensor surface harmonic (TSH) theory, is presented for symmetry-induced departures from the (n + 1) rule. A count of (n + 2) or n skeletal pairs is forced by symmetry for: (i) T- or T_d-clusters with an odd number of sets of four equivalent cage atoms, and (ii) C_m or C_mv (m ? 3) clusters with an odd number of cage atoms on the C_m-axis. Three such cases, B_nH_n or B_nH_n^4?, are found among the hypothetical supraicosahedral closo boranes proposed by Brown and Lipscomb (n = 16,19 or 22). Extended Hückel calculations show that the TSH pattern applies to the whole series of closo boranes (n$\text{?}$ 24). Charges of + 4, ?8 and ?2 are discussed for a very large B₃₂H₃₂^q cluster.