Distinctive combinatorial aspects of NMR spin invariance for icosahedral exo-cage spin cluster problems: symmetry subduction and mapping for ¹³C-fullerene and the analogous biclusters systems, [¹³C]^6-₆₀, [¹²CF]₆₀, and [¹³CF]₆₀

Abstract

The absence of any single pair of nuclei on any one of the three representative C_ℓ, ℓ = 2, 3, 5 (subduced) icosahedral rotational symmetry axes of [¹³C]₆₀, leads to a virtually unique case of the invariance hierarchy over M(SO(2))-weights reducing to a purely combinatorial problem. Aspects of the subduced mapping from the sixtyfold symmetric group onto for are considered in the context of nuclear magnetic resonance (NMR) coherence transfer networks over practical coherent superpositional (CSP)-forms of Liouville space bases within p ˇ- 4 of Liouvillian SU2 for high order multiple quantum processes. The number of equivalent sites under each type of rotational operation provides a highly structured invariance set. The use (of necessity) of a largely character-free algebra within the simply reducible (SR) {[λ]} ≡ [λ] ⊗ [n - 1, 1], or non-SR {[λ′]} ≡ [n - 2, 2] ⊗ [n - 2, 2], group aspects and additional properties allows the tertiary [n - r, r - r′, r′] () partitions to be correlated with their subduced irreps. The NMR aspects are contrasted with those arising under dodecahedral subduced spin symmetry of the fullerene analogues, [¹³C]^2- ₂₀, and the [¹²CH]₂₀, [¹³CH]₂₀ dodecahedranes.