Physical role of -invariants in dual mappings over carrier spaces for Liouville multiple-quantum NMR of spin clusters within n-automorphisms: , -cubanes and some SU(6) aspects of a -met-carb

Abstract

Abstract The nature of [A]8 MQ-NMR spin clusters and their NMR structure within cooperative (non-shell) dual mapping over carrier subspaces { H v } of Liouville space is examined in the context of the dual-group symmetry-chain and inner tensor product (ITP) algebras. The v -recoupling terms of { G G( v )} , conveniently realised as number partitions (NP) which identity the distinct carrier subspaces, are shown to play a fundamental role as Rota scalar-invariants-over-a-field in the mechanism of SU 2- to - L n cooperativity. These quasi-invariants (number partitions) from v parallel the sets of L n-irreps (chain subduced L 8 ↓ L 6 ↓ O irreps) which allow for the block-partioning of the information-rich higher-q multiple quantum aspects of MQ-NMR. The specific value of mathematical physics of induced (subduced) L n groups, ITPs and L n -invariant p-tuple (NP) models, is demonstrated for the MQ-NMR both of cubanes, under SU 2(3) × L 8 ↓ L 6 ↓ O , and of a [Ti]8−[C]12 met-carb cation under SU (6) × L 8 . The full chain subduction is strongly preferred for modest values of n, in preference to the single-step cycle-index or analogous models, on the grounds of the possibility of accidental degeneracy in the latter. Indeed, an indeterminacy in a projective approach, to the mapping from L 8 onto O ( O h ) symmetry is reported, and highlights the need to prove the retention of self-associative properties for subduced irreps derived from such features of higher L n algebras. The present L n ⊃ L n −1 chain algorithm applied to [4211] and [332] ( L 8) yields the requisite mathematical result. By contrast, the original rovibrational (RV) spin statistics fail such tests and do not constitute a faithful representation of the mapping { L 8 → Γ( O ), cubic } . The full {[λ] → Γ(( L 8 ↓ L 6 ) ↓ O )} mappings over 2⩽ m ⩽6 SU ( m ) Hilbert spaces are now given, where the improper-S2 product for the RV aspects follows directly from Galbraith's earlier work on L 6 ↓ O h .