Structure of MQ-NMR spin spaces under higher [formula omitted]n- and ( [formula omitted]n)↓ [formula omitted] symmetries. II. Γ/ [formula omitted] ( [formula omitted]6)↓ [formula omitted] subduced irreps for sixfold spin clusters pertaining to the molecular cage ion, [ 11BH] 62−

Abstract

The structures of higher n-fold spin cluster systems as irreps under the S 6/( S 6)↓ O groups are derived using combinatorial techniques over permutational fields, namely that of generalized wordlengths (GWL), to generate the invariance and irrep sets over the M ( q) subspaces for the [ A] 6 ( Ii) clusters, i.e. those derived from sets of identical nuclear spins I i whose magnitude lies between 1 2 ⩽ I i ⩽ 3 2 . The partitions and invariance properties of such monoclusters provide the background to an investigation of the structure of bicluster spin problems over both Hilbert and Liouville spaces. Hence, the [λ], [ λ ] ( S n ) partitional aspects of the NMR of the borohydride molecular cage-ion, [ 11BH] 6 2−, arise from the form of GWLs for specific primes ( p) (i.e. in S n theory sense of an index denoting the number of subfields) and the use of invariance hierarchies under the direct product group of the subduced spin symmetries. Such ( S n )↓ G spin symmetries have been presented in discussions of the symmetry of many-electron spin systems, e.g. as outlined in the seminal work of Kaplan (1975). Attention is drawn to the role of S n -inner tensor products and Cayley algebra in explicitly resolving certain problems connected with the non-simple reducibility pertaining to ( M 1- M n ( S 6) fields once p exceeds 2 (i.e. for clusters of identical higher spins). By partitioning Liouville space derived from the density operator σ(SO (3) × S 6) and its analogues under subduced spin symmetries this paper extends both the formalism and practical application of various recent multiquantum techniques to experimental NMR. The present semitheoretical “tool” to factor << kqv | C L(SO (3) × [ 6 ]) | k′ q′ v′>> and matrix representation of the Liouville operator for the subduced direct product symmetry of the total bicluster problem emphasizes Pines' 1988 argument [in Proc. C-th E. Fermi Physics Institute] that sets of selective subproblems exist which are ameniable to analysis of their information content without the need to treat the full problem; he focusses on selective q processes from an experimental viewpoint whereas we emphasize all q, [λ] forms of factoring in the analysis of spin evolution. Finally, we stress the primary theoretical importance of scalar invariants in few- and many-body spin problems in the context of SU2 × S n dual mappings and associated genealogies.