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Abstract

Analytic SU(2) × S_n dual tensorial (DT) spin dynamics over uniform NMR spins is invoked in examining the modern quantum basis for the universal non-observability rule which governs dominant intracluster J_XX' couplings of (A)[X]_n NMR systems as a specific form of (abstract) permutational spin symmetry (PSS) with well defined properties on spin-alone space. This is shown to be linked to DT constraints that apply to the cross-product ϕ¹_±1(1.1) polarisation development i.e., as being confined to [1ⁿ](S_n) (Liouvillian) salient, with the existence of [n] (rotating frame) null subspaces. Both these arise within the spin dynamics of (A)[X]₂ spin systems (or subsystems thereof) within (a hierarchy of) dominant J_XX' governing the internal L⁽⁰⁾; such spin systems provide analytic sequels to comparative spin dynamics studies of XX' PSS and AX broken-PSS systems in a Liouvillian coupled tensorial basis formalism, since both draw on (Sanctuary B. C.: Mol. Phys. 1985, 55, 1017), and on the realisation that proper PSS over a (uniform) spin-space L⁽⁰⁾ = [H⁽⁰⁾,.] zeroth-order Liouvillian and its internal (hierarchical subsets of) J_XX' (Temme F. P.: J. Mol. Struct. (THEOCHEM) 2002, 547, 153) i.e., for abstract S_n ↓ G group embeddings. The present work also examines the general irrep-structure of DT spin symmetries for the extent of unit-character irreps and the role of S_n ⊃ S_n-1 ⊃ .. ⊃ [2](S₂) group chains in defining the S_n multiple invariants under democratic recoupling of PSS of uniform spin systems. As group measures, these properties apply to both (A)[X]_n and [AX]_n PSS symmetries, with the invariant cardinality |SI|⁽ⁿ⁾ being related to time-reversal invariance (TRI) and its inherent democratic recoupling (DR) over Weyl (I • I) pairs. For [X]_2n uniform spin clusters, |SI|⁽²ⁿ⁾ is best derived via n-fold polyhedral combinatorics of the underlying DR (Temme F. P.: Proc. R. Soc. London, Ser. A 2005, 461, 321) i.e., as an augmented post-Weyl view of the essential role of TRI in (group) invariant cardinality, with the S_n-invariants represented by certain S_n subduction properties.

Keywords: NMR spin dynamics theory; Democratic recoupling; Permutation spin symmetry; Nuclear magnetic resonance; Time-reversal invariance.

References: 55 live references.