Commutator-Based (A)[X]n(SU(2)×Sn) NMR Cluster Systems: Establishment of the Universality of [n](Sn) Salients and Constraints on ϕ±11(1.1) Polarisations to the [1n] Salient: Permutational Spin Symmetry (PSS) Within NMR Spin Dynamics - an Analytic View

Analytic SU(2) × Sn 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 JXX' 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](Sn) (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 JXX' 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) JXX' (Temme F. P.: J. Mol. Struct. (THEOCHEM) 2002, 547, 153) i.e., for abstract Sn ↓ 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 Sn ⊃ Sn-1 ⊃ .. ⊃ [2](S2) group chains in defining the Sn 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 Sn-invariants represented by certain Sn subduction properties.