2026/07/26 by Dmitry A. Cheshkov, Dmitry O. Sinitsyn, Artemiy I. Nichugovskiy
#physics.chem-ph
Exact simulation of high-resolution NMR spectra requires block diagonalization of the spin Hamiltonian, whose dimension grows exponentially with the number of spins N; symmetry is the principal tool for taming this growth, yet which permutation groups can occur as the full symmetry group of a scalar-coupled spin system has lacked an exhaustive treatment. Formulating the spin system as an undirected edge-weighted complete graph, we prove an exact realizability criterion: a subgroup of SN is realizable if and only if it coincides with its symmetrized (undirected) Wielandt 2-closure. In particular, purely rotational symmetry of a single spin ring is impossible, yet chiral spin systems do exist as multi-orbit twisted stacks, and we determine the minimal spin count μ*(Cn) for every cyclic group, including the counter-intuitive realizations C8 and C9 at N = 12. A sequential symmetrization algorithm, completed by an orbit-partition decomposition, yields a provably exhaustive enumeration of all realizable symmetry types up to N = 14: the apparently new sequence a(N) = 1, 1, 3, 8, 11, 27, 36, 90, 131, 282, 394, 948, 1316, 2866 with the tower law a(N) = a(N-1) + f(N) - a catalogue of 6112 entries in all, organized by canonical identifiers and a structural grammar extending the Pople nomenclature. Finally, we present a hierarchical methodology for exact block diagonalization without physical approximations: factorization by the conserved total spin projection, Schur-Weyl contraction of magnetically equivalent composites, orbit-weight deduplication of the spin configurations, and isotypic projection over the representations of the factor group, with a uniform treatment of non-abelian groups and complex characters.