AI 中文总结
研究高分辨率NMR谱精确模拟中自旋哈密顿量维度指数增长问题，将自旋系统表述为无向边加权完全图，证明可达性准则，给出直至14个自旋的完整分类法及新序列，还提出精确块对角化方法。
AI 中文摘要
高分辨率核磁共振谱的精确模拟需要对自旋哈密顿量进行块对角化，其维度随自旋数N呈指数增长；对称性是控制这种增长的主要工具，但对于哪些置换群可以作为标量耦合自旋系统的完全对称群，一直缺乏详尽的处理。将自旋系统表述为无向边加权完全图，我们证明了一个精确的可达性准则：当且仅当$S_N$的一个子群与其对称化（无向）维兰特2-闭包重合时，它是可达的。特别地，单个自旋环的纯旋转对称性是不可能的，但手性自旋系统确实作为多轨道扭曲堆叠存在，并且我们确定了每个循环群的最小自旋数$\mu^*(C_n)$，包括在N = 12时反直觉的$C_8$和$C_9$实现。一种通过轨道划分分解完成的顺序对称化算法，产生了一个可证明的直至N = 14的所有可实现对称类型的详尽枚举：明显的新序列$a(N)=1,1,3,8,11,27,36,90,131,282,394,948,1316,2866$，具有塔式定律$a(N)=a(N - 1)+f(N)$——总共6112个条目的目录，按规范标识符和扩展波普尔命名法的结构语法组织。最后，我们提出了一种无需物理近似的精确块对角化的分层方法：通过守恒总自旋投影进行因式分解，对磁等价复合材料进行舒尔 - 外尔收缩，对自旋配置进行轨道权重去重，以及对因子群表示进行同型投影，对非阿贝尔群和复特征进行统一处理。
英文摘要
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 $S_N$ 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 $μ^{*}(C_n)$ for every cyclic group, including the counter-intuitive realizations $C_8$ and $C_9$ 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.
Comments20 pages, 2 figures, 3 tables
