The origin of mirror symmetry in high-resolution nuclear magnetic resonance spectra

A connection between the symmetry of high-field nuclear magnetic resonance (NMR) spectra, including higher-order spectra, and the properties of the spin system has been established. It is shown that, for a spectrum to be symmetric about the mid-resonance frequency (ν0), two conditions must be satisfied: (1) the resonance frequencies of the spins must be symmetrically positioned about ν0, and (2) there must exist at least one spin ordering with a monotonic increase (or decrease) in resonance frequencies such that the spectrum is invariant under the reflection of the J-coupling matrix about its anti-diagonal (one way to satisfy this condition is for the J-coupling matrix to be explicitly persymmetric). The results were validated by calculating theoretical spectra for three-, four-, five-, and six-spin systems.