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

## TL;DR

This paper explains why high-resolution NMR spectra show mirror symmetry, linking it to specific properties of spin systems.

## Contribution

The paper identifies two necessary conditions for spectral symmetry based on spin system properties and coupling matrix structure.

## Key findings

- Spectral symmetry requires symmetric resonance frequencies about ν0.
- A monotonic spin ordering ensures invariance under coupling matrix reflection.
- Theoretical validation was performed for spin systems with 3 to 6 spins.

## Abstract

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.