Mirror Symmetry of the NMR Spectrum and the Connection with the Structure of Spin Hamiltonian Matrix Representations

TL;DR

This paper develops a theoretical framework linking mirror symmetry in high-resolution NMR spectra to the underlying spin Hamiltonian matrix structures, revealing two mechanisms behind spectral symmetry.

Contribution

It introduces a comprehensive theory connecting NMR spectral mirror symmetry with Hamiltonian matrix properties and distinguishes two mechanisms causing this symmetry.

Findings

Mirror symmetry can result from Hamiltonian matrix bisymmetry in a canonical basis.

Spectral symmetry can also arise from topological isospectrality under parameter exchange.

The framework applies to various spin systems, including balanced and unbalanced types.

Abstract

This work provides a comprehensive theoretical framework for understanding the symmetry properties of High-Resolution NMR spectra. We analyze the conditions under which a spectrum exhibits mirror symmetry (palindromicity). We demonstrate that such symmetry can arise from two distinct mechanisms: (1) the direct geometric bisymmetry of the Hamiltonian matrix in a generalized canonical basis (typical for balanced systems like $A_nB_n$ or $A_nX_n$), and (2) a more fundamental property of topological isospectrality (similarity) under parameter exchange induced by the internal symmetry of the spin system, which applies even when the matrix lacks geometric symmetry (as observed in $AA'BB'$ systems).