Result: Rela 2 x: Analytic and automatic NMR relaxation theory.

Spin relaxation is modelled using the so-called relaxation superoperator Γˆˆ. Analytic forms of Γˆˆ have been derived in the literature in the simplest cases of one- or two-spin systems, with S=12 nuclei and no more than two different simultaneous relaxation mechanisms involved. Beyond that, for systems of more than two spins, with S>12 and/or multiple relaxation mechanisms at play, the derivations become notoriously complicated, which is why analytic relaxation theory has mostly been considered a dead end. Instead, numerical methods of constructing Γˆˆ have been popular. However, they lack some of the physical, chemical and pedagogical insight that can be provided by analytic expressions. To this end, we present a general, interactive and freely available Python programme, named Rela ² x, to automatically compute the analytic matrix representation of Γˆˆ for high-field NMR. Tools to analyse, approximate and visualize Γˆˆ are built into Rela ² x. As a demonstration of the functionality, Γˆˆ is presented both for the familiar dipole-dipole coupled ¹ H- ¹ H spin system and for more complicated ¹ H- ¹⁴ N and ¹ H- ¹³ C- ¹⁴ N systems with dipole-dipole coupling, chemical shift anisotropy and quadrupole interaction. We envision that the code will provide much-needed clarity in the form of a helpful tool for the study of relaxation effects, and constitute a reference source for scientists in the field of NMR.
(Copyright © 2025 The Authors. Published by Elsevier Inc. All rights reserved.)

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.