Rela2x: Analytic and automatic NMR relaxation theory

Abstract

Spin relaxation is modelled using the so-called relaxation superoperator $\widehat{\stackrel{ˆ}{\Gamma }}$. Analytic forms of $\widehat{\stackrel{ˆ}{\Gamma }}$ have been derived in the literature in the simplest cases of one- or two-spin systems, with $S=\frac{1}{2}$ nuclei and no more than two different simultaneous relaxation mechanisms involved. Beyond that, for systems of more than two spins, with $S>\frac{1}{2}$ 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 $\widehat{\stackrel{ˆ}{\Gamma }}$ 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 Rela²x, to automatically compute the analytic matrix representation of $\widehat{\stackrel{ˆ}{\Gamma }}$ for high-field NMR. Tools to analyse, approximate and visualize $\widehat{\stackrel{ˆ}{\Gamma }}$ are built into Rela²x. As a demonstration of the functionality, $\widehat{\stackrel{ˆ}{\Gamma }}$ is presented both for the familiar dipole–dipole coupled ${}^1$H-${}^1$H spin system and for more complicated ${}^1$H-¹⁴N and ${}^1$H-¹³C-¹⁴N 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.