Rela2x: Analytic and automatic NMR relaxation theory

Abstract

Abstract

Spin relaxation is modelled using the so-called relaxation superoperator \(\widehat{\widehat{\mit\Gamma}}\). Analytic forms of \(\widehat{\widehat{\mit\Gamma}}\) 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 \(\widehat{\widehat{\mit\Gamma}}\) 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 \(\widehat{\widehat{\mit\Gamma}}\) for high-field NMR. Tools to analyse, approximate and visualize \(\widehat{\widehat{\mit\Gamma}}\) are built into Rela²x. As a demonstration of the functionality, \(\widehat{\widehat{\mit\Gamma}}\) is presented both for the familiar dipole–dipole coupled \(\rm{^{1}H–^{1}H}\) spin system and for more complicated \(\rm{^{1}{H}–^{14}{N}}\) and \(\rm{^{1}H–^{13}C–^{14}N}\) 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.