Quantitative Interpretation of Transverse Spin Relaxation by Translational Diffusion in Liquids Under Arbitrary Potentials

Intermolecular spin relaxation by translational motion of spin pairs have been widely used to study properties of the biomolecules in liquids. Notably, solvent paramagnetic relaxation enhancement (sPRE) arising from paramagnetic cosolutes has gained attentions for various applications, including the structural refinement of intrinsically disordered proteins, cosolute-induced protein denaturation, and the characterization of residue-specific effective near-surface electrostatic potentials (ENS). Among these applications, the transverse sPRE rate known as Γ ₂ has been predominantly been interpreted empirically as being proportional to < r ^-6 > _norm . In this study, we present a rigorous theoretical interpretation of Γ ₂ that it is instead proportional to < r ^-4 > _norm and provide explicit formula for calculating < r ^-4 > _norm without any adjustable parameters. This interpretation is independent of the type or strength of interactions and can be broadly applied, including to the precise interpretation of ENS.