The accuracy of nuclear-magnetic-resonance (NMR)-derived nuclear magnetic dipole moments is ultimately limited by the uncertainty of the corresponding NMR shielding constants. Here, using hydrated ${\text{Tl}}^{+}$ as a demanding heavy-ion test case, we show that NMR shielding uncertainties related to hydration-shell structure and electronic-structure approximations can be reduced to the QED scale. SCAN-functional based machine-learning-force-field molecular dynamics captures the key experimentally known features of the ${\text{Tl}}^{+}$ hydration structure, while coupled-cluster-calibrated four-component Dirac-Kohn-Sham calculations provide the ensemble-averaged shielding with a controlled residual uncertainty. The resulting magnetic moments of ${}^{203}\mathrm{Tl}$ and ${}^{205}\mathrm{Tl}$ are 1.6198(3)${\mu }_N$ and 1.6357(3)${\mu }_N$, respectively. These values eliminate a systematic error of order ${10}^{-3}{\mu }_N$ and reduce the residual uncertainty to the ${10}^{-4}{\mu }_N$ level. Our workflow provides a transferable route to high-accuracy NMR-derived moments of heavy nuclei used as reference quantities in atomic and nuclear physics.

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• Atomic & molecular structure
• Electron correlation calculations for atoms & ions
• Electronic structure of atoms & molecules
• Nuclear & electron resonance
• Relativistic & quantum electrodynamic effects in atoms, molecules,& ions
• Solvation
• Magnetic moment