The helium hydride ion (${\mathrm{HeH}}^{+}$) represents the simplest molecular system composed of nuclei with unequal charges and serves as a model system of unique importance to molecular structure theory. We present high-precision calculations of the leading relativistic ($m{\alpha }^4$) and QED ($m{\alpha }^5$) corrections for the ground electronic state of ${\mathrm{HeH}}^{+}$, performed within the Born-Oppenheimer approximation and using effective operators in the nonrecoil limit. The use of explicitly correlated Gaussian (ECG) and cusp-enforced $r\mathrm{ECG}$ basis functions, combined with regularization techniques for singular operators, yields a substantial reduction in numerical uncertainty compared to previous calculations. The resulting potentials reproduce rovibrational transition energies with few-megahertz uncertainties (${10}^{-4}–{10}^{-5}{\mathrm{cm}}^{-1}$) and pure rotational transitions with submegahertz accuracy (${10}^{-5}–{10}^{-6}{\mathrm{cm}}^{-1}$), enabling comparison with the most precise spectroscopic data for ${\mathrm{HeH}}^{+}$ isotopologues. These results provide a solid foundation for including higher-order effects and establish ${\mathrm{HeH}}^{+}$ as a primary molecular system alongside ${\mathrm{H}}_2$ for high-resolution spectroscopy, astrophysical modeling, and precision tests of fundamental physics.

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• Electronic structure of atoms & molecules
• Interatomic & molecular potentials
• Molecular spectra
• Potential energy surfaces
• Relativistic & quantum electrodynamic effects in atoms, molecules,& ions
• Rotational states
• Vibrational states