The molecular hydrogen ions (H2+, HD+…) are the simplest molecules in nature. They are studied experimentally in high-resolution laser spectroscopy experiments, The comparison between experimental results and theoretical predictions allows for improved determination of fundamental constants such as the proton-electron mass ratio, and contributes to setting constraints on hypothetic beyond-standard-model interactions. The burgeoning experimental activity in this field gives strong motivation to improve the theory further by computing higher-order QED corrections. The topic of the internship and PhD is related to a "puzzle" that has emerged in recent years in the hyperfine structure. Surprisingly large deviations between theory and experiment have been observed in the hyperfine intervals of HD+. The goal will be to investigate a plausible cause of these discrepancies, which resides in the relativistic corrections to the spin-spin interaction between the electron and nuclei. This will be studied following the Non Relativistic QED (NRQED) framework, where corrections are described by effective Hamiltonians in a nonrelativistic (Schrödinger) formalism. The work is partly analytical (derivation of effective Hamiltonians), and partly numerical (computation of effective operator expectation values, using highly precise variational wavefunctions), In practice, the relative amounts of theoretical and numerical work can be adapted to the applicant’s preferences.