Recently, measurements of ro-vibrational transition frequencies in hydrogen molecular ions (H2+, HD+…), the simplest molecules in nature, have reached a record 12-digit accuracy. Comparison of experimental results with theoretical predictions has led to an improved determination of the proton-electron mass ratio 𝑚𝑝/𝑚𝑒. The burgeoning experimental activity in this field gives strong motivation to improve further the theory of hydrogen molecular ions by computing higher-order QED corrections. The internship will be focused on the hyperfine structure, with the goal of computing a new interaction term, which has been neglected so far, but is becoming highly relevant in view of the rapidly increasing experimental accuracy. In HD+, the proton and deuteron spins have an effective mutual interaction resulting from their interactions with the electron’s spin. This will be studied following the Non Relativistic QED (NRQED) framework, where QED 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.