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In the last decade, the experimental determination of the transition frequencies of Molecular Hydrogen Ions (MHIs) has improved significantly, reaching a level of precision that surpasses theoretical predictions. These advances play a crucial role in the determination of fundamental constants (e.g., $m_p/m_e$) and the testing of the Standard Model [1]. However, they also highlight the need for more accurate theoretical calculations. The main source of uncertainty in theoretical predictions arises from the evaluation of the fully relativistic one-loop self-energy correction [2].
This QED correction involves the Dirac Green function of the bound electron, which can be obtained by numerically solving the two-centre Dirac equation in a finite basis set. Our team has previously developed an approach that relies on exponential basis functions [3]. Here, we explore the Finite Element Method (FEM) using a B-Spline basis set, combined with arbitrary precision arithmetic. B-splines are often used for the calculation of QED corrections (see e.g., [4, 5, 6]). This basis has several interesting properties, such as forming a complete and orthogonal basis set, providing a potentially more accurate description of continuum states, and the possibility to adapt the grid to suit systems with different nuclear charges $Z$. The two-centre Dirac equation is constructed using the Dual Kinetic Balance (DKB) approach, which provides a symmetric description of both positive- and negative-energy states, entering the Dirac Green function. This method has been shown to improve the convergence of QED calculations in hydrogenic systems [5].
This approach is expected to contribute to an accurate description of the numerical Green function, which is an important step towards calculating the one-loop self-energy correction in a fully relativistic framework.
References :
[1] J.-Ph. Karr, S. Schiller, V. I. Korobov, and S. Alighanbari. Determination of a set of fundamental constants from molecular hydrogen ion spectroscopy: A modeling study. Phys. Rev. A, 112:022809, Aug 2025. doi:10.1103/jz54-7f7b.
[2] Vladimir I. Korobov and J.-Ph. Karr. Rovibrational spin-averaged transitions in the hydrogen molecular ions. Phys. Rev. A, 104:032806, Sep 2021. doi:10.1103/PhysRevA.104.032806.
[3] Hugo D. Nogueira and Jean-Philippe Karr. High-precision solution of the Dirac equation for the hydrogen molecular ion using a basis-set expansion. Physical Review A, 107(4):042817, April 2023. doi:10.1103/PhysRevA.107.042817.
[4] Steven A. Blundell and Neal J. Snyderman. Basis-set approach to calculating the radiative self-energy in highly ionized atoms. Physical Review A, 44(3):R1427–R1430, August 1991. doi:10.1103/PhysRevA.44.R1427.
[5] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plunien, and G. Soff. Dual Kinetic Balance Approach to Basis-Set Expansions for the Dirac Equation. Physical Review Letters, 93(13):130405, September 2004. doi:10.1103/PhysRevLett.93.130405.
[6] A. N. Artemyev and A. Surzhykov. Quantum Electrodynamical Corrections to Energy Levels of Diatomic Quasimolecules. Physical Review Letters, 114(24):243004, June 2015. doi:10.1103/PhysRevLett.114.243004