Hyperfine structure in the H$_2^+$ and HD$^+$ molecular ions at $m\alpha^6$ order

TL;DR

This paper derives a comprehensive effective Hamiltonian for relativistic hyperfine structure corrections in hydrogen molecular ions at order $m ext{alpha}^6$, including new spin-dependent interactions, enhancing theoretical accuracy.

Contribution

It introduces new corrections to electron spin-orbit and spin-spin tensor interactions at order $m ext{alpha}^6$, improving hyperfine structure modeling in H$_2^+$ and HD$^+$ ions.

Findings

Derived a complete effective Hamiltonian at $m ext{alpha}^6$ order.

Calculated improved hyperfine coefficients for specific transitions.

Enhanced the theoretical understanding of hyperfine splitting in molecular ions.

Abstract

A complete effective Hamiltonian for relativistic corrections at orders $m\alpha^6$ and $m\alpha^6(m/M)$ in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calculated for a few transitions of current experimental interest.