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Precision measurements of rovibrational energies in H$_2^+$ provide access to fundamental constants such as the proton-to-electron mass ratio or the proton charge radius, by comparison with theoretical results [1]. Because H$_2^+$ and D$_2^+$ are nonpolar, pure rotational and vibrational transitions are forbidden in the electric-dipole approximation and are very difficult to measure. As alternative method to determine the energy-level structure, spectra of Rydberg series of H$_2$ and D$_2$ converging on different spin-rovibrational states of H$_2^+$ and D$_2^+$ can be measured, from which their relative energies are obtained by Rydberg-series extrapolation [2, 3].
As application of this method, we determined the fundamental vibrational interval of H$_2^+$ by continuous-wave laser spectroscopy of Stark manifolds of Rydberg states of H$_2$ with the ion core in the ground and first vibrationally excited states [4]. From measurements of Stark manifolds at varying electric field strengths and comparison with precise calculations of the field-induced Stark shifts [5], the zero-quantum-defect positions $-R_{\textrm{H}_2}/n^2$ are determined, which yield precise ionization thresholds. We demonstrate the use of this procedure for the determination of the fundamental vibrational interval of H$_2^+$ at sub-MHz uncertainty.
This contribution also focuses on the determination of the first three rotational intervals of para-H$_2^+$ ($N^+=2,4,6$) and their spin-rotation splittings at sub-MHz accuracy by a combination of precision spectroscopy and multichannel-quantum-defect theory.
[1] V. I. Korobov et al., Phys. Rev. Lett. 118, 233001 (2017).
[2] G. Herzberg and Ch. Jungen, J. Mol. Spectrosc. 41, 425 (1972).
[3] M. Beyer et al., Phys. Rev. Lett. 123, 163002 (2019).
[4] I. Doran et al., Phys. Rev. Lett. 132, 073001 (2024).
[5] N. Hölsch et al., J. Mol. Spectrosc. 387, 111648 (2022).
