# Fundamental transitions and ionization energies of the hydrogen molecular ions at the few ppt level

## TL;DR

This paper improves calculations of ionization energies and vibrational transitions for hydrogen molecular ions using NRQED, achieving unprecedented fractional uncertainties at the 10^-12 level.

## Contribution

It introduces second-order vibrational corrections and evaluates higher-order effects, significantly enhancing the precision of theoretical predictions.

## Key findings

- Fractional uncertainty reduced to 7×10^-12 for transitions.
- Ionization energies calculated with 4×10^-12 uncertainty.
- Enhanced theoretical framework for molecular ion spectroscopy.

## Abstract

We calculate ionization energies and fundamental vibrational transitions for H$_2^+$, D$_2^+$, and HD$^+$ molecular ions. The NRQED expansion for the energy in terms of the fine structure constant $\alpha$ is used. Previous calculations of orders $m\alpha^6$ and $m\alpha^7$ are improved by including second-order contributions due to the vibrational motion of nuclei. Furthermore, we evaluate the largest corrections at the order $m\alpha^8$. That allows to reduce the fractional uncertainty to the level of $7\cdot10^{-12}$ for fundamental transitions and to $4\cdot10^{-12}$ for the ionization energies.