Precision spectroscopy of molecular hydrogen ions: an introduction

The molecular hydrogen ions (MHI) are composed of only two nuclei and a single electron. These simplest molecules are fascinating systems at the interface of atomic and molecular physics. Compared to atoms, they present the additional degrees of freedom of vibration and rotation. The spectrum of rovibrational energies provides an exceptionally large number of transitions that are potentially measurable with extremely small fractional uncertainty. After two decades of theoretical and experimental efforts, precision studies of the vibrational transition frequencies have now reached fractional uncertainties in the low- range. This is nearing the level of the most accurate experiment-theory comparisons in physics, performed on the hydrogen atom and on the g-factor of the electron. We describe the motivation for the study of MHI, present some relevant theoretical and experimental issues, indicate a few salient results and give an outlook towards future opportunities.