Introduction

The hydrogen molecular ion H₂⁺, consisting of two protons bound by an electron, is the most elementary molecule in nature. Its significance among molecules can therefore be compared to that of the hydrogen atom among atoms. Theoretically, H₂⁺ is usually treated as a two-level system, since its ground 1sσ_g and the first excited state 2pσ_u are well separated from next higher states. This rather simple molecular structure makes possible accurate fundamental quantum mechanical calculations. In our group we also study its isotopic variants HD⁺ and D₂⁺. Contrary to its theoretical simplicity the preparation of H₂⁺ is not easy and therefore the experiments on H₂⁺ are rather rare.

When a hydrogen molecular ion is exposed to light, it breaks up into a hydrogen atom and a proton:

The kinetic energy of the fragments H and H⁺ is equal to the difference between the photon energy and the bounding energy of a particular vibrational level. In our experiment we are able to distinguish fragments originating from different vibrational levels (see the experimental image).

At intensities higher than 10¹² W/cm² the coupling between the ground 1sσ_g and the first excited state 2pσ_u becomes very strong. In this regime molecule-light system is usually described by potential curves "dressed" with photons or with so-called light-induced potential curves.

Potential energy curves of the two lowest states of H₂⁺ in a weak field.

The main feature of these curves is the avoided crossing, a gap that opens up near the vibrational level that is resonant with the laser light (v=9). The gap initiates a process that is normally forbidden - dissociation of molecular vibrational levels that lie below the resonant level. This effect is known as molecular bond-softening.
At again higher intensities another gap opens due to three-photon absorption. Here a molecule absorbs three photons and re-emits one photon, which leads to effective two-photon absorption. This effect is often called above-threshold dissociation, because the molecule absorbes more photons than needed for its dissociation.

Potential energy curves of the photon dressed states.

Ionization and Coulomb explosion

When the intensity becomes high enough, also ionization of H₂⁺ may occur:

After the electron is stripped off the nuclei, the remaining protons will repel due to the Coulomb interaction acquiring kinetic energy of e²/(4πε₀R), where R is the separation between the nuclei at the moment of ionization.
Already in early experiments, the measured proton energies turned out to be much smaller than it was expected. The explanation is that the ionization doesn't occur at the equilibrium proton separation R₀ of about 2 a.u., but at a much larger R of approximately 5-12 a.u.

At intensities higher than 10¹² W/cm² the coupling between the ground 1sσ_g and the first excited state 2pσ_u becomes very strong. In this regime molecule-light system is usually described by potential curves "dressed" with photons or with so-called light-induced potential curves.

Potential seen by the  ionizing electron.