Ionization rate and Stark shift of a one-dimensional model of the hydrogen molecular ion

In this paper we study the ionization rate and the Stark shift of a one-dimensional model of the ${\mathrm{H}}_{2}^{+}$ ion. The finding of these two quantities is reduced to the solution of a complex eigenvalue problem. We solve this problem both numerically and analytically. In the latter case we consider the regime of small external electrostatic fields and small internuclear distances. We find an excellent agreement between the ionization rate computed with the two approaches, even when the approximate result is pushed beyond its expected validity. The ionization rate is very sensitive to small changes of the external electrostatic field, spanning many orders of magnitude for small changes of the intensity of the external field. The dependence of the ionization on the internuclear distance is also studied, as this has a direct connection with experimental methods in molecular physics. It is shown that for large distances the ionization rate saturates, which is a direct consequence of the behavior of the energy eigenvalue with the internuclear distance. The Stark shift is computed and from it we extract the static polarizability of ${\mathrm{H}}_{2}^{+}$ and compare our results with those found by other authors using more sophisticated methods. This work provides an extension of what is usually covered in a course of quantum mechanics where the double delta-potential is studied, while also establishing the connection to a concrete physics problem.