Ionisation rate and Stark shift of a one-dimensional model of the Hydrogen molecular ion

TL;DR

This study analyzes the ionization rate and Stark shift of a one-dimensional H₂⁺ model, using both numerical and analytical methods, revealing sensitivity to external fields and internuclear distance, and estimating the molecule's polarizability.

Contribution

It provides a combined numerical and analytical analysis of ionization and Stark shift in a simplified H₂⁺ model, including new insights into their dependence on external fields and internuclear distance.

Findings

Ionization rate varies exponentially with external field strength.

Ionization saturates at large internuclear distances.

Computed polarizability agrees with previous sophisticated methods.

Abstract

In this paper we study the ionization rate and the Stark shift of a one-dimensional model of the H$_{2}^{+}$ ion. The finding of these two quantities is reduced to the solutions 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 H$_{2}^{+}$ and compare our results with those found by other authors using more sophisticated methods.