# Interaction of the hydrogen molecule with the environment: stability of the system

## TL;DR

This study investigates the stability of the hydrogen molecule under environmental interactions modeled by non-Hermitian operators, revealing dynamic and static instabilities, metastable states, and PT symmetry breaking effects.

## Contribution

It introduces a parametric model of hydrogen-environment interaction using non-Hermitian operators and analyzes the resulting stability and symmetry properties.

## Key findings

- Hydrogen molecule is dynamically unstable under environmental interaction.
- Dissociation time decreases with increasing interaction parameter $\gamma$.
- Existence of a metastable state at a specific $\gamma$ value.

## Abstract

We study the stability of the hydrogen molecule interacting with the environment according to the balanced gain and loss energy scheme. We determined the properties of the molecule taking into account all electronic interactions, where the parameters of the Hamiltonian have been computed by using the variational method. The interaction of the hydrogen molecule with the environment was modeled parametrically ($\gamma$) with the help of the non-hermitian operator. We have shown that the hydrogen molecule is dynamically unstable. The dissociation time ($T_{D}$) decreases, if the $\gamma$ parameter increases (for $\gamma\rightarrow 0$, we get $T_{D}\rightarrow +\infty$). At the dynamic instability of the hydrogen molecule overlaps its static instability as the coupling constant $\gamma$ increases. We observed the decrease in the dissociation energy and the existence of the metastable state of the molecule ($\gamma_{MS}=0.659374$~Ry). The hydrogen molecule is statically unstable for $\gamma >\gamma_{D}=1.024638$~Ry. One can also observed the $\mathcal{PT}$ symmetry breaking effect for the electronic Hamiltonian ($\gamma_{\mathcal {PT}}=0.520873$~Ry). However, it does not affect the properties of the hydrogen molecule, such as: the electronic Hamiltonian parameters, the phonon and rotational energy, and the values of the electron-phonon coupling constants.