Charge degeneracy for a Thomas–Fermi hydrogen molecule: Bound–unbound transition

We study the relationship between quantum screening effects and charge symmetry in the hydrogen molecule. In this model, the charge symmetry of the molecule is broken in the presence of the Thomas–Fermi potential exp ( − q r ) / r . We argue that screening acts as an external field that promotes charge degeneracy in the system. In the presence of screening ( q ≠ 0 ), the charge symmetry of the H 2 + molecule is broken and thus the effective nuclear charge α splits into two, α → ( α , γ ) , where γ is the dual charge. The effective nuclear charge α is then related to the binding energy of the electron, while the dual charge γ is responsible for the screening effect in the wavefunction. Naturally, both the energy and the wavefunction are modified by this charge degeneracy. We obtain an analytical formulation for the molecular energy with a charge degeneracy effect. To test the efficiency of our model, we analyze the behavior of the molecule’s energy as a function of the Thomas–Fermi parameter q and calculate the critical parameter q ∗ , as well as the critical dual charge γ ∗ for which the bound–unbound transition occurs. We observe an interesting super-freezing screening phenomenon for the Thomas–Fermi hydrogen molecule.