Topological Study of the H₃⁺⁺ Molecular System: H₃⁺⁺ as a Cornerstone for Building Molecules during the Big Bang

The present study is devoted to the possibility that tri-atomic molecules were formed during or shortly after the Big Bang. For this purpose we consider the ordinary $H_3^{+}$ and $H_3$ and the primitive tri-atomic molecular system, $H_3^{++}$, which, as is shown, behaves differently. The study is carried out by comparing the topological features of these systems as they are reflected through their non-adiabatic coupling terms. Although the $H_3^{++}$ is not known to exist as a molecule, we found that it behaves as such at intermediate distances. However this illusion breaks down as its asymptotic region is reached. Our study indicates that whereas $H_3^{+}$ and $H_3$ dissociate smoothly, the $H_3^{++}$, does not seem to do so. Nevertheless, the fact that $H_3^{++}$ is capable of living as a molecule on borrowed time enables it to catch an electron and form a molecule via the reaction $H_3^{++} + e \to H_3^{+}$ that may dissociate properly: $H_3^{+} \to H^{+} + H_2$ or $H + H_2^+$. Thus, the two unique features acquired by $H_3^{++}$ namely, that it is the most primitive system formed by three protons and one electron and topologically, still remain for an instant a molecule, may make it the sole candidate for becoming the \bold{cornerstone} for creating the molecules.