Borromean bindings in H$_{2}^{+ }$ with screened Coulomb potentials

The stability of the bound $P$ and $D$ states of the H$_{2}^{+}$ molecular ion, where the nuclei and the electrons interacts with a screened Coulomb (Yukawa-type) potential exp(-\textit{$\mu $}r)/r, has been studied for different values of the screening parameters \textit{$\mu $}. We have determined the values of the bound $^{3}P^{o }$(\textit{$\nu $}=0, $J$=1), $^{3}P^{o }$(\textit{$\nu $}=1, $J$=1), $^{1}D^{e }$(\textit{$\nu $}=0,$ J$=2), and $^{1}D^{e }$(\textit{$\nu $}=1,$ J$=2) states energies for different values of the screening parameters using highly correlated exponential wave-functions in the framework of Ritz variational principle. The critical values of the screening parameters for the bound states are reported for which the H$_{2}^{+}$ system is stable, while all the possible fragments are unbound, that is, it shows Borromean binding for the three-body systems [1]. We have determined the range of the Borromean windows for the lower-lying $S$, $P$ and D states.\\[4pt] [1] A. Ghoshal and Y. K. Ho, \textit{Int. J. Quan. Chem.} (2011), published online; and references therein.