On highly accurate computations of the Coulomb two-center systems with unit charges

Abstract: Results of our recent highly accurate computations of the Coulomb two-center systems with the unit electrical charges $X^{+} X^{+} e<sup>{-}$ and $X^{+} Y^{+} e<sup>{-}$ are discussed. In particular, we have determined (to very high accuracy) the total energies of the ground $1 s \sigma-$states in the two-center adiabatic (or molecular) H$<em>{2}<sup>{+}$, D$_2<sup>{+}$, HD$<sup>{+}$, HT$<sup>{+}$, T$^{+}</em>{2}$ and DT$<sup>{+}$ ions. In these computations we applied the new masses of hydrogen isotopes, which have been measured in the recent high-energy experiments. We also derived (and tested) our accurate mass-interpolation formula for the total energies of the model two-center $X^{+} X^{+} e<sup>{-}$ ions with very heavy masses of the point $X<sup>{+}$ particles ($M_X \ge$ 100,000 $m_e$). By using this formula we analyze the overall accuracy and validity of the adiabatic two-center approximation in application to the three-body atomic (or Coulomb) systems.