LAUSR

The general theory of potential curves for diatomic molecules is presented. For the diatomic molecule He$_2^+$ in Born-Oppenheimer (BO) approximation it is presented the approximate analytic expression for the potential energy curves $V(R)$ for the ground state $X^2 \Sigma_u^+$ and the first excited state $A^2 \Sigma_g^+$, based on matching short and long distances behavior via two-point Pad\'e approximation. In general, it provides 3-4 s.d. correctly for internuclear distances $R \in [0, 10]$ a.u. with some irregularities for $A^2 \Sigma_g^+$ curve at small distances (much smaller than equilibrium distances) probably related to level crossings which may occur there. Solving the Schr\"odinger equation for the nuclear motion it is found that the analytic ground state potential energy curve $X^2 \Sigma_u^+$ supports 825 rovibrational states with 3-4 s.d. of accuracy in energy, which is by only 5 states less than those 830 reported in the literature where sometimes non-adiabatic corrections were considered. The analytic potential energy curve for the excited state $A^2 \Sigma_g^+$ supports all reported 9 weakly-bound rovibrational states. As for LiH dimer it is found analytic expression for the ground state $X^1\Sigma^+$, it supports 906 rovibrational states with 3-4 s.d. of accuracy in energy, which is only by 5 states more than 901 reported in the literature. For both ions difference in number of rovibrational states is related with the non-existence/existence of weakly-bound states close to threshold. Rovibrational spectra is found using the Lagrange mesh method.