{H$_2^+$, HeH and H$_2$}: approximating potential curves, calculating rovibrational states

Analytic consideration of the Bohr-Oppenheimer (BO) approximation for diatomic molecules is proposed: accurate analytic interpolation for potential curve consistent with its rovibrational spectra is found. It is shown that in the Bohr-Oppenheimer approximation for four lowest electronic states $1s\sigma_g$ and $2p\sigma_u$, $2p \pi_u$ and $3d \pi_g$ of H$_2^+$, the ground state X$^2\Sigma^+$ of HeH and the two lowest states $^1\Sigma^+_g$ and $^3\Sigma^+_u$ of H$_2$, the potential curves can be analytically interpolated in full range of internuclear distances $R$ with not less than {4-5-6} figures. Approximation based on matching the Taylor-type expansion at small $R$ and a combination of the multipole expansion with one-instanton type contribution at large distances $R$ is given by two-point Pad\'e approximant. The position of minimum, when exists, is predicted within 1$\%$ or better. For the molecular ion H$_2^+$ in the Lagrange mesh method, the spectra of vibrational, rotational and rovibrational states $(\nu,L)$ associated with $1s\sigma_g$ and $2p\sigma_u$, $2p \pi_u$ and $3d \pi_g$ potential curves is calculated. In general, $1s\sigma_g$ electronic curve contains 420 rovibrational states, which increases up to 423 when we are beyond BO approximation. For the state $2p\sigma_u$ the total number of rovibrational states (all with $\nu=0$) is equal to 3, within or beyond Bohr-Oppenheimer approximation. As for the state $2p\pi_u$ within the Bohr-Oppenheimer approximation the total number of the rovibrational bound states is equal to 284. The state $3d\pi_g$ is repulsive, no rovibrational state is found. The ground state potential curve of the heteronuclear molecule HeH does not support rovibrational states. Accurate analytical expression for the potential curves of the hydrogen molecule H$_2$ for the states $^1\Sigma^+_g$ and $^3\Sigma^+_u$ is presented.