Approximate analytic solutions of the diatomic molecules in the Schrodinger equation with hyperbolical potentials

The Schrodinger equation for the rotational-vibrational (ro-vibrational) motion of a diatomic molecule with empirical potential functions is solved approximately by means of the Nikiforov-Uvarov method. The approximate ro-vibratinal energy spectra and the corresponding normalized total wavefunctions are calculated in closed form and expressed in terms of the hypergeometric functions or Jacobi polynomials P_{n}^{(\mu,\nu)}(x), where \mu>-1, \nu>-1 and x included in [-1,+1]. The s-waves analytic solution is obtained. The numerical energy eigenvalues for selected H_{2} and Ar_{2} molecules are also calculated and compared with the previous models and experiments.