  Accurate low and high-lying bound states of Tietz-Hua oscillator potential
are presented. The radial Schr\"odinger equation is solved efficiently by means
of the generalized pseudospectral method that enables optimal spatial
discretization. Both $\ell=0$ and rotational states are considered.
Ro-vibrational levels of six diatomic molecules \emph{viz.}, H$_2$, HF, N$_2$,
NO, O$_2$, O$_2^+$ are obtained with good accuracy. Most of the states are
reported here for the first time. A detailed analysis of variation of
eigenvalues with $n, \ell$ quantum numbers is made. Results are compared with
literature data, wherever possible. These are also briefly contrasted with the
Morse potential results.
