  Bound-state spectra of shifted Deng-Fan oscillator potential are studied by
means of a generalized pseudospectral method. Very accurate results are
obtained for \emph{both low as well as high} states by a non-uniform optimal
discretization of the radial Schr\"odinger equation. Excellent agreement with
literature data is observed in \emph{both $s$-wave and rotational} states.
Detailed variation of energies with respect to potential parameters is
discussed. Application is made to the ro-vibrational levels of four
representative diatomic molecules (H$_2$, LiH, HCl, CO). Nine states having
$\{n,\ell\} =0,1,2$ are calculated with good accuracy along with 15 other
higher states for each of these molecules. Variation of energies with respect
to state indices $n$, $\ell$ show \emph{behavior} similar to that in the Morse
potential. Many new states are reported here for the first time. In short, a
simple, accurate and efficient method is presented for this and other similar
potentials in molecular physics.
