  This work presents the bound-state spectra of Morse oscillator, which remains
one of the oldest important model potentials for molecules. Accurate
ro-vibrational energies are obtained by means of a generalized pseudospectral
method that offers an optimal, non-uniform discretization of the radial grid.
Both s-wave ($\ell=0$) and rotational ($\ell \neq 0$) states for low and high
quantum numbers are calculated for four representative diatomic molecules,
namely H$_2$, LiH, HCl and CO. First nine states belonging to a maximum of $n,
\ell =2$ are computed with good accuracy, along with nine other high-lying
states for each of these molecules. Present results \emph{surpass} the accuracy
of \emph{all} hitherto published calculations found so far, except the
tridiagonal J-matrix method, which produces similar accuracy as ours. Detailed
variation of energies with respect to state indices $n,\ell$ show interesting
behavior. A host of new states including the higher ones are reported as well.
This offers a simple general efficient scheme for calculating these and other
similar potentials in molecular physics.
