  Two physically important potentials (Manning-Rosen and P\"oschl-Teller) are
considered for the ro-vibrational energy in diatomic molecules. An improved new
approximation is invoked for the centrifugal term, which is then used for their
solution within the Nikiforov-Uvarov framework. This employs a recently
proposed scheme, which combines the two widely used Greene-Aldrich and
Pekeris-type approximations. Thus, approximate analytical expressions are
derived for eigenvalues and eigenfunctions. The energies are examined with
respect to two approximation parameters, $\lambda$ and $\nu$. The original
approximations are recovered for certain specials values of these two
parameters. This offers a simple effective scheme for these and other relevant
potentials in quantum mechanics.
