  The approximate analytical bound state solution of the Schr\"odinger equation
for the Manning-Rosen potential is carried out by taking a new approximation
scheme to the orbital centrifugal term. The Nikiforov-Uvarov method is used in
the calculations. We obtain analytic forms for the energy eigenvalues and the
corresponding normalized wave functions in terms of the Jacobi polynomials or
hypergeometric functions for different screening parameters 1/b. The
rotational-vibrational energy states for a few diatomic molecules are
calculated for arbitrary quantum numbers n and l with different values of the
potential parameter {\alpha}. The present numerical results agree within five
decimal digits with the previously reported results for different 1/b values. A
few special cases of the s-wave (l=0) Manning-Rosen potential and the Hulth\'en
potential are also studied.
  Keywords: Energy eigenvalues; Manning-Rosen potential; Nikiforov-Uvarov
method, Approximation schemes.
  03.65.-w; 02.30.Gp; 03.65.Ge; 34.20.Cf
