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A new approximation scheme to the centrifugal term is proposed to obtain the$l\neq 0$ solutions of the Schr\"{o}dinger equation with the Manning-Rosenpotential. We also find the corresponding normalized wave functions in terms ofthe Jacobi polynomials. To show the accuracy of the new approximation scheme,we calculate the energy eigenvalues numerically for arbitrary quantum numbers$n$ and $l$ with two different values of the potential parameter $\alpha .$ Thebound state energies of various states for a few $% HCl,$ $CH,$ $LiH$ and $CO$diatomic molecules are also calculated. The numerical results are in goodagreement with those obtained by using program based on a numerical integrationprocedure. Our solution can be also reduced to the s-wave ($l=0$) case and tothe Hulth\'{e}n potential case.

An improved approximation to l-wave bound states of the Manning-Rosen potential by Nikiforov-Uvarov method