<p>Accurate modeling of diatomic molecular interactions requires potential functions that combine physical realism with analytical tractability. In this work, the Pekeris approximation is applied to the radial Frost-Musulin (RFM) potential and the centrifugal term of the Schrödinger equation to obtain an analytically manageable effective potential in the near-equilibrium region. Using the generalized fractional Nikiforov-Uvarov (GFNU) method, approximate analytical expressions for the bound-state energy eigenvalues are derived, with and without fractional parameters. The resulting formulation is applied to selected diatomic molecules and validated against available reference data. The Pekeris-based approach shows significantly improved agreement compared to conventional approximation schemes, with relative deviations typically below 2%. These results demonstrate that the proposed framework provides a reliable and computationally efficient method for modeling near-equilibrium bound-state energies in diatomic molecular systems.</p> Graphical Abstract <p></p>