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In this work we employed the elegant tool of parametric Nikiforov –Uvarov method to obtain an approximate bound state solution of Klein-Gordon and Schrodinger equations using Deng-Fan potential with improved approximation to the centrifugal term for some selected diatomic molecules. We obtained an unnormalized wave function and energy eigen equations for both relativistic and nonrelativistic wave equations. We applied the non-relativistic limit to the Klein-Gordon equation to obtained energy eigen equation of Schrodinger in order to affirm the high precision of analytical mathematical accuracy. We employed the mathematical tool of Hellmann-Feynman theorem to calculate expectation values <r-2>, <T> and <p2> for nine selected diatomic molecules namely: Hydrogen (H2), Hydrogen Chloride (HCl), Lithium Hydride (LiH), Iodine (I2), Chromium hydride (CrH), Titanium Hydride (TiH), Thiocynate (TiH), Carbon (II) Oxide and Scandium Flouride (ScF). The numerical bound state solutions for these diatomic molecules is in agreement to work reported in existing literature using other quantum mechanical methods. All numerical simulations were carried out by implementing a well designed MATLAB algorithm using experimentally determined spectroscopic constants for the selected diatomic molecules. The numerical bound state solutions were obtained for varying principal quantum number n with fixed orbital angular quantum number l=0, 1 and 2.