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We solved the Klein-Gordon equation with Yukawa potential usingthe Nikiforov-Uvarov method. In order to overcome the centrifugal barrier, the Greene and Aldrich approximation scheme was employed. The energy eigenvalues for relativistic, non- relativistic and the corresponding normalized wave function were obtained. A
special case of Coulomb potential was obtained. The energy eigenvalues equation were used to study some selected diatomic molecules such as N2, CO, NO, and CH.The bound state energy eigenvalues expressions and numerical computations agreed with the existing literature.