In this study, the Klein–Gordon equation was solved with the Deng–Fan potential using the Nikiforov–Uvarov-functional-analysis in higher dimensions. By employing the improved Pekeris-type approximation scheme, the relativistic and nonrelativistic energy spectra of the Deng–Fan potential were obtained in closed form. In addition, the scattering state phase shift expression of Deng–Fan potential was obtained in higher dimensions. The effects of the vibrational and rotational quantum numbers on the vibrational energies and scattering phase shift of hydrogen chloride (HCl) and lithium hydride (LiH) diatomic molecules were studied numerically and graphically at different dimensions. Interestingly, there exists inter-dimensional degeneracy symmetry for the scattering phase shift of the diatomic molecular systems considered. Our results generally were in agreement with that obtained from literatures.
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