By employing the extended Nikiforov–Uvarov (ENU) method, we solved the radial Schrodinger equation with the shifted screened Kratzer potential model. The analytical expression of the energy eigenvalues and numerical results were determined for some selected diatomic molecule systems. Variations of the energy eigenvalues obtained with potential parameters and quantum numbers were discussed graphically. Also, variations of different thermodynamic properties with temperature and maximum vibration quantum numbers were discussed extensively. Our results correspond to the results obtained in the literatures. The shifting parameters contribute a great effect to the energy results obtained. It has also been established that there exists a critical temperature at specific entropy values for the selected diatomic molecule systems.

The Schrödinger equation is solved with general molecular potential via the improved quantization rule. Expression for bound state energy eigenvalues, radial eigenfunctions, mean kinetic energy, and potential energy are obtained in compact form. In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is applied. Also, we use the Hellmann–Feynman theorem to derive the relation for expectation values. Bound state energy eigenvalues, wave functions and meanenergies of Woods–Saxon potential, Morse potential, Möbius squared and Tietz–Hua oscillators are deduced from the general molecular potential. In addition, we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz. H2, CO, HF, and O2. Results obtained are in perfect agreement with the data available from the literature for the potentials and molecules. Studies also show that as the vibrational quantum number increases, the mean kinetic energy for the system in a Tietz–Hua potential increases slowly to a threshold value and then decreases. But in a Morse potential, the mean kinetic energy increases linearly with vibrational quantum number increasing.

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