Within the framework of non-relativistic quantum mechanics, the bound state approximate solution of the SE is solved for the Coshine Yukawa potential (CYP) using the Nikiforov–Uvarov (NU) method. By employing the Greene-Aldrich-type approximation scheme, we have obtained the explicit energy-eigenvalues and corresponding normalized eigen-functions in closed form for the newly proposed CYP for hydrogen-related diatomic molecules such as hydrogen dimer (H2), lithium hydride (LiH), scandium hydride (ScH) and hydrogen chloride (HCl). Our results show that the bound state energy is highly sensitive to the spectroscopic parameters of the diatomic molecules considered. The thermodynamic properties are also evaluated including the vibrational partition function, vibrational mean energy, vibrational mean free energy, vibrational entropy and vibrational specific heat capacity. Presented also are some numerical results which show an indication of similar correlation of energies, owing to their ion-ion coupling with regards to similar atomic radii existing among the diatomic molecules.