Improved modelling for vibrational energies of diatomic molecules using the generalized fractional derivative

Abstract By using the radial Schrödinger equation with the Morse potential in the context of the generalized fractional derivative (GFD), this work provides an important improvement in modelling the vibrational energy spectrum of diatomic molecules. We have used the generalized fractional Nikiforov-Uvarov (GFNU) method to derive an analytical solution for the energy eigenvalues in D -dimensional space by applying the Pekeris-type approximation to the centrifugal term. The proposed model is thoroughly examined across many electronic states, using a diverse set of twenty-two diatomic molecules, including astrophysically important species like SiO $$^+$$ and TaO, as well as CO, Na $$_2$$ , and AlH. The potential energy curves for the selected diatomic molecules have been produced using the Morse potential with the help of molecular constants. Furthermore, the pure vibrational energy levels for several diatomic molecules have been computed in both classical and fractional models. Our calculated vibrational energies are consistent with the Rydberg-Klein-Rees (RKR) data and previous studies. Additionally, it is seen that the vibrational energy spectra of different diatomic molecules calculated with fitted fractional parameters are improved compared to those obtained in the classical case for modelling the observed RKR data. The analysis of absolute percentage deviations at each level indicates that, for all examined diatomic molecules, the fractional derivative framework produces smaller and more consistent vibrational energy errors compared to the classical limit as the quantum number increases. Consequently, this study provides strong evidence that the GFNU method is a reliable and accurate technique to obtain the pure vibrational energies of various diatomic molecules.