The main objective of this work is to determine the energy spectrum and the associated eigenfunctions of specific diatomic molecules. This study of light diatomic molecules with zero electronic spin is carried out in a relativistic framework, restricted to either weak relativistic corrections or the non-relativistic limit. To this end, we use the extended conformable fractional Nikiforov-Uvarov method together with the conformable biconfluent Heun equation method to solve the Klein-Gordon equation associated with the conformable inverse decimal power potential. The fractional parameter Formula: see text represents the exponent of deformation of the spatial metric, transforming the usual distance Formula: see text into an effective distance Formula: see text. This parameter extends the family of exactly solvable potentials and generates unprecedented fine structures in the spectra of Formula: see text, Formula: see text, and other diatomic molecules. It thus allows the spectrum to be tuned to describe systems ranging from the Kratzer potential to more general forms. The impact of the parameter Formula: see text on the quantum energy spectra is discussed in detail. Decreasing Formula: see text reduces the molecular energy and produces a narrower potential well, making the system more stable than in the classical case. This stability is accompanied by a reduction in vibrational motion, potentially allowing other types of motion to become dominant. Finally, Formula: see text does not lift degeneracy; it simply shifts all energy levels globally. The accurate reproduction of the experimental spectra of Formula: see text and Formula: see text validates the approach of using the Klein-Gordon equation with a four-term inverse-power potential to describe real molecular systems while preserving relativistic consistency. The systematic behavior as a function of the parameter Formula: see text provides a useful tool for exploring molecular properties under different bonding regimes.
Fouda et al. (Sat,) studied this question.